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pdf_name stringclasses 9 values	page_number int32 0 23	page_size sequence	layout sequence	content_list stringlengths 45 8.72k	middle stringlengths 2.2k 73.9k	model stringlengths 581 35.5k	markdown stringlengths 2 6.83k	html stringlengths 29 6.92k	html_with_coordinates stringlengths 145 7.65k	lines sequence	images sequence	equations sequence	tables sequence	pdf_info stringclasses 9 values
0003244v1	2	[ 612, 792 ]	{ "type": [ "text", "text", "title", "text", "text", "text", "interline_equation", "text", "interline_equation", "text", "text", "text", "text", "discarded", "discarded" ], "coordinates": [ [ 207, 143, 813, 252 ], [ 207, 253, 813, 329 ], [ 364, 340, 657, 356 ], [ 207, 363, 813, 474 ], [ 209, 475, 813, 539 ], [ 207, 544, 813, 590 ], [ 426, 599, 593, 614 ], [ 210, 619, 282, 632 ], [ 384, 640, 637, 655 ], [ 209, 660, 813, 740 ], [ 209, 746, 814, 793 ], [ 209, 800, 814, 849 ], [ 209, 854, 813, 906 ], [ 398, 116, 624, 128 ], [ 801, 117, 813, 126 ] ], "content": [ "We mention one last feature gleaned from the table. It follows from conditional Odlyzko bounds (assuming the Generalized Riemann Hypothesis) that those qua- dratic fields with rank and discriminant have finite class field tower; unconditional proofs are not known. Hence, conditionally, we conclude that those with discriminants , and have finite (2-)class field tower even though rank . Of course, it would be interesting to determine the length of their towers.", "The structure of this paper is as follows: we use results from group theory developed in Section 2 to pull down the condition rank from the field with degree to a subfield of with degree 8. Using the arithmetic of dihedral fields from Section 4 we then go down to the field of degree 4 occurring in Theorem 1.", "2. Group Theoretic Preliminaries", "Let be a group. If , then we let denote the commutator of and . If and are nonempty subsets of , then denotes the subgroup of generated by the set . The lower central series of is defined inductively by: and for . The derived series is defined inductively by: and for . Notice that the commutator subgroup, , of .", "Throughout this section, we assume that is a finite, nonmetacyclic, 2-group such that its abelianization is of type for some positive integer (necessarily ). Let , where mod (actually since is nonmetacyclic, cf. [1]); and for .", "Lemma 1. Let be as above (but not necessarily metabelian). Suppose that where denotes the minimal number of generators of the derived group of . Then", "", "moreover,", "", "Proof. By the Burnside Basis Theorem, , where is the Frattini subgroup of , i.e. the intersection of all maximal subgroups of , see [5]. But in the case of a 2-group, , see [8]. By Blackburn, [3], since has elementary derived group, we know that . Again, by the Burnside Basis Theorem, . 口", "Lemma 2. Let be as above. Moreover, assume is metabelian. Let be a maximal subgroup of such that is cyclic, and denote the index by . Then contains an element of order .", "Proof. Without loss of generality, let . Notice that and by our presentation of , . Thus, . But since , the order of is . This establishes the lemma. 口", "Lemma 3. Let be as above and again assume is metabelian. Let be a maximal subgroup of such that is cyclic, and assume that 0 mod 4. If , then and for .", "IMAGINARY QUADRATIC FIELDS", "3" ], "index": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 ] }	[{"type": "text", "text": "We mention one last feature gleaned from the table. It follows from conditional Odlyzko bounds (assuming the Generalized Riemann Hypothesis) that those quadratic fields with rank $\\mathrm{Cl}_{2}(k^{1})\\geq3$ and discriminant $0>d>-2000$ have finite class field tower; unconditional proofs are not known. Hence, conditionally, we conclude that those $k$ with discriminants $-1015$ , $-1595$ and $-1780$ have finite (2-)class field tower even though rank $\\mathrm{Cl}_{2}(k^{1})\\geq3$ . Of course, it would be interesting to determine the length of their towers. ", "page_idx": 2}, {"type": "text", "text": "The structure of this paper is as follows: we use results from group theory developed in Section 2 to pull down the condition rank $\\mathrm{Cl}_{2}(k^{1})=2$ from the field $k^{1}$ with degree $2^{m+2}$ to a subfield $L$ of $k^{1}$ with degree 8. Using the arithmetic of dihedral fields from Section 4 we then go down to the field $K$ of degree 4 occurring in Theorem 1. ", "page_idx": 2}, {"type": "text", "text": "2. Group Theoretic Preliminaries ", "text_level": 1, "page_idx": 2}, {"type": "text", "text": "Let $G$ be a group. If $x,y\\ \\in\\ G$ , then we let $[x,y]~=~x^{-1}y^{-1}x y$ denote the commutator of $x$ and $_y$ . If $A$ and $B$ are nonempty subsets of $G$ , then $[A,B]$ denotes the subgroup of $G$ generated by the set $\\{[a,b]:a\\in A,b\\in B\\}$ . The lower central series $\\{G_{j}\\}$ of $G$ is defined inductively by: $G_{1}\\,=\\,G$ and $G_{j+1}\\,=\\,[G,G_{j}]$ for $j~\\geq~1$ . The derived series $\\{G^{(n)}\\}$ is defined inductively by: $G^{(0)}\\ =\\ G$ and $G^{(n+1)}=[G^{(n)},G^{(n)}]$ for $n\\geq0$ . Notice that $G^{(1)}=G_{2}=[G,G]$ the commutator subgroup, $G^{\\prime}$ , of $G$ . ", "page_idx": 2}, {"type": "text", "text": "Throughout this section, we assume that $G$ is a finite, nonmetacyclic, 2-group such that its abelianization $G^{\\mathrm{ab}}=G/G^{\\prime}$ is of type $(2,2^{m})$ for some positive integer ${\\boldsymbol{r}}n$ (necessarily $\\geq2$ ). Let $G=\\langle a,b\\rangle$ , where $a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1$ mod $G_{2}$ (actually $\\mathrm{mod}G_{3}$ since $G$ is nonmetacyclic, cf. [1]); $c_{2}=[a,b]$ and $c_{j+1}=[b,c_{j}]$ for $j\\geq2$ . ", "page_idx": 2}, {"type": "text", "text": "Lemma 1. Let $G$ be as above (but not necessarily metabelian). Suppose that $d(G^{\\prime})\\,=\\,n$ where $d(G^{\\prime})$ denotes the minimal number of generators of the derived group $G^{\\prime}=G_{2}$ of $G$ . Then ", "page_idx": 2}, {"type": "equation", "text": "$$\nG^{\\prime}=\\langle c_{2},c_{3},\\cdot\\cdot\\cdot,c_{n+1}\\rangle;\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "moreover, ", "page_idx": 2}, {"type": "equation", "text": "$$\nG_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle.\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "Proof. By the Burnside Basis Theorem, $d(G_{2})=d(G_{2}/\\Phi(G))$ , where $\\Phi(G)$ is the Frattini subgroup of $G$ , i.e. the intersection of all maximal subgroups of $G$ , see [5]. But in the case of a 2-group, $\\Phi(G)=G^{2}$ , see [8]. By Blackburn, [3], since $G/G_{2}^{2}$ has elementary derived group, we know that $G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle$ . Again, by the Burnside Basis Theorem, $G_{2}=\\langle c_{2},\\cdots,c_{n+1}\\rangle$ . \u53e3 ", "page_idx": 2}, {"type": "text", "text": "Lemma 2. Let $G$ be as above. Moreover, assume $G$ is metabelian. Let $H$ be a maximal subgroup of $G$ such that $H/G^{\\prime}$ is cyclic, and denote the index $\\left(G^{\\prime}:H^{\\prime}\\right)$ by $2^{\\kappa}$ . Then $G^{\\prime}$ contains an element of order $2^{\\kappa}$ . ", "page_idx": 2}, {"type": "text", "text": "Proof. Without loss of generality, let $H\\,=\\,\\left\\langle{b,G^{\\prime}}\\right\\rangle$ . Notice that $G^{\\prime}\\,=\\,\\langle c_{2},c_{3},\\cdot\\cdot\\cdot\\rangle$ and by our presentation of $H$ , $H^{\\prime}=\\langle c_{3},c_{4},\\cdot\\cdot\\cdot\\rangle$ . Thus, $G^{\\prime}/H^{\\prime}=\\langle c_{2}H^{\\prime}\\rangle$ . But since $(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,$ , the order of $c_{2}$ is $\\geq2^{\\kappa}$ . This establishes the lemma. \u53e3 ", "page_idx": 2}, {"type": "text", "text": "Lemma 3. Let $G$ be as above and again assume $G$ is metabelian. Let $H$ be a maximal subgroup of $G$ such that $H/G^{\\prime}$ is cyclic, and assume that $(G^{\\prime}\\,:\\,H^{\\prime})\\;\\equiv\\;$ 0 mod 4. If $d(G^{\\prime})=2$ , then $G_{2}=\\langle c_{2},c_{3}\\rangle$ and $G_{j}=\\langle{c_{2}^{2^{j-2}},c_{3}^{2^{j-3}}}\\rangle$ for $j>2$ . ", "page_idx": 2}]	{"preproc_blocks": [{"type": "text", "bbox": [124, 111, 486, 195], "lines": [{"bbox": [137, 114, 486, 126], "spans": [{"bbox": [137, 114, 486, 126], "score": 1.0, "content": "We mention one last feature gleaned from the table. It follows from conditional", "type": "text"}], "index": 0}, {"bbox": [125, 126, 486, 139], "spans": [{"bbox": [125, 126, 486, 139], "score": 1.0, "content": "Odlyzko bounds (assuming the Generalized Riemann Hypothesis) that those qua-", "type": "text"}], "index": 1}, {"bbox": [126, 138, 486, 150], "spans": [{"bbox": [126, 138, 222, 150], "score": 1.0, "content": "dratic fields with rank", "type": "text"}, {"bbox": [222, 139, 273, 150], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k^{1})\\geq3", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [273, 138, 350, 150], "score": 1.0, "content": " and discriminant ", "type": "text"}, {"bbox": [351, 140, 415, 148], "score": 0.9, "content": "0>d>-2000", "type": "inline_equation", "height": 8, "width": 64}, {"bbox": [415, 138, 486, 150], "score": 1.0, "content": " have finite class", "type": "text"}], "index": 2}, {"bbox": [125, 150, 486, 162], "spans": [{"bbox": [125, 150, 486, 162], "score": 1.0, "content": "field tower; unconditional proofs are not known. Hence, conditionally, we conclude", "type": "text"}], "index": 3}, {"bbox": [126, 163, 485, 174], "spans": [{"bbox": [126, 163, 173, 174], "score": 1.0, "content": "that those ", "type": "text"}, {"bbox": [173, 164, 179, 171], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [179, 163, 266, 174], "score": 1.0, "content": " with discriminants ", "type": "text"}, {"bbox": [266, 164, 294, 172], "score": 0.47, "content": "-1015", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [294, 163, 298, 174], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [298, 164, 326, 172], "score": 0.51, "content": "-1595", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [326, 163, 348, 174], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [348, 164, 376, 172], "score": 0.81, "content": "-1780", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [376, 163, 485, 174], "score": 1.0, "content": " have finite (2-)class field", "type": "text"}], "index": 4}, {"bbox": [125, 174, 485, 186], "spans": [{"bbox": [125, 174, 228, 186], "score": 1.0, "content": "tower even though rank", "type": "text"}, {"bbox": [228, 175, 279, 186], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})\\geq3", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [279, 174, 485, 186], "score": 1.0, "content": ". Of course, it would be interesting to determine", "type": "text"}], "index": 5}, {"bbox": [126, 186, 240, 198], "spans": [{"bbox": [126, 186, 240, 198], "score": 1.0, "content": "the length of their towers.", "type": "text"}], "index": 6}], "index": 3}, {"type": "text", "bbox": [124, 196, 486, 255], "lines": [{"bbox": [137, 197, 485, 210], "spans": [{"bbox": [137, 197, 485, 210], "score": 1.0, "content": "The structure of this paper is as follows: we use results from group theory", "type": "text"}], "index": 7}, {"bbox": [125, 209, 487, 222], "spans": [{"bbox": [125, 209, 370, 222], "score": 1.0, "content": "developed in Section 2 to pull down the condition rank", "type": "text"}, {"bbox": [370, 210, 422, 221], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})=2", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [422, 209, 487, 222], "score": 1.0, "content": " from the field", "type": "text"}], "index": 8}, {"bbox": [126, 220, 488, 235], "spans": [{"bbox": [126, 222, 136, 231], "score": 0.91, "content": "k^{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 220, 194, 235], "score": 1.0, "content": " with degree ", "type": "text"}, {"bbox": [194, 223, 217, 231], "score": 0.92, "content": "2^{m+2}", "type": "inline_equation", "height": 8, "width": 23}, {"bbox": [217, 220, 278, 235], "score": 1.0, "content": " to a subfield ", "type": "text"}, {"bbox": [279, 224, 286, 231], "score": 0.89, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [286, 220, 300, 235], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [301, 222, 311, 231], "score": 0.92, "content": "k^{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [311, 220, 488, 235], "score": 1.0, "content": " with degree 8. Using the arithmetic of", "type": "text"}], "index": 9}, {"bbox": [125, 233, 486, 247], "spans": [{"bbox": [125, 233, 381, 247], "score": 1.0, "content": "dihedral fields from Section 4 we then go down to the field ", "type": "text"}, {"bbox": [381, 236, 391, 243], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [392, 233, 486, 247], "score": 1.0, "content": " of degree 4 occurring", "type": "text"}], "index": 10}, {"bbox": [126, 246, 188, 257], "spans": [{"bbox": [126, 246, 188, 257], "score": 1.0, "content": "in Theorem 1.", "type": "text"}], "index": 11}], "index": 9}, {"type": "title", "bbox": [218, 263, 393, 276], "lines": [{"bbox": [217, 266, 394, 277], "spans": [{"bbox": [217, 266, 394, 277], "score": 1.0, "content": "2. Group Theoretic Preliminaries", "type": "text"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [124, 281, 486, 367], "lines": [{"bbox": [136, 284, 486, 296], "spans": [{"bbox": [136, 284, 157, 296], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [157, 286, 165, 293], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [166, 284, 243, 296], "score": 1.0, "content": " be a group. If ", "type": "text"}, {"bbox": [244, 286, 285, 295], "score": 0.93, "content": "x,y\\ \\in\\ G", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [285, 284, 349, 296], "score": 1.0, "content": ", then we let ", "type": "text"}, {"bbox": [349, 285, 433, 296], "score": 0.92, "content": "[x,y]~=~x^{-1}y^{-1}x y", "type": "inline_equation", "height": 11, "width": 84}, {"bbox": [433, 284, 486, 296], "score": 1.0, "content": " denote the", "type": "text"}], "index": 13}, {"bbox": [125, 296, 484, 309], "spans": [{"bbox": [125, 296, 196, 309], "score": 1.0, "content": "commutator of ", "type": "text"}, {"bbox": [196, 300, 203, 305], "score": 0.89, "content": "x", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [203, 296, 228, 309], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [228, 300, 234, 307], "score": 0.88, "content": "_y", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [234, 296, 257, 309], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [257, 298, 265, 305], "score": 0.89, "content": "A", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [265, 296, 290, 309], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [290, 298, 299, 305], "score": 0.9, "content": "B", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [299, 296, 419, 309], "score": 1.0, "content": " are nonempty subsets of ", "type": "text"}, {"bbox": [419, 298, 427, 305], "score": 0.86, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [427, 296, 459, 309], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [459, 297, 484, 308], "score": 0.93, "content": "[A,B]", "type": "inline_equation", "height": 11, "width": 25}], "index": 14}, {"bbox": [126, 308, 486, 320], "spans": [{"bbox": [126, 308, 234, 320], "score": 1.0, "content": "denotes the subgroup of ", "type": "text"}, {"bbox": [235, 310, 243, 317], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [243, 308, 339, 320], "score": 1.0, "content": " generated by the set", "type": "text"}, {"bbox": [339, 309, 433, 320], "score": 0.92, "content": "\\{[a,b]:a\\in A,b\\in B\\}", "type": "inline_equation", "height": 11, "width": 94}, {"bbox": [434, 308, 486, 320], "score": 1.0, "content": ". The lower", "type": "text"}], "index": 15}, {"bbox": [124, 320, 485, 334], "spans": [{"bbox": [124, 320, 187, 334], "score": 1.0, "content": "central series ", "type": "text"}, {"bbox": [187, 321, 209, 332], "score": 0.94, "content": "\\{G_{j}\\}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [210, 320, 226, 334], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [226, 322, 234, 329], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [234, 320, 355, 334], "score": 1.0, "content": " is defined inductively by: ", "type": "text"}, {"bbox": [356, 322, 392, 331], "score": 0.92, "content": "G_{1}\\,=\\,G", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [393, 320, 416, 334], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [416, 321, 485, 332], "score": 0.92, "content": "G_{j+1}\\,=\\,[G,G_{j}]", "type": "inline_equation", "height": 11, "width": 69}], "index": 16}, {"bbox": [124, 332, 487, 345], "spans": [{"bbox": [124, 332, 142, 345], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [142, 335, 170, 344], "score": 0.92, "content": "j~\\geq~1", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [170, 332, 266, 345], "score": 1.0, "content": ". The derived series ", "type": "text"}, {"bbox": [267, 333, 297, 344], "score": 0.94, "content": "\\{G^{(n)}\\}", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [297, 332, 420, 345], "score": 1.0, "content": " is defined inductively by: ", "type": "text"}, {"bbox": [420, 333, 464, 342], "score": 0.9, "content": "G^{(0)}\\ =\\ G", "type": "inline_equation", "height": 9, "width": 44}, {"bbox": [465, 332, 487, 345], "score": 1.0, "content": " and", "type": "text"}], "index": 17}, {"bbox": [126, 342, 487, 359], "spans": [{"bbox": [126, 345, 219, 357], "score": 0.92, "content": "G^{(n+1)}=[G^{(n)},G^{(n)}]", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [219, 342, 238, 359], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [238, 347, 264, 356], "score": 0.92, "content": "n\\geq0", "type": "inline_equation", "height": 9, "width": 26}, {"bbox": [264, 342, 325, 359], "score": 1.0, "content": ". Notice that ", "type": "text"}, {"bbox": [325, 345, 410, 357], "score": 0.92, "content": "G^{(1)}=G_{2}=[G,G]", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [411, 342, 487, 359], "score": 1.0, "content": " the commutator", "type": "text"}], "index": 18}, {"bbox": [125, 357, 212, 370], "spans": [{"bbox": [125, 357, 172, 370], "score": 1.0, "content": "subgroup, ", "type": "text"}, {"bbox": [172, 358, 183, 366], "score": 0.89, "content": "G^{\\prime}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [183, 357, 200, 370], "score": 1.0, "content": ", of ", "type": "text"}, {"bbox": [200, 359, 208, 366], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [208, 357, 212, 370], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 16}, {"type": "text", "bbox": [125, 368, 486, 417], "lines": [{"bbox": [137, 368, 486, 383], "spans": [{"bbox": [137, 368, 322, 383], "score": 1.0, "content": "Throughout this section, we assume that ", "type": "text"}, {"bbox": [323, 371, 331, 378], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [331, 368, 486, 383], "score": 1.0, "content": " is a finite, nonmetacyclic, 2-group", "type": "text"}], "index": 20}, {"bbox": [125, 379, 486, 394], "spans": [{"bbox": [125, 379, 246, 394], "score": 1.0, "content": "such that its abelianization ", "type": "text"}, {"bbox": [246, 382, 300, 393], "score": 0.93, "content": "G^{\\mathrm{ab}}=G/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 54}, {"bbox": [300, 379, 346, 394], "score": 1.0, "content": " is of type ", "type": "text"}, {"bbox": [346, 382, 376, 393], "score": 0.91, "content": "(2,2^{m})", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [376, 379, 486, 394], "score": 1.0, "content": " for some positive integer", "type": "text"}], "index": 21}, {"bbox": [126, 392, 484, 406], "spans": [{"bbox": [126, 398, 135, 402], "score": 0.86, "content": "{\\boldsymbol{r}}n", "type": "inline_equation", "height": 4, "width": 9}, {"bbox": [135, 392, 191, 406], "score": 1.0, "content": " (necessarily ", "type": "text"}, {"bbox": [191, 396, 207, 403], "score": 0.88, "content": "\\geq2", "type": "inline_equation", "height": 7, "width": 16}, {"bbox": [207, 392, 235, 406], "score": 1.0, "content": "). Let ", "type": "text"}, {"bbox": [236, 394, 279, 405], "score": 0.94, "content": "G=\\langle a,b\\rangle", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [279, 392, 313, 406], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [313, 393, 370, 402], "score": 0.9, "content": "a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1", "type": "inline_equation", "height": 9, "width": 57}, {"bbox": [370, 392, 394, 406], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [394, 395, 407, 404], "score": 0.9, "content": "G_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [407, 392, 452, 406], "score": 1.0, "content": " (actually ", "type": "text"}, {"bbox": [452, 395, 484, 404], "score": 0.63, "content": "\\mathrm{mod}G_{3}", "type": "inline_equation", "height": 9, "width": 32}], "index": 22}, {"bbox": [124, 404, 440, 419], "spans": [{"bbox": [124, 404, 150, 419], "score": 1.0, "content": "since ", "type": "text"}, {"bbox": [150, 407, 158, 414], "score": 0.91, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [159, 404, 274, 419], "score": 1.0, "content": " is nonmetacyclic, cf. [1]); ", "type": "text"}, {"bbox": [275, 406, 316, 417], "score": 0.94, "content": "c_{2}=[a,b]", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [317, 404, 338, 419], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [339, 406, 394, 417], "score": 0.95, "content": "c_{j+1}=[b,c_{j}]", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [394, 404, 411, 419], "score": 1.0, "content": " for", "type": "text"}, {"bbox": [412, 407, 435, 416], "score": 0.92, "content": "j\\geq2", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [435, 404, 440, 419], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 21.5}, {"type": "text", "bbox": [124, 421, 486, 457], "lines": [{"bbox": [125, 423, 487, 435], "spans": [{"bbox": [125, 423, 199, 435], "score": 1.0, "content": "Lemma 1. Let ", "type": "text"}, {"bbox": [200, 425, 208, 432], "score": 0.87, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [208, 423, 487, 435], "score": 1.0, "content": " be as above (but not necessarily metabelian). Suppose that", "type": "text"}], "index": 24}, {"bbox": [126, 434, 487, 448], "spans": [{"bbox": [126, 436, 172, 447], "score": 0.93, "content": "d(G^{\\prime})\\,=\\,n", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [172, 434, 204, 448], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [205, 436, 228, 447], "score": 0.93, "content": "d(G^{\\prime})", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [228, 434, 487, 448], "score": 1.0, "content": " denotes the minimal number of generators of the derived", "type": "text"}], "index": 25}, {"bbox": [126, 447, 245, 460], "spans": [{"bbox": [126, 447, 153, 460], "score": 1.0, "content": "group ", "type": "text"}, {"bbox": [153, 449, 190, 458], "score": 0.93, "content": "G^{\\prime}=G_{2}", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [190, 447, 204, 460], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [205, 449, 213, 456], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [213, 447, 245, 460], "score": 1.0, "content": ". Then", "type": "text"}], "index": 26}], "index": 25}, {"type": "interline_equation", "bbox": [255, 464, 355, 475], "lines": [{"bbox": [255, 464, 355, 475], "spans": [{"bbox": [255, 464, 355, 475], "score": 0.89, "content": "G^{\\prime}=\\langle c_{2},c_{3},\\cdot\\cdot\\cdot,c_{n+1}\\rangle;", "type": "interline_equation"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [126, 479, 169, 489], "lines": [{"bbox": [126, 481, 169, 491], "spans": [{"bbox": [126, 481, 169, 491], "score": 1.0, "content": "moreover,", "type": "text"}], "index": 28}], "index": 28}, {"type": "interline_equation", "bbox": [230, 495, 381, 507], "lines": [{"bbox": [230, 495, 381, 507], "spans": [{"bbox": [230, 495, 381, 507], "score": 0.9, "content": "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle.", "type": "interline_equation"}], "index": 29}], "index": 29}, {"type": "text", "bbox": [125, 511, 486, 573], "lines": [{"bbox": [126, 514, 486, 526], "spans": [{"bbox": [126, 514, 305, 526], "score": 1.0, "content": "Proof. By the Burnside Basis Theorem, ", "type": "text"}, {"bbox": [305, 515, 398, 525], "score": 0.93, "content": "d(G_{2})=d(G_{2}/\\Phi(G))", "type": "inline_equation", "height": 10, "width": 93}, {"bbox": [398, 514, 433, 526], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [434, 515, 457, 525], "score": 0.95, "content": "\\Phi(G)", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [457, 514, 486, 526], "score": 1.0, "content": " is the", "type": "text"}], "index": 30}, {"bbox": [126, 526, 485, 538], "spans": [{"bbox": [126, 526, 216, 538], "score": 1.0, "content": "Frattini subgroup of ", "type": "text"}, {"bbox": [216, 528, 224, 535], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [225, 526, 442, 538], "score": 1.0, "content": " , i.e. the intersection of all maximal subgroups of ", "type": "text"}, {"bbox": [442, 528, 450, 535], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [451, 526, 485, 538], "score": 1.0, "content": ", see [5].", "type": "text"}], "index": 31}, {"bbox": [124, 537, 485, 551], "spans": [{"bbox": [124, 537, 258, 551], "score": 1.0, "content": "But in the case of a 2-group, ", "type": "text"}, {"bbox": [258, 538, 308, 549], "score": 0.94, "content": "\\Phi(G)=G^{2}", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [308, 537, 459, 551], "score": 1.0, "content": ", see [8]. By Blackburn, [3], since ", "type": "text"}, {"bbox": [460, 538, 485, 549], "score": 0.94, "content": "G/G_{2}^{2}", "type": "inline_equation", "height": 11, "width": 25}], "index": 32}, {"bbox": [124, 549, 485, 564], "spans": [{"bbox": [124, 549, 329, 564], "score": 1.0, "content": "has elementary derived group, we know that ", "type": "text"}, {"bbox": [329, 550, 482, 561], "score": 0.91, "content": "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle", "type": "inline_equation", "height": 11, "width": 153}, {"bbox": [483, 549, 485, 564], "score": 1.0, "content": ".", "type": "text"}], "index": 33}, {"bbox": [126, 560, 486, 575], "spans": [{"bbox": [126, 560, 301, 575], "score": 1.0, "content": "Again, by the Burnside Basis Theorem, ", "type": "text"}, {"bbox": [302, 563, 388, 573], "score": 0.9, "content": "G_{2}=\\langle c_{2},\\cdots,c_{n+1}\\rangle", "type": "inline_equation", "height": 10, "width": 86}, {"bbox": [388, 560, 392, 575], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [475, 562, 486, 572], "score": 0.9934230446815491, "content": "\u53e3", "type": "text"}], "index": 34}], "index": 32}, {"type": "text", "bbox": [125, 577, 487, 614], "lines": [{"bbox": [126, 580, 486, 592], "spans": [{"bbox": [126, 580, 198, 592], "score": 1.0, "content": "Lemma 2. Let ", "type": "text"}, {"bbox": [199, 582, 207, 589], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [207, 580, 354, 592], "score": 1.0, "content": " be as above. Moreover, assume ", "type": "text"}, {"bbox": [355, 582, 363, 589], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [363, 580, 452, 592], "score": 1.0, "content": " is metabelian. Let ", "type": "text"}, {"bbox": [453, 582, 462, 589], "score": 0.85, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [462, 580, 486, 592], "score": 1.0, "content": " be a", "type": "text"}], "index": 35}, {"bbox": [126, 592, 486, 604], "spans": [{"bbox": [126, 592, 218, 604], "score": 1.0, "content": "maximal subgroup of ", "type": "text"}, {"bbox": [218, 594, 226, 601], "score": 0.87, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [226, 592, 271, 604], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [272, 593, 295, 603], "score": 0.92, "content": "H/G^{\\prime}", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [296, 592, 434, 604], "score": 1.0, "content": " is cyclic, and denote the index ", "type": "text"}, {"bbox": [434, 592, 472, 603], "score": 0.8, "content": "\\left(G^{\\prime}:H^{\\prime}\\right)", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [472, 592, 486, 604], "score": 1.0, "content": " by", "type": "text"}], "index": 36}, {"bbox": [126, 603, 327, 615], "spans": [{"bbox": [126, 605, 136, 613], "score": 0.88, "content": "2^{\\kappa}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [137, 603, 169, 615], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [170, 605, 180, 613], "score": 0.9, "content": "G^{\\prime}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [181, 603, 312, 615], "score": 1.0, "content": " contains an element of order ", "type": "text"}, {"bbox": [312, 605, 323, 613], "score": 0.88, "content": "2^{\\kappa}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [323, 603, 327, 615], "score": 1.0, "content": ".", "type": "text"}], "index": 37}], "index": 36}, {"type": "text", "bbox": [125, 619, 487, 657], "lines": [{"bbox": [126, 621, 485, 634], "spans": [{"bbox": [126, 621, 294, 634], "score": 1.0, "content": "Proof. Without loss of generality, let ", "type": "text"}, {"bbox": [294, 622, 347, 633], "score": 0.93, "content": "H\\,=\\,\\left\\langle{b,G^{\\prime}}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 53}, {"bbox": [348, 621, 411, 634], "score": 1.0, "content": ". Notice that ", "type": "text"}, {"bbox": [412, 621, 485, 633], "score": 0.92, "content": "G^{\\prime}\\,=\\,\\langle c_{2},c_{3},\\cdot\\cdot\\cdot\\rangle", "type": "inline_equation", "height": 12, "width": 73}], "index": 38}, {"bbox": [126, 633, 487, 646], "spans": [{"bbox": [126, 633, 245, 646], "score": 1.0, "content": "and by our presentation of ", "type": "text"}, {"bbox": [245, 636, 254, 643], "score": 0.87, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [255, 633, 260, 646], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [260, 635, 332, 645], "score": 0.92, "content": "H^{\\prime}=\\langle c_{3},c_{4},\\cdot\\cdot\\cdot\\rangle", "type": "inline_equation", "height": 10, "width": 72}, {"bbox": [332, 633, 366, 646], "score": 1.0, "content": ". Thus,", "type": "text"}, {"bbox": [367, 634, 437, 645], "score": 0.93, "content": "G^{\\prime}/H^{\\prime}=\\langle c_{2}H^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [437, 633, 487, 646], "score": 1.0, "content": ". But since", "type": "text"}], "index": 39}, {"bbox": [126, 646, 486, 658], "spans": [{"bbox": [126, 647, 188, 657], "score": 0.92, "content": "(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [188, 646, 248, 658], "score": 1.0, "content": ", the order of ", "type": "text"}, {"bbox": [249, 650, 258, 656], "score": 0.88, "content": "c_{2}", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [258, 646, 270, 658], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [271, 648, 291, 656], "score": 0.9, "content": "\\geq2^{\\kappa}", "type": "inline_equation", "height": 8, "width": 20}, {"bbox": [292, 646, 421, 658], "score": 1.0, "content": ". This establishes the lemma.", "type": "text"}, {"bbox": [475, 646, 486, 656], "score": 0.991905927658081, "content": "\u53e3", "type": "text"}], "index": 40}], "index": 39}, {"type": "text", "bbox": [125, 661, 486, 701], "lines": [{"bbox": [125, 664, 487, 676], "spans": [{"bbox": [125, 664, 199, 676], "score": 1.0, "content": "Lemma 3. Let ", "type": "text"}, {"bbox": [199, 666, 207, 673], "score": 0.86, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [207, 664, 351, 676], "score": 1.0, "content": " be as above and again assume ", "type": "text"}, {"bbox": [352, 664, 361, 673], "score": 0.76, "content": "G", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [361, 664, 451, 676], "score": 1.0, "content": " is metabelian. Let ", "type": "text"}, {"bbox": [452, 664, 462, 673], "score": 0.76, "content": "H", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [462, 664, 487, 676], "score": 1.0, "content": " be a", "type": "text"}], "index": 41}, {"bbox": [126, 675, 486, 687], "spans": [{"bbox": [126, 676, 221, 687], "score": 1.0, "content": "maximal subgroup of ", "type": "text"}, {"bbox": [222, 677, 230, 685], "score": 0.85, "content": "G", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [230, 676, 279, 687], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [279, 676, 304, 687], "score": 0.89, "content": "H/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [304, 676, 430, 687], "score": 1.0, "content": " is cyclic, and assume that ", "type": "text"}, {"bbox": [430, 675, 486, 687], "score": 0.87, "content": "(G^{\\prime}\\,:\\,H^{\\prime})\\;\\equiv\\;", "type": "inline_equation", "height": 12, "width": 56}], "index": 42}, {"bbox": [124, 687, 454, 702], "spans": [{"bbox": [124, 689, 178, 702], "score": 1.0, "content": "0 mod 4. If ", "type": "text"}, {"bbox": [179, 690, 221, 701], "score": 0.93, "content": "d(G^{\\prime})=2", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [221, 689, 249, 702], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [249, 690, 305, 701], "score": 0.93, "content": "G_{2}=\\langle c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 56}, {"bbox": [305, 689, 327, 702], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [327, 687, 408, 701], "score": 0.93, "content": "G_{j}=\\langle{c_{2}^{2^{j-2}},c_{3}^{2^{j-3}}}\\rangle", "type": "inline_equation", "height": 14, "width": 81}, {"bbox": [408, 689, 426, 702], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [426, 689, 451, 700], "score": 0.88, "content": "j>2", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [451, 689, 454, 702], "score": 1.0, "content": ".", "type": "text"}], "index": 43}], "index": 42}], "layout_bboxes": [], "page_idx": 2, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [255, 464, 355, 475], "lines": [{"bbox": [255, 464, 355, 475], "spans": [{"bbox": [255, 464, 355, 475], "score": 0.89, "content": "G^{\\prime}=\\langle c_{2},c_{3},\\cdot\\cdot\\cdot,c_{n+1}\\rangle;", "type": "interline_equation"}], "index": 27}], "index": 27}, {"type": "interline_equation", "bbox": [230, 495, 381, 507], "lines": [{"bbox": [230, 495, 381, 507], "spans": [{"bbox": [230, 495, 381, 507], "score": 0.9, "content": "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle.", "type": "interline_equation"}], "index": 29}], "index": 29}], "discarded_blocks": [{"type": "discarded", "bbox": [238, 90, 373, 99], "lines": [{"bbox": [239, 93, 372, 100], "spans": [{"bbox": [239, 93, 372, 100], "score": 1.0, "content": "IMAGINARY QUADRATIC FIELDS", "type": "text"}]}]}, {"type": "discarded", "bbox": [479, 91, 486, 98], "lines": [{"bbox": [480, 93, 486, 101], "spans": [{"bbox": [480, 93, 486, 101], "score": 1.0, "content": "3", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [124, 111, 486, 195], "lines": [{"bbox": [137, 114, 486, 126], "spans": [{"bbox": [137, 114, 486, 126], "score": 1.0, "content": "We mention one last feature gleaned from the table. It follows from conditional", "type": "text"}], "index": 0}, {"bbox": [125, 126, 486, 139], "spans": [{"bbox": [125, 126, 486, 139], "score": 1.0, "content": "Odlyzko bounds (assuming the Generalized Riemann Hypothesis) that those qua-", "type": "text"}], "index": 1}, {"bbox": [126, 138, 486, 150], "spans": [{"bbox": [126, 138, 222, 150], "score": 1.0, "content": "dratic fields with rank", "type": "text"}, {"bbox": [222, 139, 273, 150], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k^{1})\\geq3", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [273, 138, 350, 150], "score": 1.0, "content": " and discriminant ", "type": "text"}, {"bbox": [351, 140, 415, 148], "score": 0.9, "content": "0>d>-2000", "type": "inline_equation", "height": 8, "width": 64}, {"bbox": [415, 138, 486, 150], "score": 1.0, "content": " have finite class", "type": "text"}], "index": 2}, {"bbox": [125, 150, 486, 162], "spans": [{"bbox": [125, 150, 486, 162], "score": 1.0, "content": "field tower; unconditional proofs are not known. Hence, conditionally, we conclude", "type": "text"}], "index": 3}, {"bbox": [126, 163, 485, 174], "spans": [{"bbox": [126, 163, 173, 174], "score": 1.0, "content": "that those ", "type": "text"}, {"bbox": [173, 164, 179, 171], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [179, 163, 266, 174], "score": 1.0, "content": " with discriminants ", "type": "text"}, {"bbox": [266, 164, 294, 172], "score": 0.47, "content": "-1015", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [294, 163, 298, 174], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [298, 164, 326, 172], "score": 0.51, "content": "-1595", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [326, 163, 348, 174], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [348, 164, 376, 172], "score": 0.81, "content": "-1780", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [376, 163, 485, 174], "score": 1.0, "content": " have finite (2-)class field", "type": "text"}], "index": 4}, {"bbox": [125, 174, 485, 186], "spans": [{"bbox": [125, 174, 228, 186], "score": 1.0, "content": "tower even though rank", "type": "text"}, {"bbox": [228, 175, 279, 186], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})\\geq3", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [279, 174, 485, 186], "score": 1.0, "content": ". Of course, it would be interesting to determine", "type": "text"}], "index": 5}, {"bbox": [126, 186, 240, 198], "spans": [{"bbox": [126, 186, 240, 198], "score": 1.0, "content": "the length of their towers.", "type": "text"}], "index": 6}], "index": 3, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [125, 114, 486, 198]}, {"type": "text", "bbox": [124, 196, 486, 255], "lines": [{"bbox": [137, 197, 485, 210], "spans": [{"bbox": [137, 197, 485, 210], "score": 1.0, "content": "The structure of this paper is as follows: we use results from group theory", "type": "text"}], "index": 7}, {"bbox": [125, 209, 487, 222], "spans": [{"bbox": [125, 209, 370, 222], "score": 1.0, "content": "developed in Section 2 to pull down the condition rank", "type": "text"}, {"bbox": [370, 210, 422, 221], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})=2", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [422, 209, 487, 222], "score": 1.0, "content": " from the field", "type": "text"}], "index": 8}, {"bbox": [126, 220, 488, 235], "spans": [{"bbox": [126, 222, 136, 231], "score": 0.91, "content": "k^{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 220, 194, 235], "score": 1.0, "content": " with degree ", "type": "text"}, {"bbox": [194, 223, 217, 231], "score": 0.92, "content": "2^{m+2}", "type": "inline_equation", "height": 8, "width": 23}, {"bbox": [217, 220, 278, 235], "score": 1.0, "content": " to a subfield ", "type": "text"}, {"bbox": [279, 224, 286, 231], "score": 0.89, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [286, 220, 300, 235], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [301, 222, 311, 231], "score": 0.92, "content": "k^{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [311, 220, 488, 235], "score": 1.0, "content": " with degree 8. Using the arithmetic of", "type": "text"}], "index": 9}, {"bbox": [125, 233, 486, 247], "spans": [{"bbox": [125, 233, 381, 247], "score": 1.0, "content": "dihedral fields from Section 4 we then go down to the field ", "type": "text"}, {"bbox": [381, 236, 391, 243], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [392, 233, 486, 247], "score": 1.0, "content": " of degree 4 occurring", "type": "text"}], "index": 10}, {"bbox": [126, 246, 188, 257], "spans": [{"bbox": [126, 246, 188, 257], "score": 1.0, "content": "in Theorem 1.", "type": "text"}], "index": 11}], "index": 9, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [125, 197, 488, 257]}, {"type": "title", "bbox": [218, 263, 393, 276], "lines": [{"bbox": [217, 266, 394, 277], "spans": [{"bbox": [217, 266, 394, 277], "score": 1.0, "content": "2. Group Theoretic Preliminaries", "type": "text"}], "index": 12}], "index": 12, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 281, 486, 367], "lines": [{"bbox": [136, 284, 486, 296], "spans": [{"bbox": [136, 284, 157, 296], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [157, 286, 165, 293], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [166, 284, 243, 296], "score": 1.0, "content": " be a group. If ", "type": "text"}, {"bbox": [244, 286, 285, 295], "score": 0.93, "content": "x,y\\ \\in\\ G", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [285, 284, 349, 296], "score": 1.0, "content": ", then we let ", "type": "text"}, {"bbox": [349, 285, 433, 296], "score": 0.92, "content": "[x,y]~=~x^{-1}y^{-1}x y", "type": "inline_equation", "height": 11, "width": 84}, {"bbox": [433, 284, 486, 296], "score": 1.0, "content": " denote the", "type": "text"}], "index": 13}, {"bbox": [125, 296, 484, 309], "spans": [{"bbox": [125, 296, 196, 309], "score": 1.0, "content": "commutator of ", "type": "text"}, {"bbox": [196, 300, 203, 305], "score": 0.89, "content": "x", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [203, 296, 228, 309], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [228, 300, 234, 307], "score": 0.88, "content": "_y", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [234, 296, 257, 309], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [257, 298, 265, 305], "score": 0.89, "content": "A", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [265, 296, 290, 309], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [290, 298, 299, 305], "score": 0.9, "content": "B", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [299, 296, 419, 309], "score": 1.0, "content": " are nonempty subsets of ", "type": "text"}, {"bbox": [419, 298, 427, 305], "score": 0.86, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [427, 296, 459, 309], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [459, 297, 484, 308], "score": 0.93, "content": "[A,B]", "type": "inline_equation", "height": 11, "width": 25}], "index": 14}, {"bbox": [126, 308, 486, 320], "spans": [{"bbox": [126, 308, 234, 320], "score": 1.0, "content": "denotes the subgroup of ", "type": "text"}, {"bbox": [235, 310, 243, 317], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [243, 308, 339, 320], "score": 1.0, "content": " generated by the set", "type": "text"}, {"bbox": [339, 309, 433, 320], "score": 0.92, "content": "\\{[a,b]:a\\in A,b\\in B\\}", "type": "inline_equation", "height": 11, "width": 94}, {"bbox": [434, 308, 486, 320], "score": 1.0, "content": ". The lower", "type": "text"}], "index": 15}, {"bbox": [124, 320, 485, 334], "spans": [{"bbox": [124, 320, 187, 334], "score": 1.0, "content": "central series ", "type": "text"}, {"bbox": [187, 321, 209, 332], "score": 0.94, "content": "\\{G_{j}\\}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [210, 320, 226, 334], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [226, 322, 234, 329], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [234, 320, 355, 334], "score": 1.0, "content": " is defined inductively by: ", "type": "text"}, {"bbox": [356, 322, 392, 331], "score": 0.92, "content": "G_{1}\\,=\\,G", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [393, 320, 416, 334], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [416, 321, 485, 332], "score": 0.92, "content": "G_{j+1}\\,=\\,[G,G_{j}]", "type": "inline_equation", "height": 11, "width": 69}], "index": 16}, {"bbox": [124, 332, 487, 345], "spans": [{"bbox": [124, 332, 142, 345], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [142, 335, 170, 344], "score": 0.92, "content": "j~\\geq~1", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [170, 332, 266, 345], "score": 1.0, "content": ". The derived series ", "type": "text"}, {"bbox": [267, 333, 297, 344], "score": 0.94, "content": "\\{G^{(n)}\\}", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [297, 332, 420, 345], "score": 1.0, "content": " is defined inductively by: ", "type": "text"}, {"bbox": [420, 333, 464, 342], "score": 0.9, "content": "G^{(0)}\\ =\\ G", "type": "inline_equation", "height": 9, "width": 44}, {"bbox": [465, 332, 487, 345], "score": 1.0, "content": " and", "type": "text"}], "index": 17}, {"bbox": [126, 342, 487, 359], "spans": [{"bbox": [126, 345, 219, 357], "score": 0.92, "content": "G^{(n+1)}=[G^{(n)},G^{(n)}]", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [219, 342, 238, 359], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [238, 347, 264, 356], "score": 0.92, "content": "n\\geq0", "type": "inline_equation", "height": 9, "width": 26}, {"bbox": [264, 342, 325, 359], "score": 1.0, "content": ". Notice that ", "type": "text"}, {"bbox": [325, 345, 410, 357], "score": 0.92, "content": "G^{(1)}=G_{2}=[G,G]", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [411, 342, 487, 359], "score": 1.0, "content": " the commutator", "type": "text"}], "index": 18}, {"bbox": [125, 357, 212, 370], "spans": [{"bbox": [125, 357, 172, 370], "score": 1.0, "content": "subgroup, ", "type": "text"}, {"bbox": [172, 358, 183, 366], "score": 0.89, "content": "G^{\\prime}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [183, 357, 200, 370], "score": 1.0, "content": ", of ", "type": "text"}, {"bbox": [200, 359, 208, 366], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [208, 357, 212, 370], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 16, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [124, 284, 487, 370]}, {"type": "text", "bbox": [125, 368, 486, 417], "lines": [{"bbox": [137, 368, 486, 383], "spans": [{"bbox": [137, 368, 322, 383], "score": 1.0, "content": "Throughout this section, we assume that ", "type": "text"}, {"bbox": [323, 371, 331, 378], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [331, 368, 486, 383], "score": 1.0, "content": " is a finite, nonmetacyclic, 2-group", "type": "text"}], "index": 20}, {"bbox": [125, 379, 486, 394], "spans": [{"bbox": [125, 379, 246, 394], "score": 1.0, "content": "such that its abelianization ", "type": "text"}, {"bbox": [246, 382, 300, 393], "score": 0.93, "content": "G^{\\mathrm{ab}}=G/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 54}, {"bbox": [300, 379, 346, 394], "score": 1.0, "content": " is of type ", "type": "text"}, {"bbox": [346, 382, 376, 393], "score": 0.91, "content": "(2,2^{m})", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [376, 379, 486, 394], "score": 1.0, "content": " for some positive integer", "type": "text"}], "index": 21}, {"bbox": [126, 392, 484, 406], "spans": [{"bbox": [126, 398, 135, 402], "score": 0.86, "content": "{\\boldsymbol{r}}n", "type": "inline_equation", "height": 4, "width": 9}, {"bbox": [135, 392, 191, 406], "score": 1.0, "content": " (necessarily ", "type": "text"}, {"bbox": [191, 396, 207, 403], "score": 0.88, "content": "\\geq2", "type": "inline_equation", "height": 7, "width": 16}, {"bbox": [207, 392, 235, 406], "score": 1.0, "content": "). Let ", "type": "text"}, {"bbox": [236, 394, 279, 405], "score": 0.94, "content": "G=\\langle a,b\\rangle", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [279, 392, 313, 406], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [313, 393, 370, 402], "score": 0.9, "content": "a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1", "type": "inline_equation", "height": 9, "width": 57}, {"bbox": [370, 392, 394, 406], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [394, 395, 407, 404], "score": 0.9, "content": "G_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [407, 392, 452, 406], "score": 1.0, "content": " (actually ", "type": "text"}, {"bbox": [452, 395, 484, 404], "score": 0.63, "content": "\\mathrm{mod}G_{3}", "type": "inline_equation", "height": 9, "width": 32}], "index": 22}, {"bbox": [124, 404, 440, 419], "spans": [{"bbox": [124, 404, 150, 419], "score": 1.0, "content": "since ", "type": "text"}, {"bbox": [150, 407, 158, 414], "score": 0.91, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [159, 404, 274, 419], "score": 1.0, "content": " is nonmetacyclic, cf. [1]); ", "type": "text"}, {"bbox": [275, 406, 316, 417], "score": 0.94, "content": "c_{2}=[a,b]", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [317, 404, 338, 419], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [339, 406, 394, 417], "score": 0.95, "content": "c_{j+1}=[b,c_{j}]", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [394, 404, 411, 419], "score": 1.0, "content": " for", "type": "text"}, {"bbox": [412, 407, 435, 416], "score": 0.92, "content": "j\\geq2", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [435, 404, 440, 419], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 21.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [124, 368, 486, 419]}, {"type": "text", "bbox": [124, 421, 486, 457], "lines": [{"bbox": [125, 423, 487, 435], "spans": [{"bbox": [125, 423, 199, 435], "score": 1.0, "content": "Lemma 1. Let ", "type": "text"}, {"bbox": [200, 425, 208, 432], "score": 0.87, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [208, 423, 487, 435], "score": 1.0, "content": " be as above (but not necessarily metabelian). Suppose that", "type": "text"}], "index": 24}, {"bbox": [126, 434, 487, 448], "spans": [{"bbox": [126, 436, 172, 447], "score": 0.93, "content": "d(G^{\\prime})\\,=\\,n", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [172, 434, 204, 448], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [205, 436, 228, 447], "score": 0.93, "content": "d(G^{\\prime})", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [228, 434, 487, 448], "score": 1.0, "content": " denotes the minimal number of generators of the derived", "type": "text"}], "index": 25}, {"bbox": [126, 447, 245, 460], "spans": [{"bbox": [126, 447, 153, 460], "score": 1.0, "content": "group ", "type": "text"}, {"bbox": [153, 449, 190, 458], "score": 0.93, "content": "G^{\\prime}=G_{2}", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [190, 447, 204, 460], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [205, 449, 213, 456], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [213, 447, 245, 460], "score": 1.0, "content": ". Then", "type": "text"}], "index": 26}], "index": 25, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [125, 423, 487, 460]}, {"type": "interline_equation", "bbox": [255, 464, 355, 475], "lines": [{"bbox": [255, 464, 355, 475], "spans": [{"bbox": [255, 464, 355, 475], "score": 0.89, "content": "G^{\\prime}=\\langle c_{2},c_{3},\\cdot\\cdot\\cdot,c_{n+1}\\rangle;", "type": "interline_equation"}], "index": 27}], "index": 27, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [126, 479, 169, 489], "lines": [{"bbox": [126, 481, 169, 491], "spans": [{"bbox": [126, 481, 169, 491], "score": 1.0, "content": "moreover,", "type": "text"}], "index": 28}], "index": 28, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [126, 481, 169, 491]}, {"type": "interline_equation", "bbox": [230, 495, 381, 507], "lines": [{"bbox": [230, 495, 381, 507], "spans": [{"bbox": [230, 495, 381, 507], "score": 0.9, "content": "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle.", "type": "interline_equation"}], "index": 29}], "index": 29, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 511, 486, 573], "lines": [{"bbox": [126, 514, 486, 526], "spans": [{"bbox": [126, 514, 305, 526], "score": 1.0, "content": "Proof. By the Burnside Basis Theorem, ", "type": "text"}, {"bbox": [305, 515, 398, 525], "score": 0.93, "content": "d(G_{2})=d(G_{2}/\\Phi(G))", "type": "inline_equation", "height": 10, "width": 93}, {"bbox": [398, 514, 433, 526], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [434, 515, 457, 525], "score": 0.95, "content": "\\Phi(G)", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [457, 514, 486, 526], "score": 1.0, "content": " is the", "type": "text"}], "index": 30}, {"bbox": [126, 526, 485, 538], "spans": [{"bbox": [126, 526, 216, 538], "score": 1.0, "content": "Frattini subgroup of ", "type": "text"}, {"bbox": [216, 528, 224, 535], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [225, 526, 442, 538], "score": 1.0, "content": " , i.e. the intersection of all maximal subgroups of ", "type": "text"}, {"bbox": [442, 528, 450, 535], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [451, 526, 485, 538], "score": 1.0, "content": ", see [5].", "type": "text"}], "index": 31}, {"bbox": [124, 537, 485, 551], "spans": [{"bbox": [124, 537, 258, 551], "score": 1.0, "content": "But in the case of a 2-group, ", "type": "text"}, {"bbox": [258, 538, 308, 549], "score": 0.94, "content": "\\Phi(G)=G^{2}", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [308, 537, 459, 551], "score": 1.0, "content": ", see [8]. By Blackburn, [3], since ", "type": "text"}, {"bbox": [460, 538, 485, 549], "score": 0.94, "content": "G/G_{2}^{2}", "type": "inline_equation", "height": 11, "width": 25}], "index": 32}, {"bbox": [124, 549, 485, 564], "spans": [{"bbox": [124, 549, 329, 564], "score": 1.0, "content": "has elementary derived group, we know that ", "type": "text"}, {"bbox": [329, 550, 482, 561], "score": 0.91, "content": "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle", "type": "inline_equation", "height": 11, "width": 153}, {"bbox": [483, 549, 485, 564], "score": 1.0, "content": ".", "type": "text"}], "index": 33}, {"bbox": [126, 560, 486, 575], "spans": [{"bbox": [126, 560, 301, 575], "score": 1.0, "content": "Again, by the Burnside Basis Theorem, ", "type": "text"}, {"bbox": [302, 563, 388, 573], "score": 0.9, "content": "G_{2}=\\langle c_{2},\\cdots,c_{n+1}\\rangle", "type": "inline_equation", "height": 10, "width": 86}, {"bbox": [388, 560, 392, 575], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [475, 562, 486, 572], "score": 0.9934230446815491, "content": "\u53e3", "type": "text"}], "index": 34}], "index": 32, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [124, 514, 486, 575]}, {"type": "text", "bbox": [125, 577, 487, 614], "lines": [{"bbox": [126, 580, 486, 592], "spans": [{"bbox": [126, 580, 198, 592], "score": 1.0, "content": "Lemma 2. Let ", "type": "text"}, {"bbox": [199, 582, 207, 589], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [207, 580, 354, 592], "score": 1.0, "content": " be as above. Moreover, assume ", "type": "text"}, {"bbox": [355, 582, 363, 589], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [363, 580, 452, 592], "score": 1.0, "content": " is metabelian. Let ", "type": "text"}, {"bbox": [453, 582, 462, 589], "score": 0.85, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [462, 580, 486, 592], "score": 1.0, "content": " be a", "type": "text"}], "index": 35}, {"bbox": [126, 592, 486, 604], "spans": [{"bbox": [126, 592, 218, 604], "score": 1.0, "content": "maximal subgroup of ", "type": "text"}, {"bbox": [218, 594, 226, 601], "score": 0.87, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [226, 592, 271, 604], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [272, 593, 295, 603], "score": 0.92, "content": "H/G^{\\prime}", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [296, 592, 434, 604], "score": 1.0, "content": " is cyclic, and denote the index ", "type": "text"}, {"bbox": [434, 592, 472, 603], "score": 0.8, "content": "\\left(G^{\\prime}:H^{\\prime}\\right)", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [472, 592, 486, 604], "score": 1.0, "content": " by", "type": "text"}], "index": 36}, {"bbox": [126, 603, 327, 615], "spans": [{"bbox": [126, 605, 136, 613], "score": 0.88, "content": "2^{\\kappa}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [137, 603, 169, 615], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [170, 605, 180, 613], "score": 0.9, "content": "G^{\\prime}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [181, 603, 312, 615], "score": 1.0, "content": " contains an element of order ", "type": "text"}, {"bbox": [312, 605, 323, 613], "score": 0.88, "content": "2^{\\kappa}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [323, 603, 327, 615], "score": 1.0, "content": ".", "type": "text"}], "index": 37}], "index": 36, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [126, 580, 486, 615]}, {"type": "text", "bbox": [125, 619, 487, 657], "lines": [{"bbox": [126, 621, 485, 634], "spans": [{"bbox": [126, 621, 294, 634], "score": 1.0, "content": "Proof. Without loss of generality, let ", "type": "text"}, {"bbox": [294, 622, 347, 633], "score": 0.93, "content": "H\\,=\\,\\left\\langle{b,G^{\\prime}}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 53}, {"bbox": [348, 621, 411, 634], "score": 1.0, "content": ". Notice that ", "type": "text"}, {"bbox": [412, 621, 485, 633], "score": 0.92, "content": "G^{\\prime}\\,=\\,\\langle c_{2},c_{3},\\cdot\\cdot\\cdot\\rangle", "type": "inline_equation", "height": 12, "width": 73}], "index": 38}, {"bbox": [126, 633, 487, 646], "spans": [{"bbox": [126, 633, 245, 646], "score": 1.0, "content": "and by our presentation of ", "type": "text"}, {"bbox": [245, 636, 254, 643], "score": 0.87, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [255, 633, 260, 646], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [260, 635, 332, 645], "score": 0.92, "content": "H^{\\prime}=\\langle c_{3},c_{4},\\cdot\\cdot\\cdot\\rangle", "type": "inline_equation", "height": 10, "width": 72}, {"bbox": [332, 633, 366, 646], "score": 1.0, "content": ". Thus,", "type": "text"}, {"bbox": [367, 634, 437, 645], "score": 0.93, "content": "G^{\\prime}/H^{\\prime}=\\langle c_{2}H^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [437, 633, 487, 646], "score": 1.0, "content": ". But since", "type": "text"}], "index": 39}, {"bbox": [126, 646, 486, 658], "spans": [{"bbox": [126, 647, 188, 657], "score": 0.92, "content": "(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [188, 646, 248, 658], "score": 1.0, "content": ", the order of ", "type": "text"}, {"bbox": [249, 650, 258, 656], "score": 0.88, "content": "c_{2}", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [258, 646, 270, 658], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [271, 648, 291, 656], "score": 0.9, "content": "\\geq2^{\\kappa}", "type": "inline_equation", "height": 8, "width": 20}, {"bbox": [292, 646, 421, 658], "score": 1.0, "content": ". This establishes the lemma.", "type": "text"}, {"bbox": [475, 646, 486, 656], "score": 0.991905927658081, "content": "\u53e3", "type": "text"}], "index": 40}], "index": 39, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [126, 621, 487, 658]}, {"type": "text", "bbox": [125, 661, 486, 701], "lines": [{"bbox": [125, 664, 487, 676], "spans": [{"bbox": [125, 664, 199, 676], "score": 1.0, "content": "Lemma 3. Let ", "type": "text"}, {"bbox": [199, 666, 207, 673], "score": 0.86, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [207, 664, 351, 676], "score": 1.0, "content": " be as above and again assume ", "type": "text"}, {"bbox": [352, 664, 361, 673], "score": 0.76, "content": "G", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [361, 664, 451, 676], "score": 1.0, "content": " is metabelian. Let ", "type": "text"}, {"bbox": [452, 664, 462, 673], "score": 0.76, "content": "H", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [462, 664, 487, 676], "score": 1.0, "content": " be a", "type": "text"}], "index": 41}, {"bbox": [126, 675, 486, 687], "spans": [{"bbox": [126, 676, 221, 687], "score": 1.0, "content": "maximal subgroup of ", "type": "text"}, {"bbox": [222, 677, 230, 685], "score": 0.85, "content": "G", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [230, 676, 279, 687], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [279, 676, 304, 687], "score": 0.89, "content": "H/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [304, 676, 430, 687], "score": 1.0, "content": " is cyclic, and assume that ", "type": "text"}, {"bbox": [430, 675, 486, 687], "score": 0.87, "content": "(G^{\\prime}\\,:\\,H^{\\prime})\\;\\equiv\\;", "type": "inline_equation", "height": 12, "width": 56}], "index": 42}, {"bbox": [124, 687, 454, 702], "spans": [{"bbox": [124, 689, 178, 702], "score": 1.0, "content": "0 mod 4. If ", "type": "text"}, {"bbox": [179, 690, 221, 701], "score": 0.93, "content": "d(G^{\\prime})=2", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [221, 689, 249, 702], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [249, 690, 305, 701], "score": 0.93, "content": "G_{2}=\\langle c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 56}, {"bbox": [305, 689, 327, 702], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [327, 687, 408, 701], "score": 0.93, "content": "G_{j}=\\langle{c_{2}^{2^{j-2}},c_{3}^{2^{j-3}}}\\rangle", "type": "inline_equation", "height": 14, "width": 81}, {"bbox": [408, 689, 426, 702], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [426, 689, 451, 700], "score": 0.88, "content": "j>2", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [451, 689, 454, 702], "score": 1.0, "content": ".", "type": "text"}], "index": 43}], "index": 42, "page_num": "page_2", "page_size": [612.0, 792.0], 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c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle."}, {"category_id": 13, "poly": [473, 1681, 502, 1681, 502, 1703, 473, 1703], "score": 0.9, "latex": "G^{\\prime}"}, {"category_id": 13, "poly": [871, 1094, 1029, 1094, 1029, 1119, 871, 1119], "score": 0.9, "latex": "a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1"}, {"category_id": 13, "poly": [1168, 925, 1291, 925, 1291, 951, 1168, 951], "score": 0.9, "latex": "G^{(0)}\\ =\\ G"}, {"category_id": 13, "poly": [839, 1565, 1078, 1565, 1078, 1594, 839, 1594], "score": 0.9, "latex": "G_{2}=\\langle c_{2},\\cdots,c_{n+1}\\rangle"}, {"category_id": 13, "poly": [975, 390, 1154, 390, 1154, 412, 975, 412], "score": 0.9, "latex": "0>d>-2000"}, {"category_id": 13, "poly": [1061, 656, 1088, 656, 1088, 676, 1061, 676], "score": 0.9, "latex": "K"}, {"category_id": 13, "poly": [808, 829, 831, 829, 831, 849, 808, 849], "score": 0.9, "latex": "B"}, {"category_id": 13, "poly": [483, 457, 498, 457, 498, 477, 483, 477], 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mention one last feature gleaned from the table. It follows from conditional Odlyzko bounds (assuming the Generalized Riemann Hypothesis) that those qua- dratic fields with rank and discriminant have finite class field tower; unconditional proofs are not known. Hence, conditionally, we conclude that those with discriminants , and have finite (2-)class field tower even though rank . Of course, it would be interesting to determine the length of their towers. The structure of this paper is as follows: we use results from group theory developed in Section 2 to pull down the condition rank from the field with degree to a subfield of with degree 8. Using the arithmetic of dihedral fields from Section 4 we then go down to the field of degree 4 occurring in Theorem 1. # 2. Group Theoretic Preliminaries Let be a group. If , then we let denote the commutator of and . If and are nonempty subsets of , then denotes the subgroup of generated by the set . The lower central series of is defined inductively by: and for . The derived series is defined inductively by: and for . Notice that the commutator subgroup, , of . Throughout this section, we assume that is a finite, nonmetacyclic, 2-group such that its abelianization is of type for some positive integer (necessarily ). Let , where mod (actually since is nonmetacyclic, cf. [1]); and for . Lemma 1. Let be as above (but not necessarily metabelian). Suppose that where denotes the minimal number of generators of the derived group of . Then $$ G^{\prime}=\langle c_{2},c_{3},\cdot\cdot\cdot,c_{n+1}\rangle; $$ moreover, $$ G_{2}/G_{2}^{2}\simeq\langle c_{2}G_{2}^{2}\rangle\oplus\cdots\oplus\langle c_{n+1}G_{2}^{2}\rangle. $$ Proof. By the Burnside Basis Theorem, , where is the Frattini subgroup of , i.e. the intersection of all maximal subgroups of , see [5]. But in the case of a 2-group, , see [8]. By Blackburn, [3], since has elementary derived group, we know that . Again, by the Burnside Basis Theorem, . 口 Lemma 2. Let be as above. Moreover, assume is metabelian. Let be a maximal subgroup of such that is cyclic, and denote the index by . Then contains an element of order . Proof. Without loss of generality, let . Notice that and by our presentation of , . Thus, . But since , the order of is . This establishes the lemma. 口 Lemma 3. Let be as above and again assume is metabelian. Let be a maximal subgroup of such that is cyclic, and assume that 0 mod 4. If , then and for .	<div class="pdf-page"> <p>We mention one last feature gleaned from the table. It follows from conditional Odlyzko bounds (assuming the Generalized Riemann Hypothesis) that those qua- dratic fields with rank and discriminant have finite class field tower; unconditional proofs are not known. Hence, conditionally, we conclude that those with discriminants , and have finite (2-)class field tower even though rank . Of course, it would be interesting to determine the length of their towers.</p> <p>The structure of this paper is as follows: we use results from group theory developed in Section 2 to pull down the condition rank from the field with degree to a subfield of with degree 8. Using the arithmetic of dihedral fields from Section 4 we then go down to the field of degree 4 occurring in Theorem 1.</p> <h1>2. Group Theoretic Preliminaries</h1> <p>Let be a group. If , then we let denote the commutator of and . If and are nonempty subsets of , then denotes the subgroup of generated by the set . The lower central series of is defined inductively by: and for . The derived series is defined inductively by: and for . Notice that the commutator subgroup, , of .</p> <p>Throughout this section, we assume that is a finite, nonmetacyclic, 2-group such that its abelianization is of type for some positive integer (necessarily ). Let , where mod (actually since is nonmetacyclic, cf. [1]); and for .</p> <p>Lemma 1. Let be as above (but not necessarily metabelian). Suppose that where denotes the minimal number of generators of the derived group of . Then</p> <p>moreover,</p> <p>Proof. By the Burnside Basis Theorem, , where is the Frattini subgroup of , i.e. the intersection of all maximal subgroups of , see [5]. But in the case of a 2-group, , see [8]. By Blackburn, [3], since has elementary derived group, we know that . Again, by the Burnside Basis Theorem, . 口</p> <p>Lemma 2. Let be as above. Moreover, assume is metabelian. Let be a maximal subgroup of such that is cyclic, and denote the index by . Then contains an element of order .</p> <p>Proof. Without loss of generality, let . Notice that and by our presentation of , . Thus, . But since , the order of is . This establishes the lemma. 口</p> <p>Lemma 3. Let be as above and again assume is metabelian. Let be a maximal subgroup of such that is cyclic, and assume that 0 mod 4. If , then and for .</p> </div>	<div class="pdf-page"> <div class="pdf-discarded" data-x="398" data-y="116" data-width="226" data-height="12" style="opacity: 0.5;">IMAGINARY QUADRATIC FIELDS</div> <div class="pdf-discarded" data-x="801" data-y="117" data-width="12" data-height="9" style="opacity: 0.5;">3</div> <p class="pdf-text" data-x="207" data-y="143" data-width="606" data-height="109">We mention one last feature gleaned from the table. It follows from conditional Odlyzko bounds (assuming the Generalized Riemann Hypothesis) that those qua- dratic fields with rank and discriminant have finite class field tower; unconditional proofs are not known. Hence, conditionally, we conclude that those with discriminants , and have finite (2-)class field tower even though rank . Of course, it would be interesting to determine the length of their towers.</p> <p class="pdf-text" data-x="207" data-y="253" data-width="606" data-height="76">The structure of this paper is as follows: we use results from group theory developed in Section 2 to pull down the condition rank from the field with degree to a subfield of with degree 8. Using the arithmetic of dihedral fields from Section 4 we then go down to the field of degree 4 occurring in Theorem 1.</p> <h1 class="pdf-title" data-x="364" data-y="340" data-width="293" data-height="16">2. Group Theoretic Preliminaries</h1> <p class="pdf-text" data-x="207" data-y="363" data-width="606" data-height="111">Let be a group. If , then we let denote the commutator of and . If and are nonempty subsets of , then denotes the subgroup of generated by the set . The lower central series of is defined inductively by: and for . The derived series is defined inductively by: and for . Notice that the commutator subgroup, , of .</p> <p class="pdf-text" data-x="209" data-y="475" data-width="604" data-height="64">Throughout this section, we assume that is a finite, nonmetacyclic, 2-group such that its abelianization is of type for some positive integer (necessarily ). Let , where mod (actually since is nonmetacyclic, cf. [1]); and for .</p> <p class="pdf-text" data-x="207" data-y="544" data-width="606" data-height="46">Lemma 1. Let be as above (but not necessarily metabelian). Suppose that where denotes the minimal number of generators of the derived group of . Then</p> <p class="pdf-text" data-x="210" data-y="619" data-width="72" data-height="13">moreover,</p> <p class="pdf-text" data-x="209" data-y="660" data-width="604" data-height="80">Proof. By the Burnside Basis Theorem, , where is the Frattini subgroup of , i.e. the intersection of all maximal subgroups of , see [5]. But in the case of a 2-group, , see [8]. By Blackburn, [3], since has elementary derived group, we know that . Again, by the Burnside Basis Theorem, . 口</p> <p class="pdf-text" data-x="209" data-y="746" data-width="605" data-height="47">Lemma 2. Let be as above. Moreover, assume is metabelian. Let be a maximal subgroup of such that is cyclic, and denote the index by . Then contains an element of order .</p> <p class="pdf-text" data-x="209" data-y="800" data-width="605" data-height="49">Proof. Without loss of generality, let . Notice that and by our presentation of , . Thus, . But since , the order of is . This establishes the lemma. 口</p> <p class="pdf-text" data-x="209" data-y="854" data-width="604" data-height="52">Lemma 3. Let be as above and again assume is metabelian. Let be a maximal subgroup of such that is cyclic, and assume that 0 mod 4. 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It follows from conditional", "Odlyzko bounds (assuming the Generalized Riemann Hypothesis) that those qua-", "dratic fields with rank \\mathrm{Cl}_{2}(k^{1})\\geq3 and discriminant 0>d>-2000 have finite class", "field tower; unconditional proofs are not known. Hence, conditionally, we conclude", "that those k with discriminants -1015 , -1595 and -1780 have finite (2-)class field", "tower even though rank \\mathrm{Cl}_{2}(k^{1})\\geq3 . Of course, it would be interesting to determine", "the length of their towers.", "The structure of this paper is as follows: we use results from group theory", "developed in Section 2 to pull down the condition rank \\mathrm{Cl}_{2}(k^{1})=2 from the field", "k^{1} with degree 2^{m+2} to a subfield L of k^{1} with degree 8. Using the arithmetic of", "dihedral fields from Section 4 we then go down to the field K of degree 4 occurring", "in Theorem 1.", "2. Group Theoretic Preliminaries", "Let G be a group. If x,y\\ \\in\\ G , then we let [x,y]~=~x^{-1}y^{-1}x y denote the", "commutator of x and _y . If A and B are nonempty subsets of G , then [A,B]", "denotes the subgroup of G generated by the set \\{[a,b]:a\\in A,b\\in B\\} . The lower", "central series \\{G_{j}\\} of G is defined inductively by: G_{1}\\,=\\,G and G_{j+1}\\,=\\,[G,G_{j}]", "for j~\\geq~1 . The derived series \\{G^{(n)}\\} is defined inductively by: G^{(0)}\\ =\\ G and", "G^{(n+1)}=[G^{(n)},G^{(n)}] for n\\geq0 . Notice that G^{(1)}=G_{2}=[G,G] the commutator", "subgroup, G^{\\prime} , of G .", "Throughout this section, we assume that G is a finite, nonmetacyclic, 2-group", "such that its abelianization G^{\\mathrm{ab}}=G/G^{\\prime} is of type (2,2^{m}) for some positive integer", "{\\boldsymbol{r}}n (necessarily \\geq2 ). Let G=\\langle a,b\\rangle , where a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1 mod G_{2} (actually \\mathrm{mod}G_{3}", "since G is nonmetacyclic, cf. [1]); c_{2}=[a,b] and c_{j+1}=[b,c_{j}] for j\\geq2 .", "Lemma 1. Let G be as above (but not necessarily metabelian). Suppose that", "d(G^{\\prime})\\,=\\,n where d(G^{\\prime}) denotes the minimal number of generators of the derived", "group G^{\\prime}=G_{2} of G . Then", "G^{\\prime}=\\langle c_{2},c_{3},\\cdot\\cdot\\cdot,c_{n+1}\\rangle;", "moreover,", "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle.", "Proof. By the Burnside Basis Theorem, d(G_{2})=d(G_{2}/\\Phi(G)) , where \\Phi(G) is the", "Frattini subgroup of G , i.e. the intersection of all maximal subgroups of G , see [5].", "But in the case of a 2-group, \\Phi(G)=G^{2} , see [8]. By Blackburn, [3], since G/G_{2}^{2}", "has elementary derived group, we know that G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle .", "Again, by the Burnside Basis Theorem, G_{2}=\\langle c_{2},\\cdots,c_{n+1}\\rangle . 口", "Lemma 2. Let G be as above. Moreover, assume G is metabelian. Let H be a", "maximal subgroup of G such that H/G^{\\prime} is cyclic, and denote the index \\left(G^{\\prime}:H^{\\prime}\\right) by", "2^{\\kappa} . Then G^{\\prime} contains an element of order 2^{\\kappa} .", "Proof. Without loss of generality, let H\\,=\\,\\left\\langle{b,G^{\\prime}}\\right\\rangle . Notice that G^{\\prime}\\,=\\,\\langle c_{2},c_{3},\\cdot\\cdot\\cdot\\rangle", "and by our presentation of H , H^{\\prime}=\\langle c_{3},c_{4},\\cdot\\cdot\\cdot\\rangle . Thus, G^{\\prime}/H^{\\prime}=\\langle c_{2}H^{\\prime}\\rangle . But since", "(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\, , the order of c_{2} is \\geq2^{\\kappa} . This establishes the lemma. 口", "Lemma 3. Let G be as above and again assume G is metabelian. Let H be a", "maximal subgroup of G such that H/G^{\\prime} is cyclic, and assume that (G^{\\prime}\\,:\\,H^{\\prime})\\;\\equiv\\;", "0 mod 4. If d(G^{\\prime})=2 , then G_{2}=\\langle c_{2},c_{3}\\rangle and G_{j}=\\langle{c_{2}^{2^{j-2}},c_{3}^{2^{j-3}}}\\rangle for j>2 ." ], "index": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 32, 33, 34, 35, 36, 37, 38, 52, 53, 54, 55, 76, 77, 78, 103, 131, 160, 190, 191, 192, 193, 194, 225, 226, 227, 263, 264, 265, 304, 305, 306 ] }	{ "coordinates": [], "index": [], "caption": [], "caption_coordinates": [] }	{ "type": [ "interline_equation", "interline_equation", "interline_equation", "interline_equation" ], "coordinates": [ [ 426, 599, 593, 614 ], [ 384, 640, 637, 655 ], [ 426, 599, 593, 614 ], [ 384, 640, 637, 655 ] ], "content": [ "", "", "G^{\\prime}=\\langle c_{2},c_{3},\\cdot\\cdot\\cdot,c_{n+1}\\rangle;", "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle." ], "index": [ 0, 1, 2, 3 ] }	{ "type": [], "coordinates": [], "content": [], "index": [] }	{"document_info": {"total_pages": 14, "parse_type": "txt", "version": "1.0.1", "total_content_list_items": 155}, "content_statistics": {"total_blocks": 182, "total_lines": 526, "total_images": 4, "total_equations": 28, "total_tables": 11, "total_characters": 29658, "total_words": 5331, "block_type_distribution": {"title": 7, "text": 126, "interline_equation": 14, "discarded": 24, "table_caption": 3, "table_body": 4, "image_body": 2, "list": 2}}, "model_statistics": {"total_layout_detections": 2826, "avg_confidence": 0.9470955414012739, "min_confidence": 0.26, "max_confidence": 1.0}, "page_statistics": [{"page_number": 0, "page_size": [612.0, 792.0], "blocks": 11, "lines": 38, "images": 0, "equations": 2, "tables": 0, "characters": 2362, "words": 398, "discarded_blocks": 1}, {"page_number": 1, "page_size": [612.0, 792.0], "blocks": 10, "lines": 30, "images": 0, "equations": 0, "tables": 3, "characters": 1688, "words": 305, "discarded_blocks": 1}, {"page_number": 2, "page_size": [612.0, 792.0], "blocks": 15, "lines": 44, "images": 0, "equations": 4, "tables": 0, "characters": 2417, "words": 416, "discarded_blocks": 2}, {"page_number": 3, "page_size": [612.0, 792.0], "blocks": 7, "lines": 27, "images": 0, "equations": 0, "tables": 3, "characters": 1591, "words": 291, "discarded_blocks": 1}, {"page_number": 4, "page_size": [612.0, 792.0], "blocks": 23, "lines": 38, "images": 0, "equations": 6, "tables": 0, "characters": 1604, "words": 303, "discarded_blocks": 4}, {"page_number": 5, "page_size": [612.0, 792.0], "blocks": 17, "lines": 40, "images": 0, "equations": 6, "tables": 0, "characters": 2131, "words": 393, "discarded_blocks": 1}, {"page_number": 6, "page_size": [612.0, 792.0], "blocks": 15, "lines": 36, "images": 2, "equations": 4, "tables": 0, "characters": 1791, "words": 324, "discarded_blocks": 2}, {"page_number": 7, "page_size": [612.0, 792.0], "blocks": 12, "lines": 49, "images": 0, "equations": 0, "tables": 0, "characters": 3049, "words": 571, "discarded_blocks": 1}, {"page_number": 8, "page_size": [612.0, 792.0], "blocks": 11, "lines": 42, "images": 0, "equations": 0, "tables": 0, "characters": 2636, "words": 479, "discarded_blocks": 2}, {"page_number": 9, "page_size": [612.0, 792.0], "blocks": 14, "lines": 35, "images": 2, "equations": 0, "tables": 0, "characters": 2036, "words": 375, "discarded_blocks": 1}, {"page_number": 10, "page_size": [612.0, 792.0], "blocks": 17, "lines": 35, "images": 0, "equations": 6, "tables": 0, "characters": 1789, "words": 327, "discarded_blocks": 2}, {"page_number": 11, "page_size": [612.0, 792.0], "blocks": 14, "lines": 40, "images": 0, "equations": 0, "tables": 2, "characters": 2094, "words": 399, "discarded_blocks": 2}, {"page_number": 12, "page_size": [612.0, 792.0], "blocks": 10, "lines": 44, "images": 0, "equations": 0, "tables": 3, "characters": 2572, "words": 487, "discarded_blocks": 2}, {"page_number": 13, "page_size": [612.0, 792.0], "blocks": 6, "lines": 28, "images": 0, "equations": 0, "tables": 0, "characters": 1898, "words": 263, "discarded_blocks": 2}]}
0003244v1	3	[ 612, 792 ]	{ "type": [ "text", "text", "table_caption", "table_body", "text", "text", "discarded" ], "coordinates": [ [ 209, 143, 814, 306 ], [ 209, 318, 814, 473 ], [ 476, 487, 547, 502 ], [ 279, 537, 742, 804 ], [ 229, 831, 752, 848 ], [ 209, 855, 813, 905 ], [ 209, 117, 219, 128 ] ], "content": [ "Proof. Assume that . By Lemma 1, and hence . Write where are positive integers. Without loss of generality, let and write for some . Since we have, mod . By the proof of Lemma 2, this implies that mod . Write for some positive integer . On the other hand, since we see that mod . If were odd, then . This, however, implies that , contrary to our assumptions. Thus is even, say . From all of this we see that c4 = c22κx1c23y1. Consequently, by induction we have cj ∈ ⟨c22j−2 , c32j−3 ⟩ for all j ≥ 4. Since Gj = ⟨c22j−2 , c23j− , cf. [1], we obtain the lemma. 口", "Let us translate the above into the field-theoretic language. Let be an imagi- nary quadratic number field of type A) or B) (see the Introduction), and let be one of the two quadratic subextensions of over which is cyclic. If and , then Lemma 2 implies that contains an element of order . Table 2 contains the relevant information for the fields occurring in Table 1. An application of the class number formula to (see e.g. Proposition 3 below) shows immediately that , where is the class number of the quadratic subfield of , where ; in particu- lar, we always have , and the assumption is always satisfied for the fields that we consider.", "Table 2", "", "We now use the above results to prove the following useful proposition.", "Proposition 1. Let be a nonmetacyclic 2-group such that ; (hence ). Let and be the two maximal subgroups of such that and are cyclic. Moreover, assume that mod 4. Finally, assume that is a subgroup of index 4 in not contained in or Then", "4" ], "index": [ 0, 1, 2, 3, 4, 5, 6 ] }	[{"type": "text", "text": "Proof. Assume that $d(G^{\\prime})=2$ . By Lemma 1, $G_{2}=\\langle c_{2},c_{3}\\rangle$ and hence $c_{4}\\in\\langle c_{2},c_{3}\\rangle$ . Write $c_{4}\\:=\\:c_{2}^{x}c_{3}^{y}$ where $x,y$ are positive integers. Without loss of generality, let $H=\\langle b,c_{2},c_{3}\\rangle$ and write $(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,$ for some $\\kappa\\geq2$ . Since $c_{3},c_{4}\\in H^{\\prime}$ we have, $c_{2}^{x}\\equiv1$ mod $H^{\\prime}$ . By the proof of Lemma 2, this implies that $x\\equiv0$ mod $2^{\\kappa}$ . Write $x\\,=\\,2^{\\kappa}x_{1}$ for some positive integer $x_{1}$ . On the other hand, since $c_{4},c_{2}^{2^{\\kappa}x_{1}}\\,\\in\\,G_{4}$ we see that $c_{3}^{y}\\equiv1$ mod $G_{4}$ . If $_y$ were odd, then $c_{3}\\in G_{4}$ . This, however, implies that $G_{2}=\\langle c_{2}\\rangle$ , contrary to our assumptions. Thus $_y$ is even, say $y=2y_{1}$ . From all of this we see that c4 = $c_{4}\\,=\\,c_{2}^{2^{\\kappa}x_{1}}c_{3}^{2y_{1}}$ c22\u03bax1c23y1. Consequently, by induction we have cj \u2208 \u27e8c22j\u22122 , c32j\u22123 \u27e9 for all j \u2265 4. Since Gj = \u27e8c22j\u22122 , c23j\u2212 $G_{j}=\\langle c_{2}^{2^{j-2}},c_{3}^{2^{j-3}},\\cdot\\cdot\\cdot,c_{j-1}^{2},c_{j},c_{j+1},\\cdot\\cdot\\cdot\\rangle$ , cf. [1], we obtain the lemma. \u53e3 ", "page_idx": 3}, {"type": "text", "text": "Let us translate the above into the field-theoretic language. Let $k$ be an imaginary quadratic number field of type A) or B) (see the Introduction), and let $M/k$ be one of the two quadratic subextensions of $k^{1}/k$ over which $k^{1}$ is cyclic. If $h_{2}(M)=2^{m+\\kappa}$ and $\\mathrm{Cl}_{2}(k)=(2,2^{m})$ , then Lemma 2 implies that $\\mathrm{Cl_{2}}(k^{1})$ contains an element of order $2^{\\kappa}$ . Table 2 contains the relevant information for the fields occurring in Table 1. An application of the class number formula to $M/\\mathbb{Q}$ (see e.g. Proposition 3 below) shows immediately that $h_{2}(M)=2^{m+\\kappa}$ , where $2^{\\kappa}$ is the class number of the quadratic subfield $\\mathbb{Q}({\\sqrt{d_{i}d_{j}}}\\,)$ of $M$ , where $(d_{i}/p_{j})=+1$ ; in particular, we always have $\\kappa\\geq2$ , and the assumption $\\left(G^{\\prime}:H^{\\prime}\\right)\\geq4$ is always satisfied for the fields that we consider. ", "page_idx": 3}, {"type": "table", "img_path": "images/20908e76a81bf659100a97a9d08d0e64405e7d94b7aaa0b27349fad02b929aae.jpg", "table_caption": ["Table 2 "], "table_footnote": [], "table_body": "\n\n<html><body><table><tr><td>M1</td><td>Cl2(M1)</td><td>M2</td><td>Cl2(M2)</td></tr><tr><td>Q(v5, V-7 . 29 ) Q(V2,\u221a-5 \u00b7 31)</td><td>(2, 16)</td><td>Q(V5 \u00b7 29, \u221a-7)</td><td>(2, 16)</td></tr><tr><td>Q(V13,\u221a-3 \u00b737)</td><td>(4, 4) (2,16)</td><td>Q(v5,\u221a-2 : 31) Q(V37,\u221a-3. 13</td><td>(2, 16) (2,16)</td></tr><tr><td></td><td></td><td></td><td>(2,16)</td></tr><tr><td>Q(V-11,v5 \u00b7 29)</td><td>(2,16)</td><td>Q(V29, \u221a-5 \u00b7 11)</td><td>(2, 16)</td></tr><tr><td>Q(V5, V-17 \u00b7 19)</td><td>(4, 4)</td><td>Q(v17, V-5 \u00b7 19 )</td><td></td></tr><tr><td>Q(V29, \u221a-2 . 7)</td><td>(2,16)</td><td>Q(V2, \u221a-7 . 29)</td><td>(2,16)</td></tr><tr><td>Q(v5 . 89, \u221a-1)</td><td>(4, 4)</td><td>Q(V5, \u221a-89 )</td><td>(2,8)</td></tr><tr><td>Q(V37, \u221a-5 \u00b7 11)</td><td>(4, 4)</td><td>Q(v5, \u221a-37 \u00b7 11)</td><td>(2,32)</td></tr><tr><td>Q(V53,\u221a-3\u00b713)</td><td>(4, 4)</td><td>Q(V13 \u00b7 53, \u221a-3)</td><td>(2,2,4)</td></tr><tr><td>Q(V37, \u221a-2 . 7)</td><td>(2, 16)</td><td>Q(v2,\u221a-7 . 37)</td><td>(2, 16)</td></tr><tr><td>Q(\u221a13,\u221a-2 \u00b7 23\uff09</td><td>(4, 4)</td><td>Q(v2,\u221a-13 \u00b7 23</td><td>(2, 16)</td></tr></table></body></html>\n\n", "page_idx": 3}, {"type": "text", "text": "We now use the above results to prove the following useful proposition. ", "page_idx": 3}, {"type": "text", "text": "Proposition 1. Let $G$ be a nonmetacyclic 2-group such that $G/G^{\\prime}\\;\\simeq\\;(2,2^{m})$ ; (hence $m>1$ ). Let $H$ and $K$ be the two maximal subgroups of $G$ such that $H/G^{\\prime}$ and $K/G^{\\prime}$ are cyclic. Moreover, assume that $(G^{\\prime}:H^{\\prime})\\equiv0$ mod 4. Finally, assume that $N$ is a subgroup of index 4 in $G$ not contained in $H$ or $K$ Then ", "page_idx": 3}]	{"preproc_blocks": [{"type": "text", "bbox": [125, 111, 487, 237], "lines": [{"bbox": [126, 114, 485, 126], "spans": [{"bbox": [126, 114, 215, 126], "score": 1.0, "content": "Proof. Assume that ", "type": "text"}, {"bbox": [215, 115, 258, 126], "score": 0.94, "content": "d(G^{\\prime})=2", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [258, 114, 326, 126], "score": 1.0, "content": ". By Lemma 1, ", "type": "text"}, {"bbox": [326, 115, 381, 126], "score": 0.95, "content": "G_{2}=\\langle c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [382, 114, 431, 126], "score": 1.0, "content": "and hence ", "type": "text"}, {"bbox": [431, 115, 482, 126], "score": 0.94, "content": "c_{4}\\in\\langle c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [483, 114, 485, 126], "score": 1.0, "content": ".", "type": "text"}], "index": 0}, {"bbox": [123, 122, 489, 143], "spans": [{"bbox": [123, 122, 154, 143], "score": 1.0, "content": "Write ", "type": "text"}, {"bbox": [155, 127, 198, 138], "score": 0.95, "content": "c_{4}\\:=\\:c_{2}^{x}c_{3}^{y}", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [199, 122, 232, 143], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [232, 131, 248, 137], "score": 0.89, "content": "x,y", "type": "inline_equation", "height": 6, "width": 16}, {"bbox": [248, 122, 489, 143], "score": 1.0, "content": " are positive integers. Without loss of generality, let", "type": "text"}], "index": 1}, {"bbox": [126, 136, 487, 152], "spans": [{"bbox": [126, 139, 187, 150], "score": 0.93, "content": "H=\\langle b,c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [187, 136, 235, 152], "score": 1.0, "content": "and write ", "type": "text"}, {"bbox": [235, 139, 297, 150], "score": 0.93, "content": "(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [297, 136, 340, 152], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [340, 140, 365, 149], "score": 0.92, "content": "\\kappa\\geq2", "type": "inline_equation", "height": 9, "width": 25}, {"bbox": [365, 136, 398, 152], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [398, 139, 444, 149], "score": 0.93, "content": "c_{3},c_{4}\\in H^{\\prime}", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [444, 136, 487, 152], "score": 1.0, "content": " we have,", "type": "text"}], "index": 2}, {"bbox": [126, 150, 487, 162], "spans": [{"bbox": [126, 152, 155, 162], "score": 0.86, "content": "c_{2}^{x}\\equiv1", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [155, 150, 179, 162], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [179, 151, 191, 159], "score": 0.84, "content": "H^{\\prime}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [191, 150, 392, 162], "score": 1.0, "content": ". By the proof of Lemma 2, this implies that ", "type": "text"}, {"bbox": [393, 152, 417, 159], "score": 0.73, "content": "x\\equiv0", "type": "inline_equation", "height": 7, "width": 24}, {"bbox": [418, 150, 442, 162], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [442, 152, 452, 159], "score": 0.89, "content": "2^{\\kappa}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [453, 150, 487, 162], "score": 1.0, "content": ". Write", "type": "text"}], "index": 3}, {"bbox": [126, 160, 482, 178], "spans": [{"bbox": [126, 165, 168, 174], "score": 0.92, "content": "x\\,=\\,2^{\\kappa}x_{1}", "type": "inline_equation", "height": 9, "width": 42}, {"bbox": [168, 160, 284, 178], "score": 1.0, "content": " for some positive integer ", "type": "text"}, {"bbox": [285, 168, 295, 174], "score": 0.89, "content": "x_{1}", "type": "inline_equation", "height": 6, "width": 10}, {"bbox": [295, 160, 420, 178], "score": 1.0, "content": ". On the other hand, since ", "type": "text"}, {"bbox": [420, 162, 482, 175], "score": 0.93, "content": "c_{4},c_{2}^{2^{\\kappa}x_{1}}\\,\\in\\,G_{4}", "type": "inline_equation", "height": 13, "width": 62}], "index": 4}, {"bbox": [125, 175, 486, 187], "spans": [{"bbox": [125, 175, 179, 187], "score": 1.0, "content": "we see that ", "type": "text"}, {"bbox": [180, 176, 209, 187], "score": 0.93, "content": "c_{3}^{y}\\equiv1", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [209, 175, 233, 187], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [233, 177, 245, 186], "score": 0.91, "content": "G_{4}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [246, 175, 264, 187], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [264, 180, 270, 186], "score": 0.89, "content": "_y", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [270, 175, 343, 187], "score": 1.0, "content": " were odd, then ", "type": "text"}, {"bbox": [343, 177, 378, 186], "score": 0.93, "content": "c_{3}\\in G_{4}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [378, 175, 486, 187], "score": 1.0, "content": ". This, however, implies", "type": "text"}], "index": 5}, {"bbox": [124, 186, 486, 200], "spans": [{"bbox": [124, 186, 147, 200], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 188, 191, 199], "score": 0.94, "content": "G_{2}=\\langle c_{2}\\rangle", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [192, 186, 356, 200], "score": 1.0, "content": ", contrary to our assumptions. Thus ", "type": "text"}, {"bbox": [356, 191, 362, 198], "score": 0.88, "content": "_y", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [362, 186, 419, 200], "score": 1.0, "content": " is even, say ", "type": "text"}, {"bbox": [419, 189, 453, 198], "score": 0.93, "content": "y=2y_{1}", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [454, 186, 486, 200], "score": 1.0, "content": ". From", "type": "text"}], "index": 6}, {"bbox": [123, 196, 487, 215], "spans": [{"bbox": [123, 197, 265, 214], "score": 1.0, "content": "all of this we see that c4 = ", "type": "text"}, {"bbox": [228, 199, 291, 212], "score": 0.94, "content": "c_{4}\\,=\\,c_{2}^{2^{\\kappa}x_{1}}c_{3}^{2y_{1}}", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [253, 196, 487, 215], "score": 1.0, "content": "c22\u03bax1c23y1. Consequently, by induction we have cj \u2208", "type": "text"}], "index": 7}, {"bbox": [123, 208, 485, 232], "spans": [{"bbox": [123, 208, 345, 232], "score": 1.0, "content": "\u27e8c22j\u22122 , c32j\u22123 \u27e9 for all j \u2265 4. Since Gj = \u27e8c22j\u22122 , c23j\u2212", "type": "text"}, {"bbox": [272, 212, 449, 227], "score": 0.92, "content": "G_{j}=\\langle c_{2}^{2^{j-2}},c_{3}^{2^{j-3}},\\cdot\\cdot\\cdot,c_{j-1}^{2},c_{j},c_{j+1},\\cdot\\cdot\\cdot\\rangle", "type": "inline_equation", "height": 15, "width": 177}, {"bbox": [450, 213, 485, 227], "score": 1.0, "content": ", cf. [1],", "type": "text"}], "index": 8}, {"bbox": [125, 226, 486, 238], "spans": [{"bbox": [125, 226, 221, 238], "score": 1.0, "content": "we obtain the lemma.", "type": "text"}, {"bbox": [475, 226, 486, 236], "score": 0.9939806461334229, "content": "\u53e3", "type": "text"}], "index": 9}], "index": 4.5}, {"type": "text", "bbox": [125, 246, 487, 366], "lines": [{"bbox": [137, 248, 485, 261], "spans": [{"bbox": [137, 248, 420, 261], "score": 1.0, "content": "Let us translate the above into the field-theoretic language. Let ", "type": "text"}, {"bbox": [421, 250, 426, 257], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [427, 248, 485, 261], "score": 1.0, "content": " be an imagi-", "type": "text"}], "index": 10}, {"bbox": [124, 261, 485, 272], "spans": [{"bbox": [124, 261, 464, 272], "score": 1.0, "content": "nary quadratic number field of type A) or B) (see the Introduction), and let ", "type": "text"}, {"bbox": [464, 262, 485, 272], "score": 0.92, "content": "M/k", "type": "inline_equation", "height": 10, "width": 21}], "index": 11}, {"bbox": [125, 273, 487, 284], "spans": [{"bbox": [125, 273, 337, 284], "score": 1.0, "content": "be one of the two quadratic subextensions of ", "type": "text"}, {"bbox": [337, 273, 358, 284], "score": 0.94, "content": "k^{1}/k", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [358, 273, 415, 284], "score": 1.0, "content": " over which ", "type": "text"}, {"bbox": [415, 273, 425, 281], "score": 0.9, "content": "k^{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [426, 273, 487, 284], "score": 1.0, "content": " is cyclic. If", "type": "text"}], "index": 12}, {"bbox": [126, 282, 487, 298], "spans": [{"bbox": [126, 285, 191, 296], "score": 0.93, "content": "h_{2}(M)=2^{m+\\kappa}", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [192, 282, 214, 298], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [214, 285, 285, 296], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k)=(2,2^{m})", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [286, 282, 413, 298], "score": 1.0, "content": ", then Lemma 2 implies that ", "type": "text"}, {"bbox": [413, 285, 446, 296], "score": 0.94, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [446, 282, 487, 298], "score": 1.0, "content": " contains", "type": "text"}], "index": 13}, {"bbox": [126, 296, 486, 308], "spans": [{"bbox": [126, 296, 218, 308], "score": 1.0, "content": "an element of order ", "type": "text"}, {"bbox": [218, 298, 228, 305], "score": 0.88, "content": "2^{\\kappa}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [228, 296, 486, 308], "score": 1.0, "content": ". Table 2 contains the relevant information for the fields", "type": "text"}], "index": 14}, {"bbox": [125, 307, 486, 321], "spans": [{"bbox": [125, 307, 424, 321], "score": 1.0, "content": "occurring in Table 1. An application of the class number formula to ", "type": "text"}, {"bbox": [424, 309, 447, 320], "score": 0.94, "content": "M/\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [447, 307, 486, 321], "score": 1.0, "content": " (see e.g.", "type": "text"}], "index": 15}, {"bbox": [124, 320, 487, 333], "spans": [{"bbox": [124, 320, 325, 333], "score": 1.0, "content": "Proposition 3 below) shows immediately that ", "type": "text"}, {"bbox": [325, 321, 390, 332], "score": 0.93, "content": "h_{2}(M)=2^{m+\\kappa}", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [391, 320, 425, 333], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [425, 322, 435, 329], "score": 0.91, "content": "2^{\\kappa}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [435, 320, 487, 333], "score": 1.0, "content": " is the class", "type": "text"}], "index": 16}, {"bbox": [124, 332, 486, 346], "spans": [{"bbox": [124, 332, 271, 346], "score": 1.0, "content": "number of the quadratic subfield ", "type": "text"}, {"bbox": [271, 332, 316, 345], "score": 0.94, "content": "\\mathbb{Q}({\\sqrt{d_{i}d_{j}}}\\,)", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [317, 332, 330, 346], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [331, 334, 342, 342], "score": 0.89, "content": "M", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [342, 332, 376, 346], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [376, 334, 433, 344], "score": 0.94, "content": "(d_{i}/p_{j})=+1", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [433, 332, 486, 346], "score": 1.0, "content": "; in particu-", "type": "text"}], "index": 17}, {"bbox": [125, 344, 486, 357], "spans": [{"bbox": [125, 344, 213, 357], "score": 1.0, "content": "lar, we always have ", "type": "text"}, {"bbox": [213, 347, 237, 354], "score": 0.92, "content": "\\kappa\\geq2", "type": "inline_equation", "height": 7, "width": 24}, {"bbox": [238, 344, 332, 357], "score": 1.0, "content": ", and the assumption ", "type": "text"}, {"bbox": [332, 345, 389, 356], "score": 0.93, "content": "\\left(G^{\\prime}:H^{\\prime}\\right)\\geq4", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [390, 344, 486, 357], "score": 1.0, "content": " is always satisfied for", "type": "text"}], "index": 18}, {"bbox": [125, 356, 244, 368], "spans": [{"bbox": [125, 356, 244, 368], "score": 1.0, "content": "the fields that we consider.", "type": "text"}], "index": 19}], "index": 14.5}, {"type": "table", "bbox": [167, 416, 444, 622], "blocks": [{"type": "table_caption", "bbox": [285, 377, 327, 389], "group_id": 0, "lines": [{"bbox": [285, 378, 327, 391], "spans": [{"bbox": [285, 378, 327, 391], "score": 1.0, "content": "Table 2", "type": "text"}], "index": 20}], "index": 20}, {"type": "table_body", "bbox": [167, 416, 444, 622], "group_id": 0, "lines": [{"bbox": [167, 416, 444, 622], "spans": [{"bbox": [167, 416, 444, 622], "score": 0.912, "html": "<html><body><table><tr><td>M1</td><td>Cl2(M1)</td><td>M2</td><td>Cl2(M2)</td></tr><tr><td>Q(v5, V-7 . 29 ) Q(V2,\u221a-5 \u00b7 31)</td><td>(2, 16)</td><td>Q(V5 \u00b7 29, \u221a-7)</td><td>(2, 16)</td></tr><tr><td>Q(V13,\u221a-3 \u00b737)</td><td>(4, 4) (2,16)</td><td>Q(v5,\u221a-2 : 31) Q(V37,\u221a-3. 13</td><td>(2, 16) (2,16)</td></tr><tr><td></td><td></td><td></td><td>(2,16)</td></tr><tr><td>Q(V-11,v5 \u00b7 29)</td><td>(2,16)</td><td>Q(V29, \u221a-5 \u00b7 11)</td><td>(2, 16)</td></tr><tr><td>Q(V5, V-17 \u00b7 19)</td><td>(4, 4)</td><td>Q(v17, V-5 \u00b7 19 )</td><td></td></tr><tr><td>Q(V29, \u221a-2 . 7)</td><td>(2,16)</td><td>Q(V2, \u221a-7 . 29)</td><td>(2,16)</td></tr><tr><td>Q(v5 . 89, \u221a-1)</td><td>(4, 4)</td><td>Q(V5, \u221a-89 )</td><td>(2,8)</td></tr><tr><td>Q(V37, \u221a-5 \u00b7 11)</td><td>(4, 4)</td><td>Q(v5, \u221a-37 \u00b7 11)</td><td>(2,32)</td></tr><tr><td>Q(V53,\u221a-3\u00b713)</td><td>(4, 4)</td><td>Q(V13 \u00b7 53, \u221a-3)</td><td>(2,2,4)</td></tr><tr><td>Q(V37, \u221a-2 . 7)</td><td>(2, 16)</td><td>Q(v2,\u221a-7 . 37)</td><td>(2, 16)</td></tr><tr><td>Q(\u221a13,\u221a-2 \u00b7 23\uff09</td><td>(4, 4)</td><td>Q(v2,\u221a-13 \u00b7 23</td><td>(2, 16)</td></tr></table></body></html>", "type": "table", "image_path": "20908e76a81bf659100a97a9d08d0e64405e7d94b7aaa0b27349fad02b929aae.jpg"}]}], "index": 28.5, "virtual_lines": [{"bbox": [167, 416, 444, 429], "spans": [], "index": 21}, {"bbox": [167, 429, 444, 442], "spans": [], "index": 22}, {"bbox": [167, 442, 444, 455], "spans": [], "index": 23}, {"bbox": [167, 455, 444, 468], "spans": [], "index": 24}, {"bbox": [167, 468, 444, 481], "spans": [], "index": 25}, {"bbox": [167, 481, 444, 494], "spans": [], "index": 26}, {"bbox": [167, 494, 444, 507], "spans": [], "index": 27}, {"bbox": [167, 507, 444, 520], "spans": [], "index": 28}, {"bbox": [167, 520, 444, 533], "spans": [], "index": 29}, {"bbox": [167, 533, 444, 546], "spans": [], "index": 30}, {"bbox": [167, 546, 444, 559], "spans": [], "index": 31}, {"bbox": [167, 559, 444, 572], "spans": [], "index": 32}, {"bbox": [167, 572, 444, 585], "spans": [], "index": 33}, {"bbox": [167, 585, 444, 598], "spans": [], "index": 34}, {"bbox": [167, 598, 444, 611], "spans": [], "index": 35}, {"bbox": [167, 611, 444, 624], "spans": [], "index": 36}]}], "index": 24.25}, {"type": "text", "bbox": [137, 643, 450, 656], "lines": [{"bbox": [138, 645, 449, 657], "spans": [{"bbox": [138, 645, 449, 657], "score": 1.0, "content": "We now use the above results to prove the following useful proposition.", "type": "text"}], "index": 37}], "index": 37}, {"type": "text", "bbox": [125, 662, 486, 700], "lines": [{"bbox": [125, 665, 486, 677], "spans": [{"bbox": [125, 665, 220, 677], "score": 1.0, "content": "Proposition 1. Let ", "type": "text"}, {"bbox": [221, 667, 229, 674], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [229, 665, 409, 677], "score": 1.0, "content": " be a nonmetacyclic 2-group such that ", "type": "text"}, {"bbox": [409, 666, 482, 677], "score": 0.92, "content": "G/G^{\\prime}\\;\\simeq\\;(2,2^{m})", "type": "inline_equation", "height": 11, "width": 73}, {"bbox": [483, 665, 486, 677], "score": 1.0, "content": ";", "type": "text"}], "index": 38}, {"bbox": [127, 677, 484, 689], "spans": [{"bbox": [127, 677, 157, 689], "score": 1.0, "content": "(hence ", "type": "text"}, {"bbox": [158, 679, 186, 687], "score": 0.83, "content": "m>1", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [186, 677, 214, 689], "score": 1.0, "content": "). Let ", "type": "text"}, {"bbox": [215, 679, 224, 686], "score": 0.84, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [225, 677, 247, 689], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [247, 679, 257, 686], "score": 0.85, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [257, 677, 405, 689], "score": 1.0, "content": " be the two maximal subgroups of ", "type": "text"}, {"bbox": [406, 679, 414, 686], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [414, 677, 460, 689], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [460, 678, 484, 689], "score": 0.92, "content": "H/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 24}], "index": 39}, {"bbox": [127, 689, 487, 701], "spans": [{"bbox": [127, 689, 144, 701], "score": 1.0, "content": "and", "type": "text"}, {"bbox": [145, 690, 169, 701], "score": 0.91, "content": "K/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [169, 689, 322, 701], "score": 1.0, "content": " are cyclic. Moreover, assume that ", "type": "text"}, {"bbox": [322, 690, 379, 701], "score": 0.91, "content": "(G^{\\prime}:H^{\\prime})\\equiv0", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [380, 689, 487, 701], "score": 1.0, "content": " mod 4. Finally, assume", "type": "text"}], "index": 40}], "index": 39}], "layout_bboxes": [], "page_idx": 3, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [{"type": "table", "bbox": [167, 416, 444, 622], "blocks": [{"type": "table_caption", "bbox": [285, 377, 327, 389], "group_id": 0, "lines": [{"bbox": [285, 378, 327, 391], "spans": [{"bbox": [285, 378, 327, 391], "score": 1.0, "content": "Table 2", "type": "text"}], "index": 20}], "index": 20}, {"type": "table_body", "bbox": [167, 416, 444, 622], "group_id": 0, "lines": [{"bbox": [167, 416, 444, 622], "spans": [{"bbox": [167, 416, 444, 622], "score": 0.912, "html": "<html><body><table><tr><td>M1</td><td>Cl2(M1)</td><td>M2</td><td>Cl2(M2)</td></tr><tr><td>Q(v5, V-7 . 29 ) Q(V2,\u221a-5 \u00b7 31)</td><td>(2, 16)</td><td>Q(V5 \u00b7 29, \u221a-7)</td><td>(2, 16)</td></tr><tr><td>Q(V13,\u221a-3 \u00b737)</td><td>(4, 4) (2,16)</td><td>Q(v5,\u221a-2 : 31) Q(V37,\u221a-3. 13</td><td>(2, 16) (2,16)</td></tr><tr><td></td><td></td><td></td><td>(2,16)</td></tr><tr><td>Q(V-11,v5 \u00b7 29)</td><td>(2,16)</td><td>Q(V29, \u221a-5 \u00b7 11)</td><td>(2, 16)</td></tr><tr><td>Q(V5, V-17 \u00b7 19)</td><td>(4, 4)</td><td>Q(v17, V-5 \u00b7 19 )</td><td></td></tr><tr><td>Q(V29, \u221a-2 . 7)</td><td>(2,16)</td><td>Q(V2, \u221a-7 . 29)</td><td>(2,16)</td></tr><tr><td>Q(v5 . 89, \u221a-1)</td><td>(4, 4)</td><td>Q(V5, \u221a-89 )</td><td>(2,8)</td></tr><tr><td>Q(V37, \u221a-5 \u00b7 11)</td><td>(4, 4)</td><td>Q(v5, \u221a-37 \u00b7 11)</td><td>(2,32)</td></tr><tr><td>Q(V53,\u221a-3\u00b713)</td><td>(4, 4)</td><td>Q(V13 \u00b7 53, \u221a-3)</td><td>(2,2,4)</td></tr><tr><td>Q(V37, \u221a-2 . 7)</td><td>(2, 16)</td><td>Q(v2,\u221a-7 . 37)</td><td>(2, 16)</td></tr><tr><td>Q(\u221a13,\u221a-2 \u00b7 23\uff09</td><td>(4, 4)</td><td>Q(v2,\u221a-13 \u00b7 23</td><td>(2, 16)</td></tr></table></body></html>", "type": "table", "image_path": "20908e76a81bf659100a97a9d08d0e64405e7d94b7aaa0b27349fad02b929aae.jpg"}]}], "index": 28.5, "virtual_lines": [{"bbox": [167, 416, 444, 429], "spans": [], "index": 21}, {"bbox": [167, 429, 444, 442], "spans": [], "index": 22}, {"bbox": [167, 442, 444, 455], "spans": [], "index": 23}, {"bbox": [167, 455, 444, 468], "spans": [], "index": 24}, {"bbox": [167, 468, 444, 481], "spans": [], "index": 25}, {"bbox": [167, 481, 444, 494], "spans": [], "index": 26}, {"bbox": [167, 494, 444, 507], "spans": [], "index": 27}, {"bbox": [167, 507, 444, 520], "spans": [], "index": 28}, {"bbox": [167, 520, 444, 533], "spans": [], "index": 29}, {"bbox": [167, 533, 444, 546], "spans": [], "index": 30}, {"bbox": [167, 546, 444, 559], "spans": [], "index": 31}, {"bbox": [167, 559, 444, 572], "spans": [], "index": 32}, {"bbox": [167, 572, 444, 585], "spans": [], "index": 33}, {"bbox": [167, 585, 444, 598], "spans": [], "index": 34}, {"bbox": [167, 598, 444, 611], "spans": [], "index": 35}, {"bbox": [167, 611, 444, 624], "spans": [], "index": 36}]}], "index": 24.25}], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [125, 91, 131, 99], "lines": [{"bbox": [125, 94, 132, 99], "spans": [{"bbox": [125, 94, 132, 99], "score": 1.0, "content": "4", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 111, 487, 237], "lines": [{"bbox": [126, 114, 485, 126], "spans": [{"bbox": [126, 114, 215, 126], "score": 1.0, "content": "Proof. Assume that ", "type": "text"}, {"bbox": [215, 115, 258, 126], "score": 0.94, "content": "d(G^{\\prime})=2", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [258, 114, 326, 126], "score": 1.0, "content": ". By Lemma 1, ", "type": "text"}, {"bbox": [326, 115, 381, 126], "score": 0.95, "content": "G_{2}=\\langle c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [382, 114, 431, 126], "score": 1.0, "content": "and hence ", "type": "text"}, {"bbox": [431, 115, 482, 126], "score": 0.94, "content": "c_{4}\\in\\langle c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [483, 114, 485, 126], "score": 1.0, "content": ".", "type": "text"}], "index": 0}, {"bbox": [123, 122, 489, 143], "spans": [{"bbox": [123, 122, 154, 143], "score": 1.0, "content": "Write ", "type": "text"}, {"bbox": [155, 127, 198, 138], "score": 0.95, "content": "c_{4}\\:=\\:c_{2}^{x}c_{3}^{y}", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [199, 122, 232, 143], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [232, 131, 248, 137], "score": 0.89, "content": "x,y", "type": "inline_equation", "height": 6, "width": 16}, {"bbox": [248, 122, 489, 143], "score": 1.0, "content": " are positive integers. Without loss of generality, let", "type": "text"}], "index": 1}, {"bbox": [126, 136, 487, 152], "spans": [{"bbox": [126, 139, 187, 150], "score": 0.93, "content": "H=\\langle b,c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [187, 136, 235, 152], "score": 1.0, "content": "and write ", "type": "text"}, {"bbox": [235, 139, 297, 150], "score": 0.93, "content": "(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [297, 136, 340, 152], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [340, 140, 365, 149], "score": 0.92, "content": "\\kappa\\geq2", "type": "inline_equation", "height": 9, "width": 25}, {"bbox": [365, 136, 398, 152], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [398, 139, 444, 149], "score": 0.93, "content": "c_{3},c_{4}\\in H^{\\prime}", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [444, 136, 487, 152], "score": 1.0, "content": " we have,", "type": "text"}], "index": 2}, {"bbox": [126, 150, 487, 162], "spans": [{"bbox": [126, 152, 155, 162], "score": 0.86, "content": "c_{2}^{x}\\equiv1", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [155, 150, 179, 162], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [179, 151, 191, 159], "score": 0.84, "content": "H^{\\prime}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [191, 150, 392, 162], "score": 1.0, "content": ". By the proof of Lemma 2, this implies that ", "type": "text"}, {"bbox": [393, 152, 417, 159], "score": 0.73, "content": "x\\equiv0", "type": "inline_equation", "height": 7, "width": 24}, {"bbox": [418, 150, 442, 162], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [442, 152, 452, 159], "score": 0.89, "content": "2^{\\kappa}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [453, 150, 487, 162], "score": 1.0, "content": ". Write", "type": "text"}], "index": 3}, {"bbox": [126, 160, 482, 178], "spans": [{"bbox": [126, 165, 168, 174], "score": 0.92, "content": "x\\,=\\,2^{\\kappa}x_{1}", "type": "inline_equation", "height": 9, "width": 42}, {"bbox": [168, 160, 284, 178], "score": 1.0, "content": " for some positive integer ", "type": "text"}, {"bbox": [285, 168, 295, 174], "score": 0.89, "content": "x_{1}", "type": "inline_equation", "height": 6, "width": 10}, {"bbox": [295, 160, 420, 178], "score": 1.0, "content": ". On the other hand, since ", "type": "text"}, {"bbox": [420, 162, 482, 175], "score": 0.93, "content": "c_{4},c_{2}^{2^{\\kappa}x_{1}}\\,\\in\\,G_{4}", "type": "inline_equation", "height": 13, "width": 62}], "index": 4}, {"bbox": [125, 175, 486, 187], "spans": [{"bbox": [125, 175, 179, 187], "score": 1.0, "content": "we see that ", "type": "text"}, {"bbox": [180, 176, 209, 187], "score": 0.93, "content": "c_{3}^{y}\\equiv1", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [209, 175, 233, 187], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [233, 177, 245, 186], "score": 0.91, "content": "G_{4}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [246, 175, 264, 187], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [264, 180, 270, 186], "score": 0.89, "content": "_y", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [270, 175, 343, 187], "score": 1.0, "content": " were odd, then ", "type": "text"}, {"bbox": [343, 177, 378, 186], "score": 0.93, "content": "c_{3}\\in G_{4}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [378, 175, 486, 187], "score": 1.0, "content": ". This, however, implies", "type": "text"}], "index": 5}, {"bbox": [124, 186, 486, 200], "spans": [{"bbox": [124, 186, 147, 200], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 188, 191, 199], "score": 0.94, "content": "G_{2}=\\langle c_{2}\\rangle", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [192, 186, 356, 200], "score": 1.0, "content": ", contrary to our assumptions. Thus ", "type": "text"}, {"bbox": [356, 191, 362, 198], "score": 0.88, "content": "_y", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [362, 186, 419, 200], "score": 1.0, "content": " is even, say ", "type": "text"}, {"bbox": [419, 189, 453, 198], "score": 0.93, "content": "y=2y_{1}", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [454, 186, 486, 200], "score": 1.0, "content": ". From", "type": "text"}], "index": 6}, {"bbox": [123, 196, 487, 215], "spans": [{"bbox": [123, 197, 265, 214], "score": 1.0, "content": "all of this we see that c4 = ", "type": "text"}, {"bbox": [228, 199, 291, 212], "score": 0.94, "content": "c_{4}\\,=\\,c_{2}^{2^{\\kappa}x_{1}}c_{3}^{2y_{1}}", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [253, 196, 487, 215], "score": 1.0, "content": "c22\u03bax1c23y1. Consequently, by induction we have cj \u2208", "type": "text"}], "index": 7}, {"bbox": [123, 208, 485, 232], "spans": [{"bbox": [123, 208, 345, 232], "score": 1.0, "content": "\u27e8c22j\u22122 , c32j\u22123 \u27e9 for all j \u2265 4. Since Gj = \u27e8c22j\u22122 , c23j\u2212", "type": "text"}, {"bbox": [272, 212, 449, 227], "score": 0.92, "content": "G_{j}=\\langle c_{2}^{2^{j-2}},c_{3}^{2^{j-3}},\\cdot\\cdot\\cdot,c_{j-1}^{2},c_{j},c_{j+1},\\cdot\\cdot\\cdot\\rangle", "type": "inline_equation", "height": 15, "width": 177}, {"bbox": [450, 213, 485, 227], "score": 1.0, "content": ", cf. [1],", "type": "text"}], "index": 8}, {"bbox": [125, 226, 486, 238], "spans": [{"bbox": [125, 226, 221, 238], "score": 1.0, "content": "we obtain the lemma.", "type": "text"}, {"bbox": [475, 226, 486, 236], "score": 0.9939806461334229, "content": "\u53e3", "type": "text"}], "index": 9}], "index": 4.5, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [123, 114, 489, 238]}, {"type": "text", "bbox": [125, 246, 487, 366], "lines": [{"bbox": [137, 248, 485, 261], "spans": [{"bbox": [137, 248, 420, 261], "score": 1.0, "content": "Let us translate the above into the field-theoretic language. Let ", "type": "text"}, {"bbox": [421, 250, 426, 257], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [427, 248, 485, 261], "score": 1.0, "content": " be an imagi-", "type": "text"}], "index": 10}, {"bbox": [124, 261, 485, 272], "spans": [{"bbox": [124, 261, 464, 272], "score": 1.0, "content": "nary quadratic number field of type A) or B) (see the Introduction), and let ", "type": "text"}, {"bbox": [464, 262, 485, 272], "score": 0.92, "content": "M/k", "type": "inline_equation", "height": 10, "width": 21}], "index": 11}, {"bbox": [125, 273, 487, 284], "spans": [{"bbox": [125, 273, 337, 284], "score": 1.0, "content": "be one of the two quadratic subextensions of ", "type": "text"}, {"bbox": [337, 273, 358, 284], "score": 0.94, "content": "k^{1}/k", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [358, 273, 415, 284], "score": 1.0, "content": " over which ", "type": "text"}, {"bbox": [415, 273, 425, 281], "score": 0.9, "content": "k^{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [426, 273, 487, 284], "score": 1.0, "content": " is cyclic. If", "type": "text"}], "index": 12}, {"bbox": [126, 282, 487, 298], "spans": [{"bbox": [126, 285, 191, 296], "score": 0.93, "content": "h_{2}(M)=2^{m+\\kappa}", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [192, 282, 214, 298], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [214, 285, 285, 296], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k)=(2,2^{m})", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [286, 282, 413, 298], "score": 1.0, "content": ", then Lemma 2 implies that ", "type": "text"}, {"bbox": [413, 285, 446, 296], "score": 0.94, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [446, 282, 487, 298], "score": 1.0, "content": " contains", "type": "text"}], "index": 13}, {"bbox": [126, 296, 486, 308], "spans": [{"bbox": [126, 296, 218, 308], "score": 1.0, "content": "an element of order ", "type": "text"}, {"bbox": [218, 298, 228, 305], "score": 0.88, "content": "2^{\\kappa}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [228, 296, 486, 308], "score": 1.0, "content": ". Table 2 contains the relevant information for the fields", "type": "text"}], "index": 14}, {"bbox": [125, 307, 486, 321], "spans": [{"bbox": [125, 307, 424, 321], "score": 1.0, "content": "occurring in Table 1. An application of the class number formula to ", "type": "text"}, {"bbox": [424, 309, 447, 320], "score": 0.94, "content": "M/\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [447, 307, 486, 321], "score": 1.0, "content": " (see e.g.", "type": "text"}], "index": 15}, {"bbox": [124, 320, 487, 333], "spans": [{"bbox": [124, 320, 325, 333], "score": 1.0, "content": "Proposition 3 below) shows immediately that ", "type": "text"}, {"bbox": [325, 321, 390, 332], "score": 0.93, "content": "h_{2}(M)=2^{m+\\kappa}", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [391, 320, 425, 333], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [425, 322, 435, 329], "score": 0.91, "content": "2^{\\kappa}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [435, 320, 487, 333], "score": 1.0, "content": " is the class", "type": "text"}], "index": 16}, {"bbox": [124, 332, 486, 346], "spans": [{"bbox": [124, 332, 271, 346], "score": 1.0, "content": "number of the quadratic subfield ", "type": "text"}, {"bbox": [271, 332, 316, 345], "score": 0.94, "content": "\\mathbb{Q}({\\sqrt{d_{i}d_{j}}}\\,)", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [317, 332, 330, 346], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [331, 334, 342, 342], "score": 0.89, "content": "M", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [342, 332, 376, 346], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [376, 334, 433, 344], "score": 0.94, "content": "(d_{i}/p_{j})=+1", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [433, 332, 486, 346], "score": 1.0, "content": "; in particu-", "type": "text"}], "index": 17}, {"bbox": [125, 344, 486, 357], "spans": [{"bbox": [125, 344, 213, 357], "score": 1.0, "content": "lar, we always have ", "type": "text"}, {"bbox": [213, 347, 237, 354], "score": 0.92, "content": "\\kappa\\geq2", "type": "inline_equation", "height": 7, "width": 24}, {"bbox": [238, 344, 332, 357], "score": 1.0, "content": ", and the assumption ", "type": "text"}, {"bbox": [332, 345, 389, 356], "score": 0.93, "content": "\\left(G^{\\prime}:H^{\\prime}\\right)\\geq4", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [390, 344, 486, 357], "score": 1.0, "content": " is always satisfied for", "type": "text"}], "index": 18}, {"bbox": [125, 356, 244, 368], "spans": [{"bbox": [125, 356, 244, 368], "score": 1.0, "content": "the fields that we consider.", "type": "text"}], "index": 19}], "index": 14.5, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [124, 248, 487, 368]}, {"type": "table", "bbox": [167, 416, 444, 622], "blocks": [{"type": "table_caption", "bbox": [285, 377, 327, 389], "group_id": 0, "lines": [{"bbox": [285, 378, 327, 391], "spans": [{"bbox": [285, 378, 327, 391], "score": 1.0, "content": "Table 2", "type": "text"}], "index": 20}], "index": 20}, {"type": "table_body", "bbox": [167, 416, 444, 622], "group_id": 0, "lines": [{"bbox": [167, 416, 444, 622], "spans": [{"bbox": [167, 416, 444, 622], "score": 0.912, "html": "<html><body><table><tr><td>M1</td><td>Cl2(M1)</td><td>M2</td><td>Cl2(M2)</td></tr><tr><td>Q(v5, V-7 . 29 ) Q(V2,\u221a-5 \u00b7 31)</td><td>(2, 16)</td><td>Q(V5 \u00b7 29, \u221a-7)</td><td>(2, 16)</td></tr><tr><td>Q(V13,\u221a-3 \u00b737)</td><td>(4, 4) (2,16)</td><td>Q(v5,\u221a-2 : 31) Q(V37,\u221a-3. 13</td><td>(2, 16) (2,16)</td></tr><tr><td></td><td></td><td></td><td>(2,16)</td></tr><tr><td>Q(V-11,v5 \u00b7 29)</td><td>(2,16)</td><td>Q(V29, \u221a-5 \u00b7 11)</td><td>(2, 16)</td></tr><tr><td>Q(V5, V-17 \u00b7 19)</td><td>(4, 4)</td><td>Q(v17, V-5 \u00b7 19 )</td><td></td></tr><tr><td>Q(V29, \u221a-2 . 7)</td><td>(2,16)</td><td>Q(V2, \u221a-7 . 29)</td><td>(2,16)</td></tr><tr><td>Q(v5 . 89, \u221a-1)</td><td>(4, 4)</td><td>Q(V5, \u221a-89 )</td><td>(2,8)</td></tr><tr><td>Q(V37, \u221a-5 \u00b7 11)</td><td>(4, 4)</td><td>Q(v5, \u221a-37 \u00b7 11)</td><td>(2,32)</td></tr><tr><td>Q(V53,\u221a-3\u00b713)</td><td>(4, 4)</td><td>Q(V13 \u00b7 53, \u221a-3)</td><td>(2,2,4)</td></tr><tr><td>Q(V37, \u221a-2 . 7)</td><td>(2, 16)</td><td>Q(v2,\u221a-7 . 37)</td><td>(2, 16)</td></tr><tr><td>Q(\u221a13,\u221a-2 \u00b7 23\uff09</td><td>(4, 4)</td><td>Q(v2,\u221a-13 \u00b7 23</td><td>(2, 16)</td></tr></table></body></html>", "type": "table", "image_path": "20908e76a81bf659100a97a9d08d0e64405e7d94b7aaa0b27349fad02b929aae.jpg"}]}], "index": 28.5, "virtual_lines": [{"bbox": [167, 416, 444, 429], "spans": [], "index": 21}, {"bbox": [167, 429, 444, 442], "spans": [], "index": 22}, {"bbox": [167, 442, 444, 455], "spans": [], "index": 23}, {"bbox": [167, 455, 444, 468], "spans": [], "index": 24}, {"bbox": [167, 468, 444, 481], "spans": [], "index": 25}, {"bbox": [167, 481, 444, 494], "spans": [], "index": 26}, {"bbox": [167, 494, 444, 507], "spans": [], "index": 27}, {"bbox": [167, 507, 444, 520], "spans": [], "index": 28}, {"bbox": [167, 520, 444, 533], "spans": [], "index": 29}, {"bbox": [167, 533, 444, 546], "spans": [], "index": 30}, {"bbox": [167, 546, 444, 559], "spans": [], "index": 31}, {"bbox": [167, 559, 444, 572], "spans": [], "index": 32}, {"bbox": [167, 572, 444, 585], "spans": [], "index": 33}, {"bbox": [167, 585, 444, 598], "spans": [], "index": 34}, {"bbox": [167, 598, 444, 611], "spans": [], "index": 35}, {"bbox": [167, 611, 444, 624], "spans": [], "index": 36}]}], "index": 24.25, "page_num": "page_3", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [137, 643, 450, 656], "lines": [{"bbox": [138, 645, 449, 657], "spans": [{"bbox": [138, 645, 449, 657], "score": 1.0, "content": "We now use the above results to prove the following useful proposition.", "type": "text"}], "index": 37}], "index": 37, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [138, 645, 449, 657]}, {"type": "text", "bbox": [125, 662, 486, 700], "lines": [{"bbox": [125, 665, 486, 677], "spans": [{"bbox": [125, 665, 220, 677], "score": 1.0, "content": "Proposition 1. Let ", "type": "text"}, {"bbox": [221, 667, 229, 674], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [229, 665, 409, 677], "score": 1.0, "content": " be a nonmetacyclic 2-group such that ", "type": "text"}, {"bbox": [409, 666, 482, 677], "score": 0.92, "content": "G/G^{\\prime}\\;\\simeq\\;(2,2^{m})", "type": "inline_equation", "height": 11, "width": 73}, {"bbox": [483, 665, 486, 677], "score": 1.0, "content": ";", "type": "text"}], "index": 38}, {"bbox": [127, 677, 484, 689], "spans": [{"bbox": [127, 677, 157, 689], "score": 1.0, "content": "(hence ", "type": "text"}, {"bbox": [158, 679, 186, 687], "score": 0.83, "content": "m>1", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [186, 677, 214, 689], "score": 1.0, "content": "). Let ", "type": "text"}, {"bbox": [215, 679, 224, 686], "score": 0.84, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [225, 677, 247, 689], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [247, 679, 257, 686], "score": 0.85, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [257, 677, 405, 689], "score": 1.0, "content": " be the two maximal subgroups of ", "type": "text"}, {"bbox": [406, 679, 414, 686], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [414, 677, 460, 689], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [460, 678, 484, 689], "score": 0.92, "content": "H/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 24}], "index": 39}, {"bbox": [127, 689, 487, 701], "spans": [{"bbox": [127, 689, 144, 701], "score": 1.0, "content": "and", "type": "text"}, {"bbox": [145, 690, 169, 701], "score": 0.91, "content": "K/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [169, 689, 322, 701], "score": 1.0, "content": " are cyclic. Moreover, assume that ", "type": "text"}, {"bbox": [322, 690, 379, 701], "score": 0.91, "content": "(G^{\\prime}:H^{\\prime})\\equiv0", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [380, 689, 487, 701], "score": 1.0, "content": " mod 4. Finally, assume", "type": "text"}], "index": 40}, {"bbox": [127, 115, 426, 126], "spans": [{"bbox": [127, 115, 146, 126], "score": 1.0, "content": "that ", "type": "text", "cross_page": true}, {"bbox": [146, 116, 156, 123], "score": 0.9, "content": "N", "type": "inline_equation", "height": 7, "width": 10, "cross_page": true}, {"bbox": [156, 115, 277, 126], "score": 1.0, "content": " is a subgroup of index 4 in", "type": "text", "cross_page": true}, {"bbox": [278, 115, 286, 124], "score": 0.8, "content": "G", "type": "inline_equation", "height": 9, "width": 8, "cross_page": true}, {"bbox": [287, 115, 363, 126], "score": 1.0, "content": " not contained in ", "type": "text", "cross_page": true}, {"bbox": [364, 115, 374, 123], "score": 0.76, "content": "H", "type": "inline_equation", "height": 8, "width": 10, "cross_page": true}, {"bbox": [374, 115, 389, 126], "score": 1.0, "content": " or ", "type": "text", "cross_page": true}, {"bbox": [389, 116, 399, 123], "score": 0.79, "content": "K", "type": "inline_equation", "height": 7, "width": 10, "cross_page": true}, {"bbox": [399, 115, 426, 126], "score": 1.0, "content": " Then", "type": "text", "cross_page": true}], "index": 0}], "index": 39, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [125, 665, 487, 701]}]}	{"layout_dets": [{"category_id": 1, "poly": [348, 684, 1353, 684, 1353, 1019, 348, 1019], "score": 0.982}, {"category_id": 1, "poly": [348, 310, 1353, 310, 1353, 659, 348, 659], "score": 0.979}, {"category_id": 1, "poly": [349, 1840, 1352, 1840, 1352, 1945, 349, 1945], "score": 0.945}, {"category_id": 5, "poly": [465, 1156, 1236, 1156, 1236, 1729, 465, 1729], "score": 0.912, "html": "<html><body><table><tr><td>M1</td><td>Cl2(M1)</td><td>M2</td><td>Cl2(M2)</td></tr><tr><td>Q(v5, V-7 . 29 ) Q(V2,\u221a-5 \u00b7 31)</td><td>(2, 16)</td><td>Q(V5 \u00b7 29, \u221a-7)</td><td>(2, 16)</td></tr><tr><td>Q(V13,\u221a-3 \u00b737)</td><td>(4, 4) (2,16)</td><td>Q(v5,\u221a-2 : 31) Q(V37,\u221a-3. 13</td><td>(2, 16) (2,16)</td></tr><tr><td></td><td></td><td></td><td>(2,16)</td></tr><tr><td>Q(V-11,v5 \u00b7 29)</td><td>(2,16)</td><td>Q(V29, \u221a-5 \u00b7 11)</td><td>(2, 16)</td></tr><tr><td>Q(V5, V-17 \u00b7 19)</td><td>(4, 4)</td><td>Q(v17, V-5 \u00b7 19 )</td><td></td></tr><tr><td>Q(V29, \u221a-2 . 7)</td><td>(2,16)</td><td>Q(V2, \u221a-7 . 29)</td><td>(2,16)</td></tr><tr><td>Q(v5 . 89, \u221a-1)</td><td>(4, 4)</td><td>Q(V5, \u221a-89 )</td><td>(2,8)</td></tr><tr><td>Q(V37, \u221a-5 \u00b7 11)</td><td>(4, 4)</td><td>Q(v5, \u221a-37 \u00b7 11)</td><td>(2,32)</td></tr><tr><td>Q(V53,\u221a-3\u00b713)</td><td>(4, 4)</td><td>Q(V13 \u00b7 53, \u221a-3)</td><td>(2,2,4)</td></tr><tr><td>Q(V37, \u221a-2 . 7)</td><td>(2, 16)</td><td>Q(v2,\u221a-7 . 37)</td><td>(2, 16)</td></tr><tr><td>Q(\u221a13,\u221a-2 \u00b7 23\uff09</td><td>(4, 4)</td><td>Q(v2,\u221a-13 \u00b7 23</td><td>(2, 16)</td></tr></table></body></html>"}, {"category_id": 1, "poly": [382, 1788, 1252, 1788, 1252, 1823, 382, 1823], "score": 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Assume that . By Lemma 1, and hence . Write where are positive integers. Without loss of generality, let and write for some . Since we have, mod . By the proof of Lemma 2, this implies that mod . Write for some positive integer . On the other hand, since we see that mod . If were odd, then . This, however, implies that , contrary to our assumptions. Thus is even, say . From all of this we see that c4 = c22κx1c23y1. Consequently, by induction we have cj ∈ ⟨c22j−2 , c32j−3 ⟩ for all j ≥ 4. Since Gj = ⟨c22j−2 , c23j− , cf. [1], we obtain the lemma. 口 Let us translate the above into the field-theoretic language. Let be an imagi- nary quadratic number field of type A) or B) (see the Introduction), and let be one of the two quadratic subextensions of over which is cyclic. If and , then Lemma 2 implies that contains an element of order . Table 2 contains the relevant information for the fields occurring in Table 1. An application of the class number formula to (see e.g. Proposition 3 below) shows immediately that , where is the class number of the quadratic subfield of , where ; in particu- lar, we always have , and the assumption is always satisfied for the fields that we consider. Table 2 ``` Table 2 ``` We now use the above results to prove the following useful proposition. Proposition 1. Let be a nonmetacyclic 2-group such that ; (hence ). Let and be the two maximal subgroups of such that and are cyclic. Moreover, assume that mod 4. Finally, assume that is a subgroup of index 4 in not contained in or Then	<div class="pdf-page"> <p>Proof. Assume that . By Lemma 1, and hence . Write where are positive integers. Without loss of generality, let and write for some . Since we have, mod . By the proof of Lemma 2, this implies that mod . Write for some positive integer . On the other hand, since we see that mod . If were odd, then . This, however, implies that , contrary to our assumptions. Thus is even, say . From all of this we see that c4 = c22κx1c23y1. Consequently, by induction we have cj ∈ ⟨c22j−2 , c32j−3 ⟩ for all j ≥ 4. Since Gj = ⟨c22j−2 , c23j− , cf. [1], we obtain the lemma. 口</p> <p>Let us translate the above into the field-theoretic language. Let be an imagi- nary quadratic number field of type A) or B) (see the Introduction), and let be one of the two quadratic subextensions of over which is cyclic. If and , then Lemma 2 implies that contains an element of order . Table 2 contains the relevant information for the fields occurring in Table 1. An application of the class number formula to (see e.g. Proposition 3 below) shows immediately that , where is the class number of the quadratic subfield of , where ; in particu- lar, we always have , and the assumption is always satisfied for the fields that we consider.</p> <h3>Table 2</h3> <p>We now use the above results to prove the following useful proposition.</p> <p>Proposition 1. Let be a nonmetacyclic 2-group such that ; (hence ). Let and be the two maximal subgroups of such that and are cyclic. Moreover, assume that mod 4. Finally, assume that is a subgroup of index 4 in not contained in or Then</p> </div>	<div class="pdf-page"> <div class="pdf-discarded" data-x="209" data-y="117" data-width="10" data-height="11" style="opacity: 0.5;">4</div> <p class="pdf-text" data-x="209" data-y="143" data-width="605" data-height="163">Proof. Assume that . By Lemma 1, and hence . Write where are positive integers. Without loss of generality, let and write for some . Since we have, mod . By the proof of Lemma 2, this implies that mod . Write for some positive integer . On the other hand, since we see that mod . If were odd, then . This, however, implies that , contrary to our assumptions. Thus is even, say . From all of this we see that c4 = c22κx1c23y1. Consequently, by induction we have cj ∈ ⟨c22j−2 , c32j−3 ⟩ for all j ≥ 4. Since Gj = ⟨c22j−2 , c23j− , cf. [1], we obtain the lemma. 口</p> <p class="pdf-text" data-x="209" data-y="318" data-width="605" data-height="155">Let us translate the above into the field-theoretic language. Let be an imagi- nary quadratic number field of type A) or B) (see the Introduction), and let be one of the two quadratic subextensions of over which is cyclic. If and , then Lemma 2 implies that contains an element of order . Table 2 contains the relevant information for the fields occurring in Table 1. An application of the class number formula to (see e.g. Proposition 3 below) shows immediately that , where is the class number of the quadratic subfield of , where ; in particu- lar, we always have , and the assumption is always satisfied for the fields that we consider.</p> <caption class="pdf-table-caption" data-x="476" data-y="487" data-width="71" data-height="15">Table 2</caption> <p class="pdf-text" data-x="229" data-y="831" data-width="523" data-height="17">We now use the above results to prove the following useful proposition.</p> <p class="pdf-text" data-x="209" data-y="855" data-width="604" data-height="50">Proposition 1. Let be a nonmetacyclic 2-group such that ; (hence ). Let and be the two maximal subgroups of such that and are cyclic. Moreover, assume that mod 4. Finally, assume that is a subgroup of index 4 in not contained in or Then</p> </div>	{ "type": [ "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "text", "text", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation" ], "coordinates": [ [ 210, 147, 811, 162 ], [ 205, 157, 818, 184 ], [ 210, 175, 814, 196 ], [ 210, 193, 814, 209 ], [ 210, 206, 806, 230 ], [ 209, 226, 813, 241 ], [ 207, 240, 813, 258 ], [ 205, 253, 814, 277 ], [ 205, 268, 811, 299 ], [ 209, 292, 813, 307 ], [ 229, 320, 811, 337 ], [ 207, 337, 811, 351 ], [ 209, 352, 814, 367 ], [ 210, 364, 814, 385 ], [ 210, 382, 813, 398 ], [ 209, 396, 813, 415 ], [ 207, 413, 814, 430 ], [ 207, 429, 813, 447 ], [ 209, 444, 813, 461 ], [ 209, 460, 408, 475 ], [ 476, 488, 547, 505 ], [ 279, 537, 742, 804 ], [ 230, 833, 751, 849 ], [ 209, 859, 813, 875 ], [ 212, 875, 809, 890 ], [ 212, 890, 814, 906 ], [ 212, 148, 712, 162 ] ], "content": [ "Proof. Assume that d(G^{\\prime})=2 . By Lemma 1, G_{2}=\\langle c_{2},c_{3}\\rangle and hence c_{4}\\in\\langle c_{2},c_{3}\\rangle .", "Write c_{4}\\:=\\:c_{2}^{x}c_{3}^{y} where x,y are positive integers. Without loss of generality, let", "H=\\langle b,c_{2},c_{3}\\rangle and write (G^{\\prime}:H^{\\prime})=2^{\\kappa}\\, for some \\kappa\\geq2 . Since c_{3},c_{4}\\in H^{\\prime} we have,", "c_{2}^{x}\\equiv1 mod H^{\\prime} . By the proof of Lemma 2, this implies that x\\equiv0 mod 2^{\\kappa} . Write", "x\\,=\\,2^{\\kappa}x_{1} for some positive integer x_{1} . On the other hand, since c_{4},c_{2}^{2^{\\kappa}x_{1}}\\,\\in\\,G_{4}", "we see that c_{3}^{y}\\equiv1 mod G_{4} . If _y were odd, then c_{3}\\in G_{4} . This, however, implies", "that G_{2}=\\langle c_{2}\\rangle , contrary to our assumptions. Thus _y is even, say y=2y_{1} . From", "all of this we see that c4 = c_{4}\\,=\\,c_{2}^{2^{\\kappa}x_{1}}c_{3}^{2y_{1}} c22κx1c23y1. Consequently, by induction we have cj ∈", "⟨c22j−2 , c32j−3 ⟩ for all j ≥ 4. Since Gj = ⟨c22j−2 , c23j− G_{j}=\\langle c_{2}^{2^{j-2}},c_{3}^{2^{j-3}},\\cdot\\cdot\\cdot,c_{j-1}^{2},c_{j},c_{j+1},\\cdot\\cdot\\cdot\\rangle , cf. [1],", "we obtain the lemma. 口", "Let us translate the above into the field-theoretic language. Let k be an imagi-", "nary quadratic number field of type A) or B) (see the Introduction), and let M/k", "be one of the two quadratic subextensions of k^{1}/k over which k^{1} is cyclic. If", "h_{2}(M)=2^{m+\\kappa} and \\mathrm{Cl}_{2}(k)=(2,2^{m}) , then Lemma 2 implies that \\mathrm{Cl_{2}}(k^{1}) contains", "an element of order 2^{\\kappa} . Table 2 contains the relevant information for the fields", "occurring in Table 1. An application of the class number formula to M/\\mathbb{Q} (see e.g.", "Proposition 3 below) shows immediately that h_{2}(M)=2^{m+\\kappa} , where 2^{\\kappa} is the class", "number of the quadratic subfield \\mathbb{Q}({\\sqrt{d_{i}d_{j}}}\\,) of M , where (d_{i}/p_{j})=+1 ; in particu-", "lar, we always have \\kappa\\geq2 , and the assumption \\left(G^{\\prime}:H^{\\prime}\\right)\\geq4 is always satisfied for", "the fields that we consider.", "Table 2", "", "We now use the above results to prove the following useful proposition.", "Proposition 1. Let G be a nonmetacyclic 2-group such that G/G^{\\prime}\\;\\simeq\\;(2,2^{m}) ;", "(hence m>1 ). Let H and K be the two maximal subgroups of G such that H/G^{\\prime}", "and K/G^{\\prime} are cyclic. Moreover, assume that (G^{\\prime}:H^{\\prime})\\equiv0 mod 4. Finally, assume", "that N is a subgroup of index 4 in G not contained in H or K Then" ], "index": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 30, 51, 73, 96, 97, 98, 99 ] }	{ "coordinates": [], "index": [], "caption": [], "caption_coordinates": [] }	{ "type": [], "coordinates": [], "content": [], "index": [] }	{ "type": [ "table", "table_caption", "table_body" ], "coordinates": [ [ 279, 537, 742, 804 ], [ 476, 487, 547, 502 ], [ 279, 537, 742, 804 ] ], "content": [ "", "Table 2", "" ], "index": [ 0, 1, 2 ] }	{"document_info": {"total_pages": 14, "parse_type": "txt", "version": "1.0.1", "total_content_list_items": 155}, "content_statistics": {"total_blocks": 182, "total_lines": 526, "total_images": 4, "total_equations": 28, "total_tables": 11, "total_characters": 29658, "total_words": 5331, "block_type_distribution": {"title": 7, "text": 126, "interline_equation": 14, "discarded": 24, "table_caption": 3, "table_body": 4, "image_body": 2, "list": 2}}, "model_statistics": {"total_layout_detections": 2826, "avg_confidence": 0.9470955414012739, "min_confidence": 0.26, "max_confidence": 1.0}, "page_statistics": [{"page_number": 0, "page_size": [612.0, 792.0], "blocks": 11, "lines": 38, "images": 0, "equations": 2, "tables": 0, "characters": 2362, "words": 398, "discarded_blocks": 1}, {"page_number": 1, "page_size": [612.0, 792.0], "blocks": 10, "lines": 30, "images": 0, "equations": 0, "tables": 3, "characters": 1688, "words": 305, "discarded_blocks": 1}, {"page_number": 2, "page_size": [612.0, 792.0], "blocks": 15, "lines": 44, "images": 0, "equations": 4, "tables": 0, "characters": 2417, "words": 416, "discarded_blocks": 2}, {"page_number": 3, "page_size": [612.0, 792.0], "blocks": 7, "lines": 27, "images": 0, "equations": 0, "tables": 3, "characters": 1591, "words": 291, "discarded_blocks": 1}, {"page_number": 4, "page_size": [612.0, 792.0], "blocks": 23, "lines": 38, "images": 0, "equations": 6, "tables": 0, "characters": 1604, "words": 303, "discarded_blocks": 4}, {"page_number": 5, "page_size": [612.0, 792.0], "blocks": 17, "lines": 40, "images": 0, "equations": 6, "tables": 0, "characters": 2131, "words": 393, "discarded_blocks": 1}, {"page_number": 6, "page_size": [612.0, 792.0], "blocks": 15, "lines": 36, "images": 2, "equations": 4, "tables": 0, "characters": 1791, "words": 324, "discarded_blocks": 2}, {"page_number": 7, "page_size": [612.0, 792.0], "blocks": 12, "lines": 49, "images": 0, "equations": 0, "tables": 0, "characters": 3049, "words": 571, "discarded_blocks": 1}, {"page_number": 8, "page_size": [612.0, 792.0], "blocks": 11, "lines": 42, "images": 0, "equations": 0, "tables": 0, "characters": 2636, "words": 479, "discarded_blocks": 2}, {"page_number": 9, "page_size": [612.0, 792.0], "blocks": 14, "lines": 35, "images": 2, "equations": 0, "tables": 0, "characters": 2036, "words": 375, "discarded_blocks": 1}, {"page_number": 10, "page_size": [612.0, 792.0], "blocks": 17, "lines": 35, "images": 0, "equations": 6, "tables": 0, "characters": 1789, "words": 327, "discarded_blocks": 2}, {"page_number": 11, "page_size": [612.0, 792.0], "blocks": 14, "lines": 40, "images": 0, "equations": 0, "tables": 2, "characters": 2094, "words": 399, "discarded_blocks": 2}, {"page_number": 12, "page_size": [612.0, 792.0], "blocks": 10, "lines": 44, "images": 0, "equations": 0, "tables": 3, "characters": 2572, "words": 487, "discarded_blocks": 2}, {"page_number": 13, "page_size": [612.0, 792.0], "blocks": 6, "lines": 28, "images": 0, "equations": 0, "tables": 0, "characters": 1898, "words": 263, "discarded_blocks": 2}]}
0003244v1	4	[ 612, 792 ]	{ "type": [ "text", "interline_equation", "text", "text", "text", "text", "text", "interline_equation", "text", "text", "text", "title", "text", "text", "text", "text", "interline_equation", "text", "text", "discarded", "discarded", "discarded", "discarded" ], "coordinates": [ [ 209, 143, 714, 160 ], [ 366, 166, 654, 218 ], [ 207, 221, 813, 268 ], [ 225, 270, 436, 284 ], [ 209, 284, 814, 330 ], [ 207, 330, 813, 408 ], [ 209, 408, 813, 439 ], [ 384, 449, 635, 464 ], [ 207, 468, 814, 545 ], [ 210, 546, 813, 577 ], [ 229, 577, 470, 593 ], [ 356, 608, 664, 625 ], [ 209, 632, 813, 680 ], [ 209, 687, 813, 704 ], [ 209, 729, 814, 761 ], [ 209, 769, 813, 800 ], [ 339, 809, 675, 826 ], [ 209, 832, 813, 880 ], [ 210, 888, 321, 905 ], [ 396, 116, 625, 129 ], [ 796, 888, 813, 902 ], [ 801, 116, 813, 128 ], [ 796, 577, 813, 592 ] ], "content": [ "", "", "Proof. Without loss of generality we assume that is metabelian. Let , where mod . Also let and (without loss of generality). Then or .", "Suppose that .", "First assume . Then and thus . But for some (cf. Lemma of [1]). Hence, , and so . Since , we get as desired.", "Next, assume that . Then by Lemma 1. Notice that and where for . Hence and so . But then by Lemma 3. Therefore, by [5], . But notice that . Thus and so which in turn implies that , as desired.", "Finally, assume . Then . Moreover there exists an exact sequence", "", "and thus . Hence it suffices to prove the result for which we now assume. and so, arguing as above, we have , where . But . Therefore, . From this we see that and thus as desired.", "Now suppose that . Then the proof is essentially the same as above once we notice that mod .", "This establishes the proposition.", "3. Number Theoretic Preliminaries", "Proposition 2. Let be a quadratic extension, and assume that the class num- ber of , , is odd. If has an unramified cyclic extension M of order 4, then is normal and .", "Proof. R´edei and Reichardt [12] proved this for ; the general case is analogous.", "We shall make extensive use of the class number formula for extensions of type :", "Proposition 3. Let be a normal quartic extension with Galois group of type , and let ) denote the quadratic subextensions. Then", "", "where denotes the unit index of is the unit group of ), is the number of infinite primes in that ramify in , is the -rank of the unit group of , and except when , where .", "Proof. See [10].", "IMAGINARY QUADRATIC FIELDS", "口", "5", "口" ], "index": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 ] }	[{"type": "text", "text": "", "page_idx": 4}, {"type": "equation", "text": "$$\n(N:N^{\\prime})\\;\\left\\{\\begin{array}{l l}{{=}}&{{2^{m}\\;\\;\\;\\;\\;i f\\,d(G^{\\prime})=1}}\\\\ {{=}}&{{2^{m+1}\\;\\;\\;\\;i f\\,d(G^{\\prime})=2}}\\\\ {{\\geq}}&{{2^{m+2}\\;\\;\\;\\;i f\\,d(G^{\\prime})\\geq3}}\\end{array}\\right..\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "Proof. Without loss of generality we assume that $G$ is metabelian. Let $G=\\langle a,b\\rangle$ , where $a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1$ mod $G_{3}$ . Also let $H=\\langle b,G^{\\prime}\\rangle$ and $K=\\langle a b,G^{\\prime}\\rangle$ (without loss of generality). Then $N=\\langle a b^{2},G^{\\prime}\\rangle$ or $N=\\langle a,b^{4},G^{\\prime}\\rangle$ . ", "page_idx": 4}, {"type": "text", "text": "Suppose that $N=\\left\\langle a b^{2},G^{\\prime}\\right\\rangle$ . ", "page_idx": 4}, {"type": "text", "text": "First assume $d(G^{\\prime})=1$ . Then $G^{\\prime}=\\langle c_{2}\\rangle$ and thus $N^{\\prime}=\\left\\langle[a b^{2},c_{2}]\\right\\rangle$ . But $[a b^{2},c_{2}]=$ $c_{2}^{2}\\eta_{4}$ for some $\\eta_{4}\\,\\in\\,G_{4}\\,=\\,\\langle c_{2}^{4}\\rangle$ (cf. Lemma $^{1}$ of [1]). Hence, $N^{\\prime}\\,=\\,\\left\\langle c_{2}^{2}\\right\\rangle$ , and so $\\left(G^{\\prime}:N^{\\prime}\\right)=2$ . Since $(N:G^{\\prime})=2^{m-1}$ , we get $(N:N^{\\prime})=2^{m}$ as desired. ", "page_idx": 4}, {"type": "text", "text": "Next, assume that $d(G^{\\prime})\\;=\\;2$ . Then $N\\,=\\,\\langle a b^{2},c_{2},c_{3}\\rangle$ by Lemma 1. Notice that $[a b^{2},c_{2}]~=~c_{2}^{2}\\eta_{4}$ and $[a b^{2},c_{3}]\\;=\\;c_{3}^{2}\\eta_{5}$ where $\\eta_{j}~\\in~G_{j}$ for $j~=~4,5$ . Hence $N^{\\prime}=\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},N_{3}\\rangle$ and so $\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5}\\rangle\\subseteq N^{\\prime}$ . But then $N^{\\prime}G_{5}\\supseteq\\langle c_{2}^{4},c_{3}^{2}\\rangle=G_{4}$ by Lemma 3. Therefore, by [5], $N^{\\prime}\\supseteq G_{4}$ . But notice that $N_{3}\\subseteq G_{4}$ . Thus $N^{\\prime}=\\left\\langle c_{2}^{2},c_{3}^{2}\\right\\rangle$ and so $(G^{\\prime}:N^{\\prime})=4$ which in turn implies that $(N:N^{\\prime})=2^{m+1}$ , as desired. ", "page_idx": 4}, {"type": "text", "text": "Finally, assume $d(G^{\\prime})\\geq3$ . Then $d(G^{\\prime}/G_{5})=3$ . Moreover there exists an exact sequence ", "page_idx": 4}, {"type": "equation", "text": "$$\nN/N^{\\prime}\\longrightarrow(N/G_{5})/(N/G_{5})^{\\prime}\\longrightarrow1,\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "and thus $\\#N^{\\mathrm{ab}}\\,\\geq\\,\\#(N/G_{5})^{\\mathrm{ab}}$ . Hence it suffices to prove the result for $G_{5}\\,=\\,1$ which we now assume. $N=\\langle a b^{2},c_{2},c_{3},c_{4}\\rangle$ and so, arguing as above, we have $N^{\\prime}=$ $\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},c_{4}^{2}\\eta_{6},N_{3}\\rangle\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2},N_{3}\\rangle$ , where $\\eta_{j}\\;\\in\\;G_{j}$ . But $N_{3}\\,=\\,\\langle[a b^{2},c_{2}^{2}\\eta_{4}]\\rangle\\,=$ $\\langle c_{2}^{4}\\rangle$ . Therefore, $N^{\\prime}\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\rangle$ . From this we see that $(G^{\\prime}\\,:\\,N^{\\prime})\\,=\\,8$ and thus $(N:N^{\\prime})=2^{m+2}$ as desired. ", "page_idx": 4}, {"type": "text", "text": "Now suppose that $N\\,=\\,\\langle a,b^{4},G^{\\prime}\\rangle$ . Then the proof is essentially the same as above once we notice that $[a,b^{4}]\\equiv c_{3}{}^{2}c_{2}{}^{-4}$ mod $G_{5}$ . ", "page_idx": 4}, {"type": "text", "text": "This establishes the proposition. ", "page_idx": 4}, {"type": "text", "text": "3. Number Theoretic Preliminaries ", "text_level": 1, "page_idx": 4}, {"type": "text", "text": "Proposition 2. Let $K/k$ be a quadratic extension, and assume that the class number of $k$ , $h(k)$ , is odd. If $K$ has an unramified cyclic extension M of order 4, then $M/k$ is normal and $\\operatorname{Gal}(M/k)\\simeq D_{4}$ . ", "page_idx": 4}, {"type": "text", "text": "Proof. R\u00b4edei and Reichardt [12] proved this for $k=\\mathbb{Q}$ ; the general case is analogous. ", "page_idx": 4}, {"type": "text", "text": "We shall make extensive use of the class number formula for extensions of type $(2,2)$ : ", "page_idx": 4}, {"type": "text", "text": "Proposition 3. Let $K/k$ be a normal quartic extension with Galois group of type $(2,2)$ , and let $k_{j}$ $(j=1,2,3)$ ) denote the quadratic subextensions. Then ", "page_idx": 4}, {"type": "equation", "text": "$$\nh(K)=2^{d-\\kappa-2-v}q(K)h(k_{1})h(k_{2})h(k_{3})/h(k)^{2},\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "where $q(K)=(E_{K}:E_{1}E_{2}E_{3})$ denotes the unit index of $K/k$ $(E_{j}$ is the unit group of $k_{j}$ ), $d$ is the number of infinite primes in $k$ that ramify in $K/k$ , $\\kappa$ is the $\\mathbb{Z}$ -rank of the unit group $E_{k}$ of $k$ , and $\\upsilon=0$ except when $K\\subseteq k(\\sqrt{E_{k}}\\,)$ , where $\\upsilon=1$ . ", "page_idx": 4}, {"type": "text", "text": "Proof. See [10]. ", "page_idx": 4}]	{"preproc_blocks": [{"type": "text", "bbox": [125, 111, 427, 124], "lines": [{"bbox": [127, 115, 426, 126], "spans": [{"bbox": [127, 115, 146, 126], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [146, 116, 156, 123], "score": 0.9, "content": "N", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [156, 115, 277, 126], "score": 1.0, "content": " is a subgroup of index 4 in", "type": "text"}, {"bbox": [278, 115, 286, 124], "score": 0.8, "content": "G", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [287, 115, 363, 126], "score": 1.0, "content": " not contained in ", "type": "text"}, {"bbox": [364, 115, 374, 123], "score": 0.76, "content": "H", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [374, 115, 389, 126], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [389, 116, 399, 123], "score": 0.79, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [399, 115, 426, 126], "score": 1.0, "content": " Then", "type": "text"}], "index": 0}], "index": 0}, {"type": "interline_equation", "bbox": [219, 129, 391, 169], "lines": [{"bbox": [219, 129, 391, 169], "spans": [{"bbox": [219, 129, 391, 169], "score": 0.92, "content": "(N:N^{\\prime})\\;\\left\\{\\begin{array}{l l}{{=}}&{{2^{m}\\;\\;\\;\\;\\;i f\\,d(G^{\\prime})=1}}\\\\ {{=}}&{{2^{m+1}\\;\\;\\;\\;i f\\,d(G^{\\prime})=2}}\\\\ {{\\geq}}&{{2^{m+2}\\;\\;\\;\\;i f\\,d(G^{\\prime})\\geq3}}\\end{array}\\right..", "type": "interline_equation"}], "index": 1}], "index": 1}, {"type": "text", "bbox": [124, 171, 486, 208], "lines": [{"bbox": [126, 174, 485, 187], "spans": [{"bbox": [126, 174, 344, 187], "score": 1.0, "content": "Proof. Without loss of generality we assume that ", "type": "text"}, {"bbox": [344, 176, 352, 183], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [353, 174, 438, 187], "score": 1.0, "content": " is metabelian. Let ", "type": "text"}, {"bbox": [439, 175, 482, 186], "score": 0.94, "content": "G=\\langle a,b\\rangle", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [482, 174, 485, 187], "score": 1.0, "content": ",", "type": "text"}], "index": 2}, {"bbox": [126, 186, 487, 199], "spans": [{"bbox": [126, 186, 154, 199], "score": 1.0, "content": "where", "type": "text"}, {"bbox": [155, 186, 213, 195], "score": 0.92, "content": "a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1", "type": "inline_equation", "height": 9, "width": 58}, {"bbox": [214, 186, 237, 199], "score": 1.0, "content": " mod", "type": "text"}, {"bbox": [238, 188, 250, 197], "score": 0.9, "content": "G_{3}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [250, 186, 295, 199], "score": 1.0, "content": ". Also let ", "type": "text"}, {"bbox": [295, 187, 345, 198], "score": 0.92, "content": "H=\\langle b,G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [346, 186, 368, 199], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [369, 187, 424, 198], "score": 0.93, "content": "K=\\langle a b,G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [425, 186, 487, 199], "score": 1.0, "content": "(without loss", "type": "text"}], "index": 3}, {"bbox": [126, 198, 359, 210], "spans": [{"bbox": [126, 198, 217, 210], "score": 1.0, "content": "of generality). Then ", "type": "text"}, {"bbox": [217, 199, 276, 210], "score": 0.95, "content": "N=\\langle a b^{2},G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [277, 198, 291, 210], "score": 1.0, "content": "or ", "type": "text"}, {"bbox": [292, 199, 356, 210], "score": 0.94, "content": "N=\\langle a,b^{4},G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 64}, {"bbox": [356, 198, 359, 210], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 3}, {"type": "text", "bbox": [135, 209, 261, 220], "lines": [{"bbox": [137, 209, 261, 222], "spans": [{"bbox": [137, 209, 198, 221], "score": 1.0, "content": "Suppose that ", "type": "text"}, {"bbox": [198, 211, 258, 222], "score": 0.93, "content": "N=\\left\\langle a b^{2},G^{\\prime}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 60}, {"bbox": [258, 209, 261, 221], "score": 1.0, "content": ".", "type": "text"}], "index": 5}], "index": 5}, {"type": "text", "bbox": [125, 220, 487, 256], "lines": [{"bbox": [137, 220, 487, 235], "spans": [{"bbox": [137, 220, 194, 235], "score": 1.0, "content": "First assume", "type": "text"}, {"bbox": [195, 223, 236, 234], "score": 0.92, "content": "d(G^{\\prime})=1", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [237, 220, 268, 235], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [268, 223, 308, 234], "score": 0.93, "content": "G^{\\prime}=\\langle c_{2}\\rangle", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [309, 220, 349, 235], "score": 1.0, "content": "and thus", "type": "text"}, {"bbox": [350, 222, 416, 234], "score": 0.92, "content": "N^{\\prime}=\\left\\langle[a b^{2},c_{2}]\\right\\rangle", "type": "inline_equation", "height": 12, "width": 66}, {"bbox": [416, 220, 441, 235], "score": 1.0, "content": ". But ", "type": "text"}, {"bbox": [442, 223, 487, 234], "score": 0.9, "content": "[a b^{2},c_{2}]=", "type": "inline_equation", "height": 11, "width": 45}], "index": 6}, {"bbox": [126, 234, 486, 246], "spans": [{"bbox": [126, 235, 144, 246], "score": 0.92, "content": "c_{2}^{2}\\eta_{4}", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [145, 234, 190, 246], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [191, 235, 261, 246], "score": 0.92, "content": "\\eta_{4}\\,\\in\\,G_{4}\\,=\\,\\langle c_{2}^{4}\\rangle", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [262, 234, 323, 246], "score": 1.0, "content": "(cf. Lemma ", "type": "text"}, {"bbox": [323, 236, 329, 243], "score": 0.26, "content": "^{1}", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [329, 234, 402, 246], "score": 1.0, "content": " of [1]). Hence, ", "type": "text"}, {"bbox": [403, 235, 448, 246], "score": 0.93, "content": "N^{\\prime}\\,=\\,\\left\\langle c_{2}^{2}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [448, 234, 486, 246], "score": 1.0, "content": ", and so", "type": "text"}], "index": 7}, {"bbox": [126, 244, 438, 259], "spans": [{"bbox": [126, 247, 183, 258], "score": 0.92, "content": "\\left(G^{\\prime}:N^{\\prime}\\right)=2", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [183, 244, 216, 259], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [216, 247, 288, 258], "score": 0.91, "content": "(N:G^{\\prime})=2^{m-1}", "type": "inline_equation", "height": 11, "width": 72}, {"bbox": [288, 244, 325, 259], "score": 1.0, "content": ", we get ", "type": "text"}, {"bbox": [325, 247, 388, 258], "score": 0.91, "content": "(N:N^{\\prime})=2^{m}", "type": "inline_equation", "height": 11, "width": 63}, {"bbox": [388, 244, 438, 259], "score": 1.0, "content": " as desired.", "type": "text"}], "index": 8}], "index": 7}, {"type": "text", "bbox": [124, 256, 486, 316], "lines": [{"bbox": [136, 256, 487, 270], "spans": [{"bbox": [136, 256, 225, 270], "score": 1.0, "content": "Next, assume that ", "type": "text"}, {"bbox": [225, 259, 271, 270], "score": 0.93, "content": "d(G^{\\prime})\\;=\\;2", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [272, 256, 309, 270], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [309, 258, 385, 270], "score": 0.93, "content": "N\\,=\\,\\langle a b^{2},c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 12, "width": 76}, {"bbox": [385, 256, 487, 270], "score": 1.0, "content": "by Lemma 1. Notice", "type": "text"}], "index": 9}, {"bbox": [124, 268, 487, 284], "spans": [{"bbox": [124, 268, 149, 284], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [149, 270, 218, 282], "score": 0.92, "content": "[a b^{2},c_{2}]~=~c_{2}^{2}\\eta_{4}", "type": "inline_equation", "height": 12, "width": 69}, {"bbox": [219, 268, 244, 284], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [244, 270, 314, 282], "score": 0.92, "content": "[a b^{2},c_{3}]\\;=\\;c_{3}^{2}\\eta_{5}", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [314, 268, 348, 284], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [349, 272, 387, 282], "score": 0.93, "content": "\\eta_{j}~\\in~G_{j}", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [388, 268, 408, 284], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [409, 272, 447, 281], "score": 0.91, "content": "j~=~4,5", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [447, 268, 487, 284], "score": 1.0, "content": ". Hence", "type": "text"}], "index": 10}, {"bbox": [126, 280, 487, 295], "spans": [{"bbox": [126, 283, 218, 294], "score": 0.92, "content": "N^{\\prime}=\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},N_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [218, 280, 252, 295], "score": 1.0, "content": "and so ", "type": "text"}, {"bbox": [253, 282, 326, 294], "score": 0.92, "content": "\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5}\\rangle\\subseteq N^{\\prime}", "type": "inline_equation", "height": 12, "width": 73}, {"bbox": [327, 280, 376, 295], "score": 1.0, "content": ". But then ", "type": "text"}, {"bbox": [376, 282, 471, 294], "score": 0.91, "content": "N^{\\prime}G_{5}\\supseteq\\langle c_{2}^{4},c_{3}^{2}\\rangle=G_{4}", "type": "inline_equation", "height": 12, "width": 95}, {"bbox": [471, 280, 487, 295], "score": 1.0, "content": " by", "type": "text"}], "index": 11}, {"bbox": [124, 293, 485, 307], "spans": [{"bbox": [124, 293, 248, 307], "score": 1.0, "content": "Lemma 3. Therefore, by [5], ", "type": "text"}, {"bbox": [248, 295, 286, 304], "score": 0.92, "content": "N^{\\prime}\\supseteq G_{4}", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [286, 293, 360, 307], "score": 1.0, "content": ". But notice that", "type": "text"}, {"bbox": [361, 296, 399, 304], "score": 0.94, "content": "N_{3}\\subseteq G_{4}", "type": "inline_equation", "height": 8, "width": 38}, {"bbox": [399, 293, 429, 307], "score": 1.0, "content": ". Thus", "type": "text"}, {"bbox": [430, 295, 485, 306], "score": 0.93, "content": "N^{\\prime}=\\left\\langle c_{2}^{2},c_{3}^{2}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 55}], "index": 12}, {"bbox": [124, 304, 462, 318], "spans": [{"bbox": [124, 304, 157, 318], "score": 1.0, "content": "and so ", "type": "text"}, {"bbox": [157, 307, 214, 317], "score": 0.93, "content": "(G^{\\prime}:N^{\\prime})=4", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [215, 304, 335, 318], "score": 1.0, "content": " which in turn implies that ", "type": "text"}, {"bbox": [335, 307, 408, 317], "score": 0.92, "content": "(N:N^{\\prime})=2^{m+1}", "type": "inline_equation", "height": 10, "width": 73}, {"bbox": [409, 304, 462, 318], "score": 1.0, "content": ", as desired.", "type": "text"}], "index": 13}], "index": 11}, {"type": "text", "bbox": [125, 316, 486, 340], "lines": [{"bbox": [137, 318, 487, 330], "spans": [{"bbox": [137, 318, 208, 330], "score": 1.0, "content": "Finally, assume ", "type": "text"}, {"bbox": [208, 319, 250, 329], "score": 0.94, "content": "d(G^{\\prime})\\geq3", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [251, 318, 284, 330], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [284, 319, 343, 329], "score": 0.94, "content": "d(G^{\\prime}/G_{5})=3", "type": "inline_equation", "height": 10, "width": 59}, {"bbox": [344, 318, 487, 330], "score": 1.0, "content": ". Moreover there exists an exact", "type": "text"}], "index": 14}, {"bbox": [125, 331, 167, 343], "spans": [{"bbox": [125, 331, 167, 343], "score": 1.0, "content": "sequence", "type": "text"}], "index": 15}], "index": 14.5}, {"type": "interline_equation", "bbox": [230, 348, 380, 359], "lines": [{"bbox": [230, 348, 380, 359], "spans": [{"bbox": [230, 348, 380, 359], "score": 0.91, "content": "N/N^{\\prime}\\longrightarrow(N/G_{5})/(N/G_{5})^{\\prime}\\longrightarrow1,", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [124, 362, 487, 422], "lines": [{"bbox": [125, 363, 485, 378], "spans": [{"bbox": [125, 363, 168, 378], "score": 1.0, "content": "and thus ", "type": "text"}, {"bbox": [169, 366, 263, 377], "score": 0.91, "content": "\\#N^{\\mathrm{ab}}\\,\\geq\\,\\#(N/G_{5})^{\\mathrm{ab}}", "type": "inline_equation", "height": 11, "width": 94}, {"bbox": [263, 363, 451, 378], "score": 1.0, "content": ". Hence it suffices to prove the result for ", "type": "text"}, {"bbox": [451, 367, 485, 376], "score": 0.92, "content": "G_{5}\\,=\\,1", "type": "inline_equation", "height": 9, "width": 34}], "index": 17}, {"bbox": [126, 376, 487, 390], "spans": [{"bbox": [126, 376, 227, 390], "score": 1.0, "content": "which we now assume. ", "type": "text"}, {"bbox": [227, 378, 311, 389], "score": 0.91, "content": "N=\\langle a b^{2},c_{2},c_{3},c_{4}\\rangle", "type": "inline_equation", "height": 11, "width": 84}, {"bbox": [311, 376, 462, 390], "score": 1.0, "content": "and so, arguing as above, we have ", "type": "text"}, {"bbox": [462, 378, 487, 388], "score": 0.87, "content": "N^{\\prime}=", "type": "inline_equation", "height": 10, "width": 25}], "index": 18}, {"bbox": [126, 388, 485, 402], "spans": [{"bbox": [126, 389, 287, 401], "score": 0.9, "content": "\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},c_{4}^{2}\\eta_{6},N_{3}\\rangle\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2},N_{3}\\rangle", "type": "inline_equation", "height": 12, "width": 161}, {"bbox": [287, 388, 324, 402], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [325, 391, 361, 401], "score": 0.94, "content": "\\eta_{j}\\;\\in\\;G_{j}", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [362, 388, 393, 402], "score": 1.0, "content": ". But ", "type": "text"}, {"bbox": [393, 389, 485, 401], "score": 0.91, "content": "N_{3}\\,=\\,\\langle[a b^{2},c_{2}^{2}\\eta_{4}]\\rangle\\,=", "type": "inline_equation", "height": 12, "width": 92}], "index": 19}, {"bbox": [126, 399, 487, 415], "spans": [{"bbox": [126, 402, 142, 412], "score": 0.92, "content": "\\langle c_{2}^{4}\\rangle", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [143, 399, 201, 415], "score": 1.0, "content": ". Therefore, ", "type": "text"}, {"bbox": [201, 401, 268, 412], "score": 0.93, "content": "N^{\\prime}\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\rangle", "type": "inline_equation", "height": 11, "width": 67}, {"bbox": [268, 399, 379, 415], "score": 1.0, "content": ". From this we see that ", "type": "text"}, {"bbox": [379, 402, 442, 412], "score": 0.9, "content": "(G^{\\prime}\\,:\\,N^{\\prime})\\,=\\,8", "type": "inline_equation", "height": 10, "width": 63}, {"bbox": [442, 399, 487, 415], "score": 1.0, "content": " and thus", "type": "text"}], "index": 20}, {"bbox": [126, 410, 250, 426], "spans": [{"bbox": [126, 414, 199, 425], "score": 0.92, "content": "(N:N^{\\prime})=2^{m+2}", "type": "inline_equation", "height": 11, "width": 73}, {"bbox": [199, 410, 250, 426], "score": 1.0, "content": " as desired.", "type": "text"}], "index": 21}], "index": 19}, {"type": "text", "bbox": [126, 423, 486, 447], "lines": [{"bbox": [136, 424, 487, 437], "spans": [{"bbox": [136, 424, 222, 437], "score": 1.0, "content": "Now suppose that ", "type": "text"}, {"bbox": [222, 425, 290, 437], "score": 0.94, "content": "N\\,=\\,\\langle a,b^{4},G^{\\prime}\\rangle", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [290, 424, 487, 437], "score": 1.0, "content": ". Then the proof is essentially the same as", "type": "text"}], "index": 22}, {"bbox": [124, 434, 355, 450], "spans": [{"bbox": [124, 434, 242, 450], "score": 1.0, "content": "above once we notice that ", "type": "text"}, {"bbox": [243, 437, 312, 448], "score": 0.91, "content": "[a,b^{4}]\\equiv c_{3}{}^{2}c_{2}{}^{-4}", "type": "inline_equation", "height": 11, "width": 69}, {"bbox": [313, 434, 337, 450], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [337, 439, 349, 448], "score": 0.91, "content": "G_{5}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [350, 434, 355, 450], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22.5}, {"type": "text", "bbox": [137, 447, 281, 459], "lines": [{"bbox": [137, 448, 279, 460], "spans": [{"bbox": [137, 448, 279, 460], "score": 1.0, "content": "This establishes the proposition.", "type": "text"}], "index": 24}], "index": 24}, {"type": "title", "bbox": [213, 471, 397, 484], "lines": [{"bbox": [214, 474, 397, 484], "spans": [{"bbox": [214, 474, 397, 484], "score": 1.0, "content": "3. Number Theoretic Preliminaries", "type": "text"}], "index": 25}], "index": 25}, {"type": "text", "bbox": [125, 489, 486, 526], "lines": [{"bbox": [126, 492, 486, 505], "spans": [{"bbox": [126, 492, 218, 505], "score": 1.0, "content": "Proposition 2. Let ", "type": "text"}, {"bbox": [218, 493, 238, 503], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [238, 492, 486, 505], "score": 1.0, "content": " be a quadratic extension, and assume that the class num-", "type": "text"}], "index": 26}, {"bbox": [126, 502, 487, 517], "spans": [{"bbox": [126, 502, 154, 517], "score": 1.0, "content": "ber of ", "type": "text"}, {"bbox": [154, 505, 160, 513], "score": 0.76, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [160, 502, 166, 517], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [166, 505, 185, 515], "score": 0.92, "content": "h(k)", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [186, 502, 235, 517], "score": 1.0, "content": ", is odd. If ", "type": "text"}, {"bbox": [236, 505, 245, 513], "score": 0.82, "content": "K", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [245, 502, 487, 517], "score": 1.0, "content": " has an unramified cyclic extension M of order 4, then", "type": "text"}], "index": 27}, {"bbox": [126, 515, 288, 528], "spans": [{"bbox": [126, 516, 147, 527], "score": 0.87, "content": "M/k", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [147, 515, 214, 528], "score": 1.0, "content": " is normal and ", "type": "text"}, {"bbox": [215, 516, 285, 527], "score": 0.93, "content": "\\operatorname{Gal}(M/k)\\simeq D_{4}", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [285, 515, 288, 528], "score": 1.0, "content": ".", "type": "text"}], "index": 28}], "index": 27}, {"type": "text", "bbox": [125, 532, 486, 545], "lines": [{"bbox": [126, 534, 486, 547], "spans": [{"bbox": [126, 534, 329, 547], "score": 1.0, "content": "Proof. R\u00b4edei and Reichardt [12] proved this for ", "type": "text"}, {"bbox": [329, 536, 356, 545], "score": 0.92, "content": "k=\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [356, 534, 486, 547], "score": 1.0, "content": "; the general case is analogous.", "type": "text"}], "index": 29}], "index": 29}, {"type": "text", "bbox": [125, 564, 487, 589], "lines": [{"bbox": [137, 565, 486, 579], "spans": [{"bbox": [137, 565, 486, 579], "score": 1.0, "content": "We shall make extensive use of the class number formula for extensions of type", "type": "text"}], "index": 30}, {"bbox": [126, 578, 153, 591], "spans": [{"bbox": [126, 579, 148, 590], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 578, 153, 591], "score": 1.0, "content": ":", "type": "text"}], "index": 31}], "index": 30.5}, {"type": "text", "bbox": [125, 595, 486, 619], "lines": [{"bbox": [125, 597, 486, 609], "spans": [{"bbox": [125, 597, 218, 609], "score": 1.0, "content": "Proposition 3. Let", "type": "text"}, {"bbox": [219, 597, 239, 609], "score": 0.87, "content": "K/k", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [239, 597, 486, 609], "score": 1.0, "content": " be a normal quartic extension with Galois group of type", "type": "text"}], "index": 32}, {"bbox": [126, 610, 437, 621], "spans": [{"bbox": [126, 610, 148, 621], "score": 0.91, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 610, 187, 621], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [188, 610, 198, 621], "score": 0.86, "content": "k_{j}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [198, 610, 202, 621], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [202, 610, 248, 621], "score": 0.77, "content": "(j=1,2,3)", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [248, 610, 437, 621], "score": 1.0, "content": ") denote the quadratic subextensions. Then", "type": "text"}], "index": 33}], "index": 32.5}, {"type": "interline_equation", "bbox": [203, 626, 404, 639], "lines": [{"bbox": [203, 626, 404, 639], "spans": [{"bbox": [203, 626, 404, 639], "score": 0.9, "content": "h(K)=2^{d-\\kappa-2-v}q(K)h(k_{1})h(k_{2})h(k_{3})/h(k)^{2},", "type": "interline_equation"}], "index": 34}], "index": 34}, {"type": "text", "bbox": [125, 644, 486, 681], "lines": [{"bbox": [127, 646, 487, 658], "spans": [{"bbox": [127, 646, 154, 658], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 647, 255, 658], "score": 0.91, "content": "q(K)=(E_{K}:E_{1}E_{2}E_{3})", "type": "inline_equation", "height": 11, "width": 101}, {"bbox": [256, 646, 370, 658], "score": 1.0, "content": " denotes the unit index of ", "type": "text"}, {"bbox": [371, 647, 390, 658], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [391, 646, 396, 658], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [396, 647, 409, 658], "score": 0.69, "content": "(E_{j}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [410, 646, 487, 658], "score": 1.0, "content": " is the unit group", "type": "text"}], "index": 35}, {"bbox": [126, 658, 487, 670], "spans": [{"bbox": [126, 658, 137, 669], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [137, 659, 147, 670], "score": 0.8, "content": "k_{j}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [147, 658, 157, 669], "score": 1.0, "content": "), ", "type": "text"}, {"bbox": [158, 659, 163, 667], "score": 0.8, "content": "d", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [164, 658, 319, 669], "score": 1.0, "content": " is the number of infinite primes in", "type": "text"}, {"bbox": [320, 658, 326, 667], "score": 0.74, "content": "k", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [327, 658, 393, 669], "score": 1.0, "content": " that ramify in ", "type": "text"}, {"bbox": [393, 659, 413, 669], "score": 0.88, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [413, 658, 418, 669], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [418, 660, 425, 667], "score": 0.44, "content": "\\kappa", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [425, 658, 455, 669], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [455, 659, 462, 667], "score": 0.8, "content": "\\mathbb{Z}", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [463, 658, 487, 669], "score": 1.0, "content": "-rank", "type": "text"}], "index": 36}, {"bbox": [126, 670, 466, 682], "spans": [{"bbox": [126, 670, 202, 682], "score": 1.0, "content": "of the unit group ", "type": "text"}, {"bbox": [202, 671, 215, 681], "score": 0.87, "content": "E_{k}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [215, 670, 230, 682], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [230, 671, 236, 680], "score": 0.77, "content": "k", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [236, 670, 261, 682], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [261, 670, 286, 680], "score": 0.88, "content": "\\upsilon=0", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [286, 670, 344, 682], "score": 1.0, "content": " except when ", "type": "text"}, {"bbox": [344, 670, 403, 682], "score": 0.93, "content": "K\\subseteq k(\\sqrt{E_{k}}\\,)", "type": "inline_equation", "height": 12, "width": 59}, {"bbox": [403, 670, 436, 682], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [437, 672, 461, 680], "score": 0.88, "content": "\\upsilon=1", "type": "inline_equation", "height": 8, "width": 24}, {"bbox": [462, 670, 466, 682], "score": 1.0, "content": ".", "type": "text"}], "index": 37}], "index": 36}, {"type": "text", "bbox": [126, 687, 192, 700], "lines": [{"bbox": [126, 687, 194, 702], "spans": [{"bbox": [126, 687, 194, 702], "score": 1.0, "content": "Proof. See [10].", "type": "text"}], "index": 38}], "index": 38}], "layout_bboxes": [], "page_idx": 4, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [219, 129, 391, 169], "lines": [{"bbox": [219, 129, 391, 169], "spans": [{"bbox": [219, 129, 391, 169], "score": 0.92, "content": "(N:N^{\\prime})\\;\\left\\{\\begin{array}{l l}{{=}}&{{2^{m}\\;\\;\\;\\;\\;i f\\,d(G^{\\prime})=1}}\\\\ {{=}}&{{2^{m+1}\\;\\;\\;\\;i f\\,d(G^{\\prime})=2}}\\\\ {{\\geq}}&{{2^{m+2}\\;\\;\\;\\;i f\\,d(G^{\\prime})\\geq3}}\\end{array}\\right..", "type": "interline_equation"}], "index": 1}], "index": 1}, {"type": "interline_equation", "bbox": [230, 348, 380, 359], "lines": [{"bbox": [230, 348, 380, 359], "spans": [{"bbox": [230, 348, 380, 359], "score": 0.91, "content": "N/N^{\\prime}\\longrightarrow(N/G_{5})/(N/G_{5})^{\\prime}\\longrightarrow1,", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "interline_equation", "bbox": [203, 626, 404, 639], "lines": [{"bbox": [203, 626, 404, 639], "spans": [{"bbox": [203, 626, 404, 639], "score": 0.9, "content": "h(K)=2^{d-\\kappa-2-v}q(K)h(k_{1})h(k_{2})h(k_{3})/h(k)^{2},", "type": "interline_equation"}], "index": 34}], "index": 34}], "discarded_blocks": [{"type": "discarded", "bbox": [237, 90, 374, 100], "lines": [{"bbox": [239, 92, 372, 101], "spans": [{"bbox": [239, 92, 372, 101], "score": 1.0, "content": "IMAGINARY QUADRATIC FIELDS", "type": "text"}]}]}, {"type": "discarded", "bbox": [476, 687, 486, 698], "lines": [{"bbox": [475, 689, 487, 700], "spans": [{"bbox": [475, 689, 487, 700], "score": 0.9910128712654114, "content": "\u53e3", "type": "text"}]}]}, {"type": "discarded", "bbox": [479, 90, 486, 99], "lines": [{"bbox": [480, 93, 486, 101], "spans": [{"bbox": [480, 93, 486, 101], "score": 1.0, "content": "5", "type": "text"}]}]}, {"type": "discarded", "bbox": [476, 447, 486, 458], "lines": [{"bbox": [475, 450, 487, 460], "spans": [{"bbox": [475, 450, 487, 460], "score": 0.9908492565155029, "content": "\u53e3", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 111, 427, 124], "lines": [], "index": 0, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [127, 115, 426, 126], "lines_deleted": true}, {"type": "interline_equation", "bbox": [219, 129, 391, 169], "lines": [{"bbox": [219, 129, 391, 169], "spans": [{"bbox": [219, 129, 391, 169], "score": 0.92, "content": "(N:N^{\\prime})\\;\\left\\{\\begin{array}{l l}{{=}}&{{2^{m}\\;\\;\\;\\;\\;i f\\,d(G^{\\prime})=1}}\\\\ {{=}}&{{2^{m+1}\\;\\;\\;\\;i f\\,d(G^{\\prime})=2}}\\\\ {{\\geq}}&{{2^{m+2}\\;\\;\\;\\;i f\\,d(G^{\\prime})\\geq3}}\\end{array}\\right..", "type": "interline_equation"}], "index": 1}], "index": 1, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 171, 486, 208], "lines": [{"bbox": [126, 174, 485, 187], "spans": [{"bbox": [126, 174, 344, 187], "score": 1.0, "content": "Proof. Without loss of generality we assume that ", "type": "text"}, {"bbox": [344, 176, 352, 183], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [353, 174, 438, 187], "score": 1.0, "content": " is metabelian. Let ", "type": "text"}, {"bbox": [439, 175, 482, 186], "score": 0.94, "content": "G=\\langle a,b\\rangle", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [482, 174, 485, 187], "score": 1.0, "content": ",", "type": "text"}], "index": 2}, {"bbox": [126, 186, 487, 199], "spans": [{"bbox": [126, 186, 154, 199], "score": 1.0, "content": "where", "type": "text"}, {"bbox": [155, 186, 213, 195], "score": 0.92, "content": "a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1", "type": "inline_equation", "height": 9, "width": 58}, {"bbox": [214, 186, 237, 199], "score": 1.0, "content": " mod", "type": "text"}, {"bbox": [238, 188, 250, 197], "score": 0.9, "content": "G_{3}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [250, 186, 295, 199], "score": 1.0, "content": ". Also let ", "type": "text"}, {"bbox": [295, 187, 345, 198], "score": 0.92, "content": "H=\\langle b,G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [346, 186, 368, 199], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [369, 187, 424, 198], "score": 0.93, "content": "K=\\langle a b,G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [425, 186, 487, 199], "score": 1.0, "content": "(without loss", "type": "text"}], "index": 3}, {"bbox": [126, 198, 359, 210], "spans": [{"bbox": [126, 198, 217, 210], "score": 1.0, "content": "of generality). Then ", "type": "text"}, {"bbox": [217, 199, 276, 210], "score": 0.95, "content": "N=\\langle a b^{2},G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [277, 198, 291, 210], "score": 1.0, "content": "or ", "type": "text"}, {"bbox": [292, 199, 356, 210], "score": 0.94, "content": "N=\\langle a,b^{4},G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 64}, {"bbox": [356, 198, 359, 210], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 3, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [126, 174, 487, 210]}, {"type": "text", "bbox": [135, 209, 261, 220], "lines": [{"bbox": [137, 209, 261, 222], "spans": [{"bbox": [137, 209, 198, 221], "score": 1.0, "content": "Suppose that ", "type": "text"}, {"bbox": [198, 211, 258, 222], "score": 0.93, "content": "N=\\left\\langle a b^{2},G^{\\prime}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 60}, {"bbox": [258, 209, 261, 221], "score": 1.0, "content": ".", "type": "text"}], "index": 5}], "index": 5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [137, 209, 261, 222]}, {"type": "text", "bbox": [125, 220, 487, 256], "lines": [{"bbox": [137, 220, 487, 235], "spans": [{"bbox": [137, 220, 194, 235], "score": 1.0, "content": "First assume", "type": "text"}, {"bbox": [195, 223, 236, 234], "score": 0.92, "content": "d(G^{\\prime})=1", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [237, 220, 268, 235], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [268, 223, 308, 234], "score": 0.93, "content": "G^{\\prime}=\\langle c_{2}\\rangle", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [309, 220, 349, 235], "score": 1.0, "content": "and thus", "type": "text"}, {"bbox": [350, 222, 416, 234], "score": 0.92, "content": "N^{\\prime}=\\left\\langle[a b^{2},c_{2}]\\right\\rangle", "type": "inline_equation", "height": 12, "width": 66}, {"bbox": [416, 220, 441, 235], "score": 1.0, "content": ". But ", "type": "text"}, {"bbox": [442, 223, 487, 234], "score": 0.9, "content": "[a b^{2},c_{2}]=", "type": "inline_equation", "height": 11, "width": 45}], "index": 6}, {"bbox": [126, 234, 486, 246], "spans": [{"bbox": [126, 235, 144, 246], "score": 0.92, "content": "c_{2}^{2}\\eta_{4}", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [145, 234, 190, 246], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [191, 235, 261, 246], "score": 0.92, "content": "\\eta_{4}\\,\\in\\,G_{4}\\,=\\,\\langle c_{2}^{4}\\rangle", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [262, 234, 323, 246], "score": 1.0, "content": "(cf. Lemma ", "type": "text"}, {"bbox": [323, 236, 329, 243], "score": 0.26, "content": "^{1}", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [329, 234, 402, 246], "score": 1.0, "content": " of [1]). Hence, ", "type": "text"}, {"bbox": [403, 235, 448, 246], "score": 0.93, "content": "N^{\\prime}\\,=\\,\\left\\langle c_{2}^{2}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [448, 234, 486, 246], "score": 1.0, "content": ", and so", "type": "text"}], "index": 7}, {"bbox": [126, 244, 438, 259], "spans": [{"bbox": [126, 247, 183, 258], "score": 0.92, "content": "\\left(G^{\\prime}:N^{\\prime}\\right)=2", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [183, 244, 216, 259], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [216, 247, 288, 258], "score": 0.91, "content": "(N:G^{\\prime})=2^{m-1}", "type": "inline_equation", "height": 11, "width": 72}, {"bbox": [288, 244, 325, 259], "score": 1.0, "content": ", we get ", "type": "text"}, {"bbox": [325, 247, 388, 258], "score": 0.91, "content": "(N:N^{\\prime})=2^{m}", "type": "inline_equation", "height": 11, "width": 63}, {"bbox": [388, 244, 438, 259], "score": 1.0, "content": " as desired.", "type": "text"}], "index": 8}], "index": 7, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [126, 220, 487, 259]}, {"type": "text", "bbox": [124, 256, 486, 316], "lines": [{"bbox": [136, 256, 487, 270], "spans": [{"bbox": [136, 256, 225, 270], "score": 1.0, "content": "Next, assume that ", "type": "text"}, {"bbox": [225, 259, 271, 270], "score": 0.93, "content": "d(G^{\\prime})\\;=\\;2", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [272, 256, 309, 270], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [309, 258, 385, 270], "score": 0.93, "content": "N\\,=\\,\\langle a b^{2},c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 12, "width": 76}, {"bbox": [385, 256, 487, 270], "score": 1.0, "content": "by Lemma 1. Notice", "type": "text"}], "index": 9}, {"bbox": [124, 268, 487, 284], "spans": [{"bbox": [124, 268, 149, 284], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [149, 270, 218, 282], "score": 0.92, "content": "[a b^{2},c_{2}]~=~c_{2}^{2}\\eta_{4}", "type": "inline_equation", "height": 12, "width": 69}, {"bbox": [219, 268, 244, 284], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [244, 270, 314, 282], "score": 0.92, "content": "[a b^{2},c_{3}]\\;=\\;c_{3}^{2}\\eta_{5}", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [314, 268, 348, 284], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [349, 272, 387, 282], "score": 0.93, "content": "\\eta_{j}~\\in~G_{j}", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [388, 268, 408, 284], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [409, 272, 447, 281], "score": 0.91, "content": "j~=~4,5", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [447, 268, 487, 284], "score": 1.0, "content": ". Hence", "type": "text"}], "index": 10}, {"bbox": [126, 280, 487, 295], "spans": [{"bbox": [126, 283, 218, 294], "score": 0.92, "content": "N^{\\prime}=\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},N_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [218, 280, 252, 295], "score": 1.0, "content": "and so ", "type": "text"}, {"bbox": [253, 282, 326, 294], "score": 0.92, "content": "\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5}\\rangle\\subseteq N^{\\prime}", "type": "inline_equation", "height": 12, "width": 73}, {"bbox": [327, 280, 376, 295], "score": 1.0, "content": ". But then ", "type": "text"}, {"bbox": [376, 282, 471, 294], "score": 0.91, "content": "N^{\\prime}G_{5}\\supseteq\\langle c_{2}^{4},c_{3}^{2}\\rangle=G_{4}", "type": "inline_equation", "height": 12, "width": 95}, {"bbox": [471, 280, 487, 295], "score": 1.0, "content": " by", "type": "text"}], "index": 11}, {"bbox": [124, 293, 485, 307], "spans": [{"bbox": [124, 293, 248, 307], "score": 1.0, "content": "Lemma 3. Therefore, by [5], ", "type": "text"}, {"bbox": [248, 295, 286, 304], "score": 0.92, "content": "N^{\\prime}\\supseteq G_{4}", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [286, 293, 360, 307], "score": 1.0, "content": ". But notice that", "type": "text"}, {"bbox": [361, 296, 399, 304], "score": 0.94, "content": "N_{3}\\subseteq G_{4}", "type": "inline_equation", "height": 8, "width": 38}, {"bbox": [399, 293, 429, 307], "score": 1.0, "content": ". Thus", "type": "text"}, {"bbox": [430, 295, 485, 306], "score": 0.93, "content": "N^{\\prime}=\\left\\langle c_{2}^{2},c_{3}^{2}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 55}], "index": 12}, {"bbox": [124, 304, 462, 318], "spans": [{"bbox": [124, 304, 157, 318], "score": 1.0, "content": "and so ", "type": "text"}, {"bbox": [157, 307, 214, 317], "score": 0.93, "content": "(G^{\\prime}:N^{\\prime})=4", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [215, 304, 335, 318], "score": 1.0, "content": " which in turn implies that ", "type": "text"}, {"bbox": [335, 307, 408, 317], "score": 0.92, "content": "(N:N^{\\prime})=2^{m+1}", "type": "inline_equation", "height": 10, "width": 73}, {"bbox": [409, 304, 462, 318], "score": 1.0, "content": ", as desired.", "type": "text"}], "index": 13}], "index": 11, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [124, 256, 487, 318]}, {"type": "text", "bbox": [125, 316, 486, 340], "lines": [{"bbox": [137, 318, 487, 330], "spans": [{"bbox": [137, 318, 208, 330], "score": 1.0, "content": "Finally, assume ", "type": "text"}, {"bbox": [208, 319, 250, 329], "score": 0.94, "content": "d(G^{\\prime})\\geq3", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [251, 318, 284, 330], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [284, 319, 343, 329], "score": 0.94, "content": "d(G^{\\prime}/G_{5})=3", "type": "inline_equation", "height": 10, "width": 59}, {"bbox": [344, 318, 487, 330], "score": 1.0, "content": ". Moreover there exists an exact", "type": "text"}], "index": 14}, {"bbox": [125, 331, 167, 343], "spans": [{"bbox": [125, 331, 167, 343], "score": 1.0, "content": "sequence", "type": "text"}], "index": 15}], "index": 14.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [125, 318, 487, 343]}, {"type": "interline_equation", "bbox": [230, 348, 380, 359], "lines": [{"bbox": [230, 348, 380, 359], "spans": [{"bbox": [230, 348, 380, 359], "score": 0.91, "content": "N/N^{\\prime}\\longrightarrow(N/G_{5})/(N/G_{5})^{\\prime}\\longrightarrow1,", "type": "interline_equation"}], "index": 16}], "index": 16, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 362, 487, 422], "lines": [{"bbox": [125, 363, 485, 378], "spans": [{"bbox": [125, 363, 168, 378], "score": 1.0, "content": "and thus ", "type": "text"}, {"bbox": [169, 366, 263, 377], "score": 0.91, "content": "\\#N^{\\mathrm{ab}}\\,\\geq\\,\\#(N/G_{5})^{\\mathrm{ab}}", "type": "inline_equation", "height": 11, "width": 94}, {"bbox": [263, 363, 451, 378], "score": 1.0, "content": ". Hence it suffices to prove the result for ", "type": "text"}, {"bbox": [451, 367, 485, 376], "score": 0.92, "content": "G_{5}\\,=\\,1", "type": "inline_equation", "height": 9, "width": 34}], "index": 17}, {"bbox": [126, 376, 487, 390], "spans": [{"bbox": [126, 376, 227, 390], "score": 1.0, "content": "which we now assume. ", "type": "text"}, {"bbox": [227, 378, 311, 389], "score": 0.91, "content": "N=\\langle a b^{2},c_{2},c_{3},c_{4}\\rangle", "type": "inline_equation", "height": 11, "width": 84}, {"bbox": [311, 376, 462, 390], "score": 1.0, "content": "and so, arguing as above, we have ", "type": "text"}, {"bbox": [462, 378, 487, 388], "score": 0.87, "content": "N^{\\prime}=", "type": "inline_equation", "height": 10, "width": 25}], "index": 18}, {"bbox": [126, 388, 485, 402], "spans": [{"bbox": [126, 389, 287, 401], "score": 0.9, "content": "\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},c_{4}^{2}\\eta_{6},N_{3}\\rangle\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2},N_{3}\\rangle", "type": "inline_equation", "height": 12, "width": 161}, {"bbox": [287, 388, 324, 402], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [325, 391, 361, 401], "score": 0.94, "content": "\\eta_{j}\\;\\in\\;G_{j}", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [362, 388, 393, 402], "score": 1.0, "content": ". But ", "type": "text"}, {"bbox": [393, 389, 485, 401], "score": 0.91, "content": "N_{3}\\,=\\,\\langle[a b^{2},c_{2}^{2}\\eta_{4}]\\rangle\\,=", "type": "inline_equation", "height": 12, "width": 92}], "index": 19}, {"bbox": [126, 399, 487, 415], "spans": [{"bbox": [126, 402, 142, 412], "score": 0.92, "content": "\\langle c_{2}^{4}\\rangle", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [143, 399, 201, 415], "score": 1.0, "content": ". Therefore, ", "type": "text"}, {"bbox": [201, 401, 268, 412], "score": 0.93, "content": "N^{\\prime}\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\rangle", "type": "inline_equation", "height": 11, "width": 67}, {"bbox": [268, 399, 379, 415], "score": 1.0, "content": ". From this we see that ", "type": "text"}, {"bbox": [379, 402, 442, 412], "score": 0.9, "content": "(G^{\\prime}\\,:\\,N^{\\prime})\\,=\\,8", "type": "inline_equation", "height": 10, "width": 63}, {"bbox": [442, 399, 487, 415], "score": 1.0, "content": " and thus", "type": "text"}], "index": 20}, {"bbox": [126, 410, 250, 426], "spans": [{"bbox": [126, 414, 199, 425], "score": 0.92, "content": "(N:N^{\\prime})=2^{m+2}", "type": "inline_equation", "height": 11, "width": 73}, {"bbox": [199, 410, 250, 426], "score": 1.0, "content": " as desired.", "type": "text"}], "index": 21}], "index": 19, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [125, 363, 487, 426]}, {"type": "text", "bbox": [126, 423, 486, 447], "lines": [{"bbox": [136, 424, 487, 437], "spans": [{"bbox": [136, 424, 222, 437], "score": 1.0, "content": "Now suppose that ", "type": "text"}, {"bbox": [222, 425, 290, 437], "score": 0.94, "content": "N\\,=\\,\\langle a,b^{4},G^{\\prime}\\rangle", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [290, 424, 487, 437], "score": 1.0, "content": ". Then the proof is essentially the same as", "type": "text"}], "index": 22}, {"bbox": [124, 434, 355, 450], "spans": [{"bbox": [124, 434, 242, 450], "score": 1.0, "content": "above once we notice that ", "type": "text"}, {"bbox": [243, 437, 312, 448], "score": 0.91, "content": "[a,b^{4}]\\equiv c_{3}{}^{2}c_{2}{}^{-4}", "type": "inline_equation", "height": 11, "width": 69}, {"bbox": [313, 434, 337, 450], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [337, 439, 349, 448], "score": 0.91, "content": "G_{5}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [350, 434, 355, 450], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [124, 424, 487, 450]}, {"type": "text", "bbox": [137, 447, 281, 459], "lines": [{"bbox": [137, 448, 279, 460], "spans": [{"bbox": [137, 448, 279, 460], "score": 1.0, "content": "This establishes the proposition.", "type": "text"}], "index": 24}], "index": 24, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [137, 448, 279, 460]}, {"type": "title", "bbox": [213, 471, 397, 484], "lines": [{"bbox": [214, 474, 397, 484], "spans": [{"bbox": [214, 474, 397, 484], "score": 1.0, "content": "3. Number Theoretic Preliminaries", "type": "text"}], "index": 25}], "index": 25, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 489, 486, 526], "lines": [{"bbox": [126, 492, 486, 505], "spans": [{"bbox": [126, 492, 218, 505], "score": 1.0, "content": "Proposition 2. Let ", "type": "text"}, {"bbox": [218, 493, 238, 503], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [238, 492, 486, 505], "score": 1.0, "content": " be a quadratic extension, and assume that the class num-", "type": "text"}], "index": 26}, {"bbox": [126, 502, 487, 517], "spans": [{"bbox": [126, 502, 154, 517], "score": 1.0, "content": "ber of ", "type": "text"}, {"bbox": [154, 505, 160, 513], "score": 0.76, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [160, 502, 166, 517], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [166, 505, 185, 515], "score": 0.92, "content": "h(k)", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [186, 502, 235, 517], "score": 1.0, "content": ", is odd. If ", "type": "text"}, {"bbox": [236, 505, 245, 513], "score": 0.82, "content": "K", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [245, 502, 487, 517], "score": 1.0, "content": " has an unramified cyclic extension M of order 4, then", "type": "text"}], "index": 27}, {"bbox": [126, 515, 288, 528], "spans": [{"bbox": [126, 516, 147, 527], "score": 0.87, "content": "M/k", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [147, 515, 214, 528], "score": 1.0, "content": " is normal and ", "type": "text"}, {"bbox": [215, 516, 285, 527], "score": 0.93, "content": "\\operatorname{Gal}(M/k)\\simeq D_{4}", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [285, 515, 288, 528], "score": 1.0, "content": ".", "type": "text"}], "index": 28}], "index": 27, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [126, 492, 487, 528]}, {"type": "text", "bbox": [125, 532, 486, 545], "lines": [{"bbox": [126, 534, 486, 547], "spans": [{"bbox": [126, 534, 329, 547], "score": 1.0, "content": "Proof. R\u00b4edei and Reichardt [12] proved this for ", "type": "text"}, {"bbox": [329, 536, 356, 545], "score": 0.92, "content": "k=\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [356, 534, 486, 547], "score": 1.0, "content": "; the general case is analogous.", "type": "text"}], "index": 29}], "index": 29, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [126, 534, 486, 547]}, {"type": "text", "bbox": [125, 564, 487, 589], "lines": [{"bbox": [137, 565, 486, 579], "spans": [{"bbox": [137, 565, 486, 579], "score": 1.0, "content": "We shall make extensive use of the class number formula for extensions of type", "type": "text"}], "index": 30}, {"bbox": [126, 578, 153, 591], "spans": [{"bbox": [126, 579, 148, 590], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 578, 153, 591], "score": 1.0, "content": ":", "type": "text"}], "index": 31}], "index": 30.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [126, 565, 486, 591]}, {"type": "text", "bbox": [125, 595, 486, 619], "lines": [{"bbox": [125, 597, 486, 609], "spans": [{"bbox": [125, 597, 218, 609], "score": 1.0, "content": "Proposition 3. Let", "type": "text"}, {"bbox": [219, 597, 239, 609], "score": 0.87, "content": "K/k", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [239, 597, 486, 609], "score": 1.0, "content": " be a normal quartic extension with Galois group of type", "type": "text"}], "index": 32}, {"bbox": [126, 610, 437, 621], "spans": [{"bbox": [126, 610, 148, 621], "score": 0.91, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 610, 187, 621], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [188, 610, 198, 621], "score": 0.86, "content": "k_{j}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [198, 610, 202, 621], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [202, 610, 248, 621], "score": 0.77, "content": "(j=1,2,3)", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [248, 610, 437, 621], "score": 1.0, "content": ") denote the quadratic subextensions. Then", "type": "text"}], "index": 33}], "index": 32.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [125, 597, 486, 621]}, {"type": "interline_equation", "bbox": [203, 626, 404, 639], "lines": [{"bbox": [203, 626, 404, 639], "spans": [{"bbox": [203, 626, 404, 639], "score": 0.9, "content": "h(K)=2^{d-\\kappa-2-v}q(K)h(k_{1})h(k_{2})h(k_{3})/h(k)^{2},", "type": "interline_equation"}], "index": 34}], "index": 34, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 644, 486, 681], "lines": [{"bbox": [127, 646, 487, 658], "spans": [{"bbox": [127, 646, 154, 658], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 647, 255, 658], "score": 0.91, "content": "q(K)=(E_{K}:E_{1}E_{2}E_{3})", "type": "inline_equation", "height": 11, "width": 101}, {"bbox": [256, 646, 370, 658], "score": 1.0, "content": " denotes the unit index of ", "type": "text"}, {"bbox": [371, 647, 390, 658], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [391, 646, 396, 658], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [396, 647, 409, 658], "score": 0.69, "content": "(E_{j}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [410, 646, 487, 658], "score": 1.0, "content": " is the unit group", "type": "text"}], "index": 35}, {"bbox": [126, 658, 487, 670], "spans": [{"bbox": [126, 658, 137, 669], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [137, 659, 147, 670], "score": 0.8, "content": "k_{j}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [147, 658, 157, 669], "score": 1.0, "content": "), ", "type": "text"}, {"bbox": [158, 659, 163, 667], "score": 0.8, "content": "d", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [164, 658, 319, 669], "score": 1.0, "content": " is the number of infinite primes in", "type": "text"}, {"bbox": [320, 658, 326, 667], "score": 0.74, "content": "k", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [327, 658, 393, 669], "score": 1.0, "content": " that ramify in ", "type": "text"}, {"bbox": [393, 659, 413, 669], "score": 0.88, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [413, 658, 418, 669], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [418, 660, 425, 667], "score": 0.44, "content": "\\kappa", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [425, 658, 455, 669], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [455, 659, 462, 667], "score": 0.8, "content": "\\mathbb{Z}", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [463, 658, 487, 669], "score": 1.0, "content": "-rank", "type": "text"}], "index": 36}, {"bbox": [126, 670, 466, 682], "spans": [{"bbox": [126, 670, 202, 682], "score": 1.0, "content": "of the unit group ", "type": "text"}, {"bbox": [202, 671, 215, 681], "score": 0.87, "content": "E_{k}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [215, 670, 230, 682], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [230, 671, 236, 680], "score": 0.77, "content": "k", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [236, 670, 261, 682], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [261, 670, 286, 680], "score": 0.88, "content": "\\upsilon=0", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [286, 670, 344, 682], "score": 1.0, "content": " except when ", "type": "text"}, {"bbox": [344, 670, 403, 682], "score": 0.93, "content": "K\\subseteq k(\\sqrt{E_{k}}\\,)", "type": "inline_equation", "height": 12, "width": 59}, {"bbox": [403, 670, 436, 682], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [437, 672, 461, 680], "score": 0.88, "content": "\\upsilon=1", "type": "inline_equation", "height": 8, "width": 24}, {"bbox": [462, 670, 466, 682], "score": 1.0, "content": ".", "type": "text"}], "index": 37}], "index": 36, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [126, 646, 487, 682]}, {"type": "text", "bbox": [126, 687, 192, 700], "lines": [{"bbox": [126, 687, 194, 702], "spans": [{"bbox": [126, 687, 194, 702], "score": 1.0, "content": "Proof. See [10].", "type": "text"}], "index": 38}], "index": 38, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [126, 687, 194, 702]}]}	{"layout_dets": [{"category_id": 1, "poly": [346, 1008, 1353, 1008, 1353, 1174, 346, 1174], "score": 0.972}, {"category_id": 1, "poly": [347, 712, 1352, 712, 1352, 878, 347, 878], "score": 0.97}, {"category_id": 1, "poly": [347, 477, 1352, 477, 1352, 580, 347, 580], "score": 0.965}, {"category_id": 1, "poly": [348, 1361, 1351, 1361, 1351, 1462, 348, 1462], "score": 0.963}, {"category_id": 1, "poly": [348, 612, 1353, 612, 1353, 712, 348, 712], "score": 0.959}, {"category_id": 1, "poly": [349, 1789, 1351, 1789, 1351, 1893, 349, 1893], "score": 0.957}, {"category_id": 1, "poly": [348, 1653, 1350, 1653, 1350, 1722, 348, 1722], "score": 0.955}, {"category_id": 8, "poly": [607, 355, 1085, 355, 1085, 464, 607, 464], "score": 0.948}, {"category_id": 1, "poly": [348, 879, 1351, 879, 1351, 946, 348, 946], "score": 0.946}, {"category_id": 1, "poly": [350, 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1484.0, 914.0, 1484.0, 914.0, 1521.0, 352.0, 1521.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [990.0, 1484.0, 1351.0, 1484.0, 1351.0, 1521.0, 990.0, 1521.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1336.0, 260.0, 1350.0, 260.0, 1350.0, 282.0, 1336.0, 282.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [381.0, 582.0, 551.0, 582.0, 551.0, 615.0, 381.0, 615.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [719.0, 582.0, 727.0, 582.0, 727.0, 615.0, 719.0, 615.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1322.0, 1250.0, 1353.0, 1250.0, 1353.0, 1279.0, 1322.0, 1279.0], "score": 1.0, "text": ""}], "page_info": {"page_no": 4, "height": 2200, "width": 1700}}	IMAGINARY QUADRATIC FIELDS 5 $$ (N:N^{\prime})\;\left\{\begin{array}{l l}{{=}}&{{2^{m}\;\;\;\;\;i f\,d(G^{\prime})=1}}\\ {{=}}&{{2^{m+1}\;\;\;\;i f\,d(G^{\prime})=2}}\\ {{\geq}}&{{2^{m+2}\;\;\;\;i f\,d(G^{\prime})\geq3}}\end{array}\right.. $$ Proof. Without loss of generality we assume that is metabelian. Let , where mod . Also let and (without loss of generality). Then or . Suppose that . First assume . Then and thus . But for some (cf. Lemma of [1]). Hence, , and so . Since , we get as desired. Next, assume that . Then by Lemma 1. Notice that and where for . Hence and so . But then by Lemma 3. Therefore, by [5], . But notice that . Thus and so which in turn implies that , as desired. Finally, assume . Then . Moreover there exists an exact sequence $$ N/N^{\prime}\longrightarrow(N/G_{5})/(N/G_{5})^{\prime}\longrightarrow1, $$ and thus . Hence it suffices to prove the result for which we now assume. and so, arguing as above, we have , where . But . Therefore, . From this we see that and thus as desired. Now suppose that . Then the proof is essentially the same as above once we notice that mod . This establishes the proposition. 口 # 3. Number Theoretic Preliminaries Proposition 2. Let be a quadratic extension, and assume that the class num- ber of , , is odd. If has an unramified cyclic extension M of order 4, then is normal and . Proof. R´edei and Reichardt [12] proved this for ; the general case is analogous. We shall make extensive use of the class number formula for extensions of type : Proposition 3. Let be a normal quartic extension with Galois group of type , and let ) denote the quadratic subextensions. Then $$ h(K)=2^{d-\kappa-2-v}q(K)h(k_{1})h(k_{2})h(k_{3})/h(k)^{2}, $$ where denotes the unit index of is the unit group of ), is the number of infinite primes in that ramify in , is the -rank of the unit group of , and except when , where . Proof. See [10]. 口	<div class="pdf-page"> <p>Proof. Without loss of generality we assume that is metabelian. Let , where mod . Also let and (without loss of generality). Then or .</p> <p>Suppose that .</p> <p>First assume . Then and thus . But for some (cf. Lemma of [1]). Hence, , and so . Since , we get as desired.</p> <p>Next, assume that . Then by Lemma 1. Notice that and where for . Hence and so . But then by Lemma 3. Therefore, by [5], . But notice that . Thus and so which in turn implies that , as desired.</p> <p>Finally, assume . Then . Moreover there exists an exact sequence</p> <p>and thus . Hence it suffices to prove the result for which we now assume. and so, arguing as above, we have , where . But . Therefore, . From this we see that and thus as desired.</p> <p>Now suppose that . Then the proof is essentially the same as above once we notice that mod .</p> <p>This establishes the proposition.</p> <h1>3. Number Theoretic Preliminaries</h1> <p>Proposition 2. Let be a quadratic extension, and assume that the class num- ber of , , is odd. If has an unramified cyclic extension M of order 4, then is normal and .</p> <p>Proof. R´edei and Reichardt [12] proved this for ; the general case is analogous.</p> <p>We shall make extensive use of the class number formula for extensions of type :</p> <p>Proposition 3. Let be a normal quartic extension with Galois group of type , and let ) denote the quadratic subextensions. Then</p> <p>where denotes the unit index of is the unit group of ), is the number of infinite primes in that ramify in , is the -rank of the unit group of , and except when , where .</p> <p>Proof. See [10].</p> </div>	<div class="pdf-page"> <div class="pdf-discarded" data-x="396" data-y="116" data-width="229" data-height="13" style="opacity: 0.5;">IMAGINARY QUADRATIC FIELDS</div> <div class="pdf-discarded" data-x="801" data-y="116" data-width="12" data-height="12" style="opacity: 0.5;">5</div> <p class="pdf-text" data-x="207" data-y="221" data-width="606" data-height="47">Proof. Without loss of generality we assume that is metabelian. Let , where mod . Also let and (without loss of generality). Then or .</p> <p class="pdf-text" data-x="225" data-y="270" data-width="211" data-height="14">Suppose that .</p> <p class="pdf-text" data-x="209" data-y="284" data-width="605" data-height="46">First assume . Then and thus . But for some (cf. Lemma of [1]). Hence, , and so . Since , we get as desired.</p> <p class="pdf-text" data-x="207" data-y="330" data-width="606" data-height="78">Next, assume that . Then by Lemma 1. Notice that and where for . Hence and so . But then by Lemma 3. Therefore, by [5], . But notice that . Thus and so which in turn implies that , as desired.</p> <p class="pdf-text" data-x="209" data-y="408" data-width="604" data-height="31">Finally, assume . Then . Moreover there exists an exact sequence</p> <p class="pdf-text" data-x="207" data-y="468" data-width="607" data-height="77">and thus . Hence it suffices to prove the result for which we now assume. and so, arguing as above, we have , where . But . Therefore, . From this we see that and thus as desired.</p> <p class="pdf-text" data-x="210" data-y="546" data-width="603" data-height="31">Now suppose that . Then the proof is essentially the same as above once we notice that mod .</p> <p class="pdf-text" data-x="229" data-y="577" data-width="241" data-height="16">This establishes the proposition.</p> <div class="pdf-discarded" data-x="796" data-y="577" data-width="17" data-height="15" style="opacity: 0.5;">口</div> <h1 class="pdf-title" data-x="356" data-y="608" data-width="308" data-height="17">3. Number Theoretic Preliminaries</h1> <p class="pdf-text" data-x="209" data-y="632" data-width="604" data-height="48">Proposition 2. Let be a quadratic extension, and assume that the class num- ber of , , is odd. If has an unramified cyclic extension M of order 4, then is normal and .</p> <p class="pdf-text" data-x="209" data-y="687" data-width="604" data-height="17">Proof. R´edei and Reichardt [12] proved this for ; the general case is analogous.</p> <p class="pdf-text" data-x="209" data-y="729" data-width="605" data-height="32">We shall make extensive use of the class number formula for extensions of type :</p> <p class="pdf-text" data-x="209" data-y="769" data-width="604" data-height="31">Proposition 3. Let be a normal quartic extension with Galois group of type , and let ) denote the quadratic subextensions. Then</p> <p class="pdf-text" data-x="209" data-y="832" data-width="604" data-height="48">where denotes the unit index of is the unit group of ), is the number of infinite primes in that ramify in , is the -rank of the unit group of , and except when , where .</p> <p class="pdf-text" data-x="210" data-y="888" data-width="111" data-height="17">Proof. See [10].</p> <div class="pdf-discarded" data-x="796" data-y="888" data-width="17" data-height="14" style="opacity: 0.5;">口</div> </div>	{ "type": [ "interline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "interline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "inline_equation", "inline_equation", "inline_equation", "interline_equation", "inline_equation", "inline_equation", "inline_equation", "text" ], "coordinates": [ [ 366, 166, 654, 218 ], [ 210, 224, 811, 241 ], [ 210, 240, 814, 257 ], [ 210, 256, 600, 271 ], [ 229, 270, 436, 287 ], [ 229, 284, 814, 303 ], [ 210, 302, 813, 318 ], [ 210, 315, 732, 334 ], [ 227, 330, 814, 349 ], [ 207, 346, 814, 367 ], [ 210, 362, 814, 381 ], [ 207, 378, 811, 396 ], [ 207, 393, 773, 411 ], [ 229, 411, 814, 426 ], [ 209, 427, 279, 443 ], [ 384, 449, 635, 464 ], [ 209, 469, 811, 488 ], [ 210, 486, 814, 504 ], [ 210, 501, 811, 519 ], [ 210, 515, 814, 536 ], [ 210, 530, 418, 550 ], [ 227, 548, 814, 565 ], [ 207, 561, 593, 581 ], [ 229, 579, 466, 594 ], [ 358, 612, 664, 625 ], [ 210, 636, 813, 652 ], [ 210, 649, 814, 668 ], [ 210, 665, 481, 682 ], [ 210, 690, 813, 707 ], [ 229, 730, 813, 748 ], [ 210, 747, 256, 764 ], [ 209, 771, 813, 787 ], [ 210, 788, 731, 802 ], [ 339, 809, 675, 826 ], [ 212, 835, 814, 850 ], [ 210, 850, 814, 866 ], [ 210, 866, 779, 881 ], [ 210, 888, 324, 907 ] ], "content": [ "(N:N^{\\prime})\\;\\left\\{\\begin{array}{l l}{{=}}&{{2^{m}\\;\\;\\;\\;\\;i f\\,d(G^{\\prime})=1}}\\\\ {{=}}&{{2^{m+1}\\;\\;\\;\\;i f\\,d(G^{\\prime})=2}}\\\\ {{\\geq}}&{{2^{m+2}\\;\\;\\;\\;i f\\,d(G^{\\prime})\\geq3}}\\end{array}\\right..", "Proof. Without loss of generality we assume that G is metabelian. Let G=\\langle a,b\\rangle ,", "where a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1 mod G_{3} . Also let H=\\langle b,G^{\\prime}\\rangle and K=\\langle a b,G^{\\prime}\\rangle (without loss", "of generality). Then N=\\langle a b^{2},G^{\\prime}\\rangle or N=\\langle a,b^{4},G^{\\prime}\\rangle .", "Suppose that N=\\left\\langle a b^{2},G^{\\prime}\\right\\rangle .", "First assume d(G^{\\prime})=1 . Then G^{\\prime}=\\langle c_{2}\\rangle and thus N^{\\prime}=\\left\\langle[a b^{2},c_{2}]\\right\\rangle . But [a b^{2},c_{2}]=", "c_{2}^{2}\\eta_{4} for some \\eta_{4}\\,\\in\\,G_{4}\\,=\\,\\langle c_{2}^{4}\\rangle (cf. Lemma ^{1} of [1]). Hence, N^{\\prime}\\,=\\,\\left\\langle c_{2}^{2}\\right\\rangle , and so", "\\left(G^{\\prime}:N^{\\prime}\\right)=2 . Since (N:G^{\\prime})=2^{m-1} , we get (N:N^{\\prime})=2^{m} as desired.", "Next, assume that d(G^{\\prime})\\;=\\;2 . Then N\\,=\\,\\langle a b^{2},c_{2},c_{3}\\rangle by Lemma 1. Notice", "that [a b^{2},c_{2}]~=~c_{2}^{2}\\eta_{4} and [a b^{2},c_{3}]\\;=\\;c_{3}^{2}\\eta_{5} where \\eta_{j}~\\in~G_{j} for j~=~4,5 . Hence", "N^{\\prime}=\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},N_{3}\\rangle and so \\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5}\\rangle\\subseteq N^{\\prime} . But then N^{\\prime}G_{5}\\supseteq\\langle c_{2}^{4},c_{3}^{2}\\rangle=G_{4} by", "Lemma 3. Therefore, by [5], N^{\\prime}\\supseteq G_{4} . But notice that N_{3}\\subseteq G_{4} . Thus N^{\\prime}=\\left\\langle c_{2}^{2},c_{3}^{2}\\right\\rangle", "and so (G^{\\prime}:N^{\\prime})=4 which in turn implies that (N:N^{\\prime})=2^{m+1} , as desired.", "Finally, assume d(G^{\\prime})\\geq3 . Then d(G^{\\prime}/G_{5})=3 . Moreover there exists an exact", "sequence", "N/N^{\\prime}\\longrightarrow(N/G_{5})/(N/G_{5})^{\\prime}\\longrightarrow1,", "and thus \\#N^{\\mathrm{ab}}\\,\\geq\\,\\#(N/G_{5})^{\\mathrm{ab}} . Hence it suffices to prove the result for G_{5}\\,=\\,1", "which we now assume. N=\\langle a b^{2},c_{2},c_{3},c_{4}\\rangle and so, arguing as above, we have N^{\\prime}=", "\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},c_{4}^{2}\\eta_{6},N_{3}\\rangle\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2},N_{3}\\rangle , where \\eta_{j}\\;\\in\\;G_{j} . But N_{3}\\,=\\,\\langle[a b^{2},c_{2}^{2}\\eta_{4}]\\rangle\\,=", "\\langle c_{2}^{4}\\rangle . Therefore, N^{\\prime}\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\rangle . From this we see that (G^{\\prime}\\,:\\,N^{\\prime})\\,=\\,8 and thus", "(N:N^{\\prime})=2^{m+2} as desired.", "Now suppose that N\\,=\\,\\langle a,b^{4},G^{\\prime}\\rangle . Then the proof is essentially the same as", "above once we notice that [a,b^{4}]\\equiv c_{3}{}^{2}c_{2}{}^{-4} mod G_{5} .", "This establishes the proposition.", "3. Number Theoretic Preliminaries", "Proposition 2. Let K/k be a quadratic extension, and assume that the class num-", "ber of k , h(k) , is odd. If K has an unramified cyclic extension M of order 4, then", "M/k is normal and \\operatorname{Gal}(M/k)\\simeq D_{4} .", "Proof. R´edei and Reichardt [12] proved this for k=\\mathbb{Q} ; the general case is analogous.", "We shall make extensive use of the class number formula for extensions of type", "(2,2) :", "Proposition 3. Let K/k be a normal quartic extension with Galois group of type", "(2,2) , and let k_{j} (j=1,2,3) ) denote the quadratic subextensions. Then", "h(K)=2^{d-\\kappa-2-v}q(K)h(k_{1})h(k_{2})h(k_{3})/h(k)^{2},", "where q(K)=(E_{K}:E_{1}E_{2}E_{3}) denotes the unit index of K/k (E_{j} is the unit group", "of k_{j} ), d is the number of infinite primes in k that ramify in K/k , \\kappa is the \\mathbb{Z} -rank", "of the unit group E_{k} of k , and \\upsilon=0 except when K\\subseteq k(\\sqrt{E_{k}}\\,) , where \\upsilon=1 .", "Proof. See [10]." ], "index": [ 0, 1, 2, 3, 5, 10, 11, 12, 18, 19, 20, 21, 22, 31, 32, 46, 62, 63, 64, 65, 66, 83, 84, 106, 130, 155, 156, 157, 183, 212, 213, 243, 244, 276, 310, 311, 312, 347 ] }	{ "coordinates": [], "index": [], "caption": [], "caption_coordinates": [] }	{ "type": [ "interline_equation", "interline_equation", "interline_equation", "interline_equation", "interline_equation", "interline_equation" ], "coordinates": [ [ 366, 166, 654, 218 ], [ 384, 449, 635, 464 ], [ 339, 809, 675, 826 ], [ 366, 166, 654, 218 ], [ 384, 449, 635, 464 ], [ 339, 809, 675, 826 ] ], "content": [ "", "", "", "(N:N^{\\prime})\\;\\left\\{\\begin{array}{l l}{{=}}&{{2^{m}\\;\\;\\;\\;\\;i f\\,d(G^{\\prime})=1}}\\\\ {{=}}&{{2^{m+1}\\;\\;\\;\\;i f\\,d(G^{\\prime})=2}}\\\\ {{\\geq}}&{{2^{m+2}\\;\\;\\;\\;i f\\,d(G^{\\prime})\\geq3}}\\end{array}\\right..", "N/N^{\\prime}\\longrightarrow(N/G_{5})/(N/G_{5})^{\\prime}\\longrightarrow1,", "h(K)=2^{d-\\kappa-2-v}q(K)h(k_{1})h(k_{2})h(k_{3})/h(k)^{2}," ], "index": [ 0, 1, 2, 3, 4, 5 ] }	{ "type": [], "coordinates": [], "content": [], "index": [] }	{"document_info": {"total_pages": 14, "parse_type": "txt", "version": "1.0.1", "total_content_list_items": 155}, "content_statistics": {"total_blocks": 182, "total_lines": 526, "total_images": 4, "total_equations": 28, "total_tables": 11, "total_characters": 29658, "total_words": 5331, "block_type_distribution": {"title": 7, "text": 126, "interline_equation": 14, "discarded": 24, "table_caption": 3, "table_body": 4, "image_body": 2, "list": 2}}, "model_statistics": {"total_layout_detections": 2826, "avg_confidence": 0.9470955414012739, "min_confidence": 0.26, "max_confidence": 1.0}, "page_statistics": [{"page_number": 0, "page_size": [612.0, 792.0], "blocks": 11, "lines": 38, "images": 0, "equations": 2, "tables": 0, "characters": 2362, "words": 398, "discarded_blocks": 1}, {"page_number": 1, "page_size": [612.0, 792.0], "blocks": 10, "lines": 30, "images": 0, "equations": 0, "tables": 3, "characters": 1688, "words": 305, "discarded_blocks": 1}, {"page_number": 2, "page_size": [612.0, 792.0], "blocks": 15, "lines": 44, "images": 0, "equations": 4, "tables": 0, "characters": 2417, "words": 416, "discarded_blocks": 2}, {"page_number": 3, "page_size": [612.0, 792.0], "blocks": 7, "lines": 27, "images": 0, "equations": 0, "tables": 3, "characters": 1591, "words": 291, "discarded_blocks": 1}, {"page_number": 4, "page_size": [612.0, 792.0], "blocks": 23, "lines": 38, "images": 0, "equations": 6, "tables": 0, "characters": 1604, "words": 303, "discarded_blocks": 4}, {"page_number": 5, "page_size": [612.0, 792.0], "blocks": 17, "lines": 40, "images": 0, "equations": 6, "tables": 0, "characters": 2131, "words": 393, "discarded_blocks": 1}, {"page_number": 6, "page_size": [612.0, 792.0], "blocks": 15, "lines": 36, "images": 2, "equations": 4, "tables": 0, "characters": 1791, "words": 324, "discarded_blocks": 2}, {"page_number": 7, "page_size": [612.0, 792.0], "blocks": 12, "lines": 49, "images": 0, "equations": 0, "tables": 0, "characters": 3049, "words": 571, "discarded_blocks": 1}, {"page_number": 8, "page_size": [612.0, 792.0], "blocks": 11, "lines": 42, "images": 0, "equations": 0, "tables": 0, "characters": 2636, "words": 479, "discarded_blocks": 2}, {"page_number": 9, "page_size": [612.0, 792.0], "blocks": 14, "lines": 35, "images": 2, "equations": 0, "tables": 0, "characters": 2036, "words": 375, "discarded_blocks": 1}, {"page_number": 10, "page_size": [612.0, 792.0], "blocks": 17, "lines": 35, "images": 0, "equations": 6, "tables": 0, "characters": 1789, "words": 327, "discarded_blocks": 2}, {"page_number": 11, "page_size": [612.0, 792.0], "blocks": 14, "lines": 40, "images": 0, "equations": 0, "tables": 2, "characters": 2094, "words": 399, "discarded_blocks": 2}, {"page_number": 12, "page_size": [612.0, 792.0], "blocks": 10, "lines": 44, "images": 0, "equations": 0, "tables": 3, "characters": 2572, "words": 487, "discarded_blocks": 2}, {"page_number": 13, "page_size": [612.0, 792.0], "blocks": 6, "lines": 28, "images": 0, "equations": 0, "tables": 0, "characters": 1898, "words": 263, "discarded_blocks": 2}]}
0003244v1	5	[ 612, 792 ]	{ "type": [ "text", "text", "interline_equation", "text", "text", "interline_equation", "text", "text", "text", "text", "interline_equation", "text", "text", "text", "text", "text", "discarded" ], "coordinates": [ [ 207, 143, 814, 192 ], [ 209, 199, 814, 230 ], [ 404, 236, 614, 270 ], [ 209, 272, 813, 320 ], [ 210, 325, 814, 358 ], [ 271, 367, 747, 381 ], [ 209, 385, 814, 448 ], [ 209, 456, 813, 502 ], [ 229, 508, 610, 524 ], [ 209, 531, 813, 563 ], [ 316, 570, 704, 611 ], [ 210, 614, 813, 645 ], [ 209, 651, 813, 699 ], [ 209, 700, 814, 795 ], [ 207, 800, 814, 850 ], [ 207, 855, 814, 906 ], [ 207, 116, 219, 128 ] ], "content": [ "Another important result is the ambiguous class number formula. For cyclic extensions , let denote the group of ideal classes in fixed by , i.e. the ambiguous ideal class group of , and its 2-Sylow subgroup.", "Proposition 4. Let be a cyclic extension of prime degree ; then the number of ambiguous ideal classes is given by", "", "where is the number of primes (including those at ) of that ramify in , is the unit group of , and is its subgroup consisting of norms of elements from . Moreover, is trivial if and only if .", "Proof. See Lang [9, part II] for the formula. For a proof of the second assertion (see e.g. Moriya [11]), note that is defined by the exact sequence", "", "where generates . Taking -parts we see that is equiv- alent to . By induction we get , but since mod in the group ring , this implies . But then must be trivial. 口", "We make one further remark concerning the ambiguous class number formula that will be useful below. If the class number is odd, then it is known that where .", "We also need a result essentially due to G. Gras [4]:", "Proposition 5. Let be a quadratic extension of number fields and assume that . Then is ramified and", "", "where denotes the set of ideal classes of that become principal (capitulate) in .", "Proof. We first notice that is ramified. If the extension were unramified, then would be the 2-class field of , and since is cyclic, it would follow that , contrary to assumption.", "Before we start with the rest of the proof, we cite the results of Gras that we need (we could also give a slightly longer direct proof without referring to his results). Let be a cyclic extension of prime power order , and let be a generator of . For any -group on which acts we put . Moreover, let be the algebraic norm, that is, exponentiation by . Then [4, Cor. 4.3] reads", "Lemma 4. Suppose that ; let be the smallest positive integer such that and write with integers and . If for , then .", "We claim that if , then satisfies the assumptions of Lemma 4: in fact, let denote the transfer of ideal classes. Then for any ideal class , hence . Moreover,", "6" ], "index": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 ] }	[{"type": "text", "text": "Another important result is the ambiguous class number formula. For cyclic extensions $K/k$ , let $\\operatorname{Am}(K/k)$ denote the group of ideal classes in $K$ fixed by $\\operatorname{Gal}(K/k)$ , i.e. the ambiguous ideal class group of $K$ , and $\\mathrm{{Am}_{2}}$ its 2-Sylow subgroup. ", "page_idx": 5}, {"type": "text", "text": "Proposition 4. Let $K/k$ be a cyclic extension of prime degree $p$ ; then the number of ambiguous ideal classes is given by ", "page_idx": 5}, {"type": "equation", "text": "$$\n\\#\\operatorname{Am}(K/k)=h(k)\\,{\\frac{p^{t-1}}{(E:H)}},\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "where $t$ is the number of primes (including those at $\\infty$ ) of $k$ that ramify in $K/k$ , $E$ is the unit group of $k$ , and $H$ is its subgroup consisting of norms of elements from $K^{\\times}$ . Moreover, $\\mathrm{Cl}_{p}(K)$ is trivial if and only if $p\\nmid\\#\\operatorname{Am}(K/k)$ . ", "page_idx": 5}, {"type": "text", "text": "Proof. See Lang [9, part II] for the formula. For a proof of the second assertion (see e.g. Moriya [11]), note that $\\operatorname{Am}(K/k)$ is defined by the exact sequence ", "page_idx": 5}, {"type": "equation", "text": "$$\n1\\;\\longrightarrow\\;\\mathrm{Am}(K/k)\\;\\longrightarrow\\;\\mathrm{Cl}(K)\\;\\longrightarrow\\;\\mathrm{Cl}(K)^{1-\\sigma}\\;\\longrightarrow\\;1,\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "where $\\sigma$ generates $\\operatorname{Gal}(K/k)$ . Taking $p$ -parts we see that $p\\nmid\\#\\operatorname{Am}(K/k)$ is equivalent to $\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{1-\\sigma}$ . By induction we get $\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{(1-\\sigma)^{\\nu}}$ , but since $(1-\\sigma)^{p}\\equiv0$ mod $p$ in the group ring $\\mathbb{Z}[G]$ , this implies $\\operatorname{Cl}_{p}(K)\\subseteq\\operatorname{Cl}_{p}(K)^{p}$ . But then $\\mathrm{Cl}_{p}(K)$ must be trivial. \u53e3 ", "page_idx": 5}, {"type": "text", "text": "We make one further remark concerning the ambiguous class number formula that will be useful below. If the class number $h(k)$ is odd, then it is known that $\\#\\operatorname{Am}_{2}(K/k)=2^{r}$ where $r=\\mathrm{rank}\\,\\mathrm{Cl}_{2}(K)$ . ", "page_idx": 5}, {"type": "text", "text": "We also need a result essentially due to G. Gras [4]: ", "page_idx": 5}, {"type": "text", "text": "Proposition 5. Let $K/k$ be a quadratic extension of number fields and assume that $h_{2}(k)=\\#\\operatorname{Am}_{2}(K/k)=2$ . Then $K/k$ is ramified and ", "page_idx": 5}, {"type": "equation", "text": "$$\n\\operatorname{Cl}_{2}(K)\\simeq\\left\\{\\!\\!\\begin{array}{l l}{{(2,2)~o r~\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq3)~}}&{{i f\\#\\kappa_{K/k}=1,}}\\\\ {{\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq1)~}}&{{i f\\#\\kappa_{K/k}=2,}}\\end{array}\\!\\!\\right.\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "where $\\kappa_{K/k}$ denotes the set of ideal classes of $k$ that become principal (capitulate) in $K$ . ", "page_idx": 5}, {"type": "text", "text": "Proof. We first notice that $K/k$ is ramified. If the extension were unramified, then $K$ would be the 2-class field of $k$ , and since $\\mathrm{Cl_{2}}(k)$ is cyclic, it would follow that $\\mathrm{Cl}_{2}(K)=1$ , contrary to assumption. ", "page_idx": 5}, {"type": "text", "text": "Before we start with the rest of the proof, we cite the results of Gras that we need (we could also give a slightly longer direct proof without referring to his results). Let $K/k$ be a cyclic extension of prime power order $p^{r}$ , and let $\\sigma$ be a generator of $G\\,=\\,\\operatorname{Gal}(K/k)$ . For any $p$ -group $M$ on which $G$ acts we put $M_{i}\\,=\\,\\{m\\,\\in\\,M\\,:\\,m^{(1-\\sigma)^{i}}\\,=\\,1\\}$ . Moreover, let $\\nu$ be the algebraic norm, that is, exponentiation by $1+\\sigma+\\sigma^{2}+...+\\sigma^{p^{\\intercal}-1}$ . Then [4, Cor. 4.3] reads ", "page_idx": 5}, {"type": "text", "text": "Lemma 4. Suppose that $M^{\\nu}=1$ ; let $n$ be the smallest positive integer such that $M_{n}\\,=\\,M$ and write $n=a(p-1)+b$ with integers $a\\geq0$ and $0\\,\\leq\\,b\\,\\leq\\,p\\,-\\,2$ . If $\\#\\,M_{i+1}/M_{i}=p$ for $i=0,1,\\dots,n-1$ , then $M\\simeq(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}$ . ", "page_idx": 5}, {"type": "text", "text": "We claim that if $\\kappa_{K/k}~=~2$ , then $M\\ =\\ \\mathrm{Cl}_{2}(K)$ satisfies the assumptions of Lemma 4: in fact, let $j=j_{k\\rightarrow K}$ denote the transfer of ideal classes. Then $c^{1+\\sigma}=$ $j(N_{K/k}c)$ for any ideal class $c\\,\\in\\,\\mathrm{Cl}_{2}(K)$ , hence $M^{\\nu}=j(\\mathrm{Cl}_{2}(k))\\,=\\,1$ . Moreover, ", "page_idx": 5}]	{"preproc_blocks": [{"type": "text", "bbox": [124, 111, 487, 149], "lines": [{"bbox": [138, 114, 486, 126], "spans": [{"bbox": [138, 114, 486, 126], "score": 1.0, "content": "Another important result is the ambiguous class number formula. For cyclic", "type": "text"}], "index": 0}, {"bbox": [126, 127, 485, 138], "spans": [{"bbox": [126, 127, 175, 138], "score": 1.0, "content": "extensions ", "type": "text"}, {"bbox": [175, 127, 195, 138], "score": 0.92, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [195, 127, 218, 138], "score": 1.0, "content": ", let ", "type": "text"}, {"bbox": [219, 127, 262, 138], "score": 0.92, "content": "\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [262, 127, 434, 138], "score": 1.0, "content": " denote the group of ideal classes in ", "type": "text"}, {"bbox": [434, 128, 444, 135], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [444, 127, 485, 138], "score": 1.0, "content": " fixed by", "type": "text"}], "index": 1}, {"bbox": [126, 138, 486, 151], "spans": [{"bbox": [126, 139, 168, 150], "score": 0.89, "content": "\\operatorname{Gal}(K/k)", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [169, 138, 337, 151], "score": 1.0, "content": ", i.e. the ambiguous ideal class group of", "type": "text"}, {"bbox": [338, 140, 347, 147], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [348, 138, 370, 151], "score": 1.0, "content": ", and", "type": "text"}, {"bbox": [370, 140, 390, 149], "score": 0.48, "content": "\\mathrm{{Am}_{2}}", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [391, 138, 486, 151], "score": 1.0, "content": " its 2-Sylow subgroup.", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [125, 154, 487, 178], "lines": [{"bbox": [125, 156, 486, 169], "spans": [{"bbox": [125, 156, 218, 169], "score": 1.0, "content": "Proposition 4. Let ", "type": "text"}, {"bbox": [218, 158, 238, 168], "score": 0.89, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [239, 156, 402, 169], "score": 1.0, "content": " be a cyclic extension of prime degree ", "type": "text"}, {"bbox": [403, 161, 408, 167], "score": 0.86, "content": "p", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [408, 156, 486, 169], "score": 1.0, "content": "; then the number", "type": "text"}], "index": 3}, {"bbox": [126, 168, 291, 181], "spans": [{"bbox": [126, 168, 291, 181], "score": 1.0, "content": "of ambiguous ideal classes is given by", "type": "text"}], "index": 4}], "index": 3.5}, {"type": "interline_equation", "bbox": [242, 183, 367, 209], "lines": [{"bbox": [242, 183, 367, 209], "spans": [{"bbox": [242, 183, 367, 209], "score": 0.94, "content": "\\#\\operatorname{Am}(K/k)=h(k)\\,{\\frac{p^{t-1}}{(E:H)}},", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "text", "bbox": [125, 211, 486, 248], "lines": [{"bbox": [126, 213, 485, 225], "spans": [{"bbox": [126, 213, 153, 225], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 215, 158, 222], "score": 0.34, "content": "t", "type": "inline_equation", "height": 7, "width": 4}, {"bbox": [158, 213, 350, 225], "score": 1.0, "content": " is the number of primes (including those at ", "type": "text"}, {"bbox": [351, 217, 361, 222], "score": 0.8, "content": "\\infty", "type": "inline_equation", "height": 5, "width": 10}, {"bbox": [361, 213, 379, 225], "score": 1.0, "content": ") of ", "type": "text"}, {"bbox": [379, 214, 385, 222], "score": 0.79, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [385, 213, 451, 225], "score": 1.0, "content": " that ramify in ", "type": "text"}, {"bbox": [451, 214, 471, 225], "score": 0.89, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [471, 213, 476, 225], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [477, 215, 485, 222], "score": 0.82, "content": "E", "type": "inline_equation", "height": 7, "width": 8}], "index": 6}, {"bbox": [125, 225, 487, 237], "spans": [{"bbox": [125, 225, 213, 237], "score": 1.0, "content": "is the unit group of ", "type": "text"}, {"bbox": [213, 227, 219, 234], "score": 0.85, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [219, 225, 245, 237], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [245, 227, 254, 234], "score": 0.85, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [255, 225, 487, 237], "score": 1.0, "content": " is its subgroup consisting of norms of elements from", "type": "text"}], "index": 7}, {"bbox": [126, 237, 401, 250], "spans": [{"bbox": [126, 238, 142, 246], "score": 0.89, "content": "K^{\\times}", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [142, 237, 196, 250], "score": 1.0, "content": ". Moreover, ", "type": "text"}, {"bbox": [196, 238, 228, 249], "score": 0.94, "content": "\\mathrm{Cl}_{p}(K)", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [228, 237, 331, 250], "score": 1.0, "content": " is trivial if and only if ", "type": "text"}, {"bbox": [331, 237, 398, 249], "score": 0.9, "content": "p\\nmid\\#\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 12, "width": 67}, {"bbox": [399, 237, 401, 250], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 7}, {"type": "text", "bbox": [126, 252, 487, 277], "lines": [{"bbox": [126, 254, 487, 269], "spans": [{"bbox": [126, 254, 487, 269], "score": 1.0, "content": "Proof. See Lang [9, part II] for the formula. For a proof of the second assertion", "type": "text"}], "index": 9}, {"bbox": [125, 266, 456, 280], "spans": [{"bbox": [125, 266, 268, 280], "score": 1.0, "content": "(see e.g. Moriya [11]), note that ", "type": "text"}, {"bbox": [268, 268, 311, 279], "score": 0.91, "content": "\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [312, 266, 456, 280], "score": 1.0, "content": " is defined by the exact sequence", "type": "text"}], "index": 10}], "index": 9.5}, {"type": "interline_equation", "bbox": [162, 284, 447, 295], "lines": [{"bbox": [162, 284, 447, 295], "spans": [{"bbox": [162, 284, 447, 295], "score": 0.86, "content": "1\\;\\longrightarrow\\;\\mathrm{Am}(K/k)\\;\\longrightarrow\\;\\mathrm{Cl}(K)\\;\\longrightarrow\\;\\mathrm{Cl}(K)^{1-\\sigma}\\;\\longrightarrow\\;1,", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [125, 298, 487, 347], "lines": [{"bbox": [126, 299, 485, 313], "spans": [{"bbox": [126, 299, 154, 313], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [155, 304, 161, 309], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [162, 299, 208, 313], "score": 1.0, "content": " generates ", "type": "text"}, {"bbox": [208, 301, 251, 312], "score": 0.93, "content": "\\operatorname{Gal}(K/k)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [252, 299, 291, 313], "score": 1.0, "content": ". Taking ", "type": "text"}, {"bbox": [291, 304, 297, 311], "score": 0.88, "content": "p", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [297, 299, 378, 313], "score": 1.0, "content": "-parts we see that ", "type": "text"}, {"bbox": [378, 301, 445, 312], "score": 0.9, "content": "p\\nmid\\#\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 11, "width": 67}, {"bbox": [445, 299, 485, 313], "score": 1.0, "content": " is equiv-", "type": "text"}], "index": 12}, {"bbox": [124, 311, 487, 327], "spans": [{"bbox": [124, 311, 163, 327], "score": 1.0, "content": "alent to ", "type": "text"}, {"bbox": [164, 313, 257, 325], "score": 0.9, "content": "\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{1-\\sigma}", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [258, 311, 358, 327], "score": 1.0, "content": ". By induction we get ", "type": "text"}, {"bbox": [359, 313, 463, 325], "score": 0.92, "content": "\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{(1-\\sigma)^{\\nu}}", "type": "inline_equation", "height": 12, "width": 104}, {"bbox": [463, 311, 487, 327], "score": 1.0, "content": ", but", "type": "text"}], "index": 13}, {"bbox": [125, 324, 485, 338], "spans": [{"bbox": [125, 324, 151, 338], "score": 1.0, "content": "since ", "type": "text"}, {"bbox": [151, 325, 206, 336], "score": 0.94, "content": "(1-\\sigma)^{p}\\equiv0", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [207, 324, 230, 338], "score": 1.0, "content": " mod", "type": "text"}, {"bbox": [231, 329, 236, 336], "score": 0.84, "content": "p", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [236, 324, 318, 338], "score": 1.0, "content": " in the group ring ", "type": "text"}, {"bbox": [318, 326, 339, 336], "score": 0.93, "content": "\\mathbb{Z}[G]", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [339, 324, 399, 338], "score": 1.0, "content": ", this implies ", "type": "text"}, {"bbox": [399, 326, 482, 336], "score": 0.91, "content": "\\operatorname{Cl}_{p}(K)\\subseteq\\operatorname{Cl}_{p}(K)^{p}", "type": "inline_equation", "height": 10, "width": 83}, {"bbox": [483, 324, 485, 338], "score": 1.0, "content": ".", "type": "text"}], "index": 14}, {"bbox": [125, 336, 486, 349], "spans": [{"bbox": [125, 336, 168, 349], "score": 1.0, "content": "But then ", "type": "text"}, {"bbox": [168, 338, 200, 348], "score": 0.93, "content": "\\mathrm{Cl}_{p}(K)", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [200, 336, 271, 349], "score": 1.0, "content": " must be trivial.", "type": "text"}, {"bbox": [476, 338, 486, 347], "score": 0.9896161556243896, "content": "\u53e3", "type": "text"}], "index": 15}], "index": 13.5}, {"type": "text", "bbox": [125, 353, 486, 389], "lines": [{"bbox": [125, 355, 486, 367], "spans": [{"bbox": [125, 355, 486, 367], "score": 1.0, "content": "We make one further remark concerning the ambiguous class number formula that", "type": "text"}], "index": 16}, {"bbox": [126, 367, 486, 379], "spans": [{"bbox": [126, 367, 324, 379], "score": 1.0, "content": "will be useful below. If the class number ", "type": "text"}, {"bbox": [324, 368, 343, 379], "score": 0.92, "content": "h(k)", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [344, 367, 486, 379], "score": 1.0, "content": " is odd, then it is known that", "type": "text"}], "index": 17}, {"bbox": [126, 379, 312, 391], "spans": [{"bbox": [126, 380, 205, 390], "score": 0.93, "content": "\\#\\operatorname{Am}_{2}(K/k)=2^{r}", "type": "inline_equation", "height": 10, "width": 79}, {"bbox": [206, 379, 237, 391], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [238, 380, 309, 390], "score": 0.92, "content": "r=\\mathrm{rank}\\,\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 71}, {"bbox": [309, 379, 312, 391], "score": 1.0, "content": ".", "type": "text"}], "index": 18}], "index": 17}, {"type": "text", "bbox": [137, 393, 365, 406], "lines": [{"bbox": [137, 395, 364, 408], "spans": [{"bbox": [137, 395, 364, 408], "score": 1.0, "content": "We also need a result essentially due to G. Gras [4]:", "type": "text"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [125, 411, 486, 436], "lines": [{"bbox": [126, 414, 487, 426], "spans": [{"bbox": [126, 414, 219, 426], "score": 1.0, "content": "Proposition 5. Let ", "type": "text"}, {"bbox": [220, 415, 240, 425], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [240, 414, 487, 426], "score": 1.0, "content": " be a quadratic extension of number fields and assume", "type": "text"}], "index": 20}, {"bbox": [126, 426, 382, 437], "spans": [{"bbox": [126, 426, 146, 437], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [146, 427, 259, 437], "score": 0.93, "content": "h_{2}(k)=\\#\\operatorname{Am}_{2}(K/k)=2", "type": "inline_equation", "height": 10, "width": 113}, {"bbox": [259, 426, 291, 437], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [292, 426, 312, 437], "score": 0.88, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [312, 426, 382, 437], "score": 1.0, "content": " is ramified and", "type": "text"}], "index": 21}], "index": 20.5}, {"type": "interline_equation", "bbox": [189, 441, 421, 473], "lines": [{"bbox": [189, 441, 421, 473], "spans": [{"bbox": [189, 441, 421, 473], "score": 0.92, "content": "\\operatorname{Cl}_{2}(K)\\simeq\\left\\{\\!\\!\\begin{array}{l l}{{(2,2)~o r~\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq3)~}}&{{i f\\#\\kappa_{K/k}=1,}}\\\\ {{\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq1)~}}&{{i f\\#\\kappa_{K/k}=2,}}\\end{array}\\!\\!\\right.", "type": "interline_equation"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [126, 475, 486, 499], "lines": [{"bbox": [125, 475, 487, 492], "spans": [{"bbox": [125, 475, 154, 492], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 481, 176, 489], "score": 0.88, "content": "\\kappa_{K/k}", "type": "inline_equation", "height": 8, "width": 22}, {"bbox": [176, 475, 329, 492], "score": 1.0, "content": " denotes the set of ideal classes of ", "type": "text"}, {"bbox": [329, 478, 335, 486], "score": 0.81, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [335, 475, 487, 492], "score": 1.0, "content": " that become principal (capitulate)", "type": "text"}], "index": 23}, {"bbox": [126, 489, 152, 500], "spans": [{"bbox": [126, 489, 137, 500], "score": 1.0, "content": "in", "type": "text"}, {"bbox": [138, 491, 148, 498], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [148, 489, 152, 500], "score": 1.0, "content": ".", "type": "text"}], "index": 24}], "index": 23.5}, {"type": "text", "bbox": [125, 504, 486, 541], "lines": [{"bbox": [127, 507, 485, 519], "spans": [{"bbox": [127, 507, 245, 519], "score": 1.0, "content": "Proof. We first notice that ", "type": "text"}, {"bbox": [245, 508, 264, 518], "score": 0.92, "content": "K/k", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [264, 507, 485, 519], "score": 1.0, "content": " is ramified. If the extension were unramified, then", "type": "text"}], "index": 25}, {"bbox": [126, 518, 486, 531], "spans": [{"bbox": [126, 520, 135, 528], "score": 0.9, "content": "K", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [136, 518, 266, 531], "score": 1.0, "content": " would be the 2-class field of ", "type": "text"}, {"bbox": [267, 520, 272, 528], "score": 0.87, "content": "k", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [273, 518, 323, 531], "score": 1.0, "content": ", and since ", "type": "text"}, {"bbox": [324, 520, 352, 530], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [352, 518, 486, 531], "score": 1.0, "content": " is cyclic, it would follow that", "type": "text"}], "index": 26}, {"bbox": [126, 530, 286, 545], "spans": [{"bbox": [126, 532, 176, 542], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K)=1", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [176, 530, 286, 545], "score": 1.0, "content": ", contrary to assumption.", "type": "text"}], "index": 27}], "index": 26}, {"type": "text", "bbox": [125, 542, 487, 615], "lines": [{"bbox": [137, 542, 486, 555], "spans": [{"bbox": [137, 542, 486, 555], "score": 1.0, "content": "Before we start with the rest of the proof, we cite the results of Gras that", "type": "text"}], "index": 28}, {"bbox": [126, 555, 486, 568], "spans": [{"bbox": [126, 555, 486, 568], "score": 1.0, "content": "we need (we could also give a slightly longer direct proof without referring to", "type": "text"}], "index": 29}, {"bbox": [125, 565, 484, 580], "spans": [{"bbox": [125, 565, 204, 580], "score": 1.0, "content": "his results). Let ", "type": "text"}, {"bbox": [204, 568, 224, 578], "score": 0.93, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [224, 565, 425, 580], "score": 1.0, "content": " be a cyclic extension of prime power order ", "type": "text"}, {"bbox": [425, 568, 435, 577], "score": 0.92, "content": "p^{r}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [435, 565, 478, 580], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [478, 571, 484, 576], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 6}], "index": 30}, {"bbox": [124, 578, 486, 592], "spans": [{"bbox": [124, 578, 207, 592], "score": 1.0, "content": "be a generator of ", "type": "text"}, {"bbox": [207, 579, 275, 590], "score": 0.91, "content": "G\\,=\\,\\operatorname{Gal}(K/k)", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [275, 578, 323, 592], "score": 1.0, "content": ". For any ", "type": "text"}, {"bbox": [323, 583, 329, 589], "score": 0.87, "content": "p", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [329, 578, 361, 592], "score": 1.0, "content": "-group ", "type": "text"}, {"bbox": [361, 580, 372, 587], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [372, 578, 420, 592], "score": 1.0, "content": " on which ", "type": "text"}, {"bbox": [420, 580, 428, 588], "score": 0.9, "content": "G", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [429, 578, 486, 592], "score": 1.0, "content": " acts we put", "type": "text"}], "index": 31}, {"bbox": [126, 589, 487, 605], "spans": [{"bbox": [126, 590, 265, 604], "score": 0.9, "content": "M_{i}\\,=\\,\\{m\\,\\in\\,M\\,:\\,m^{(1-\\sigma)^{i}}\\,=\\,1\\}", "type": "inline_equation", "height": 14, "width": 139}, {"bbox": [266, 589, 337, 605], "score": 1.0, "content": ". Moreover, let ", "type": "text"}, {"bbox": [338, 596, 344, 601], "score": 0.86, "content": "\\nu", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [344, 589, 487, 605], "score": 1.0, "content": " be the algebraic norm, that is,", "type": "text"}], "index": 32}, {"bbox": [124, 602, 429, 618], "spans": [{"bbox": [124, 602, 207, 618], "score": 1.0, "content": "exponentiation by ", "type": "text"}, {"bbox": [207, 604, 314, 614], "score": 0.9, "content": "1+\\sigma+\\sigma^{2}+...+\\sigma^{p^{\\intercal}-1}", "type": "inline_equation", "height": 10, "width": 107}, {"bbox": [315, 602, 429, 618], "score": 1.0, "content": ". Then [4, Cor. 4.3] reads", "type": "text"}], "index": 33}], "index": 30.5}, {"type": "text", "bbox": [124, 619, 487, 658], "lines": [{"bbox": [124, 622, 487, 635], "spans": [{"bbox": [124, 622, 239, 635], "score": 1.0, "content": "Lemma 4. Suppose that ", "type": "text"}, {"bbox": [239, 623, 275, 632], "score": 0.85, "content": "M^{\\nu}=1", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [276, 622, 296, 635], "score": 1.0, "content": "; let ", "type": "text"}, {"bbox": [297, 627, 303, 631], "score": 0.71, "content": "n", "type": "inline_equation", "height": 4, "width": 6}, {"bbox": [303, 622, 487, 635], "score": 1.0, "content": " be the smallest positive integer such that", "type": "text"}], "index": 34}, {"bbox": [126, 634, 487, 647], "spans": [{"bbox": [126, 635, 168, 645], "score": 0.86, "content": "M_{n}\\,=\\,M", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [168, 634, 216, 647], "score": 1.0, "content": " and write ", "type": "text"}, {"bbox": [217, 635, 292, 646], "score": 0.92, "content": "n=a(p-1)+b", "type": "inline_equation", "height": 11, "width": 75}, {"bbox": [293, 634, 355, 647], "score": 1.0, "content": " with integers ", "type": "text"}, {"bbox": [356, 636, 381, 644], "score": 0.89, "content": "a\\geq0", "type": "inline_equation", "height": 8, "width": 25}, {"bbox": [382, 634, 405, 647], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [405, 636, 469, 645], "score": 0.91, "content": "0\\,\\leq\\,b\\,\\leq\\,p\\,-\\,2", "type": "inline_equation", "height": 9, "width": 64}, {"bbox": [469, 634, 487, 647], "score": 1.0, "content": ". If", "type": "text"}], "index": 35}, {"bbox": [125, 645, 472, 660], "spans": [{"bbox": [125, 646, 195, 658], "score": 0.91, "content": "\\#\\,M_{i+1}/M_{i}=p", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [196, 645, 213, 660], "score": 1.0, "content": " for", "type": "text"}, {"bbox": [214, 648, 293, 657], "score": 0.87, "content": "i=0,1,\\dots,n-1", "type": "inline_equation", "height": 9, "width": 79}, {"bbox": [293, 645, 320, 660], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [321, 646, 468, 658], "score": 0.93, "content": "M\\simeq(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}", "type": "inline_equation", "height": 12, "width": 147}, {"bbox": [468, 645, 472, 660], "score": 1.0, "content": ".", "type": "text"}], "index": 36}], "index": 35}, {"type": "text", "bbox": [124, 662, 487, 701], "lines": [{"bbox": [137, 663, 487, 677], "spans": [{"bbox": [137, 663, 218, 677], "score": 1.0, "content": "We claim that if ", "type": "text"}, {"bbox": [218, 665, 264, 677], "score": 0.89, "content": "\\kappa_{K/k}~=~2", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [264, 663, 295, 677], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [296, 666, 357, 676], "score": 0.93, "content": "M\\ =\\ \\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 61}, {"bbox": [357, 663, 487, 677], "score": 1.0, "content": " satisfies the assumptions of", "type": "text"}], "index": 37}, {"bbox": [123, 676, 486, 690], "spans": [{"bbox": [123, 676, 222, 690], "score": 1.0, "content": "Lemma 4: in fact, let", "type": "text"}, {"bbox": [223, 678, 266, 688], "score": 0.91, "content": "j=j_{k\\rightarrow K}", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [266, 676, 454, 690], "score": 1.0, "content": " denote the transfer of ideal classes. Then ", "type": "text"}, {"bbox": [454, 677, 486, 687], "score": 0.87, "content": "c^{1+\\sigma}=", "type": "inline_equation", "height": 10, "width": 32}], "index": 38}, {"bbox": [126, 689, 486, 702], "spans": [{"bbox": [126, 690, 167, 702], "score": 0.92, "content": "j(N_{K/k}c)", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [167, 690, 254, 702], "score": 1.0, "content": " for any ideal class ", "type": "text"}, {"bbox": [254, 689, 305, 701], "score": 0.92, "content": "c\\,\\in\\,\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [305, 690, 339, 702], "score": 1.0, "content": ", hence ", "type": "text"}, {"bbox": [340, 690, 432, 701], "score": 0.92, "content": "M^{\\nu}=j(\\mathrm{Cl}_{2}(k))\\,=\\,1", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [433, 690, 486, 702], "score": 1.0, "content": ". Moreover,", "type": "text"}], "index": 39}], "index": 38}], "layout_bboxes": [], "page_idx": 5, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [242, 183, 367, 209], "lines": [{"bbox": [242, 183, 367, 209], "spans": [{"bbox": [242, 183, 367, 209], "score": 0.94, "content": "\\#\\operatorname{Am}(K/k)=h(k)\\,{\\frac{p^{t-1}}{(E:H)}},", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [162, 284, 447, 295], "lines": [{"bbox": [162, 284, 447, 295], "spans": [{"bbox": [162, 284, 447, 295], "score": 0.86, "content": "1\\;\\longrightarrow\\;\\mathrm{Am}(K/k)\\;\\longrightarrow\\;\\mathrm{Cl}(K)\\;\\longrightarrow\\;\\mathrm{Cl}(K)^{1-\\sigma}\\;\\longrightarrow\\;1,", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "interline_equation", "bbox": [189, 441, 421, 473], "lines": [{"bbox": [189, 441, 421, 473], "spans": [{"bbox": [189, 441, 421, 473], "score": 0.92, "content": "\\operatorname{Cl}_{2}(K)\\simeq\\left\\{\\!\\!\\begin{array}{l l}{{(2,2)~o r~\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq3)~}}&{{i f\\#\\kappa_{K/k}=1,}}\\\\ {{\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq1)~}}&{{i f\\#\\kappa_{K/k}=2,}}\\end{array}\\!\\!\\right.", "type": "interline_equation"}], "index": 22}], "index": 22}], "discarded_blocks": [{"type": "discarded", "bbox": [124, 90, 131, 99], "lines": [{"bbox": [126, 93, 132, 101], "spans": [{"bbox": [126, 93, 132, 101], "score": 1.0, "content": "6", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [124, 111, 487, 149], "lines": [{"bbox": [138, 114, 486, 126], "spans": [{"bbox": [138, 114, 486, 126], "score": 1.0, "content": "Another important result is the ambiguous class number formula. For cyclic", "type": "text"}], "index": 0}, {"bbox": [126, 127, 485, 138], "spans": [{"bbox": [126, 127, 175, 138], "score": 1.0, "content": "extensions ", "type": "text"}, {"bbox": [175, 127, 195, 138], "score": 0.92, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [195, 127, 218, 138], "score": 1.0, "content": ", let ", "type": "text"}, {"bbox": [219, 127, 262, 138], "score": 0.92, "content": "\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [262, 127, 434, 138], "score": 1.0, "content": " denote the group of ideal classes in ", "type": "text"}, {"bbox": [434, 128, 444, 135], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [444, 127, 485, 138], "score": 1.0, "content": " fixed by", "type": "text"}], "index": 1}, {"bbox": [126, 138, 486, 151], "spans": [{"bbox": [126, 139, 168, 150], "score": 0.89, "content": "\\operatorname{Gal}(K/k)", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [169, 138, 337, 151], "score": 1.0, "content": ", i.e. the ambiguous ideal class group of", "type": "text"}, {"bbox": [338, 140, 347, 147], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [348, 138, 370, 151], "score": 1.0, "content": ", and", "type": "text"}, {"bbox": [370, 140, 390, 149], "score": 0.48, "content": "\\mathrm{{Am}_{2}}", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [391, 138, 486, 151], "score": 1.0, "content": " its 2-Sylow subgroup.", "type": "text"}], "index": 2}], "index": 1, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [126, 114, 486, 151]}, {"type": "text", "bbox": [125, 154, 487, 178], "lines": [{"bbox": [125, 156, 486, 169], "spans": [{"bbox": [125, 156, 218, 169], "score": 1.0, "content": "Proposition 4. Let ", "type": "text"}, {"bbox": [218, 158, 238, 168], "score": 0.89, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [239, 156, 402, 169], "score": 1.0, "content": " be a cyclic extension of prime degree ", "type": "text"}, {"bbox": [403, 161, 408, 167], "score": 0.86, "content": "p", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [408, 156, 486, 169], "score": 1.0, "content": "; then the number", "type": "text"}], "index": 3}, {"bbox": [126, 168, 291, 181], "spans": [{"bbox": [126, 168, 291, 181], "score": 1.0, "content": "of ambiguous ideal classes is given by", "type": "text"}], "index": 4}], "index": 3.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [125, 156, 486, 181]}, {"type": "interline_equation", "bbox": [242, 183, 367, 209], "lines": [{"bbox": [242, 183, 367, 209], "spans": [{"bbox": [242, 183, 367, 209], "score": 0.94, "content": "\\#\\operatorname{Am}(K/k)=h(k)\\,{\\frac{p^{t-1}}{(E:H)}},", "type": "interline_equation"}], "index": 5}], "index": 5, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 211, 486, 248], "lines": [{"bbox": [126, 213, 485, 225], "spans": [{"bbox": [126, 213, 153, 225], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 215, 158, 222], "score": 0.34, "content": "t", "type": "inline_equation", "height": 7, "width": 4}, {"bbox": [158, 213, 350, 225], "score": 1.0, "content": " is the number of primes (including those at ", "type": "text"}, {"bbox": [351, 217, 361, 222], "score": 0.8, "content": "\\infty", "type": "inline_equation", "height": 5, "width": 10}, {"bbox": [361, 213, 379, 225], "score": 1.0, "content": ") of ", "type": "text"}, {"bbox": [379, 214, 385, 222], "score": 0.79, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [385, 213, 451, 225], "score": 1.0, "content": " that ramify in ", "type": "text"}, {"bbox": [451, 214, 471, 225], "score": 0.89, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [471, 213, 476, 225], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [477, 215, 485, 222], "score": 0.82, "content": "E", "type": "inline_equation", "height": 7, "width": 8}], "index": 6}, {"bbox": [125, 225, 487, 237], "spans": [{"bbox": [125, 225, 213, 237], "score": 1.0, "content": "is the unit group of ", "type": "text"}, {"bbox": [213, 227, 219, 234], "score": 0.85, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [219, 225, 245, 237], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [245, 227, 254, 234], "score": 0.85, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [255, 225, 487, 237], "score": 1.0, "content": " is its subgroup consisting of norms of elements from", "type": "text"}], "index": 7}, {"bbox": [126, 237, 401, 250], "spans": [{"bbox": [126, 238, 142, 246], "score": 0.89, "content": "K^{\\times}", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [142, 237, 196, 250], "score": 1.0, "content": ". Moreover, ", "type": "text"}, {"bbox": [196, 238, 228, 249], "score": 0.94, "content": "\\mathrm{Cl}_{p}(K)", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [228, 237, 331, 250], "score": 1.0, "content": " is trivial if and only if ", "type": "text"}, {"bbox": [331, 237, 398, 249], "score": 0.9, "content": "p\\nmid\\#\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 12, "width": 67}, {"bbox": [399, 237, 401, 250], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 7, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [125, 213, 487, 250]}, {"type": "text", "bbox": [126, 252, 487, 277], "lines": [{"bbox": [126, 254, 487, 269], "spans": [{"bbox": [126, 254, 487, 269], "score": 1.0, "content": "Proof. See Lang [9, part II] for the formula. For a proof of the second assertion", "type": "text"}], "index": 9}, {"bbox": [125, 266, 456, 280], "spans": [{"bbox": [125, 266, 268, 280], "score": 1.0, "content": "(see e.g. Moriya [11]), note that ", "type": "text"}, {"bbox": [268, 268, 311, 279], "score": 0.91, "content": "\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [312, 266, 456, 280], "score": 1.0, "content": " is defined by the exact sequence", "type": "text"}], "index": 10}], "index": 9.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [125, 254, 487, 280]}, {"type": "interline_equation", "bbox": [162, 284, 447, 295], "lines": [{"bbox": [162, 284, 447, 295], "spans": [{"bbox": [162, 284, 447, 295], "score": 0.86, "content": "1\\;\\longrightarrow\\;\\mathrm{Am}(K/k)\\;\\longrightarrow\\;\\mathrm{Cl}(K)\\;\\longrightarrow\\;\\mathrm{Cl}(K)^{1-\\sigma}\\;\\longrightarrow\\;1,", "type": "interline_equation"}], "index": 11}], "index": 11, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 298, 487, 347], "lines": [{"bbox": [126, 299, 485, 313], "spans": [{"bbox": [126, 299, 154, 313], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [155, 304, 161, 309], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [162, 299, 208, 313], "score": 1.0, "content": " generates ", "type": "text"}, {"bbox": [208, 301, 251, 312], "score": 0.93, "content": "\\operatorname{Gal}(K/k)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [252, 299, 291, 313], "score": 1.0, "content": ". Taking ", "type": "text"}, {"bbox": [291, 304, 297, 311], "score": 0.88, "content": "p", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [297, 299, 378, 313], "score": 1.0, "content": "-parts we see that ", "type": "text"}, {"bbox": [378, 301, 445, 312], "score": 0.9, "content": "p\\nmid\\#\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 11, "width": 67}, {"bbox": [445, 299, 485, 313], "score": 1.0, "content": " is equiv-", "type": "text"}], "index": 12}, {"bbox": [124, 311, 487, 327], "spans": [{"bbox": [124, 311, 163, 327], "score": 1.0, "content": "alent to ", "type": "text"}, {"bbox": [164, 313, 257, 325], "score": 0.9, "content": "\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{1-\\sigma}", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [258, 311, 358, 327], "score": 1.0, "content": ". By induction we get ", "type": "text"}, {"bbox": [359, 313, 463, 325], "score": 0.92, "content": "\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{(1-\\sigma)^{\\nu}}", "type": "inline_equation", "height": 12, "width": 104}, {"bbox": [463, 311, 487, 327], "score": 1.0, "content": ", but", "type": "text"}], "index": 13}, {"bbox": [125, 324, 485, 338], "spans": [{"bbox": [125, 324, 151, 338], "score": 1.0, "content": "since ", "type": "text"}, {"bbox": [151, 325, 206, 336], "score": 0.94, "content": "(1-\\sigma)^{p}\\equiv0", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [207, 324, 230, 338], "score": 1.0, "content": " mod", "type": "text"}, {"bbox": [231, 329, 236, 336], "score": 0.84, "content": "p", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [236, 324, 318, 338], "score": 1.0, "content": " in the group ring ", "type": "text"}, {"bbox": [318, 326, 339, 336], "score": 0.93, "content": "\\mathbb{Z}[G]", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [339, 324, 399, 338], "score": 1.0, "content": ", this implies ", "type": "text"}, {"bbox": [399, 326, 482, 336], "score": 0.91, "content": "\\operatorname{Cl}_{p}(K)\\subseteq\\operatorname{Cl}_{p}(K)^{p}", "type": "inline_equation", "height": 10, "width": 83}, {"bbox": [483, 324, 485, 338], "score": 1.0, "content": ".", "type": "text"}], "index": 14}, {"bbox": [125, 336, 486, 349], "spans": [{"bbox": [125, 336, 168, 349], "score": 1.0, "content": "But then ", "type": "text"}, {"bbox": [168, 338, 200, 348], "score": 0.93, "content": "\\mathrm{Cl}_{p}(K)", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [200, 336, 271, 349], "score": 1.0, "content": " must be trivial.", "type": "text"}, {"bbox": [476, 338, 486, 347], "score": 0.9896161556243896, "content": "\u53e3", "type": "text"}], "index": 15}], "index": 13.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [124, 299, 487, 349]}, {"type": "text", "bbox": [125, 353, 486, 389], "lines": [{"bbox": [125, 355, 486, 367], "spans": [{"bbox": [125, 355, 486, 367], "score": 1.0, "content": "We make one further remark concerning the ambiguous class number formula that", "type": "text"}], "index": 16}, {"bbox": [126, 367, 486, 379], "spans": [{"bbox": [126, 367, 324, 379], "score": 1.0, "content": "will be useful below. If the class number ", "type": "text"}, {"bbox": [324, 368, 343, 379], "score": 0.92, "content": "h(k)", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [344, 367, 486, 379], "score": 1.0, "content": " is odd, then it is known that", "type": "text"}], "index": 17}, {"bbox": [126, 379, 312, 391], "spans": [{"bbox": [126, 380, 205, 390], "score": 0.93, "content": "\\#\\operatorname{Am}_{2}(K/k)=2^{r}", "type": "inline_equation", "height": 10, "width": 79}, {"bbox": [206, 379, 237, 391], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [238, 380, 309, 390], "score": 0.92, "content": "r=\\mathrm{rank}\\,\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 71}, {"bbox": [309, 379, 312, 391], "score": 1.0, "content": ".", "type": "text"}], "index": 18}], "index": 17, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [125, 355, 486, 391]}, {"type": "text", "bbox": [137, 393, 365, 406], "lines": [{"bbox": [137, 395, 364, 408], "spans": [{"bbox": [137, 395, 364, 408], "score": 1.0, "content": "We also need a result essentially due to G. Gras [4]:", "type": "text"}], "index": 19}], "index": 19, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [137, 395, 364, 408]}, {"type": "text", "bbox": [125, 411, 486, 436], "lines": [{"bbox": [126, 414, 487, 426], "spans": [{"bbox": [126, 414, 219, 426], "score": 1.0, "content": "Proposition 5. Let ", "type": "text"}, {"bbox": [220, 415, 240, 425], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [240, 414, 487, 426], "score": 1.0, "content": " be a quadratic extension of number fields and assume", "type": "text"}], "index": 20}, {"bbox": [126, 426, 382, 437], "spans": [{"bbox": [126, 426, 146, 437], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [146, 427, 259, 437], "score": 0.93, "content": "h_{2}(k)=\\#\\operatorname{Am}_{2}(K/k)=2", "type": "inline_equation", "height": 10, "width": 113}, {"bbox": [259, 426, 291, 437], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [292, 426, 312, 437], "score": 0.88, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [312, 426, 382, 437], "score": 1.0, "content": " is ramified and", "type": "text"}], "index": 21}], "index": 20.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [126, 414, 487, 437]}, {"type": "interline_equation", "bbox": [189, 441, 421, 473], "lines": [{"bbox": [189, 441, 421, 473], "spans": [{"bbox": [189, 441, 421, 473], "score": 0.92, "content": "\\operatorname{Cl}_{2}(K)\\simeq\\left\\{\\!\\!\\begin{array}{l l}{{(2,2)~o r~\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq3)~}}&{{i f\\#\\kappa_{K/k}=1,}}\\\\ {{\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq1)~}}&{{i f\\#\\kappa_{K/k}=2,}}\\end{array}\\!\\!\\right.", "type": "interline_equation"}], "index": 22}], "index": 22, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [126, 475, 486, 499], "lines": [{"bbox": [125, 475, 487, 492], "spans": [{"bbox": [125, 475, 154, 492], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 481, 176, 489], "score": 0.88, "content": "\\kappa_{K/k}", "type": "inline_equation", "height": 8, "width": 22}, {"bbox": [176, 475, 329, 492], "score": 1.0, "content": " denotes the set of ideal classes of ", "type": "text"}, {"bbox": [329, 478, 335, 486], "score": 0.81, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [335, 475, 487, 492], "score": 1.0, "content": " that become principal (capitulate)", "type": "text"}], "index": 23}, {"bbox": [126, 489, 152, 500], "spans": [{"bbox": [126, 489, 137, 500], "score": 1.0, "content": "in", "type": "text"}, {"bbox": [138, 491, 148, 498], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [148, 489, 152, 500], "score": 1.0, "content": ".", "type": "text"}], "index": 24}], "index": 23.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [125, 475, 487, 500]}, {"type": "text", "bbox": [125, 504, 486, 541], "lines": [{"bbox": [127, 507, 485, 519], "spans": [{"bbox": [127, 507, 245, 519], "score": 1.0, "content": "Proof. We first notice that ", "type": "text"}, {"bbox": [245, 508, 264, 518], "score": 0.92, "content": "K/k", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [264, 507, 485, 519], "score": 1.0, "content": " is ramified. If the extension were unramified, then", "type": "text"}], "index": 25}, {"bbox": [126, 518, 486, 531], "spans": [{"bbox": [126, 520, 135, 528], "score": 0.9, "content": "K", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [136, 518, 266, 531], "score": 1.0, "content": " would be the 2-class field of ", "type": "text"}, {"bbox": [267, 520, 272, 528], "score": 0.87, "content": "k", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [273, 518, 323, 531], "score": 1.0, "content": ", and since ", "type": "text"}, {"bbox": [324, 520, 352, 530], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [352, 518, 486, 531], "score": 1.0, "content": " is cyclic, it would follow that", "type": "text"}], "index": 26}, {"bbox": [126, 530, 286, 545], "spans": [{"bbox": [126, 532, 176, 542], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K)=1", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [176, 530, 286, 545], "score": 1.0, "content": ", contrary to assumption.", "type": "text"}], "index": 27}], "index": 26, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [126, 507, 486, 545]}, {"type": "text", "bbox": [125, 542, 487, 615], "lines": [{"bbox": [137, 542, 486, 555], "spans": [{"bbox": [137, 542, 486, 555], "score": 1.0, "content": "Before we start with the rest of the proof, we cite the results of Gras that", "type": "text"}], "index": 28}, {"bbox": [126, 555, 486, 568], "spans": [{"bbox": [126, 555, 486, 568], "score": 1.0, "content": "we need (we could also give a slightly longer direct proof without referring to", "type": "text"}], "index": 29}, {"bbox": [125, 565, 484, 580], "spans": [{"bbox": [125, 565, 204, 580], "score": 1.0, "content": "his results). Let ", "type": "text"}, {"bbox": [204, 568, 224, 578], "score": 0.93, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [224, 565, 425, 580], "score": 1.0, "content": " be a cyclic extension of prime power order ", "type": "text"}, {"bbox": [425, 568, 435, 577], "score": 0.92, "content": "p^{r}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [435, 565, 478, 580], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [478, 571, 484, 576], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 6}], "index": 30}, {"bbox": [124, 578, 486, 592], "spans": [{"bbox": [124, 578, 207, 592], "score": 1.0, "content": "be a generator of ", "type": "text"}, {"bbox": [207, 579, 275, 590], "score": 0.91, "content": "G\\,=\\,\\operatorname{Gal}(K/k)", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [275, 578, 323, 592], "score": 1.0, "content": ". For any ", "type": "text"}, {"bbox": [323, 583, 329, 589], "score": 0.87, "content": "p", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [329, 578, 361, 592], "score": 1.0, "content": "-group ", "type": "text"}, {"bbox": [361, 580, 372, 587], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [372, 578, 420, 592], "score": 1.0, "content": " on which ", "type": "text"}, {"bbox": [420, 580, 428, 588], "score": 0.9, "content": "G", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [429, 578, 486, 592], "score": 1.0, "content": " acts we put", "type": "text"}], "index": 31}, {"bbox": [126, 589, 487, 605], "spans": [{"bbox": [126, 590, 265, 604], "score": 0.9, "content": "M_{i}\\,=\\,\\{m\\,\\in\\,M\\,:\\,m^{(1-\\sigma)^{i}}\\,=\\,1\\}", "type": "inline_equation", "height": 14, "width": 139}, {"bbox": [266, 589, 337, 605], "score": 1.0, "content": ". Moreover, let ", "type": "text"}, {"bbox": [338, 596, 344, 601], "score": 0.86, "content": "\\nu", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [344, 589, 487, 605], "score": 1.0, "content": " be the algebraic norm, that is,", "type": "text"}], "index": 32}, {"bbox": [124, 602, 429, 618], "spans": [{"bbox": [124, 602, 207, 618], "score": 1.0, "content": "exponentiation by ", "type": "text"}, {"bbox": [207, 604, 314, 614], "score": 0.9, "content": "1+\\sigma+\\sigma^{2}+...+\\sigma^{p^{\\intercal}-1}", "type": "inline_equation", "height": 10, "width": 107}, {"bbox": [315, 602, 429, 618], "score": 1.0, "content": ". Then [4, Cor. 4.3] reads", "type": "text"}], "index": 33}], "index": 30.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [124, 542, 487, 618]}, {"type": "text", "bbox": [124, 619, 487, 658], "lines": [{"bbox": [124, 622, 487, 635], "spans": [{"bbox": [124, 622, 239, 635], "score": 1.0, "content": "Lemma 4. Suppose that ", "type": "text"}, {"bbox": [239, 623, 275, 632], "score": 0.85, "content": "M^{\\nu}=1", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [276, 622, 296, 635], "score": 1.0, "content": "; let ", "type": "text"}, {"bbox": [297, 627, 303, 631], "score": 0.71, "content": "n", "type": "inline_equation", "height": 4, "width": 6}, {"bbox": [303, 622, 487, 635], "score": 1.0, "content": " be the smallest positive integer such that", "type": "text"}], "index": 34}, {"bbox": [126, 634, 487, 647], "spans": [{"bbox": [126, 635, 168, 645], "score": 0.86, "content": "M_{n}\\,=\\,M", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [168, 634, 216, 647], "score": 1.0, "content": " and write ", "type": "text"}, {"bbox": [217, 635, 292, 646], "score": 0.92, "content": "n=a(p-1)+b", "type": "inline_equation", "height": 11, "width": 75}, {"bbox": [293, 634, 355, 647], "score": 1.0, "content": " with integers ", "type": "text"}, {"bbox": [356, 636, 381, 644], "score": 0.89, "content": "a\\geq0", "type": "inline_equation", "height": 8, "width": 25}, {"bbox": [382, 634, 405, 647], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [405, 636, 469, 645], "score": 0.91, "content": "0\\,\\leq\\,b\\,\\leq\\,p\\,-\\,2", "type": "inline_equation", "height": 9, "width": 64}, {"bbox": [469, 634, 487, 647], "score": 1.0, "content": ". If", "type": "text"}], "index": 35}, {"bbox": [125, 645, 472, 660], "spans": [{"bbox": [125, 646, 195, 658], "score": 0.91, "content": "\\#\\,M_{i+1}/M_{i}=p", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [196, 645, 213, 660], "score": 1.0, "content": " for", "type": "text"}, {"bbox": [214, 648, 293, 657], "score": 0.87, "content": "i=0,1,\\dots,n-1", "type": "inline_equation", "height": 9, "width": 79}, {"bbox": [293, 645, 320, 660], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [321, 646, 468, 658], "score": 0.93, "content": "M\\simeq(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}", "type": "inline_equation", "height": 12, "width": 147}, {"bbox": [468, 645, 472, 660], "score": 1.0, "content": ".", "type": "text"}], "index": 36}], "index": 35, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [124, 622, 487, 660]}, {"type": "text", "bbox": [124, 662, 487, 701], "lines": [{"bbox": [137, 663, 487, 677], "spans": [{"bbox": [137, 663, 218, 677], "score": 1.0, "content": "We claim that if ", "type": "text"}, {"bbox": [218, 665, 264, 677], "score": 0.89, "content": "\\kappa_{K/k}~=~2", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [264, 663, 295, 677], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [296, 666, 357, 676], "score": 0.93, "content": "M\\ =\\ \\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 61}, {"bbox": [357, 663, 487, 677], "score": 1.0, "content": " satisfies the assumptions of", "type": "text"}], "index": 37}, {"bbox": [123, 676, 486, 690], "spans": [{"bbox": [123, 676, 222, 690], "score": 1.0, "content": "Lemma 4: in fact, let", "type": "text"}, {"bbox": [223, 678, 266, 688], "score": 0.91, "content": "j=j_{k\\rightarrow K}", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [266, 676, 454, 690], "score": 1.0, "content": " denote the transfer of ideal classes. Then ", "type": "text"}, {"bbox": [454, 677, 486, 687], "score": 0.87, "content": "c^{1+\\sigma}=", "type": "inline_equation", "height": 10, "width": 32}], "index": 38}, {"bbox": [126, 689, 486, 702], "spans": [{"bbox": [126, 690, 167, 702], "score": 0.92, "content": "j(N_{K/k}c)", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [167, 690, 254, 702], "score": 1.0, "content": " for any ideal class ", "type": "text"}, {"bbox": [254, 689, 305, 701], "score": 0.92, "content": "c\\,\\in\\,\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [305, 690, 339, 702], "score": 1.0, "content": ", hence ", "type": "text"}, {"bbox": [340, 690, 432, 701], "score": 0.92, "content": "M^{\\nu}=j(\\mathrm{Cl}_{2}(k))\\,=\\,1", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [433, 690, 486, 702], "score": 1.0, "content": ". 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For cyclic extensions , let denote the group of ideal classes in fixed by , i.e. the ambiguous ideal class group of , and its 2-Sylow subgroup. Proposition 4. Let be a cyclic extension of prime degree ; then the number of ambiguous ideal classes is given by $$ \#\operatorname{Am}(K/k)=h(k)\,{\frac{p^{t-1}}{(E:H)}}, $$ where is the number of primes (including those at ) of that ramify in , is the unit group of , and is its subgroup consisting of norms of elements from . Moreover, is trivial if and only if . Proof. See Lang [9, part II] for the formula. For a proof of the second assertion (see e.g. Moriya [11]), note that is defined by the exact sequence $$ 1\;\longrightarrow\;\mathrm{Am}(K/k)\;\longrightarrow\;\mathrm{Cl}(K)\;\longrightarrow\;\mathrm{Cl}(K)^{1-\sigma}\;\longrightarrow\;1, $$ where generates . Taking -parts we see that is equiv- alent to . By induction we get , but since mod in the group ring , this implies . But then must be trivial. 口 We make one further remark concerning the ambiguous class number formula that will be useful below. If the class number is odd, then it is known that where . We also need a result essentially due to G. Gras [4]: Proposition 5. Let be a quadratic extension of number fields and assume that . Then is ramified and $$ \operatorname{Cl}_{2}(K)\simeq\left\{\!\!\begin{array}{l l}{{(2,2)~o r~\mathbb{Z}/2^{n}\mathbb{Z}~(n\geq3)~}}&{{i f\#\kappa_{K/k}=1,}}\\ {{\mathbb{Z}/2^{n}\mathbb{Z}~(n\geq1)~}}&{{i f\#\kappa_{K/k}=2,}}\end{array}\!\!\right. $$ where denotes the set of ideal classes of that become principal (capitulate) in . Proof. We first notice that is ramified. If the extension were unramified, then would be the 2-class field of , and since is cyclic, it would follow that , contrary to assumption. Before we start with the rest of the proof, we cite the results of Gras that we need (we could also give a slightly longer direct proof without referring to his results). Let be a cyclic extension of prime power order , and let be a generator of . For any -group on which acts we put . Moreover, let be the algebraic norm, that is, exponentiation by . Then [4, Cor. 4.3] reads Lemma 4. Suppose that ; let be the smallest positive integer such that and write with integers and . If for , then . We claim that if , then satisfies the assumptions of Lemma 4: in fact, let denote the transfer of ideal classes. Then for any ideal class , hence . Moreover,	<div class="pdf-page"> <p>Another important result is the ambiguous class number formula. For cyclic extensions , let denote the group of ideal classes in fixed by , i.e. the ambiguous ideal class group of , and its 2-Sylow subgroup.</p> <p>Proposition 4. Let be a cyclic extension of prime degree ; then the number of ambiguous ideal classes is given by</p> <p>where is the number of primes (including those at ) of that ramify in , is the unit group of , and is its subgroup consisting of norms of elements from . Moreover, is trivial if and only if .</p> <p>Proof. See Lang [9, part II] for the formula. For a proof of the second assertion (see e.g. Moriya [11]), note that is defined by the exact sequence</p> <p>where generates . Taking -parts we see that is equiv- alent to . By induction we get , but since mod in the group ring , this implies . But then must be trivial. 口</p> <p>We make one further remark concerning the ambiguous class number formula that will be useful below. If the class number is odd, then it is known that where .</p> <p>We also need a result essentially due to G. Gras [4]:</p> <p>Proposition 5. Let be a quadratic extension of number fields and assume that . Then is ramified and</p> <p>where denotes the set of ideal classes of that become principal (capitulate) in .</p> <p>Proof. We first notice that is ramified. If the extension were unramified, then would be the 2-class field of , and since is cyclic, it would follow that , contrary to assumption.</p> <p>Before we start with the rest of the proof, we cite the results of Gras that we need (we could also give a slightly longer direct proof without referring to his results). Let be a cyclic extension of prime power order , and let be a generator of . For any -group on which acts we put . Moreover, let be the algebraic norm, that is, exponentiation by . Then [4, Cor. 4.3] reads</p> <p>Lemma 4. Suppose that ; let be the smallest positive integer such that and write with integers and . If for , then .</p> <p>We claim that if , then satisfies the assumptions of Lemma 4: in fact, let denote the transfer of ideal classes. Then for any ideal class , hence . Moreover,</p> </div>	<div class="pdf-page"> <div class="pdf-discarded" data-x="207" data-y="116" data-width="12" data-height="12" style="opacity: 0.5;">6</div> <p class="pdf-text" data-x="207" data-y="143" data-width="607" data-height="49">Another important result is the ambiguous class number formula. For cyclic extensions , let denote the group of ideal classes in fixed by , i.e. the ambiguous ideal class group of , and its 2-Sylow subgroup.</p> <p class="pdf-text" data-x="209" data-y="199" data-width="605" data-height="31">Proposition 4. Let be a cyclic extension of prime degree ; then the number of ambiguous ideal classes is given by</p> <p class="pdf-text" data-x="209" data-y="272" data-width="604" data-height="48">where is the number of primes (including those at ) of that ramify in , is the unit group of , and is its subgroup consisting of norms of elements from . Moreover, is trivial if and only if .</p> <p class="pdf-text" data-x="210" data-y="325" data-width="604" data-height="33">Proof. See Lang [9, part II] for the formula. For a proof of the second assertion (see e.g. Moriya [11]), note that is defined by the exact sequence</p> <p class="pdf-text" data-x="209" data-y="385" data-width="605" data-height="63">where generates . Taking -parts we see that is equiv- alent to . By induction we get , but since mod in the group ring , this implies . But then must be trivial. 口</p> <p class="pdf-text" data-x="209" data-y="456" data-width="604" data-height="46">We make one further remark concerning the ambiguous class number formula that will be useful below. If the class number is odd, then it is known that where .</p> <p class="pdf-text" data-x="229" data-y="508" data-width="381" data-height="16">We also need a result essentially due to G. Gras [4]:</p> <p class="pdf-text" data-x="209" data-y="531" data-width="604" data-height="32">Proposition 5. Let be a quadratic extension of number fields and assume that . Then is ramified and</p> <p class="pdf-text" data-x="210" data-y="614" data-width="603" data-height="31">where denotes the set of ideal classes of that become principal (capitulate) in .</p> <p class="pdf-text" data-x="209" data-y="651" data-width="604" data-height="48">Proof. We first notice that is ramified. If the extension were unramified, then would be the 2-class field of , and since is cyclic, it would follow that , contrary to assumption.</p> <p class="pdf-text" data-x="209" data-y="700" data-width="605" data-height="95">Before we start with the rest of the proof, we cite the results of Gras that we need (we could also give a slightly longer direct proof without referring to his results). Let be a cyclic extension of prime power order , and let be a generator of . For any -group on which acts we put . Moreover, let be the algebraic norm, that is, exponentiation by . Then [4, Cor. 4.3] reads</p> <p class="pdf-text" data-x="207" data-y="800" data-width="607" data-height="50">Lemma 4. Suppose that ; let be the smallest positive integer such that and write with integers and . If for , then .</p> <p class="pdf-text" data-x="207" data-y="855" data-width="607" data-height="51">We claim that if , then satisfies the assumptions of Lemma 4: in fact, let denote the transfer of ideal classes. Then for any ideal class , hence . Moreover,</p> </div>	{ "type": [ "text", "inline_equation", "inline_equation", "inline_equation", "text", "interline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "inline_equation", "interline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "inline_equation", "inline_equation", "text", "inline_equation", "inline_equation", "interline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation" ], "coordinates": [ [ 230, 147, 813, 162 ], [ 210, 164, 811, 178 ], [ 210, 178, 813, 195 ], [ 209, 201, 813, 218 ], [ 210, 217, 486, 234 ], [ 404, 236, 614, 270 ], [ 210, 275, 811, 290 ], [ 209, 290, 814, 306 ], [ 210, 306, 670, 323 ], [ 210, 328, 814, 347 ], [ 209, 343, 762, 362 ], [ 271, 367, 747, 381 ], [ 210, 386, 811, 404 ], [ 207, 402, 814, 422 ], [ 209, 418, 811, 437 ], [ 209, 434, 813, 451 ], [ 209, 458, 813, 474 ], [ 210, 474, 813, 490 ], [ 210, 490, 522, 505 ], [ 229, 510, 609, 527 ], [ 210, 535, 814, 550 ], [ 210, 550, 639, 565 ], [ 316, 570, 704, 611 ], [ 209, 614, 814, 636 ], [ 210, 632, 254, 646 ], [ 212, 655, 811, 671 ], [ 210, 669, 813, 686 ], [ 210, 685, 478, 704 ], [ 229, 700, 813, 717 ], [ 210, 717, 813, 734 ], [ 209, 730, 809, 749 ], [ 207, 747, 813, 765 ], [ 210, 761, 814, 782 ], [ 207, 778, 717, 799 ], [ 207, 804, 814, 821 ], [ 210, 819, 814, 836 ], [ 209, 833, 789, 853 ], [ 229, 857, 814, 875 ], [ 205, 874, 813, 892 ], [ 210, 890, 813, 907 ] ], "content": [ "Another important result is the ambiguous class number formula. For cyclic", "extensions K/k , let \\operatorname{Am}(K/k) denote the group of ideal classes in K fixed by", "\\operatorname{Gal}(K/k) , i.e. the ambiguous ideal class group of K , and \\mathrm{{Am}_{2}} its 2-Sylow subgroup.", "Proposition 4. Let K/k be a cyclic extension of prime degree p ; then the number", "of ambiguous ideal classes is given by", "\\#\\operatorname{Am}(K/k)=h(k)\\,{\\frac{p^{t-1}}{(E:H)}},", "where t is the number of primes (including those at \\infty ) of k that ramify in K/k , E", "is the unit group of k , and H is its subgroup consisting of norms of elements from", "K^{\\times} . Moreover, \\mathrm{Cl}_{p}(K) is trivial if and only if p\\nmid\\#\\operatorname{Am}(K/k) .", "Proof. See Lang [9, part II] for the formula. For a proof of the second assertion", "(see e.g. Moriya [11]), note that \\operatorname{Am}(K/k) is defined by the exact sequence", "1\\;\\longrightarrow\\;\\mathrm{Am}(K/k)\\;\\longrightarrow\\;\\mathrm{Cl}(K)\\;\\longrightarrow\\;\\mathrm{Cl}(K)^{1-\\sigma}\\;\\longrightarrow\\;1,", "where \\sigma generates \\operatorname{Gal}(K/k) . Taking p -parts we see that p\\nmid\\#\\operatorname{Am}(K/k) is equiv-", "alent to \\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{1-\\sigma} . By induction we get \\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{(1-\\sigma)^{\\nu}} , but", "since (1-\\sigma)^{p}\\equiv0 mod p in the group ring \\mathbb{Z}[G] , this implies \\operatorname{Cl}_{p}(K)\\subseteq\\operatorname{Cl}_{p}(K)^{p} .", "But then \\mathrm{Cl}_{p}(K) must be trivial. 口", "We make one further remark concerning the ambiguous class number formula that", "will be useful below. If the class number h(k) is odd, then it is known that", "\\#\\operatorname{Am}_{2}(K/k)=2^{r} where r=\\mathrm{rank}\\,\\mathrm{Cl}_{2}(K) .", "We also need a result essentially due to G. Gras [4]:", "Proposition 5. Let K/k be a quadratic extension of number fields and assume", "that h_{2}(k)=\\#\\operatorname{Am}_{2}(K/k)=2 . Then K/k is ramified and", "\\operatorname{Cl}_{2}(K)\\simeq\\left\\{\\!\\!\\begin{array}{l l}{{(2,2)~o r~\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq3)~}}&{{i f\\#\\kappa_{K/k}=1,}}\\\\ {{\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq1)~}}&{{i f\\#\\kappa_{K/k}=2,}}\\end{array}\\!\\!\\right.", "where \\kappa_{K/k} denotes the set of ideal classes of k that become principal (capitulate)", "in K .", "Proof. We first notice that K/k is ramified. If the extension were unramified, then", "K would be the 2-class field of k , and since \\mathrm{Cl_{2}}(k) is cyclic, it would follow that", "\\mathrm{Cl}_{2}(K)=1 , contrary to assumption.", "Before we start with the rest of the proof, we cite the results of Gras that", "we need (we could also give a slightly longer direct proof without referring to", "his results). Let K/k be a cyclic extension of prime power order p^{r} , and let \\sigma", "be a generator of G\\,=\\,\\operatorname{Gal}(K/k) . For any p -group M on which G acts we put", "M_{i}\\,=\\,\\{m\\,\\in\\,M\\,:\\,m^{(1-\\sigma)^{i}}\\,=\\,1\\} . Moreover, let \\nu be the algebraic norm, that is,", "exponentiation by 1+\\sigma+\\sigma^{2}+...+\\sigma^{p^{\\intercal}-1} . Then [4, Cor. 4.3] reads", "Lemma 4. Suppose that M^{\\nu}=1 ; let n be the smallest positive integer such that", "M_{n}\\,=\\,M and write n=a(p-1)+b with integers a\\geq0 and 0\\,\\leq\\,b\\,\\leq\\,p\\,-\\,2 . If", "\\#\\,M_{i+1}/M_{i}=p for i=0,1,\\dots,n-1 , then M\\simeq(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b} .", "We claim that if \\kappa_{K/k}~=~2 , then M\\ =\\ \\mathrm{Cl}_{2}(K) satisfies the assumptions of", "Lemma 4: in fact, let j=j_{k\\rightarrow K} denote the transfer of ideal classes. Then c^{1+\\sigma}=", "j(N_{K/k}c) for any ideal class c\\,\\in\\,\\mathrm{Cl}_{2}(K) , hence M^{\\nu}=j(\\mathrm{Cl}_{2}(k))\\,=\\,1 . Moreover," ], "index": [ 0, 1, 2, 3, 4, 8, 14, 15, 16, 23, 24, 34, 46, 47, 48, 49, 62, 63, 64, 81, 101, 102, 123, 146, 147, 171, 172, 173, 199, 200, 201, 202, 203, 204, 233, 234, 235, 270, 271, 272 ] }	{ "coordinates": [], "index": [], "caption": [], "caption_coordinates": [] }	{ "type": [ "interline_equation", "interline_equation", "interline_equation", "interline_equation", "interline_equation", "interline_equation" ], "coordinates": [ [ 404, 236, 614, 270 ], [ 271, 367, 747, 381 ], [ 316, 570, 704, 611 ], [ 404, 236, 614, 270 ], [ 271, 367, 747, 381 ], [ 316, 570, 704, 611 ] ], "content": [ "", "", "", "\\#\\operatorname{Am}(K/k)=h(k)\\,{\\frac{p^{t-1}}{(E:H)}},", "1\\;\\longrightarrow\\;\\mathrm{Am}(K/k)\\;\\longrightarrow\\;\\mathrm{Cl}(K)\\;\\longrightarrow\\;\\mathrm{Cl}(K)^{1-\\sigma}\\;\\longrightarrow\\;1,", "\\operatorname{Cl}_{2}(K)\\simeq\\left\\{\\!\\!\\begin{array}{l l}{{(2,2)~o r~\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq3)~}}&{{i f\\#\\kappa_{K/k}=1,}}\\\\ {{\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq1)~}}&{{i f\\#\\kappa_{K/k}=2,}}\\end{array}\\!\\!\\right." ], "index": [ 0, 1, 2, 3, 4, 5 ] }	{ "type": [], "coordinates": [], "content": [], "index": [] }	{"document_info": {"total_pages": 14, "parse_type": "txt", "version": "1.0.1", "total_content_list_items": 155}, "content_statistics": {"total_blocks": 182, "total_lines": 526, "total_images": 4, "total_equations": 28, "total_tables": 11, "total_characters": 29658, "total_words": 5331, "block_type_distribution": {"title": 7, "text": 126, "interline_equation": 14, "discarded": 24, "table_caption": 3, "table_body": 4, "image_body": 2, "list": 2}}, "model_statistics": {"total_layout_detections": 2826, "avg_confidence": 0.9470955414012739, "min_confidence": 0.26, "max_confidence": 1.0}, "page_statistics": [{"page_number": 0, "page_size": [612.0, 792.0], "blocks": 11, "lines": 38, "images": 0, "equations": 2, "tables": 0, "characters": 2362, "words": 398, "discarded_blocks": 1}, {"page_number": 1, "page_size": [612.0, 792.0], "blocks": 10, "lines": 30, "images": 0, "equations": 0, "tables": 3, "characters": 1688, "words": 305, "discarded_blocks": 1}, {"page_number": 2, "page_size": [612.0, 792.0], "blocks": 15, "lines": 44, "images": 0, "equations": 4, "tables": 0, "characters": 2417, "words": 416, "discarded_blocks": 2}, {"page_number": 3, "page_size": [612.0, 792.0], "blocks": 7, "lines": 27, "images": 0, "equations": 0, "tables": 3, "characters": 1591, "words": 291, "discarded_blocks": 1}, {"page_number": 4, "page_size": [612.0, 792.0], "blocks": 23, "lines": 38, "images": 0, "equations": 6, "tables": 0, "characters": 1604, "words": 303, "discarded_blocks": 4}, {"page_number": 5, "page_size": [612.0, 792.0], "blocks": 17, "lines": 40, "images": 0, "equations": 6, "tables": 0, "characters": 2131, "words": 393, "discarded_blocks": 1}, {"page_number": 6, "page_size": [612.0, 792.0], "blocks": 15, "lines": 36, "images": 2, "equations": 4, "tables": 0, "characters": 1791, "words": 324, "discarded_blocks": 2}, {"page_number": 7, "page_size": [612.0, 792.0], "blocks": 12, "lines": 49, "images": 0, "equations": 0, "tables": 0, "characters": 3049, "words": 571, "discarded_blocks": 1}, {"page_number": 8, "page_size": [612.0, 792.0], "blocks": 11, "lines": 42, "images": 0, "equations": 0, "tables": 0, "characters": 2636, "words": 479, "discarded_blocks": 2}, {"page_number": 9, "page_size": [612.0, 792.0], "blocks": 14, "lines": 35, "images": 2, "equations": 0, "tables": 0, "characters": 2036, "words": 375, "discarded_blocks": 1}, {"page_number": 10, "page_size": [612.0, 792.0], "blocks": 17, "lines": 35, "images": 0, "equations": 6, "tables": 0, "characters": 1789, "words": 327, "discarded_blocks": 2}, {"page_number": 11, "page_size": [612.0, 792.0], "blocks": 14, "lines": 40, "images": 0, "equations": 0, "tables": 2, "characters": 2094, "words": 399, "discarded_blocks": 2}, {"page_number": 12, "page_size": [612.0, 792.0], "blocks": 10, "lines": 44, "images": 0, "equations": 0, "tables": 3, "characters": 2572, "words": 487, "discarded_blocks": 2}, {"page_number": 13, "page_size": [612.0, 792.0], "blocks": 6, "lines": 28, "images": 0, "equations": 0, "tables": 0, "characters": 1898, "words": 263, "discarded_blocks": 2}]}
0003244v1	6	[ 612, 792 ]	{ "type": [ "text", "text", "text", "interline_equation", "text", "text", "title", "text", "image_body", "text", "text", "interline_equation", "text", "discarded", "discarded" ], "coordinates": [ [ 209, 143, 814, 206 ], [ 227, 206, 635, 222 ], [ 207, 228, 813, 259 ], [ 319, 265, 702, 324 ], [ 207, 327, 814, 405 ], [ 209, 405, 813, 500 ], [ 322, 508, 699, 524 ], [ 207, 532, 813, 594 ], [ 219, 605, 799, 716 ], [ 209, 718, 814, 782 ], [ 209, 783, 813, 844 ], [ 237, 850, 779, 868 ], [ 207, 871, 813, 905 ], [ 396, 116, 625, 129 ], [ 801, 116, 813, 128 ] ], "content": [ "in our case, hence has order 2. Since the orders of decrease towards as grows (Gras [4, Prop. 4.1.ii)]), we conclude that # for all . Since and when , Lemma 4 now implies that , that is, the 2-class group is cyclic.", "The second result of Gras that we need is [4, Prop. 4.3]", "Lemma 5. Suppose that but assume the other conditions in Lemma 4. Then and", "", "If , then this lemma shows that is either cyclic of order or of type . (Notice that the hypothesis of the lemma is satisfied since is ramified implying that the norm is onto; and so the argument above this lemma applies.) It remains to show that the case cannot occur here.", "Now assume that ; since is ramified, the norm is onto, and using once more we find , where is the nontrivial ideal class from . On the other hand, still has order 2 in , hence we must also have . But this implies that , i.e. that each ideal class in is ambiguous, contradicting our assumption that . 口", "4. Arithmetic of some Dihedral Extensions", "In this section we study the arithmetic of some dihedral extensions , that is, normal extensions of with Galois group , the dihedral group of order 8. Hence may be presented as . Now consider the following diagrams (Galois correspondence):", "", "In this situation, we let and denote the unit indices of the bicyclic extensions and , where and are the unit groups in and respectively. Finally, let denote the kernel of the transfer of ideal classes for .", "The following remark will be used several times: if for some , then , where is the norm of . To see this, let ; then , since . Clearly and hence fixed by . Furthermore,", "", "implying that . Finally notice that , since otherwise implying that is normal, which is not the case.", "IMAGINARY QUADRATIC FIELDS", "7" ], "index": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 ] }	[{"type": "text", "text": "$M_{1}\\;=\\;\\mathrm{Am}_{2}(K/k)$ in our case, hence $M_{1}/M_{0}$ has order 2. Since the orders of $M_{i+1}/M_{i}$ decrease towards $^{1}$ as $i$ grows (Gras [4, Prop. 4.1.ii)]), we conclude that # $:M_{i+1}/M_{i}=2$ for all $i<n$ . Since $a=n$ and $b=0$ when $p=2$ , Lemma 4 now implies that $\\mathrm{Cl}_{2}(K)\\simeq\\mathbb{Z}/2^{n}\\mathbb{Z}$ , that is, the 2-class group is cyclic. ", "page_idx": 6}, {"type": "text", "text": "The second result of Gras that we need is [4, Prop. 4.3] ", "page_idx": 6}, {"type": "text", "text": "Lemma 5. Suppose that $M^{\\nu}\\ne1$ but assume the other conditions in Lemma 4. Then $n\\geq2$ and ", "page_idx": 6}, {"type": "equation", "text": "$$\nM\\simeq\\left\\{\\!\\!\\begin{array}{l l}{(\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n<p;}\\\\ {(\\mathbb{Z}/p\\mathbb{Z})^{p}\\ \\mathrm{or}\\ (\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n=p;}\\\\ {(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}}&{\\mathrm{if}\\ n>p.}\\end{array}\\right.\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "If $\\kappa_{K/k}=1$ , then this lemma shows that $\\mathrm{Cl}_{2}(K)$ is either cyclic of order $\\geq4$ or of type $(2,2)$ . (Notice that the hypothesis of the lemma is satisfied since $K/k$ is ramified implying that the norm $N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)$ is onto; and so the argument above this lemma applies.) It remains to show that the case $\\mathrm{Cl}_{2}(K)\\simeq$ $\\mathbb{Z}/4\\mathbb{Z}$ cannot occur here. ", "page_idx": 6}, {"type": "text", "text": "Now assume that $\\operatorname{Cl}_{2}(K)~=~\\langle{\\cal C}\\rangle~\\simeq~\\mathbb{Z}/4\\mathbb{Z}$ ; since $K/k$ is ramified, the norm $N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)$ is onto, and using $\\kappa_{K/k}=1$ once more we find $C^{1+\\sigma}=c$ , where $c$ is the nontrivial ideal class from $\\mathrm{Cl_{2}}(k)$ . On the other hand, $c\\in\\mathrm{Cl}_{2}(k)$ still has order 2 in $\\mathrm{Cl}_{2}(K)$ , hence we must also have $C^{2}=C^{1+\\sigma}$ . But this implies that $C^{\\sigma}=C$ , i.e. that each ideal class in $K$ is ambiguous, contradicting our assumption that $\\#\\operatorname{Am}_{2}(K/k)=2$ . \u53e3 ", "page_idx": 6}, {"type": "text", "text": "4. Arithmetic of some Dihedral Extensions ", "text_level": 1, "page_idx": 6}, {"type": "text", "text": "In this section we study the arithmetic of some dihedral extensions $L/\\mathbb{Q}$ , that is, normal extensions $L$ of $\\mathbb{Q}$ with Galois group $\\operatorname{Gal}(L/\\mathbb{Q})\\simeq D_{4}$ , the dihedral group of order 8. Hence $D_{4}$ may be presented as $\\langle\\tau,\\sigma\|\\tau^{2}=\\sigma^{4}=1,\\tau\\sigma\\tau=\\sigma^{-1}\\rangle$ . Now consider the following diagrams (Galois correspondence): ", "page_idx": 6}, {"type": "image", "img_path": "images/110ff188220c8bb5d817468ee0d73bccaf61d8e5373f09de7f29c9cd995490d3.jpg", "img_caption": [], "img_footnote": [], "page_idx": 6}, {"type": "text", "text": "In this situation, we let $q_{1}=(E_{L}:E_{1}E_{1}^{\\prime}E_{K})$ and $q_{2}=(E_{L}:E_{2}E_{2}^{\\prime}E_{K})$ denote the unit indices of the bicyclic extensions $L/k_{1}$ and $L/k_{2}$ , where $E_{i}$ and $E_{i}^{\\prime}$ are the unit groups in $K_{i}$ and $K_{i}^{\\prime}$ respectively. Finally, let $\\kappa_{i}$ denote the kernel of the transfer of ideal classes $j_{k_{i}\\to K_{i}}:\\mathrm{Cl}_{2}(k_{i})\\longrightarrow\\mathrm{Cl}_{2}(K_{i})$ for $i=1,2$ . ", "page_idx": 6}, {"type": "text", "text": "The following remark will be used several times: if $K_{1}\\;=\\;k_{1}(\\sqrt{\\alpha}\\,)$ for some $\\alpha\\in k_{1}$ , then $k_{2}=\\mathbb{Q}({\\sqrt{a}}\\,)$ , where $a=\\alpha\\alpha^{\\prime}$ is the norm of $\\alpha$ . To see this, let $\\gamma=\\sqrt{\\alpha}$ ; then $\\gamma^{\\tau}\\,=\\,\\gamma$ , since $\\gamma\\ \\in\\ K_{1}$ . Clearly $\\gamma^{1+\\sigma}\\,=\\,\\sqrt{a}\\,\\in\\,K$ and hence fixed by $\\sigma^{2}$ . Furthermore, ", "page_idx": 6}, {"type": "equation", "text": "$$\n(\\gamma^{1+\\sigma})^{\\sigma\\tau}=\\gamma^{\\sigma\\tau+\\sigma^{2}\\tau}=\\gamma^{\\tau\\sigma^{3}+\\tau\\sigma^{2}}=(\\gamma^{\\tau})^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{(1+\\sigma)\\sigma^{2}}=\\gamma^{1+\\sigma},\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "implying that ${\\sqrt{a}}\\ \\in\\ k_{2}$ . Finally notice that ${\\sqrt{a}}\\ \\not\\in\\ \\mathbb{Q}$ , since otherwise $\\sqrt{\\alpha^{\\prime}}\\,=$ ${\\sqrt{a}}/{\\sqrt{\\alpha}}\\in K_{1}$ implying that $K_{1}/\\mathbb{Q}$ is normal, which is not the case. ", "page_idx": 6}]	{"preproc_blocks": [{"type": "text", "bbox": [125, 111, 487, 160], "lines": [{"bbox": [126, 114, 487, 126], "spans": [{"bbox": [126, 115, 206, 126], "score": 0.92, "content": "M_{1}\\;=\\;\\mathrm{Am}_{2}(K/k)", "type": "inline_equation", "height": 11, "width": 80}, {"bbox": [207, 114, 297, 126], "score": 1.0, "content": " in our case, hence ", "type": "text"}, {"bbox": [298, 115, 330, 126], "score": 0.94, "content": "M_{1}/M_{0}", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [331, 114, 487, 126], "score": 1.0, "content": " has order 2. Since the orders of", "type": "text"}], "index": 0}, {"bbox": [126, 126, 486, 138], "spans": [{"bbox": [126, 128, 167, 138], "score": 0.93, "content": "M_{i+1}/M_{i}", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [167, 126, 246, 138], "score": 1.0, "content": " decrease towards ", "type": "text"}, {"bbox": [246, 128, 252, 135], "score": 0.35, "content": "^{1}", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [252, 126, 267, 138], "score": 1.0, "content": " as ", "type": "text"}, {"bbox": [267, 128, 271, 135], "score": 0.85, "content": "i", "type": "inline_equation", "height": 7, "width": 4}, {"bbox": [271, 126, 486, 138], "score": 1.0, "content": " grows (Gras [4, Prop. 4.1.ii)]), we conclude that", "type": "text"}], "index": 1}, {"bbox": [126, 138, 486, 150], "spans": [{"bbox": [126, 138, 133, 150], "score": 1.0, "content": "#", "type": "text"}, {"bbox": [133, 139, 196, 150], "score": 0.9, "content": ":M_{i+1}/M_{i}=2", "type": "inline_equation", "height": 11, "width": 63}, {"bbox": [196, 138, 229, 150], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [230, 140, 254, 147], "score": 0.91, "content": "i<n", "type": "inline_equation", "height": 7, "width": 24}, {"bbox": [254, 138, 288, 150], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [288, 142, 314, 147], "score": 0.88, "content": "a=n", "type": "inline_equation", "height": 5, "width": 26}, {"bbox": [315, 138, 337, 150], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [338, 140, 361, 147], "score": 0.91, "content": "b=0", "type": "inline_equation", "height": 7, "width": 23}, {"bbox": [362, 138, 391, 150], "score": 1.0, "content": " when ", "type": "text"}, {"bbox": [392, 140, 416, 149], "score": 0.92, "content": "p=2", "type": "inline_equation", "height": 9, "width": 24}, {"bbox": [416, 138, 486, 150], "score": 1.0, "content": ", Lemma 4 now", "type": "text"}], "index": 2}, {"bbox": [126, 150, 410, 162], "spans": [{"bbox": [126, 150, 181, 162], "score": 1.0, "content": "implies that ", "type": "text"}, {"bbox": [181, 151, 255, 162], "score": 0.94, "content": "\\mathrm{Cl}_{2}(K)\\simeq\\mathbb{Z}/2^{n}\\mathbb{Z}", "type": "inline_equation", "height": 11, "width": 74}, {"bbox": [255, 150, 410, 162], "score": 1.0, "content": ", that is, the 2-class group is cyclic.", "type": "text"}], "index": 3}], "index": 1.5}, {"type": "text", "bbox": [136, 160, 380, 172], "lines": [{"bbox": [137, 162, 381, 173], "spans": [{"bbox": [137, 162, 381, 173], "score": 1.0, "content": "The second result of Gras that we need is [4, Prop. 4.3]", "type": "text"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [124, 177, 486, 201], "lines": [{"bbox": [125, 180, 486, 192], "spans": [{"bbox": [125, 180, 240, 192], "score": 1.0, "content": "Lemma 5. Suppose that ", "type": "text"}, {"bbox": [240, 181, 277, 191], "score": 0.92, "content": "M^{\\nu}\\ne1", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [278, 180, 486, 192], "score": 1.0, "content": " but assume the other conditions in Lemma 4.", "type": "text"}], "index": 5}, {"bbox": [127, 192, 199, 204], "spans": [{"bbox": [127, 192, 151, 204], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [152, 194, 176, 203], "score": 0.91, "content": "n\\geq2", "type": "inline_equation", "height": 9, "width": 24}, {"bbox": [177, 192, 199, 204], "score": 1.0, "content": " and", "type": "text"}], "index": 6}], "index": 5.5}, {"type": "interline_equation", "bbox": [191, 205, 420, 251], "lines": [{"bbox": [191, 205, 420, 251], "spans": [{"bbox": [191, 205, 420, 251], "score": 0.92, "content": "M\\simeq\\left\\{\\!\\!\\begin{array}{l l}{(\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n<p;}\\\\ {(\\mathbb{Z}/p\\mathbb{Z})^{p}\\ \\mathrm{or}\\ (\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n=p;}\\\\ {(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}}&{\\mathrm{if}\\ n>p.}\\end{array}\\right.", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [124, 253, 487, 314], "lines": [{"bbox": [136, 255, 485, 269], "spans": [{"bbox": [136, 255, 148, 269], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [149, 258, 191, 268], "score": 0.92, "content": "\\kappa_{K/k}=1", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [191, 255, 325, 269], "score": 1.0, "content": ", then this lemma shows that ", "type": "text"}, {"bbox": [326, 257, 357, 267], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [358, 255, 468, 269], "score": 1.0, "content": " is either cyclic of order ", "type": "text"}, {"bbox": [468, 258, 485, 266], "score": 0.84, "content": "\\geq4", "type": "inline_equation", "height": 8, "width": 17}], "index": 8}, {"bbox": [125, 268, 485, 280], "spans": [{"bbox": [125, 268, 173, 280], "score": 1.0, "content": "or of type ", "type": "text"}, {"bbox": [173, 269, 196, 280], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [196, 268, 465, 280], "score": 1.0, "content": ". (Notice that the hypothesis of the lemma is satisfied since ", "type": "text"}, {"bbox": [465, 269, 485, 280], "score": 0.93, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}], "index": 9}, {"bbox": [124, 279, 487, 294], "spans": [{"bbox": [124, 279, 282, 294], "score": 1.0, "content": "is ramified implying that the norm", "type": "text"}, {"bbox": [283, 281, 398, 292], "score": 0.9, "content": "N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)", "type": "inline_equation", "height": 11, "width": 115}, {"bbox": [399, 279, 487, 294], "score": 1.0, "content": " is onto; and so the", "type": "text"}], "index": 10}, {"bbox": [125, 292, 486, 304], "spans": [{"bbox": [125, 292, 442, 304], "score": 1.0, "content": "argument above this lemma applies.) It remains to show that the case ", "type": "text"}, {"bbox": [442, 293, 486, 303], "score": 0.92, "content": "\\mathrm{Cl}_{2}(K)\\simeq", "type": "inline_equation", "height": 10, "width": 44}], "index": 11}, {"bbox": [126, 303, 234, 317], "spans": [{"bbox": [126, 305, 150, 316], "score": 0.94, "content": "\\mathbb{Z}/4\\mathbb{Z}", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [150, 303, 234, 317], "score": 1.0, "content": " cannot occur here.", "type": "text"}], "index": 12}], "index": 10}, {"type": "text", "bbox": [125, 314, 486, 387], "lines": [{"bbox": [135, 315, 487, 329], "spans": [{"bbox": [135, 315, 221, 329], "score": 1.0, "content": "Now assume that ", "type": "text"}, {"bbox": [221, 317, 330, 327], "score": 0.91, "content": "\\operatorname{Cl}_{2}(K)~=~\\langle{\\cal C}\\rangle~\\simeq~\\mathbb{Z}/4\\mathbb{Z}", "type": "inline_equation", "height": 10, "width": 109}, {"bbox": [330, 315, 363, 329], "score": 1.0, "content": "; since ", "type": "text"}, {"bbox": [363, 317, 383, 327], "score": 0.93, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [384, 315, 487, 329], "score": 1.0, "content": " is ramified, the norm", "type": "text"}], "index": 13}, {"bbox": [126, 327, 487, 342], "spans": [{"bbox": [126, 329, 239, 340], "score": 0.91, "content": "N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)", "type": "inline_equation", "height": 11, "width": 113}, {"bbox": [239, 327, 319, 342], "score": 1.0, "content": " is onto, and using ", "type": "text"}, {"bbox": [319, 330, 359, 340], "score": 0.93, "content": "\\kappa_{K/k}=1", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [360, 327, 441, 342], "score": 1.0, "content": " once more we find ", "type": "text"}, {"bbox": [441, 328, 482, 337], "score": 0.92, "content": "C^{1+\\sigma}=c", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [483, 327, 487, 342], "score": 1.0, "content": ",", "type": "text"}], "index": 14}, {"bbox": [126, 339, 486, 352], "spans": [{"bbox": [126, 339, 154, 352], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 344, 159, 349], "score": 0.88, "content": "c", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [159, 339, 301, 352], "score": 1.0, "content": " is the nontrivial ideal class from ", "type": "text"}, {"bbox": [302, 341, 329, 351], "score": 0.92, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [330, 339, 421, 352], "score": 1.0, "content": ". On the other hand, ", "type": "text"}, {"bbox": [421, 341, 466, 351], "score": 0.93, "content": "c\\in\\mathrm{Cl}_{2}(k)", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [466, 339, 486, 352], "score": 1.0, "content": " still", "type": "text"}], "index": 15}, {"bbox": [124, 350, 487, 365], "spans": [{"bbox": [124, 350, 189, 365], "score": 1.0, "content": "has order 2 in ", "type": "text"}, {"bbox": [190, 353, 221, 363], "score": 0.91, "content": "\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [222, 350, 337, 365], "score": 1.0, "content": ", hence we must also have ", "type": "text"}, {"bbox": [337, 352, 386, 361], "score": 0.93, "content": "C^{2}=C^{1+\\sigma}", "type": "inline_equation", "height": 9, "width": 49}, {"bbox": [387, 350, 487, 365], "score": 1.0, "content": ". But this implies that", "type": "text"}], "index": 16}, {"bbox": [126, 363, 485, 376], "spans": [{"bbox": [126, 366, 160, 373], "score": 0.91, "content": "C^{\\sigma}=C", "type": "inline_equation", "height": 7, "width": 34}, {"bbox": [161, 363, 284, 376], "score": 1.0, "content": ", i.e. that each ideal class in ", "type": "text"}, {"bbox": [284, 366, 293, 373], "score": 0.92, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [294, 363, 485, 376], "score": 1.0, "content": " is ambiguous, contradicting our assumption", "type": "text"}], "index": 17}, {"bbox": [126, 376, 487, 388], "spans": [{"bbox": [126, 376, 148, 388], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 377, 223, 387], "score": 0.94, "content": "\\#\\operatorname{Am}_{2}(K/k)=2", "type": "inline_equation", "height": 10, "width": 75}, {"bbox": [223, 376, 227, 388], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [475, 376, 487, 386], "score": 0.9919366836547852, "content": "\u53e3", "type": "text"}], "index": 18}], "index": 15.5}, {"type": "title", "bbox": [193, 393, 418, 406], "lines": [{"bbox": [193, 396, 418, 407], "spans": [{"bbox": [193, 396, 418, 407], "score": 1.0, "content": "4. Arithmetic of some Dihedral Extensions", "type": "text"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [124, 412, 486, 460], "lines": [{"bbox": [137, 413, 485, 426], "spans": [{"bbox": [137, 413, 428, 426], "score": 1.0, "content": "In this section we study the arithmetic of some dihedral extensions", "type": "text"}, {"bbox": [429, 415, 449, 425], "score": 0.94, "content": "L/\\mathbb{Q}", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [449, 413, 485, 426], "score": 1.0, "content": ", that is,", "type": "text"}], "index": 20}, {"bbox": [125, 425, 487, 439], "spans": [{"bbox": [125, 425, 208, 439], "score": 1.0, "content": "normal extensions ", "type": "text"}, {"bbox": [208, 428, 216, 435], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [216, 425, 231, 439], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [231, 428, 239, 437], "score": 0.91, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [240, 425, 325, 439], "score": 1.0, "content": " with Galois group ", "type": "text"}, {"bbox": [326, 427, 396, 437], "score": 0.93, "content": "\\operatorname{Gal}(L/\\mathbb{Q})\\simeq D_{4}", "type": "inline_equation", "height": 10, "width": 70}, {"bbox": [397, 425, 487, 439], "score": 1.0, "content": ", the dihedral group", "type": "text"}], "index": 21}, {"bbox": [125, 437, 487, 451], "spans": [{"bbox": [125, 437, 208, 451], "score": 1.0, "content": "of order 8. Hence ", "type": "text"}, {"bbox": [209, 440, 222, 448], "score": 0.91, "content": "D_{4}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [222, 437, 320, 451], "score": 1.0, "content": " may be presented as ", "type": "text"}, {"bbox": [320, 438, 455, 450], "score": 0.91, "content": "\\langle\\tau,\\sigma\|\\tau^{2}=\\sigma^{4}=1,\\tau\\sigma\\tau=\\sigma^{-1}\\rangle", "type": "inline_equation", "height": 12, "width": 135}, {"bbox": [456, 437, 487, 451], "score": 1.0, "content": ". Now", "type": "text"}], "index": 22}, {"bbox": [126, 450, 375, 462], "spans": [{"bbox": [126, 450, 375, 462], "score": 1.0, "content": "consider the following diagrams (Galois correspondence):", "type": "text"}], "index": 23}], "index": 21.5}, {"type": "image", "bbox": [131, 468, 478, 554], "blocks": [{"type": "image_body", "bbox": [131, 468, 478, 554], "group_id": 0, "lines": [{"bbox": [131, 468, 478, 554], "spans": [{"bbox": [131, 468, 478, 554], "score": 0.921, "type": "image", "image_path": "110ff188220c8bb5d817468ee0d73bccaf61d8e5373f09de7f29c9cd995490d3.jpg"}]}], "index": 25, "virtual_lines": [{"bbox": [131, 468, 478, 496.6666666666667], "spans": [], "index": 24}, {"bbox": [131, 496.6666666666667, 478, 525.3333333333334], "spans": [], "index": 25}, {"bbox": [131, 525.3333333333334, 478, 554.0], "spans": [], "index": 26}]}], "index": 25}, {"type": "text", "bbox": [125, 556, 487, 605], "lines": [{"bbox": [137, 559, 486, 571], "spans": [{"bbox": [137, 559, 244, 571], "score": 1.0, "content": "In this situation, we let ", "type": "text"}, {"bbox": [245, 560, 337, 570], "score": 0.91, "content": "q_{1}=(E_{L}:E_{1}E_{1}^{\\prime}E_{K})", "type": "inline_equation", "height": 10, "width": 92}, {"bbox": [338, 559, 360, 571], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [360, 560, 452, 571], "score": 0.91, "content": "q_{2}=(E_{L}:E_{2}E_{2}^{\\prime}E_{K})", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [453, 559, 486, 571], "score": 1.0, "content": " denote", "type": "text"}], "index": 27}, {"bbox": [126, 571, 486, 584], "spans": [{"bbox": [126, 571, 316, 584], "score": 1.0, "content": "the unit indices of the bicyclic extensions ", "type": "text"}, {"bbox": [316, 572, 338, 583], "score": 0.93, "content": "L/k_{1}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [338, 571, 362, 584], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [362, 572, 384, 583], "score": 0.93, "content": "L/k_{2}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [384, 571, 420, 584], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [421, 573, 432, 582], "score": 0.91, "content": "E_{i}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [432, 571, 456, 584], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [456, 572, 467, 583], "score": 0.92, "content": "E_{i}^{\\prime}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [468, 571, 486, 584], "score": 1.0, "content": " are", "type": "text"}], "index": 28}, {"bbox": [126, 583, 486, 596], "spans": [{"bbox": [126, 583, 208, 596], "score": 1.0, "content": "the unit groups in ", "type": "text"}, {"bbox": [208, 585, 221, 594], "score": 0.92, "content": "K_{i}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [221, 583, 243, 596], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [243, 584, 255, 595], "score": 0.92, "content": "K_{i}^{\\prime}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [256, 583, 367, 596], "score": 1.0, "content": " respectively. Finally, let ", "type": "text"}, {"bbox": [367, 587, 376, 594], "score": 0.9, "content": "\\kappa_{i}", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [377, 583, 486, 596], "score": 1.0, "content": " denote the kernel of the", "type": "text"}], "index": 29}, {"bbox": [124, 594, 408, 608], "spans": [{"bbox": [124, 594, 230, 608], "score": 1.0, "content": "transfer of ideal classes ", "type": "text"}, {"bbox": [230, 596, 354, 606], "score": 0.91, "content": "j_{k_{i}\\to K_{i}}:\\mathrm{Cl}_{2}(k_{i})\\longrightarrow\\mathrm{Cl}_{2}(K_{i})", "type": "inline_equation", "height": 10, "width": 124}, {"bbox": [354, 594, 372, 608], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [372, 597, 404, 606], "score": 0.92, "content": "i=1,2", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [404, 594, 408, 608], "score": 1.0, "content": ".", "type": "text"}], "index": 30}], "index": 28.5}, {"type": "text", "bbox": [125, 606, 486, 653], "lines": [{"bbox": [137, 605, 487, 619], "spans": [{"bbox": [137, 605, 376, 619], "score": 1.0, "content": "The following remark will be used several times: if ", "type": "text"}, {"bbox": [377, 608, 442, 618], "score": 0.94, "content": "K_{1}\\;=\\;k_{1}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 10, "width": 65}, {"bbox": [442, 605, 487, 619], "score": 1.0, "content": " for some", "type": "text"}], "index": 31}, {"bbox": [126, 618, 485, 632], "spans": [{"bbox": [126, 621, 154, 630], "score": 0.92, "content": "\\alpha\\in k_{1}", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [155, 618, 182, 632], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [182, 619, 236, 631], "score": 0.94, "content": "k_{2}=\\mathbb{Q}({\\sqrt{a}}\\,)", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [236, 618, 270, 632], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [270, 620, 304, 628], "score": 0.93, "content": "a=\\alpha\\alpha^{\\prime}", "type": "inline_equation", "height": 8, "width": 34}, {"bbox": [304, 618, 369, 632], "score": 1.0, "content": " is the norm of ", "type": "text"}, {"bbox": [369, 623, 376, 628], "score": 0.87, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [376, 618, 447, 632], "score": 1.0, "content": ". To see this, let", "type": "text"}, {"bbox": [448, 619, 482, 630], "score": 0.92, "content": "\\gamma=\\sqrt{\\alpha}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [482, 618, 485, 632], "score": 1.0, "content": ";", "type": "text"}], "index": 32}, {"bbox": [124, 630, 487, 644], "spans": [{"bbox": [124, 630, 149, 644], "score": 1.0, "content": "then ", "type": "text"}, {"bbox": [150, 633, 184, 642], "score": 0.93, "content": "\\gamma^{\\tau}\\,=\\,\\gamma", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [184, 630, 216, 644], "score": 1.0, "content": ", since ", "type": "text"}, {"bbox": [217, 633, 252, 642], "score": 0.93, "content": "\\gamma\\ \\in\\ K_{1}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [253, 630, 299, 644], "score": 1.0, "content": ". Clearly ", "type": "text"}, {"bbox": [299, 631, 377, 642], "score": 0.94, "content": "\\gamma^{1+\\sigma}\\,=\\,\\sqrt{a}\\,\\in\\,K", "type": "inline_equation", "height": 11, "width": 78}, {"bbox": [377, 630, 471, 644], "score": 1.0, "content": " and hence fixed by ", "type": "text"}, {"bbox": [471, 631, 482, 640], "score": 0.9, "content": "\\sigma^{2}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [482, 630, 487, 644], "score": 1.0, "content": ".", "type": "text"}], "index": 33}, {"bbox": [125, 642, 185, 655], "spans": [{"bbox": [125, 642, 185, 655], "score": 1.0, "content": "Furthermore,", "type": "text"}], "index": 34}], "index": 32.5}, {"type": "interline_equation", "bbox": [142, 658, 466, 672], "lines": [{"bbox": [142, 658, 466, 672], "spans": [{"bbox": [142, 658, 466, 672], "score": 0.88, "content": "(\\gamma^{1+\\sigma})^{\\sigma\\tau}=\\gamma^{\\sigma\\tau+\\sigma^{2}\\tau}=\\gamma^{\\tau\\sigma^{3}+\\tau\\sigma^{2}}=(\\gamma^{\\tau})^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{(1+\\sigma)\\sigma^{2}}=\\gamma^{1+\\sigma},", "type": "interline_equation"}], "index": 35}], "index": 35}, {"type": "text", "bbox": [124, 674, 486, 700], "lines": [{"bbox": [125, 676, 486, 689], "spans": [{"bbox": [125, 677, 191, 689], "score": 1.0, "content": "implying that ", "type": "text"}, {"bbox": [192, 678, 233, 689], "score": 0.93, "content": "{\\sqrt{a}}\\ \\in\\ k_{2}", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [233, 677, 334, 689], "score": 1.0, "content": ". Finally notice that ", "type": "text"}, {"bbox": [335, 678, 374, 689], "score": 0.94, "content": "{\\sqrt{a}}\\ \\not\\in\\ \\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [375, 677, 453, 689], "score": 1.0, "content": ", since otherwise", "type": "text"}, {"bbox": [453, 676, 486, 688], "score": 0.88, "content": "\\sqrt{\\alpha^{\\prime}}\\,=", "type": "inline_equation", "height": 12, "width": 33}], "index": 36}, {"bbox": [126, 689, 422, 702], "spans": [{"bbox": [126, 690, 185, 701], "score": 0.94, "content": "{\\sqrt{a}}/{\\sqrt{\\alpha}}\\in K_{1}", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [185, 689, 250, 702], "score": 1.0, "content": " implying that ", "type": "text"}, {"bbox": [250, 690, 276, 701], "score": 0.94, "content": "K_{1}/\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [277, 689, 422, 702], "score": 1.0, "content": " is normal, which is not the case.", "type": "text"}], "index": 37}], "index": 36.5}], "layout_bboxes": [], "page_idx": 6, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [{"type": "image", "bbox": [131, 468, 478, 554], "blocks": [{"type": "image_body", "bbox": [131, 468, 478, 554], "group_id": 0, "lines": [{"bbox": [131, 468, 478, 554], "spans": [{"bbox": [131, 468, 478, 554], "score": 0.921, "type": "image", "image_path": "110ff188220c8bb5d817468ee0d73bccaf61d8e5373f09de7f29c9cd995490d3.jpg"}]}], "index": 25, "virtual_lines": [{"bbox": [131, 468, 478, 496.6666666666667], "spans": [], "index": 24}, {"bbox": [131, 496.6666666666667, 478, 525.3333333333334], "spans": [], "index": 25}, {"bbox": [131, 525.3333333333334, 478, 554.0], "spans": [], "index": 26}]}], "index": 25}], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [191, 205, 420, 251], "lines": [{"bbox": [191, 205, 420, 251], "spans": [{"bbox": [191, 205, 420, 251], "score": 0.92, "content": "M\\simeq\\left\\{\\!\\!\\begin{array}{l l}{(\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n<p;}\\\\ {(\\mathbb{Z}/p\\mathbb{Z})^{p}\\ \\mathrm{or}\\ (\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n=p;}\\\\ {(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}}&{\\mathrm{if}\\ n>p.}\\end{array}\\right.", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "interline_equation", "bbox": [142, 658, 466, 672], "lines": [{"bbox": [142, 658, 466, 672], "spans": [{"bbox": [142, 658, 466, 672], "score": 0.88, "content": "(\\gamma^{1+\\sigma})^{\\sigma\\tau}=\\gamma^{\\sigma\\tau+\\sigma^{2}\\tau}=\\gamma^{\\tau\\sigma^{3}+\\tau\\sigma^{2}}=(\\gamma^{\\tau})^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{(1+\\sigma)\\sigma^{2}}=\\gamma^{1+\\sigma},", "type": "interline_equation"}], "index": 35}], "index": 35}], "discarded_blocks": [{"type": "discarded", "bbox": [237, 90, 374, 100], "lines": [{"bbox": [239, 93, 372, 100], "spans": [{"bbox": [239, 93, 372, 100], "score": 1.0, "content": "IMAGINARY QUADRATIC FIELDS", "type": "text"}]}]}, {"type": "discarded", "bbox": [479, 90, 486, 99], "lines": [{"bbox": [481, 94, 485, 100], "spans": [{"bbox": [481, 94, 485, 100], "score": 1.0, "content": "7", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 111, 487, 160], "lines": [{"bbox": [126, 114, 487, 126], "spans": [{"bbox": [126, 115, 206, 126], "score": 0.92, "content": "M_{1}\\;=\\;\\mathrm{Am}_{2}(K/k)", "type": "inline_equation", "height": 11, "width": 80}, {"bbox": [207, 114, 297, 126], "score": 1.0, "content": " in our case, hence ", "type": "text"}, {"bbox": [298, 115, 330, 126], "score": 0.94, "content": "M_{1}/M_{0}", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [331, 114, 487, 126], "score": 1.0, "content": " has order 2. Since the orders of", "type": "text"}], "index": 0}, {"bbox": [126, 126, 486, 138], "spans": [{"bbox": [126, 128, 167, 138], "score": 0.93, "content": "M_{i+1}/M_{i}", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [167, 126, 246, 138], "score": 1.0, "content": " decrease towards ", "type": "text"}, {"bbox": [246, 128, 252, 135], "score": 0.35, "content": "^{1}", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [252, 126, 267, 138], "score": 1.0, "content": " as ", "type": "text"}, {"bbox": [267, 128, 271, 135], "score": 0.85, "content": "i", "type": "inline_equation", "height": 7, "width": 4}, {"bbox": [271, 126, 486, 138], "score": 1.0, "content": " grows (Gras [4, Prop. 4.1.ii)]), we conclude that", "type": "text"}], "index": 1}, {"bbox": [126, 138, 486, 150], "spans": [{"bbox": [126, 138, 133, 150], "score": 1.0, "content": "#", "type": "text"}, {"bbox": [133, 139, 196, 150], "score": 0.9, "content": ":M_{i+1}/M_{i}=2", "type": "inline_equation", "height": 11, "width": 63}, {"bbox": [196, 138, 229, 150], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [230, 140, 254, 147], "score": 0.91, "content": "i<n", "type": "inline_equation", "height": 7, "width": 24}, {"bbox": [254, 138, 288, 150], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [288, 142, 314, 147], "score": 0.88, "content": "a=n", "type": "inline_equation", "height": 5, "width": 26}, {"bbox": [315, 138, 337, 150], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [338, 140, 361, 147], "score": 0.91, "content": "b=0", "type": "inline_equation", "height": 7, "width": 23}, {"bbox": [362, 138, 391, 150], "score": 1.0, "content": " when ", "type": "text"}, {"bbox": [392, 140, 416, 149], "score": 0.92, "content": "p=2", "type": "inline_equation", "height": 9, "width": 24}, {"bbox": [416, 138, 486, 150], "score": 1.0, "content": ", Lemma 4 now", "type": "text"}], "index": 2}, {"bbox": [126, 150, 410, 162], "spans": [{"bbox": [126, 150, 181, 162], "score": 1.0, "content": "implies that ", "type": "text"}, {"bbox": [181, 151, 255, 162], "score": 0.94, "content": "\\mathrm{Cl}_{2}(K)\\simeq\\mathbb{Z}/2^{n}\\mathbb{Z}", "type": "inline_equation", "height": 11, "width": 74}, {"bbox": [255, 150, 410, 162], "score": 1.0, "content": ", that is, the 2-class group is cyclic.", "type": "text"}], "index": 3}], "index": 1.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [126, 114, 487, 162]}, {"type": "text", "bbox": [136, 160, 380, 172], "lines": [{"bbox": [137, 162, 381, 173], "spans": [{"bbox": [137, 162, 381, 173], "score": 1.0, "content": "The second result of Gras that we need is [4, Prop. 4.3]", "type": "text"}], "index": 4}], "index": 4, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [137, 162, 381, 173]}, {"type": "text", "bbox": [124, 177, 486, 201], "lines": [{"bbox": [125, 180, 486, 192], "spans": [{"bbox": [125, 180, 240, 192], "score": 1.0, "content": "Lemma 5. Suppose that ", "type": "text"}, {"bbox": [240, 181, 277, 191], "score": 0.92, "content": "M^{\\nu}\\ne1", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [278, 180, 486, 192], "score": 1.0, "content": " but assume the other conditions in Lemma 4.", "type": "text"}], "index": 5}, {"bbox": [127, 192, 199, 204], "spans": [{"bbox": [127, 192, 151, 204], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [152, 194, 176, 203], "score": 0.91, "content": "n\\geq2", "type": "inline_equation", "height": 9, "width": 24}, {"bbox": [177, 192, 199, 204], "score": 1.0, "content": " and", "type": "text"}], "index": 6}], "index": 5.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [125, 180, 486, 204]}, {"type": "interline_equation", "bbox": [191, 205, 420, 251], "lines": [{"bbox": [191, 205, 420, 251], "spans": [{"bbox": [191, 205, 420, 251], "score": 0.92, "content": "M\\simeq\\left\\{\\!\\!\\begin{array}{l l}{(\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n<p;}\\\\ {(\\mathbb{Z}/p\\mathbb{Z})^{p}\\ \\mathrm{or}\\ (\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n=p;}\\\\ {(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}}&{\\mathrm{if}\\ n>p.}\\end{array}\\right.", "type": "interline_equation"}], "index": 7}], "index": 7, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 253, 487, 314], "lines": [{"bbox": [136, 255, 485, 269], "spans": [{"bbox": [136, 255, 148, 269], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [149, 258, 191, 268], "score": 0.92, "content": "\\kappa_{K/k}=1", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [191, 255, 325, 269], "score": 1.0, "content": ", then this lemma shows that ", "type": "text"}, {"bbox": [326, 257, 357, 267], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [358, 255, 468, 269], "score": 1.0, "content": " is either cyclic of order ", "type": "text"}, {"bbox": [468, 258, 485, 266], "score": 0.84, "content": "\\geq4", "type": "inline_equation", "height": 8, "width": 17}], "index": 8}, {"bbox": [125, 268, 485, 280], "spans": [{"bbox": [125, 268, 173, 280], "score": 1.0, "content": "or of type ", "type": "text"}, {"bbox": [173, 269, 196, 280], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [196, 268, 465, 280], "score": 1.0, "content": ". (Notice that the hypothesis of the lemma is satisfied since ", "type": "text"}, {"bbox": [465, 269, 485, 280], "score": 0.93, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}], "index": 9}, {"bbox": [124, 279, 487, 294], "spans": [{"bbox": [124, 279, 282, 294], "score": 1.0, "content": "is ramified implying that the norm", "type": "text"}, {"bbox": [283, 281, 398, 292], "score": 0.9, "content": "N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)", "type": "inline_equation", "height": 11, "width": 115}, {"bbox": [399, 279, 487, 294], "score": 1.0, "content": " is onto; and so the", "type": "text"}], "index": 10}, {"bbox": [125, 292, 486, 304], "spans": [{"bbox": [125, 292, 442, 304], "score": 1.0, "content": "argument above this lemma applies.) It remains to show that the case ", "type": "text"}, {"bbox": [442, 293, 486, 303], "score": 0.92, "content": "\\mathrm{Cl}_{2}(K)\\simeq", "type": "inline_equation", "height": 10, "width": 44}], "index": 11}, {"bbox": [126, 303, 234, 317], "spans": [{"bbox": [126, 305, 150, 316], "score": 0.94, "content": "\\mathbb{Z}/4\\mathbb{Z}", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [150, 303, 234, 317], "score": 1.0, "content": " cannot occur here.", "type": "text"}], "index": 12}], "index": 10, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [124, 255, 487, 317]}, {"type": "text", "bbox": [125, 314, 486, 387], "lines": [{"bbox": [135, 315, 487, 329], "spans": [{"bbox": [135, 315, 221, 329], "score": 1.0, "content": "Now assume that ", "type": "text"}, {"bbox": [221, 317, 330, 327], "score": 0.91, "content": "\\operatorname{Cl}_{2}(K)~=~\\langle{\\cal C}\\rangle~\\simeq~\\mathbb{Z}/4\\mathbb{Z}", "type": "inline_equation", "height": 10, "width": 109}, {"bbox": [330, 315, 363, 329], "score": 1.0, "content": "; since ", "type": "text"}, {"bbox": [363, 317, 383, 327], "score": 0.93, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [384, 315, 487, 329], "score": 1.0, "content": " is ramified, the norm", "type": "text"}], "index": 13}, {"bbox": [126, 327, 487, 342], "spans": [{"bbox": [126, 329, 239, 340], "score": 0.91, "content": "N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)", "type": "inline_equation", "height": 11, "width": 113}, {"bbox": [239, 327, 319, 342], "score": 1.0, "content": " is onto, and using ", "type": "text"}, {"bbox": [319, 330, 359, 340], "score": 0.93, "content": "\\kappa_{K/k}=1", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [360, 327, 441, 342], "score": 1.0, "content": " once more we find ", "type": "text"}, {"bbox": [441, 328, 482, 337], "score": 0.92, "content": "C^{1+\\sigma}=c", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [483, 327, 487, 342], "score": 1.0, "content": ",", "type": "text"}], "index": 14}, {"bbox": [126, 339, 486, 352], "spans": [{"bbox": [126, 339, 154, 352], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 344, 159, 349], "score": 0.88, "content": "c", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [159, 339, 301, 352], "score": 1.0, "content": " is the nontrivial ideal class from ", "type": "text"}, {"bbox": [302, 341, 329, 351], "score": 0.92, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [330, 339, 421, 352], "score": 1.0, "content": ". On the other hand, ", "type": "text"}, {"bbox": [421, 341, 466, 351], "score": 0.93, "content": "c\\in\\mathrm{Cl}_{2}(k)", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [466, 339, 486, 352], "score": 1.0, "content": " still", "type": "text"}], "index": 15}, {"bbox": [124, 350, 487, 365], "spans": [{"bbox": [124, 350, 189, 365], "score": 1.0, "content": "has order 2 in ", "type": "text"}, {"bbox": [190, 353, 221, 363], "score": 0.91, "content": "\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [222, 350, 337, 365], "score": 1.0, "content": ", hence we must also have ", "type": "text"}, {"bbox": [337, 352, 386, 361], "score": 0.93, "content": "C^{2}=C^{1+\\sigma}", "type": "inline_equation", "height": 9, "width": 49}, {"bbox": [387, 350, 487, 365], "score": 1.0, "content": ". But this implies that", "type": "text"}], "index": 16}, {"bbox": [126, 363, 485, 376], "spans": [{"bbox": [126, 366, 160, 373], "score": 0.91, "content": "C^{\\sigma}=C", "type": "inline_equation", "height": 7, "width": 34}, {"bbox": [161, 363, 284, 376], "score": 1.0, "content": ", i.e. that each ideal class in ", "type": "text"}, {"bbox": [284, 366, 293, 373], "score": 0.92, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [294, 363, 485, 376], "score": 1.0, "content": " is ambiguous, contradicting our assumption", "type": "text"}], "index": 17}, {"bbox": [126, 376, 487, 388], "spans": [{"bbox": [126, 376, 148, 388], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 377, 223, 387], "score": 0.94, "content": "\\#\\operatorname{Am}_{2}(K/k)=2", "type": "inline_equation", "height": 10, "width": 75}, {"bbox": [223, 376, 227, 388], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [475, 376, 487, 386], "score": 0.9919366836547852, "content": "\u53e3", "type": "text"}], "index": 18}], "index": 15.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [124, 315, 487, 388]}, {"type": "title", "bbox": [193, 393, 418, 406], "lines": [{"bbox": [193, 396, 418, 407], "spans": [{"bbox": [193, 396, 418, 407], "score": 1.0, "content": "4. Arithmetic of some Dihedral Extensions", "type": "text"}], "index": 19}], "index": 19, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 412, 486, 460], "lines": [{"bbox": [137, 413, 485, 426], "spans": [{"bbox": [137, 413, 428, 426], "score": 1.0, "content": "In this section we study the arithmetic of some dihedral extensions", "type": "text"}, {"bbox": [429, 415, 449, 425], "score": 0.94, "content": "L/\\mathbb{Q}", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [449, 413, 485, 426], "score": 1.0, "content": ", that is,", "type": "text"}], "index": 20}, {"bbox": [125, 425, 487, 439], "spans": [{"bbox": [125, 425, 208, 439], "score": 1.0, "content": "normal extensions ", "type": "text"}, {"bbox": [208, 428, 216, 435], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [216, 425, 231, 439], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [231, 428, 239, 437], "score": 0.91, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [240, 425, 325, 439], "score": 1.0, "content": " with Galois group ", "type": "text"}, {"bbox": [326, 427, 396, 437], "score": 0.93, "content": "\\operatorname{Gal}(L/\\mathbb{Q})\\simeq D_{4}", "type": "inline_equation", "height": 10, "width": 70}, {"bbox": [397, 425, 487, 439], "score": 1.0, "content": ", the dihedral group", "type": "text"}], "index": 21}, {"bbox": [125, 437, 487, 451], "spans": [{"bbox": [125, 437, 208, 451], "score": 1.0, "content": "of order 8. Hence ", "type": "text"}, {"bbox": [209, 440, 222, 448], "score": 0.91, "content": "D_{4}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [222, 437, 320, 451], "score": 1.0, "content": " may be presented as ", "type": "text"}, {"bbox": [320, 438, 455, 450], "score": 0.91, "content": "\\langle\\tau,\\sigma\|\\tau^{2}=\\sigma^{4}=1,\\tau\\sigma\\tau=\\sigma^{-1}\\rangle", "type": "inline_equation", "height": 12, "width": 135}, {"bbox": [456, 437, 487, 451], "score": 1.0, "content": ". Now", "type": "text"}], "index": 22}, {"bbox": [126, 450, 375, 462], "spans": [{"bbox": [126, 450, 375, 462], "score": 1.0, "content": "consider the following diagrams (Galois correspondence):", "type": "text"}], "index": 23}], "index": 21.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [125, 413, 487, 462]}, {"type": "image", "bbox": [131, 468, 478, 554], "blocks": [{"type": "image_body", "bbox": [131, 468, 478, 554], "group_id": 0, "lines": [{"bbox": [131, 468, 478, 554], "spans": [{"bbox": [131, 468, 478, 554], "score": 0.921, "type": "image", "image_path": "110ff188220c8bb5d817468ee0d73bccaf61d8e5373f09de7f29c9cd995490d3.jpg"}]}], "index": 25, "virtual_lines": [{"bbox": [131, 468, 478, 496.6666666666667], "spans": [], "index": 24}, {"bbox": [131, 496.6666666666667, 478, 525.3333333333334], "spans": [], "index": 25}, {"bbox": [131, 525.3333333333334, 478, 554.0], "spans": [], "index": 26}]}], "index": 25, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 556, 487, 605], "lines": [{"bbox": [137, 559, 486, 571], "spans": [{"bbox": [137, 559, 244, 571], "score": 1.0, "content": "In this situation, we let ", "type": "text"}, {"bbox": [245, 560, 337, 570], "score": 0.91, "content": "q_{1}=(E_{L}:E_{1}E_{1}^{\\prime}E_{K})", "type": "inline_equation", "height": 10, "width": 92}, {"bbox": [338, 559, 360, 571], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [360, 560, 452, 571], "score": 0.91, "content": "q_{2}=(E_{L}:E_{2}E_{2}^{\\prime}E_{K})", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [453, 559, 486, 571], "score": 1.0, "content": " denote", "type": "text"}], "index": 27}, {"bbox": [126, 571, 486, 584], "spans": [{"bbox": [126, 571, 316, 584], "score": 1.0, "content": "the unit indices of the bicyclic extensions ", "type": "text"}, {"bbox": [316, 572, 338, 583], "score": 0.93, "content": "L/k_{1}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [338, 571, 362, 584], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [362, 572, 384, 583], "score": 0.93, "content": "L/k_{2}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [384, 571, 420, 584], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [421, 573, 432, 582], "score": 0.91, "content": "E_{i}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [432, 571, 456, 584], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [456, 572, 467, 583], "score": 0.92, "content": "E_{i}^{\\prime}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [468, 571, 486, 584], "score": 1.0, "content": " are", "type": "text"}], "index": 28}, {"bbox": [126, 583, 486, 596], "spans": [{"bbox": [126, 583, 208, 596], "score": 1.0, "content": "the unit groups in ", "type": "text"}, {"bbox": [208, 585, 221, 594], "score": 0.92, "content": "K_{i}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [221, 583, 243, 596], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [243, 584, 255, 595], "score": 0.92, "content": "K_{i}^{\\prime}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [256, 583, 367, 596], "score": 1.0, "content": " respectively. Finally, let ", "type": "text"}, {"bbox": [367, 587, 376, 594], "score": 0.9, "content": "\\kappa_{i}", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [377, 583, 486, 596], "score": 1.0, "content": " denote the kernel of the", "type": "text"}], "index": 29}, {"bbox": [124, 594, 408, 608], "spans": [{"bbox": [124, 594, 230, 608], "score": 1.0, "content": "transfer of ideal classes ", "type": "text"}, {"bbox": [230, 596, 354, 606], "score": 0.91, "content": "j_{k_{i}\\to K_{i}}:\\mathrm{Cl}_{2}(k_{i})\\longrightarrow\\mathrm{Cl}_{2}(K_{i})", "type": "inline_equation", "height": 10, "width": 124}, {"bbox": [354, 594, 372, 608], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [372, 597, 404, 606], "score": 0.92, "content": "i=1,2", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [404, 594, 408, 608], "score": 1.0, "content": ".", "type": "text"}], "index": 30}], "index": 28.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [124, 559, 486, 608]}, {"type": "text", "bbox": [125, 606, 486, 653], "lines": [{"bbox": [137, 605, 487, 619], "spans": [{"bbox": [137, 605, 376, 619], "score": 1.0, "content": "The following remark will be used several times: if ", "type": "text"}, {"bbox": [377, 608, 442, 618], "score": 0.94, "content": "K_{1}\\;=\\;k_{1}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 10, "width": 65}, {"bbox": [442, 605, 487, 619], "score": 1.0, "content": " for some", "type": "text"}], "index": 31}, {"bbox": [126, 618, 485, 632], "spans": [{"bbox": [126, 621, 154, 630], "score": 0.92, "content": "\\alpha\\in k_{1}", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [155, 618, 182, 632], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [182, 619, 236, 631], "score": 0.94, "content": "k_{2}=\\mathbb{Q}({\\sqrt{a}}\\,)", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [236, 618, 270, 632], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [270, 620, 304, 628], "score": 0.93, "content": "a=\\alpha\\alpha^{\\prime}", "type": "inline_equation", "height": 8, "width": 34}, {"bbox": [304, 618, 369, 632], "score": 1.0, "content": " is the norm of ", "type": "text"}, {"bbox": [369, 623, 376, 628], "score": 0.87, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [376, 618, 447, 632], "score": 1.0, "content": ". To see this, let", "type": "text"}, {"bbox": [448, 619, 482, 630], "score": 0.92, "content": "\\gamma=\\sqrt{\\alpha}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [482, 618, 485, 632], "score": 1.0, "content": ";", "type": "text"}], "index": 32}, {"bbox": [124, 630, 487, 644], "spans": [{"bbox": [124, 630, 149, 644], "score": 1.0, "content": "then ", "type": "text"}, {"bbox": [150, 633, 184, 642], "score": 0.93, "content": "\\gamma^{\\tau}\\,=\\,\\gamma", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [184, 630, 216, 644], "score": 1.0, "content": ", since ", "type": "text"}, {"bbox": [217, 633, 252, 642], "score": 0.93, "content": "\\gamma\\ \\in\\ K_{1}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [253, 630, 299, 644], "score": 1.0, "content": ". Clearly ", "type": "text"}, {"bbox": [299, 631, 377, 642], "score": 0.94, "content": "\\gamma^{1+\\sigma}\\,=\\,\\sqrt{a}\\,\\in\\,K", "type": "inline_equation", "height": 11, "width": 78}, {"bbox": [377, 630, 471, 644], "score": 1.0, "content": " and hence fixed by ", "type": "text"}, {"bbox": [471, 631, 482, 640], "score": 0.9, "content": "\\sigma^{2}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [482, 630, 487, 644], "score": 1.0, "content": ".", "type": "text"}], "index": 33}, {"bbox": [125, 642, 185, 655], "spans": [{"bbox": [125, 642, 185, 655], "score": 1.0, "content": "Furthermore,", "type": "text"}], "index": 34}], "index": 32.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [124, 605, 487, 655]}, {"type": "interline_equation", "bbox": [142, 658, 466, 672], "lines": [{"bbox": [142, 658, 466, 672], "spans": [{"bbox": [142, 658, 466, 672], "score": 0.88, "content": "(\\gamma^{1+\\sigma})^{\\sigma\\tau}=\\gamma^{\\sigma\\tau+\\sigma^{2}\\tau}=\\gamma^{\\tau\\sigma^{3}+\\tau\\sigma^{2}}=(\\gamma^{\\tau})^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{(1+\\sigma)\\sigma^{2}}=\\gamma^{1+\\sigma},", "type": "interline_equation"}], "index": 35}], "index": 35, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 674, 486, 700], "lines": [{"bbox": [125, 676, 486, 689], "spans": [{"bbox": [125, 677, 191, 689], "score": 1.0, "content": "implying that ", "type": "text"}, {"bbox": [192, 678, 233, 689], "score": 0.93, "content": "{\\sqrt{a}}\\ \\in\\ k_{2}", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [233, 677, 334, 689], "score": 1.0, "content": ". Finally notice that ", "type": "text"}, {"bbox": [335, 678, 374, 689], "score": 0.94, "content": "{\\sqrt{a}}\\ \\not\\in\\ \\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [375, 677, 453, 689], "score": 1.0, "content": ", since otherwise", "type": "text"}, {"bbox": [453, 676, 486, 688], "score": 0.88, "content": "\\sqrt{\\alpha^{\\prime}}\\,=", "type": "inline_equation", "height": 12, "width": 33}], "index": 36}, {"bbox": [126, 689, 422, 702], "spans": [{"bbox": [126, 690, 185, 701], "score": 0.94, "content": "{\\sqrt{a}}/{\\sqrt{\\alpha}}\\in K_{1}", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [185, 689, 250, 702], "score": 1.0, "content": " implying that ", "type": "text"}, {"bbox": [250, 690, 276, 701], "score": 0.94, "content": "K_{1}/\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [277, 689, 422, 702], "score": 1.0, "content": " is normal, which is not the case.", "type": "text"}], "index": 37}], 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Since the orders of decrease towards as grows (Gras [4, Prop. 4.1.ii)]), we conclude that # for all . Since and when , Lemma 4 now implies that , that is, the 2-class group is cyclic. The second result of Gras that we need is [4, Prop. 4.3] Lemma 5. Suppose that but assume the other conditions in Lemma 4. Then and $$ M\simeq\left\{\!\!\begin{array}{l l}{(\mathbb{Z}/p^{2}\mathbb{Z})\times(\mathbb{Z}/p\mathbb{Z})^{n-2}}&{\mathrm{if}\ n<p;}\\ {(\mathbb{Z}/p\mathbb{Z})^{p}\ \mathrm{or}\ (\mathbb{Z}/p^{2}\mathbb{Z})\times(\mathbb{Z}/p\mathbb{Z})^{n-2}}&{\mathrm{if}\ n=p;}\\ {(\mathbb{Z}/p^{a+1}\mathbb{Z})^{b}\times(\mathbb{Z}/p^{a}\mathbb{Z})^{p-1-b}}&{\mathrm{if}\ n>p.}\end{array}\right. $$ If , then this lemma shows that is either cyclic of order or of type . (Notice that the hypothesis of the lemma is satisfied since is ramified implying that the norm is onto; and so the argument above this lemma applies.) It remains to show that the case cannot occur here. Now assume that ; since is ramified, the norm is onto, and using once more we find , where is the nontrivial ideal class from . On the other hand, still has order 2 in , hence we must also have . But this implies that , i.e. that each ideal class in is ambiguous, contradicting our assumption that . 口 # 4. Arithmetic of some Dihedral Extensions In this section we study the arithmetic of some dihedral extensions , that is, normal extensions of with Galois group , the dihedral group of order 8. Hence may be presented as . Now consider the following diagrams (Galois correspondence): In this situation, we let and denote the unit indices of the bicyclic extensions and , where and are the unit groups in and respectively. Finally, let denote the kernel of the transfer of ideal classes for . The following remark will be used several times: if for some , then , where is the norm of . To see this, let ; then , since . Clearly and hence fixed by . Furthermore, $$ (\gamma^{1+\sigma})^{\sigma\tau}=\gamma^{\sigma\tau+\sigma^{2}\tau}=\gamma^{\tau\sigma^{3}+\tau\sigma^{2}}=(\gamma^{\tau})^{\sigma^{3}+\sigma^{2}}=\gamma^{\sigma^{3}+\sigma^{2}}=\gamma^{(1+\sigma)\sigma^{2}}=\gamma^{1+\sigma}, $$ implying that . Finally notice that , since otherwise implying that is normal, which is not the case.	<div class="pdf-page"> <p>in our case, hence has order 2. Since the orders of decrease towards as grows (Gras [4, Prop. 4.1.ii)]), we conclude that # for all . Since and when , Lemma 4 now implies that , that is, the 2-class group is cyclic.</p> <p>The second result of Gras that we need is [4, Prop. 4.3]</p> <p>Lemma 5. Suppose that but assume the other conditions in Lemma 4. Then and</p> <p>If , then this lemma shows that is either cyclic of order or of type . (Notice that the hypothesis of the lemma is satisfied since is ramified implying that the norm is onto; and so the argument above this lemma applies.) It remains to show that the case cannot occur here.</p> <p>Now assume that ; since is ramified, the norm is onto, and using once more we find , where is the nontrivial ideal class from . On the other hand, still has order 2 in , hence we must also have . But this implies that , i.e. that each ideal class in is ambiguous, contradicting our assumption that . 口</p> <h1>4. Arithmetic of some Dihedral Extensions</h1> <p>In this section we study the arithmetic of some dihedral extensions , that is, normal extensions of with Galois group , the dihedral group of order 8. Hence may be presented as . Now consider the following diagrams (Galois correspondence):</p> <p>In this situation, we let and denote the unit indices of the bicyclic extensions and , where and are the unit groups in and respectively. Finally, let denote the kernel of the transfer of ideal classes for .</p> <p>The following remark will be used several times: if for some , then , where is the norm of . To see this, let ; then , since . Clearly and hence fixed by . Furthermore,</p> <p>implying that . Finally notice that , since otherwise implying that is normal, which is not the case.</p> </div>	<div class="pdf-page"> <div class="pdf-discarded" data-x="396" data-y="116" data-width="229" data-height="13" style="opacity: 0.5;">IMAGINARY QUADRATIC FIELDS</div> <div class="pdf-discarded" data-x="801" data-y="116" data-width="12" data-height="12" style="opacity: 0.5;">7</div> <p class="pdf-text" data-x="209" data-y="143" data-width="605" data-height="63">in our case, hence has order 2. Since the orders of decrease towards as grows (Gras [4, Prop. 4.1.ii)]), we conclude that # for all . Since and when , Lemma 4 now implies that , that is, the 2-class group is cyclic.</p> <p class="pdf-text" data-x="227" data-y="206" data-width="408" data-height="16">The second result of Gras that we need is [4, Prop. 4.3]</p> <p class="pdf-text" data-x="207" data-y="228" data-width="606" data-height="31">Lemma 5. Suppose that but assume the other conditions in Lemma 4. Then and</p> <p class="pdf-text" data-x="207" data-y="327" data-width="607" data-height="78">If , then this lemma shows that is either cyclic of order or of type . (Notice that the hypothesis of the lemma is satisfied since is ramified implying that the norm is onto; and so the argument above this lemma applies.) It remains to show that the case cannot occur here.</p> <p class="pdf-text" data-x="209" data-y="405" data-width="604" data-height="95">Now assume that ; since is ramified, the norm is onto, and using once more we find , where is the nontrivial ideal class from . On the other hand, still has order 2 in , hence we must also have . But this implies that , i.e. that each ideal class in is ambiguous, contradicting our assumption that . 口</p> <h1 class="pdf-title" data-x="322" data-y="508" data-width="377" data-height="16">4. Arithmetic of some Dihedral Extensions</h1> <p class="pdf-text" data-x="207" data-y="532" data-width="606" data-height="62">In this section we study the arithmetic of some dihedral extensions , that is, normal extensions of with Galois group , the dihedral group of order 8. Hence may be presented as . Now consider the following diagrams (Galois correspondence):</p> <p class="pdf-text" data-x="209" data-y="718" data-width="605" data-height="64">In this situation, we let and denote the unit indices of the bicyclic extensions and , where and are the unit groups in and respectively. Finally, let denote the kernel of the transfer of ideal classes for .</p> <p class="pdf-text" data-x="209" data-y="783" data-width="604" data-height="61">The following remark will be used several times: if for some , then , where is the norm of . To see this, let ; then , since . Clearly and hence fixed by . Furthermore,</p> <p class="pdf-text" data-x="207" data-y="871" data-width="606" data-height="34">implying that . Finally notice that , since otherwise implying that is normal, which is not the case.</p> </div>	{ "type": [ "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "inline_equation", "inline_equation", "interline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "inline_equation", "inline_equation", "inline_equation", "text", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "interline_equation", "inline_equation", "inline_equation" ], "coordinates": [ [ 210, 147, 814, 162 ], [ 210, 162, 813, 178 ], [ 210, 178, 813, 193 ], [ 210, 193, 686, 209 ], [ 229, 209, 637, 223 ], [ 209, 232, 813, 248 ], [ 212, 248, 332, 263 ], [ 319, 265, 702, 324 ], [ 227, 329, 811, 347 ], [ 209, 346, 811, 362 ], [ 207, 360, 814, 380 ], [ 209, 377, 813, 393 ], [ 210, 391, 391, 409 ], [ 225, 407, 814, 425 ], [ 210, 422, 814, 442 ], [ 210, 438, 813, 455 ], [ 207, 452, 814, 471 ], [ 210, 469, 811, 486 ], [ 210, 486, 814, 501 ], [ 322, 512, 699, 526 ], [ 229, 533, 811, 550 ], [ 209, 549, 814, 567 ], [ 209, 565, 814, 583 ], [ 210, 581, 627, 597 ], [ 219, 605, 799, 716 ], [ 229, 722, 813, 738 ], [ 210, 738, 813, 755 ], [ 210, 753, 813, 770 ], [ 207, 768, 682, 786 ], [ 229, 782, 814, 800 ], [ 210, 799, 811, 817 ], [ 207, 814, 814, 832 ], [ 209, 830, 309, 846 ], [ 237, 850, 779, 868 ], [ 209, 874, 813, 890 ], [ 210, 890, 706, 907 ] ], "content": [ "M_{1}\\;=\\;\\mathrm{Am}_{2}(K/k) in our case, hence M_{1}/M_{0} has order 2. Since the orders of", "M_{i+1}/M_{i} decrease towards ^{1} as i grows (Gras [4, Prop. 4.1.ii)]), we conclude that", "# :M_{i+1}/M_{i}=2 for all i<n . Since a=n and b=0 when p=2 , Lemma 4 now", "implies that \\mathrm{Cl}_{2}(K)\\simeq\\mathbb{Z}/2^{n}\\mathbb{Z} , that is, the 2-class group is cyclic.", "The second result of Gras that we need is [4, Prop. 4.3]", "Lemma 5. Suppose that M^{\\nu}\\ne1 but assume the other conditions in Lemma 4.", "Then n\\geq2 and", "M\\simeq\\left\\{\\!\\!\\begin{array}{l l}{(\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n<p;}\\\\ {(\\mathbb{Z}/p\\mathbb{Z})^{p}\\ \\mathrm{or}\\ (\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n=p;}\\\\ {(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}}&{\\mathrm{if}\\ n>p.}\\end{array}\\right.", "If \\kappa_{K/k}=1 , then this lemma shows that \\mathrm{Cl}_{2}(K) is either cyclic of order \\geq4", "or of type (2,2) . (Notice that the hypothesis of the lemma is satisfied since K/k", "is ramified implying that the norm N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k) is onto; and so the", "argument above this lemma applies.) It remains to show that the case \\mathrm{Cl}_{2}(K)\\simeq", "\\mathbb{Z}/4\\mathbb{Z} cannot occur here.", "Now assume that \\operatorname{Cl}_{2}(K)~=~\\langle{\\cal C}\\rangle~\\simeq~\\mathbb{Z}/4\\mathbb{Z} ; since K/k is ramified, the norm", "N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k) is onto, and using \\kappa_{K/k}=1 once more we find C^{1+\\sigma}=c ,", "where c is the nontrivial ideal class from \\mathrm{Cl_{2}}(k) . On the other hand, c\\in\\mathrm{Cl}_{2}(k) still", "has order 2 in \\mathrm{Cl}_{2}(K) , hence we must also have C^{2}=C^{1+\\sigma} . But this implies that", "C^{\\sigma}=C , i.e. that each ideal class in K is ambiguous, contradicting our assumption", "that \\#\\operatorname{Am}_{2}(K/k)=2 . 口", "4. Arithmetic of some Dihedral Extensions", "In this section we study the arithmetic of some dihedral extensions L/\\mathbb{Q} , that is,", "normal extensions L of \\mathbb{Q} with Galois group \\operatorname{Gal}(L/\\mathbb{Q})\\simeq D_{4} , the dihedral group", "of order 8. Hence D_{4} may be presented as \\langle\\tau,\\sigma\|\\tau^{2}=\\sigma^{4}=1,\\tau\\sigma\\tau=\\sigma^{-1}\\rangle . Now", "consider the following diagrams (Galois correspondence):", "", "In this situation, we let q_{1}=(E_{L}:E_{1}E_{1}^{\\prime}E_{K}) and q_{2}=(E_{L}:E_{2}E_{2}^{\\prime}E_{K}) denote", "the unit indices of the bicyclic extensions L/k_{1} and L/k_{2} , where E_{i} and E_{i}^{\\prime} are", "the unit groups in K_{i} and K_{i}^{\\prime} respectively. Finally, let \\kappa_{i} denote the kernel of the", "transfer of ideal classes j_{k_{i}\\to K_{i}}:\\mathrm{Cl}_{2}(k_{i})\\longrightarrow\\mathrm{Cl}_{2}(K_{i}) for i=1,2 .", "The following remark will be used several times: if K_{1}\\;=\\;k_{1}(\\sqrt{\\alpha}\\,) for some", "\\alpha\\in k_{1} , then k_{2}=\\mathbb{Q}({\\sqrt{a}}\\,) , where a=\\alpha\\alpha^{\\prime} is the norm of \\alpha . To see this, let \\gamma=\\sqrt{\\alpha} ;", "then \\gamma^{\\tau}\\,=\\,\\gamma , since \\gamma\\ \\in\\ K_{1} . Clearly \\gamma^{1+\\sigma}\\,=\\,\\sqrt{a}\\,\\in\\,K and hence fixed by \\sigma^{2} .", "Furthermore,", "(\\gamma^{1+\\sigma})^{\\sigma\\tau}=\\gamma^{\\sigma\\tau+\\sigma^{2}\\tau}=\\gamma^{\\tau\\sigma^{3}+\\tau\\sigma^{2}}=(\\gamma^{\\tau})^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{(1+\\sigma)\\sigma^{2}}=\\gamma^{1+\\sigma},", "implying that {\\sqrt{a}}\\ \\in\\ k_{2} . Finally notice that {\\sqrt{a}}\\ \\not\\in\\ \\mathbb{Q} , since otherwise \\sqrt{\\alpha^{\\prime}}\\,=", "{\\sqrt{a}}/{\\sqrt{\\alpha}}\\in K_{1} implying that K_{1}/\\mathbb{Q} is normal, which is not the case." ], "index": [ 0, 1, 2, 3, 4, 9, 10, 16, 24, 25, 26, 27, 28, 37, 38, 39, 40, 41, 42, 56, 76, 77, 78, 79, 100, 125, 126, 127, 128, 154, 155, 156, 157, 187, 221, 222 ] }	{ "coordinates": [ [ 219, 605, 799, 716 ], [ 219, 605, 799, 716 ] ], "index": [ 0, 1 ], "caption": [ "", "" ], "caption_coordinates": [ [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ] ] }	{ "type": [ "interline_equation", "interline_equation", "interline_equation", "interline_equation" ], "coordinates": [ [ 319, 265, 702, 324 ], [ 237, 850, 779, 868 ], [ 319, 265, 702, 324 ], [ 237, 850, 779, 868 ] ], "content": [ "", "", "M\\simeq\\left\\{\\!\\!\\begin{array}{l l}{(\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n<p;}\\\\ {(\\mathbb{Z}/p\\mathbb{Z})^{p}\\ \\mathrm{or}\\ (\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n=p;}\\\\ {(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}}&{\\mathrm{if}\\ n>p.}\\end{array}\\right.", "(\\gamma^{1+\\sigma})^{\\sigma\\tau}=\\gamma^{\\sigma\\tau+\\sigma^{2}\\tau}=\\gamma^{\\tau\\sigma^{3}+\\tau\\sigma^{2}}=(\\gamma^{\\tau})^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{(1+\\sigma)\\sigma^{2}}=\\gamma^{1+\\sigma}," ], "index": [ 0, 1, 2, 3 ] }	{ "type": [], "coordinates": [], "content": [], "index": [] }	{"document_info": {"total_pages": 14, "parse_type": "txt", "version": "1.0.1", "total_content_list_items": 155}, "content_statistics": {"total_blocks": 182, "total_lines": 526, "total_images": 4, "total_equations": 28, "total_tables": 11, "total_characters": 29658, "total_words": 5331, "block_type_distribution": {"title": 7, "text": 126, "interline_equation": 14, "discarded": 24, "table_caption": 3, "table_body": 4, "image_body": 2, "list": 2}}, "model_statistics": {"total_layout_detections": 2826, "avg_confidence": 0.9470955414012739, "min_confidence": 0.26, "max_confidence": 1.0}, "page_statistics": [{"page_number": 0, "page_size": [612.0, 792.0], "blocks": 11, "lines": 38, "images": 0, "equations": 2, "tables": 0, "characters": 2362, "words": 398, "discarded_blocks": 1}, {"page_number": 1, "page_size": [612.0, 792.0], "blocks": 10, "lines": 30, "images": 0, "equations": 0, "tables": 3, "characters": 1688, "words": 305, "discarded_blocks": 1}, {"page_number": 2, "page_size": [612.0, 792.0], "blocks": 15, "lines": 44, "images": 0, "equations": 4, "tables": 0, "characters": 2417, "words": 416, "discarded_blocks": 2}, {"page_number": 3, "page_size": [612.0, 792.0], "blocks": 7, "lines": 27, "images": 0, "equations": 0, "tables": 3, "characters": 1591, "words": 291, "discarded_blocks": 1}, {"page_number": 4, "page_size": [612.0, 792.0], "blocks": 23, "lines": 38, "images": 0, "equations": 6, "tables": 0, "characters": 1604, "words": 303, "discarded_blocks": 4}, {"page_number": 5, "page_size": [612.0, 792.0], "blocks": 17, "lines": 40, "images": 0, "equations": 6, "tables": 0, "characters": 2131, "words": 393, "discarded_blocks": 1}, {"page_number": 6, "page_size": [612.0, 792.0], "blocks": 15, "lines": 36, "images": 2, "equations": 4, "tables": 0, "characters": 1791, "words": 324, "discarded_blocks": 2}, {"page_number": 7, "page_size": [612.0, 792.0], "blocks": 12, "lines": 49, "images": 0, "equations": 0, "tables": 0, "characters": 3049, "words": 571, "discarded_blocks": 1}, {"page_number": 8, "page_size": [612.0, 792.0], "blocks": 11, "lines": 42, "images": 0, "equations": 0, "tables": 0, "characters": 2636, "words": 479, "discarded_blocks": 2}, {"page_number": 9, "page_size": [612.0, 792.0], "blocks": 14, "lines": 35, "images": 2, "equations": 0, "tables": 0, "characters": 2036, "words": 375, "discarded_blocks": 1}, {"page_number": 10, "page_size": [612.0, 792.0], "blocks": 17, "lines": 35, "images": 0, "equations": 6, "tables": 0, "characters": 1789, "words": 327, "discarded_blocks": 2}, {"page_number": 11, "page_size": [612.0, 792.0], "blocks": 14, "lines": 40, "images": 0, "equations": 0, "tables": 2, "characters": 2094, "words": 399, "discarded_blocks": 2}, {"page_number": 12, "page_size": [612.0, 792.0], "blocks": 10, "lines": 44, "images": 0, "equations": 0, "tables": 3, "characters": 2572, "words": 487, "discarded_blocks": 2}, {"page_number": 13, "page_size": [612.0, 792.0], "blocks": 6, "lines": 28, "images": 0, "equations": 0, "tables": 0, "characters": 1898, "words": 263, "discarded_blocks": 2}]}
0003244v1	7	[ 612, 792 ]	{ "type": [ "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "discarded" ], "coordinates": [ [ 209, 143, 813, 175 ], [ 209, 183, 813, 231 ], [ 209, 239, 813, 347 ], [ 209, 349, 813, 395 ], [ 209, 395, 813, 504 ], [ 209, 504, 813, 581 ], [ 209, 583, 813, 612 ], [ 209, 614, 813, 674 ], [ 209, 685, 813, 733 ], [ 209, 740, 813, 817 ], [ 207, 818, 813, 905 ], [ 209, 117, 219, 128 ] ], "content": [ "Recall that a quadratic extension is called essentially ramified if is not an ideal square. This definition is independent of the choice of .", "Proposition 6. Let be a non-CM totally complex dihedral extension not con- taining , and assume that and are essentially ramified. If the fundamental unit of the real quadratic subfield of has norm , then .", "Proof. Notice first that cannot be real (in fact, is not totally real by assumption, and since is a cyclic quartic extension, no infinite prime can ramify in ); thus exactly one of , is real, and the other is complex. Multiplying the class number formulas, Proposition 3, for and (note that since both and are essentially ramified) we find that is a square. If we can prove that , then is a square between 2 and 8, which implies that we must have and as claimed.", "We start by remarking that if becomes a square in , where is a root of unity in , then so does one of . This follows from the fact that the only non-trivial roots of unity that can be in are the sixth roots of unity , and here .", "Now we prove that under the assumptions we made; the claim will then follow by symmetry. Assume first that is real and let be the fundamental unit of . We claim that . Suppose otherwise; then is one of , or . If , then and . (Here and below .) This however cannot occur since by assumption implying that , a contradiction. Similarly, if , then again .", "Thus , and for some unit . Suppose that for some unit . Then , contradicting our assumption that is essentially ramified. The same argument shows that , hence either and or for some unit and . Here is a root of unity generating the torsion subgroup of .", "Next consider the case where is complex, and let denote the fundamental unit of . Then stays fundamental in by the argument above.", "Let be a fundamental unit in . If became a square in , then clearly could not be essentially ramified. Thus if we have , then is a square in . Applying to this relation we find that is a square in , contradicting the assumption that does not contain . 口", "Proposition 7. Suppose that . Then is essentially ramified if and only if ; if is not essentially ramified, then , where and .", "Proof. First notice that if is not essentially ramified, then : in fact, in this case we have , and if we had , then would have to be principal, say . This implies that for some unit , which in view of implies that must be a square. But then would be a square, and this is impossible.", "Conversely, suppose . Let be a nonprincipal ideal in of absolute norm , and assume that in . Then for some unit , and similarly , where is a unit in . But then in , where means equal up to a square in . Thus is a square in , so our assumption that implies that must be a square in . The same argument show that is a square in , hence we find . Thus is fixed by and so . This gives , hence is not essentially ramified, and moreover, . 冏口", "8" ], "index": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ] }	[{"type": "text", "text": "Recall that a quadratic extension $K\\,=\\,k({\\sqrt{\\alpha}}\\,)$ is called essentially ramified if $\\alpha{\\mathcal{O}}_{k}$ is not an ideal square. This definition is independent of the choice of $\\alpha$ . ", "page_idx": 7}, {"type": "text", "text": "Proposition 6. Let $L/\\mathbb{Q}$ be a non-CM totally complex dihedral extension not containing $\\sqrt{-1}$ , and assume that $L/K_{1}$ and $L/K_{2}$ are essentially ramified. If the fundamental unit of the real quadratic subfield of $K$ has norm $^{-1}$ , then $q_{1}q_{2}=2$ . ", "page_idx": 7}, {"type": "text", "text": "Proof. Notice first that $k$ cannot be real (in fact, $K$ is not totally real by assumption, and since $L/k$ is a cyclic quartic extension, no infinite prime can ramify in $K/k$ ); thus exactly one of $k_{1}$ , $k_{2}$ is real, and the other is complex. Multiplying the class number formulas, Proposition 3, for $L/k_{1}$ and $L/k_{2}$ (note that $\\upsilon=0$ since both $L/K_{1}$ and $L/K_{2}$ are essentially ramified) we find that $2q_{1}q_{2}$ is a square. If we can prove that $q_{1},q_{2}\\leq2$ , then $2q_{1}q_{2}$ is a square between 2 and 8, which implies that we must have $2q_{1}q_{2}=4$ and $q_{1}q_{2}=2$ as claimed. ", "page_idx": 7}, {"type": "text", "text": "We start by remarking that if $\\zeta\\eta$ becomes a square in $L$ , where $\\zeta$ is a root of unity in $L$ , then so does one of $\\pm\\eta$ . This follows from the fact that the only non-trivial roots of unity that can be in $L$ are the sixth roots of unity $\\langle\\zeta_{6}\\rangle$ , and here $\\zeta_{6}=-\\zeta_{3}^{2}$ . ", "page_idx": 7}, {"type": "text", "text": "Now we prove that $q_{1}\\leq2$ under the assumptions we made; the claim $q_{2}\\leq2$ will then follow by symmetry. Assume first that $k_{1}$ is real and let $\\varepsilon$ be the fundamental unit of $k_{1}$ . We claim that $\\sqrt{\\pm\\varepsilon}\\notin L$ . Suppose otherwise; then $k_{1}(\\sqrt{\\pm\\varepsilon})$ is one of $K_{1}$ , $K_{1}^{\\prime}$ or $K$ . If $k_{1}(\\sqrt{\\pm\\varepsilon}\\,)=K_{1}$ , then $K_{1}^{\\prime}=k_{1}(\\sqrt{\\pm\\varepsilon^{\\prime}}\\,)$ and $K=k_{1}\\big(\\sqrt{\\varepsilon\\varepsilon^{\\prime}}\\big)$ . (Here and below $x^{\\prime}\\,=\\,x^{\\sigma}$ .) This however cannot occur since by assumption $\\varepsilon\\varepsilon^{\\prime}\\,=\\,-1$ implying that $\\sqrt{-1}\\in L$ , a contradiction. Similarly, if $k_{1}({\\sqrt{\\pm\\varepsilon}}\\,)\\,=\\,K$ , then again $\\sqrt{-1}\\in L$ . ", "page_idx": 7}, {"type": "text", "text": "Thus $\\sqrt{\\pm\\varepsilon}~\\notin~L$ , and $E_{1}~=~\\langle-1,\\varepsilon,\\eta\\rangle$ for some unit $\\eta\\ \\in\\ E_{1}$ . Suppose that $\\sqrt{u\\eta}\\in L$ for some unit $u\\in k_{1}$ . Then $L=K_{1}(\\sqrt{u\\eta}\\,)$ , contradicting our assumption that $L/K_{1}$ is essentially ramified. The same argument shows that $\\sqrt{u\\eta^{\\prime}}\\notin L$ , hence either $E_{L}\\,=\\,\\langle\\zeta,\\varepsilon,\\eta,\\eta^{\\prime}\\rangle$ and $q_{1}=1$ or $E_{L}=\\langle\\zeta,\\varepsilon,\\eta,\\sqrt{u\\eta\\eta^{\\prime}}\\rangle$ for some unit $u\\in k_{1}$ and $q_{1}=2$ . Here $\\zeta$ is a root of unity generating the torsion subgroup $W_{L}$ of $E_{L}$ . ", "page_idx": 7}, {"type": "text", "text": "Next consider the case where $k_{1}$ is complex, and let $\\varepsilon$ denote the fundamental unit of $k_{2}$ . Then $\\pm\\varepsilon$ stays fundamental in $L$ by the argument above. ", "page_idx": 7}, {"type": "text", "text": "Let $\\eta$ be a fundamental unit in $K_{1}$ . If $\\pm\\eta$ became a square in $L$ , then clearly $L/K_{1}$ could not be essentially ramified. Thus if we have $q_{1}\\geq4$ , then $\\pm\\varepsilon\\eta=\\alpha^{2}$ is a square in $L$ . Applying $\\tau$ to this relation we find that $-1=\\varepsilon\\varepsilon^{\\prime}$ is a square in $L$ , contradicting the assumption that $L$ does not contain $\\sqrt{-1}$ . \u53e3 ", "page_idx": 7}, {"type": "text", "text": "Proposition 7. Suppose that $q_{2}=1$ . Then $K_{2}/k_{2}$ is essentially ramified if and only if $\\kappa_{2}=1$ ; if $K_{2}/k_{2}$ is not essentially ramified, then $\\kappa_{2}=\\langle[6]\\rangle$ , where $K_{2}=$ $k_{2}(\\sqrt{\\beta}\\,)$ and $(\\beta)=\\mathfrak{b}^{2}$ . ", "page_idx": 7}, {"type": "text", "text": "Proof. First notice that if $K_{2}/k_{2}$ is not essentially ramified, then $\\kappa_{2}\\neq1$ : in fact, in this case we have $(\\beta)\\;=\\;6^{2}$ , and if we had $\\,\\kappa_{2}\\,=\\,1$ , then $\\mathfrak{b}$ would have to be principal, say ${\\mathfrak{b}}=(\\gamma)$ . This implies that $\\beta\\,=\\,\\varepsilon\\gamma^{2}$ for some unit $\\varepsilon\\in k_{2}$ , which in view of $q_{2}=1$ implies that $\\varepsilon$ must be a square. But then $\\beta$ would be a square, and this is impossible. ", "page_idx": 7}, {"type": "text", "text": "Conversely, suppose $\\kappa_{2}\\neq1$ . Let $\\mathfrak{a}$ be a nonprincipal ideal in $k_{2}$ of absolute norm $a$ , and assume that ${\\mathfrak{a}}=(\\alpha)$ in $K_{2}$ . Then $\\alpha^{1-\\sigma^{2}}=\\eta$ for some unit $\\eta\\in E_{2}$ , and similarly $\\alpha^{\\sigma-\\sigma^{3}}\\,=\\,\\eta^{\\prime}$ , where $\\eta^{\\prime}$ is a unit in $E_{2}^{\\prime}$ . But then $\\eta\\eta^{\\prime}\\,=\\,\\alpha^{1+\\sigma-\\sigma^{2}-\\sigma^{3}}\\,\\stackrel{\\cdot2}{=}$ $N_{L/k}\\alpha\\,=\\,\\pm N_{L/k}\\mathfrak{a}\\,=\\,\\pm a^{2}\\,\\stackrel{?}{=}\\,\\pm1$ in $L^{\\times}$ , where $\\underline{{\\underline{{2}}}}$ means equal up to a square in $L^{\\times}$ . Thus $\\pm\\eta\\eta^{\\prime}$ is a square in $L$ , so our assumption that $q_{2}=1$ implies that $\\pm\\eta\\eta^{\\prime}$ must be a square in $k_{2}$ . The same argument show that $\\pm\\eta/\\eta^{\\prime}$ is a square in $k_{2}$ , hence we find $\\eta\\in k_{2}$ . Thus $\\alpha^{1-\\sigma^{2}}$ is fixed by $\\sigma^{2}$ and so $\\beta:=\\alpha^{2}\\in k_{2}$ . This gives $K_{2}=k_{2}(\\sqrt{\\beta}\\,)$ , hence $K_{2}/k_{2}$ is not essentially ramified, and moreover, $a\\sim{\\mathfrak{b}}$ . \u518f\u53e3 ", "page_idx": 7}]	{"preproc_blocks": [{"type": "text", "bbox": [125, 111, 486, 136], "lines": [{"bbox": [137, 114, 486, 126], "spans": [{"bbox": [137, 114, 291, 126], "score": 1.0, "content": "Recall that a quadratic extension ", "type": "text"}, {"bbox": [291, 115, 346, 126], "score": 0.95, "content": "K\\,=\\,k({\\sqrt{\\alpha}}\\,)", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [347, 114, 486, 126], "score": 1.0, "content": " is called essentially ramified if", "type": "text"}], "index": 0}, {"bbox": [126, 126, 462, 138], "spans": [{"bbox": [126, 128, 145, 137], "score": 0.92, "content": "\\alpha{\\mathcal{O}}_{k}", "type": "inline_equation", "height": 9, "width": 19}, {"bbox": [145, 126, 452, 138], "score": 1.0, "content": " is not an ideal square. This definition is independent of the choice of ", "type": "text"}, {"bbox": [452, 131, 459, 135], "score": 0.88, "content": "\\alpha", "type": "inline_equation", "height": 4, "width": 7}, {"bbox": [459, 126, 462, 138], "score": 1.0, "content": ".", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [125, 142, 486, 179], "lines": [{"bbox": [126, 145, 487, 158], "spans": [{"bbox": [126, 145, 218, 158], "score": 1.0, "content": "Proposition 6. Let ", "type": "text"}, {"bbox": [218, 146, 239, 156], "score": 0.92, "content": "L/\\mathbb{Q}", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [239, 145, 487, 158], "score": 1.0, "content": " be a non-CM totally complex dihedral extension not con-", "type": "text"}], "index": 2}, {"bbox": [126, 157, 487, 170], "spans": [{"bbox": [126, 157, 160, 170], "score": 1.0, "content": "taining ", "type": "text"}, {"bbox": [161, 157, 182, 168], "score": 0.91, "content": "\\sqrt{-1}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [183, 157, 267, 170], "score": 1.0, "content": ", and assume that ", "type": "text"}, {"bbox": [268, 158, 293, 169], "score": 0.93, "content": "L/K_{1}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [293, 157, 317, 170], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [317, 158, 342, 169], "score": 0.93, "content": "L/K_{2}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [343, 157, 487, 170], "score": 1.0, "content": " are essentially ramified. If the", "type": "text"}], "index": 3}, {"bbox": [126, 169, 481, 182], "spans": [{"bbox": [126, 169, 341, 182], "score": 1.0, "content": "fundamental unit of the real quadratic subfield of ", "type": "text"}, {"bbox": [342, 171, 351, 178], "score": 0.87, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [352, 169, 398, 182], "score": 1.0, "content": " has norm ", "type": "text"}, {"bbox": [399, 171, 412, 179], "score": 0.85, "content": "^{-1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [412, 169, 440, 182], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [440, 171, 477, 180], "score": 0.91, "content": "q_{1}q_{2}=2", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [477, 169, 481, 182], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 3}, {"type": "text", "bbox": [125, 185, 486, 269], "lines": [{"bbox": [127, 188, 485, 200], "spans": [{"bbox": [127, 188, 227, 200], "score": 1.0, "content": "Proof. Notice first that ", "type": "text"}, {"bbox": [227, 189, 233, 197], "score": 0.9, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [233, 188, 333, 200], "score": 1.0, "content": " cannot be real (in fact,", "type": "text"}, {"bbox": [334, 190, 343, 197], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [343, 188, 485, 200], "score": 1.0, "content": " is not totally real by assumption,", "type": "text"}], "index": 5}, {"bbox": [126, 199, 486, 212], "spans": [{"bbox": [126, 199, 170, 212], "score": 1.0, "content": "and since ", "type": "text"}, {"bbox": [170, 201, 188, 211], "score": 0.93, "content": "L/k", "type": "inline_equation", "height": 10, "width": 18}, {"bbox": [188, 199, 459, 212], "score": 1.0, "content": " is a cyclic quartic extension, no infinite prime can ramify in ", "type": "text"}, {"bbox": [459, 201, 478, 211], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [479, 199, 486, 212], "score": 1.0, "content": ");", "type": "text"}], "index": 6}, {"bbox": [126, 212, 486, 224], "spans": [{"bbox": [126, 212, 213, 224], "score": 1.0, "content": "thus exactly one of ", "type": "text"}, {"bbox": [213, 213, 223, 222], "score": 0.9, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [223, 212, 228, 224], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [229, 213, 239, 222], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [239, 212, 486, 224], "score": 1.0, "content": " is real, and the other is complex. Multiplying the class", "type": "text"}], "index": 7}, {"bbox": [125, 224, 486, 236], "spans": [{"bbox": [125, 224, 289, 236], "score": 1.0, "content": "number formulas, Proposition 3, for ", "type": "text"}, {"bbox": [289, 225, 311, 235], "score": 0.94, "content": "L/k_{1}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [311, 224, 334, 236], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [334, 225, 356, 235], "score": 0.94, "content": "L/k_{2}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [357, 224, 408, 236], "score": 1.0, "content": " (note that ", "type": "text"}, {"bbox": [409, 226, 435, 233], "score": 0.91, "content": "\\upsilon=0", "type": "inline_equation", "height": 7, "width": 26}, {"bbox": [436, 224, 486, 236], "score": 1.0, "content": " since both", "type": "text"}], "index": 8}, {"bbox": [126, 236, 487, 248], "spans": [{"bbox": [126, 237, 151, 247], "score": 0.94, "content": "L/K_{1}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [151, 236, 173, 248], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [173, 237, 198, 247], "score": 0.93, "content": "L/K_{2}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [198, 236, 365, 248], "score": 1.0, "content": " are essentially ramified) we find that ", "type": "text"}, {"bbox": [365, 238, 388, 247], "score": 0.91, "content": "2q_{1}q_{2}", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [388, 236, 487, 248], "score": 1.0, "content": " is a square. If we can", "type": "text"}], "index": 9}, {"bbox": [125, 248, 486, 260], "spans": [{"bbox": [125, 248, 174, 260], "score": 1.0, "content": "prove that", "type": "text"}, {"bbox": [175, 250, 217, 259], "score": 0.92, "content": "q_{1},q_{2}\\leq2", "type": "inline_equation", "height": 9, "width": 42}, {"bbox": [218, 248, 246, 260], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [247, 250, 270, 259], "score": 0.92, "content": "2q_{1}q_{2}", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [270, 248, 486, 260], "score": 1.0, "content": " is a square between 2 and 8, which implies that", "type": "text"}], "index": 10}, {"bbox": [125, 259, 340, 272], "spans": [{"bbox": [125, 259, 188, 272], "score": 1.0, "content": "we must have ", "type": "text"}, {"bbox": [189, 262, 230, 271], "score": 0.93, "content": "2q_{1}q_{2}=4", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [230, 259, 252, 272], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [252, 262, 288, 270], "score": 0.93, "content": "q_{1}q_{2}=2", "type": "inline_equation", "height": 8, "width": 36}, {"bbox": [289, 259, 340, 272], "score": 1.0, "content": " as claimed.", "type": "text"}], "index": 11}], "index": 8}, {"type": "text", "bbox": [125, 270, 486, 306], "lines": [{"bbox": [137, 271, 485, 284], "spans": [{"bbox": [137, 271, 266, 284], "score": 1.0, "content": "We start by remarking that if ", "type": "text"}, {"bbox": [266, 273, 276, 282], "score": 0.92, "content": "\\zeta\\eta", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [277, 271, 366, 284], "score": 1.0, "content": " becomes a square in ", "type": "text"}, {"bbox": [366, 273, 374, 280], "score": 0.88, "content": "L", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [374, 271, 406, 284], "score": 1.0, "content": ", where", "type": "text"}, {"bbox": [407, 273, 412, 282], "score": 0.9, "content": "\\zeta", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [412, 271, 485, 284], "score": 1.0, "content": " is a root of unity", "type": "text"}], "index": 12}, {"bbox": [126, 284, 486, 295], "spans": [{"bbox": [126, 284, 137, 295], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [138, 285, 145, 293], "score": 0.89, "content": "L", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [145, 284, 239, 295], "score": 1.0, "content": ", then so does one of", "type": "text"}, {"bbox": [239, 286, 253, 294], "score": 0.9, "content": "\\pm\\eta", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [253, 284, 486, 295], "score": 1.0, "content": ". This follows from the fact that the only non-trivial", "type": "text"}], "index": 13}, {"bbox": [125, 295, 485, 307], "spans": [{"bbox": [125, 295, 251, 307], "score": 1.0, "content": "roots of unity that can be in ", "type": "text"}, {"bbox": [251, 297, 258, 304], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [258, 295, 380, 307], "score": 1.0, "content": " are the sixth roots of unity", "type": "text"}, {"bbox": [380, 297, 397, 307], "score": 0.93, "content": "\\langle\\zeta_{6}\\rangle", "type": "inline_equation", "height": 10, "width": 17}, {"bbox": [397, 295, 442, 307], "score": 1.0, "content": ", and here ", "type": "text"}, {"bbox": [442, 296, 482, 307], "score": 0.93, "content": "\\zeta_{6}=-\\zeta_{3}^{2}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [482, 295, 485, 307], "score": 1.0, "content": ".", "type": "text"}], "index": 14}], "index": 13}, {"type": "text", "bbox": [125, 306, 486, 390], "lines": [{"bbox": [137, 307, 486, 319], "spans": [{"bbox": [137, 307, 222, 319], "score": 1.0, "content": "Now we prove that ", "type": "text"}, {"bbox": [222, 309, 249, 318], "score": 0.92, "content": "q_{1}\\leq2", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [250, 307, 439, 319], "score": 1.0, "content": " under the assumptions we made; the claim ", "type": "text"}, {"bbox": [439, 309, 466, 318], "score": 0.92, "content": "q_{2}\\leq2", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [467, 307, 486, 319], "score": 1.0, "content": " will", "type": "text"}], "index": 15}, {"bbox": [125, 320, 487, 330], "spans": [{"bbox": [125, 320, 318, 330], "score": 1.0, "content": "then follow by symmetry. Assume first that ", "type": "text"}, {"bbox": [318, 321, 327, 330], "score": 0.92, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [328, 320, 392, 330], "score": 1.0, "content": " is real and let ", "type": "text"}, {"bbox": [393, 324, 397, 328], "score": 0.88, "content": "\\varepsilon", "type": "inline_equation", "height": 4, "width": 4}, {"bbox": [398, 320, 487, 330], "score": 1.0, "content": " be the fundamental", "type": "text"}], "index": 16}, {"bbox": [126, 331, 487, 344], "spans": [{"bbox": [126, 331, 159, 344], "score": 1.0, "content": "unit of ", "type": "text"}, {"bbox": [159, 333, 169, 342], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [169, 331, 244, 344], "score": 1.0, "content": ". We claim that ", "type": "text"}, {"bbox": [244, 331, 285, 342], "score": 0.93, "content": "\\sqrt{\\pm\\varepsilon}\\notin L", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [286, 331, 403, 344], "score": 1.0, "content": ". Suppose otherwise; then ", "type": "text"}, {"bbox": [404, 331, 444, 343], "score": 0.93, "content": "k_{1}(\\sqrt{\\pm\\varepsilon})", "type": "inline_equation", "height": 12, "width": 40}, {"bbox": [444, 331, 487, 344], "score": 1.0, "content": " is one of", "type": "text"}], "index": 17}, {"bbox": [126, 343, 487, 356], "spans": [{"bbox": [126, 346, 139, 354], "score": 0.89, "content": "K_{1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [139, 343, 145, 356], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [145, 345, 158, 356], "score": 0.91, "content": "K_{1}^{\\prime}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [158, 343, 174, 356], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [174, 346, 183, 353], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [184, 343, 201, 356], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [201, 344, 268, 356], "score": 0.94, "content": "k_{1}(\\sqrt{\\pm\\varepsilon}\\,)=K_{1}", "type": "inline_equation", "height": 12, "width": 67}, {"bbox": [268, 343, 297, 356], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [297, 344, 367, 356], "score": 0.94, "content": "K_{1}^{\\prime}=k_{1}(\\sqrt{\\pm\\varepsilon^{\\prime}}\\,)", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [367, 343, 389, 356], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [390, 344, 453, 356], "score": 0.93, "content": "K=k_{1}\\big(\\sqrt{\\varepsilon\\varepsilon^{\\prime}}\\big)", "type": "inline_equation", "height": 12, "width": 63}, {"bbox": [453, 343, 487, 356], "score": 1.0, "content": ". (Here", "type": "text"}], "index": 18}, {"bbox": [125, 355, 485, 369], "spans": [{"bbox": [125, 355, 175, 369], "score": 1.0, "content": "and below ", "type": "text"}, {"bbox": [175, 357, 212, 365], "score": 0.91, "content": "x^{\\prime}\\,=\\,x^{\\sigma}", "type": "inline_equation", "height": 8, "width": 37}, {"bbox": [212, 355, 443, 369], "score": 1.0, "content": ".) This however cannot occur since by assumption ", "type": "text"}, {"bbox": [443, 357, 485, 366], "score": 0.92, "content": "\\varepsilon\\varepsilon^{\\prime}\\,=\\,-1", "type": "inline_equation", "height": 9, "width": 42}], "index": 19}, {"bbox": [125, 368, 487, 380], "spans": [{"bbox": [125, 368, 189, 380], "score": 1.0, "content": "implying that ", "type": "text"}, {"bbox": [190, 368, 232, 379], "score": 0.92, "content": "\\sqrt{-1}\\in L", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [232, 368, 367, 380], "score": 1.0, "content": ", a contradiction. Similarly, if ", "type": "text"}, {"bbox": [367, 369, 432, 380], "score": 0.94, "content": "k_{1}({\\sqrt{\\pm\\varepsilon}}\\,)\\,=\\,K", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [432, 368, 487, 380], "score": 1.0, "content": ", then again", "type": "text"}], "index": 20}, {"bbox": [126, 380, 170, 392], "spans": [{"bbox": [126, 380, 166, 391], "score": 0.92, "content": "\\sqrt{-1}\\in L", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [167, 380, 170, 392], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 18}, {"type": "text", "bbox": [125, 390, 486, 450], "lines": [{"bbox": [137, 390, 486, 405], "spans": [{"bbox": [137, 390, 164, 405], "score": 1.0, "content": "Thus ", "type": "text"}, {"bbox": [164, 392, 210, 403], "score": 0.94, "content": "\\sqrt{\\pm\\varepsilon}~\\notin~L", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [210, 390, 238, 405], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [239, 393, 308, 403], "score": 0.94, "content": "E_{1}~=~\\langle-1,\\varepsilon,\\eta\\rangle", "type": "inline_equation", "height": 10, "width": 69}, {"bbox": [309, 390, 379, 405], "score": 1.0, "content": "for some unit ", "type": "text"}, {"bbox": [379, 394, 414, 403], "score": 0.93, "content": "\\eta\\ \\in\\ E_{1}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [414, 390, 486, 405], "score": 1.0, "content": ". Suppose that", "type": "text"}], "index": 22}, {"bbox": [126, 404, 485, 417], "spans": [{"bbox": [126, 406, 164, 416], "score": 0.93, "content": "\\sqrt{u\\eta}\\in L", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [165, 404, 228, 417], "score": 1.0, "content": " for some unit ", "type": "text"}, {"bbox": [228, 406, 256, 415], "score": 0.92, "content": "u\\in k_{1}", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [257, 404, 289, 417], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [289, 405, 351, 416], "score": 0.94, "content": "L=K_{1}(\\sqrt{u\\eta}\\,)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [352, 404, 485, 417], "score": 1.0, "content": ", contradicting our assumption", "type": "text"}], "index": 23}, {"bbox": [126, 417, 486, 429], "spans": [{"bbox": [126, 417, 147, 429], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [147, 418, 172, 428], "score": 0.94, "content": "L/K_{1}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [172, 417, 413, 429], "score": 1.0, "content": " is essentially ramified. The same argument shows that ", "type": "text"}, {"bbox": [414, 417, 455, 428], "score": 0.94, "content": "\\sqrt{u\\eta^{\\prime}}\\notin L", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [455, 417, 486, 429], "score": 1.0, "content": ", hence", "type": "text"}], "index": 24}, {"bbox": [126, 428, 484, 442], "spans": [{"bbox": [126, 428, 154, 442], "score": 1.0, "content": "either ", "type": "text"}, {"bbox": [155, 430, 227, 440], "score": 0.94, "content": "E_{L}\\,=\\,\\langle\\zeta,\\varepsilon,\\eta,\\eta^{\\prime}\\rangle", "type": "inline_equation", "height": 10, "width": 72}, {"bbox": [227, 428, 250, 442], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [251, 431, 279, 439], "score": 0.93, "content": "q_{1}=1", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [280, 428, 295, 442], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [296, 429, 389, 440], "score": 0.93, "content": "E_{L}=\\langle\\zeta,\\varepsilon,\\eta,\\sqrt{u\\eta\\eta^{\\prime}}\\rangle", "type": "inline_equation", "height": 11, "width": 93}, {"bbox": [390, 428, 455, 442], "score": 1.0, "content": "for some unit ", "type": "text"}, {"bbox": [455, 430, 484, 439], "score": 0.92, "content": "u\\in k_{1}", "type": "inline_equation", "height": 9, "width": 29}], "index": 25}, {"bbox": [126, 441, 479, 452], "spans": [{"bbox": [126, 441, 145, 452], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [145, 443, 173, 451], "score": 0.92, "content": "q_{1}=2", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [173, 441, 203, 452], "score": 1.0, "content": ". Here ", "type": "text"}, {"bbox": [203, 442, 208, 451], "score": 0.88, "content": "\\zeta", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [209, 441, 431, 452], "score": 1.0, "content": " is a root of unity generating the torsion subgroup ", "type": "text"}, {"bbox": [431, 442, 446, 451], "score": 0.92, "content": "W_{L}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [447, 441, 461, 452], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [461, 442, 475, 451], "score": 0.91, "content": "E_{L}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [475, 441, 479, 452], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 24}, {"type": "text", "bbox": [125, 451, 486, 474], "lines": [{"bbox": [137, 451, 486, 465], "spans": [{"bbox": [137, 451, 270, 465], "score": 1.0, "content": "Next consider the case where ", "type": "text"}, {"bbox": [271, 454, 280, 463], "score": 0.92, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [280, 451, 371, 465], "score": 1.0, "content": " is complex, and let ", "type": "text"}, {"bbox": [371, 457, 376, 461], "score": 0.87, "content": "\\varepsilon", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [377, 451, 486, 465], "score": 1.0, "content": " denote the fundamental", "type": "text"}], "index": 27}, {"bbox": [126, 464, 425, 476], "spans": [{"bbox": [126, 464, 158, 476], "score": 1.0, "content": "unit of ", "type": "text"}, {"bbox": [158, 466, 168, 475], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [168, 464, 201, 476], "score": 1.0, "content": ". Then", "type": "text"}, {"bbox": [201, 466, 214, 474], "score": 0.86, "content": "\\pm\\varepsilon", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [214, 464, 311, 476], "score": 1.0, "content": " stays fundamental in ", "type": "text"}, {"bbox": [311, 466, 318, 473], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [318, 464, 425, 476], "score": 1.0, "content": " by the argument above.", "type": "text"}], "index": 28}], "index": 27.5}, {"type": "text", "bbox": [125, 475, 486, 522], "lines": [{"bbox": [137, 477, 485, 488], "spans": [{"bbox": [137, 477, 156, 488], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [156, 481, 162, 488], "score": 0.9, "content": "\\eta", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [162, 477, 280, 488], "score": 1.0, "content": " be a fundamental unit in ", "type": "text"}, {"bbox": [280, 478, 294, 487], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [294, 477, 313, 488], "score": 1.0, "content": ". If", "type": "text"}, {"bbox": [313, 479, 326, 488], "score": 0.89, "content": "\\pm\\eta", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [327, 477, 419, 488], "score": 1.0, "content": " became a square in ", "type": "text"}, {"bbox": [419, 478, 426, 486], "score": 0.89, "content": "L", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [426, 477, 485, 488], "score": 1.0, "content": ", then clearly", "type": "text"}], "index": 29}, {"bbox": [126, 488, 487, 501], "spans": [{"bbox": [126, 489, 151, 500], "score": 0.93, "content": "L/K_{1}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [151, 488, 375, 501], "score": 1.0, "content": " could not be essentially ramified. Thus if we have ", "type": "text"}, {"bbox": [376, 491, 403, 499], "score": 0.91, "content": "q_{1}\\geq4", "type": "inline_equation", "height": 8, "width": 27}, {"bbox": [404, 488, 432, 501], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [432, 489, 474, 500], "score": 0.93, "content": "\\pm\\varepsilon\\eta=\\alpha^{2}", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [475, 488, 487, 501], "score": 1.0, "content": " is", "type": "text"}], "index": 30}, {"bbox": [125, 500, 487, 513], "spans": [{"bbox": [125, 500, 177, 513], "score": 1.0, "content": "a square in ", "type": "text"}, {"bbox": [178, 502, 185, 509], "score": 0.89, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [185, 500, 235, 513], "score": 1.0, "content": ". Applying ", "type": "text"}, {"bbox": [236, 505, 241, 509], "score": 0.87, "content": "\\tau", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [242, 500, 370, 513], "score": 1.0, "content": " to this relation we find that ", "type": "text"}, {"bbox": [371, 501, 409, 510], "score": 0.9, "content": "-1=\\varepsilon\\varepsilon^{\\prime}", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [410, 500, 475, 513], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [475, 502, 482, 509], "score": 0.87, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [483, 500, 487, 513], "score": 1.0, "content": ",", "type": "text"}], "index": 31}, {"bbox": [125, 512, 486, 524], "spans": [{"bbox": [125, 512, 277, 524], "score": 1.0, "content": "contradicting the assumption that", "type": "text"}, {"bbox": [278, 514, 285, 521], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [286, 512, 363, 524], "score": 1.0, "content": " does not contain ", "type": "text"}, {"bbox": [363, 513, 385, 523], "score": 0.92, "content": "\\sqrt{-1}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [385, 512, 388, 524], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [476, 513, 486, 522], "score": 0.9836314916610718, "content": "\u53e3", "type": "text"}], "index": 32}], "index": 30.5}, {"type": "text", "bbox": [125, 530, 486, 567], "lines": [{"bbox": [126, 532, 487, 545], "spans": [{"bbox": [126, 532, 261, 545], "score": 1.0, "content": "Proposition 7. Suppose that ", "type": "text"}, {"bbox": [261, 534, 291, 543], "score": 0.92, "content": "q_{2}=1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [292, 532, 327, 545], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [327, 533, 355, 544], "score": 0.94, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [355, 532, 487, 545], "score": 1.0, "content": " is essentially ramified if and", "type": "text"}], "index": 33}, {"bbox": [127, 543, 486, 558], "spans": [{"bbox": [127, 543, 158, 558], "score": 1.0, "content": "only if ", "type": "text"}, {"bbox": [158, 546, 189, 555], "score": 0.84, "content": "\\kappa_{2}=1", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [190, 543, 205, 558], "score": 1.0, "content": "; if ", "type": "text"}, {"bbox": [206, 545, 234, 556], "score": 0.9, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [234, 543, 380, 558], "score": 1.0, "content": " is not essentially ramified, then ", "type": "text"}, {"bbox": [380, 545, 424, 556], "score": 0.93, "content": "\\kappa_{2}=\\langle[6]\\rangle", "type": "inline_equation", "height": 11, "width": 44}, {"bbox": [425, 543, 460, 558], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [460, 545, 486, 555], "score": 0.87, "content": "K_{2}=", "type": "inline_equation", "height": 10, "width": 26}], "index": 34}, {"bbox": [126, 556, 223, 568], "spans": [{"bbox": [126, 556, 160, 568], "score": 0.92, "content": "k_{2}(\\sqrt{\\beta}\\,)", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [160, 556, 182, 568], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [182, 556, 219, 568], "score": 0.92, "content": "(\\beta)=\\mathfrak{b}^{2}", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [220, 556, 223, 568], "score": 1.0, "content": ".", "type": "text"}], "index": 35}], "index": 34}, {"type": "text", "bbox": [125, 573, 486, 632], "lines": [{"bbox": [126, 574, 485, 587], "spans": [{"bbox": [126, 574, 243, 587], "score": 1.0, "content": "Proof. First notice that if ", "type": "text"}, {"bbox": [243, 576, 271, 586], "score": 0.93, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [271, 574, 415, 587], "score": 1.0, "content": " is not essentially ramified, then ", "type": "text"}, {"bbox": [416, 577, 446, 586], "score": 0.92, "content": "\\kappa_{2}\\neq1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [446, 574, 485, 587], "score": 1.0, "content": ": in fact,", "type": "text"}], "index": 36}, {"bbox": [124, 586, 486, 599], "spans": [{"bbox": [124, 586, 221, 599], "score": 1.0, "content": "in this case we have ", "type": "text"}, {"bbox": [221, 587, 261, 598], "score": 0.93, "content": "(\\beta)\\;=\\;6^{2}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [261, 586, 334, 599], "score": 1.0, "content": ", and if we had ", "type": "text"}, {"bbox": [335, 589, 367, 597], "score": 0.91, "content": "\\,\\kappa_{2}\\,=\\,1", "type": "inline_equation", "height": 8, "width": 32}, {"bbox": [367, 586, 397, 599], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [398, 588, 403, 596], "score": 0.83, "content": "\\mathfrak{b}", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [403, 586, 486, 599], "score": 1.0, "content": " would have to be", "type": "text"}], "index": 37}, {"bbox": [126, 599, 486, 611], "spans": [{"bbox": [126, 599, 188, 611], "score": 1.0, "content": "principal, say ", "type": "text"}, {"bbox": [189, 600, 222, 610], "score": 0.93, "content": "{\\mathfrak{b}}=(\\gamma)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [222, 599, 309, 611], "score": 1.0, "content": ". This implies that", "type": "text"}, {"bbox": [310, 600, 346, 610], "score": 0.93, "content": "\\beta\\,=\\,\\varepsilon\\gamma^{2}", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [347, 599, 412, 611], "score": 1.0, "content": " for some unit ", "type": "text"}, {"bbox": [412, 601, 441, 609], "score": 0.92, "content": "\\varepsilon\\in k_{2}", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [441, 599, 486, 611], "score": 1.0, "content": ", which in", "type": "text"}], "index": 38}, {"bbox": [125, 611, 487, 623], "spans": [{"bbox": [125, 611, 159, 623], "score": 1.0, "content": "view of ", "type": "text"}, {"bbox": [160, 613, 187, 622], "score": 0.92, "content": "q_{2}=1", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [187, 611, 244, 623], "score": 1.0, "content": " implies that", "type": "text"}, {"bbox": [245, 615, 249, 620], "score": 0.89, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 4}, {"bbox": [250, 611, 375, 623], "score": 1.0, "content": " must be a square. But then ", "type": "text"}, {"bbox": [375, 613, 381, 622], "score": 0.91, "content": "\\beta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [382, 611, 487, 623], "score": 1.0, "content": " would be a square, and", "type": "text"}], "index": 39}, {"bbox": [125, 623, 204, 635], "spans": [{"bbox": [125, 623, 204, 635], "score": 1.0, "content": "this is impossible.", "type": "text"}], "index": 40}], "index": 38}, {"type": "text", "bbox": [124, 633, 486, 700], "lines": [{"bbox": [138, 635, 487, 647], "spans": [{"bbox": [138, 635, 226, 647], "score": 1.0, "content": "Conversely, suppose", "type": "text"}, {"bbox": [227, 636, 255, 646], "score": 0.92, "content": "\\kappa_{2}\\neq1", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [256, 635, 280, 647], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [280, 639, 285, 644], "score": 0.87, "content": "\\mathfrak{a}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [285, 635, 399, 647], "score": 1.0, "content": " be a nonprincipal ideal in", "type": "text"}, {"bbox": [400, 636, 409, 645], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [410, 635, 487, 647], "score": 1.0, "content": " of absolute norm", "type": "text"}], "index": 41}, {"bbox": [126, 646, 486, 660], "spans": [{"bbox": [126, 652, 132, 657], "score": 0.85, "content": "a", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [132, 647, 216, 660], "score": 1.0, "content": ", and assume that ", "type": "text"}, {"bbox": [216, 649, 250, 659], "score": 0.94, "content": "{\\mathfrak{a}}=(\\alpha)", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [251, 647, 266, 660], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [267, 649, 280, 658], "score": 0.92, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [280, 647, 315, 660], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [316, 646, 363, 659], "score": 0.95, "content": "\\alpha^{1-\\sigma^{2}}=\\eta", "type": "inline_equation", "height": 13, "width": 47}, {"bbox": [363, 647, 429, 660], "score": 1.0, "content": " for some unit ", "type": "text"}, {"bbox": [430, 649, 462, 659], "score": 0.93, "content": "\\eta\\in E_{2}", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [462, 647, 486, 660], "score": 1.0, "content": ", and", "type": "text"}], "index": 42}, {"bbox": [123, 658, 485, 676], "spans": [{"bbox": [123, 658, 167, 676], "score": 1.0, "content": "similarly ", "type": "text"}, {"bbox": [168, 661, 219, 673], "score": 0.93, "content": "\\alpha^{\\sigma-\\sigma^{3}}\\,=\\,\\eta^{\\prime}", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [220, 658, 256, 676], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [256, 663, 264, 673], "score": 0.9, "content": "\\eta^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [265, 658, 324, 676], "score": 1.0, "content": " is a unit in ", "type": "text"}, {"bbox": [324, 663, 336, 673], "score": 0.92, "content": "E_{2}^{\\prime}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [336, 658, 390, 676], "score": 1.0, "content": ". But then ", "type": "text"}, {"bbox": [390, 660, 485, 673], "score": 0.9, "content": "\\eta\\eta^{\\prime}\\,=\\,\\alpha^{1+\\sigma-\\sigma^{2}-\\sigma^{3}}\\,\\stackrel{\\cdot2}{=}", "type": "inline_equation", "height": 13, "width": 95}], "index": 43}, {"bbox": [125, 674, 487, 690], "spans": [{"bbox": [125, 675, 269, 689], "score": 0.89, "content": "N_{L/k}\\alpha\\,=\\,\\pm N_{L/k}\\mathfrak{a}\\,=\\,\\pm a^{2}\\,\\stackrel{?}{=}\\,\\pm1", "type": "inline_equation", "height": 14, "width": 144}, {"bbox": [270, 674, 285, 690], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [285, 677, 299, 686], "score": 0.89, "content": "L^{\\times}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [299, 674, 335, 690], "score": 1.0, "content": ", where", "type": "text"}, {"bbox": [336, 675, 344, 686], "score": 0.71, "content": "\\underline{{\\underline{{2}}}}", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [344, 674, 487, 690], "score": 1.0, "content": " means equal up to a square in", "type": "text"}], "index": 44}, {"bbox": [126, 689, 484, 702], "spans": [{"bbox": [126, 690, 140, 698], "score": 0.89, "content": "L^{\\times}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [140, 689, 172, 702], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [172, 690, 194, 700], "score": 0.92, "content": "\\pm\\eta\\eta^{\\prime}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [194, 689, 258, 702], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [259, 691, 266, 698], "score": 0.88, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [266, 689, 376, 702], "score": 1.0, "content": ", so our assumption that ", "type": "text"}, {"bbox": [376, 691, 405, 700], "score": 0.93, "content": "q_{2}=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [405, 689, 463, 702], "score": 1.0, "content": " implies that ", "type": "text"}, {"bbox": [464, 690, 484, 700], "score": 0.91, "content": "\\pm\\eta\\eta^{\\prime}", "type": "inline_equation", "height": 10, "width": 20}], "index": 45}], "index": 43}], "layout_bboxes": [], "page_idx": 7, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [125, 91, 131, 99], "lines": [{"bbox": [126, 93, 131, 101], "spans": [{"bbox": [126, 93, 131, 101], "score": 1.0, "content": "8", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 111, 486, 136], "lines": [{"bbox": [137, 114, 486, 126], "spans": [{"bbox": [137, 114, 291, 126], "score": 1.0, "content": "Recall that a quadratic extension ", "type": "text"}, {"bbox": [291, 115, 346, 126], "score": 0.95, "content": "K\\,=\\,k({\\sqrt{\\alpha}}\\,)", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [347, 114, 486, 126], "score": 1.0, "content": " is called essentially ramified if", "type": "text"}], "index": 0}, {"bbox": [126, 126, 462, 138], "spans": [{"bbox": [126, 128, 145, 137], "score": 0.92, "content": "\\alpha{\\mathcal{O}}_{k}", "type": "inline_equation", "height": 9, "width": 19}, {"bbox": [145, 126, 452, 138], "score": 1.0, "content": " is not an ideal square. This definition is independent of the choice of ", "type": "text"}, {"bbox": [452, 131, 459, 135], "score": 0.88, "content": "\\alpha", "type": "inline_equation", "height": 4, "width": 7}, {"bbox": [459, 126, 462, 138], "score": 1.0, "content": ".", "type": "text"}], "index": 1}], "index": 0.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [126, 114, 486, 138]}, {"type": "text", "bbox": [125, 142, 486, 179], "lines": [{"bbox": [126, 145, 487, 158], "spans": [{"bbox": [126, 145, 218, 158], "score": 1.0, "content": "Proposition 6. Let ", "type": "text"}, {"bbox": [218, 146, 239, 156], "score": 0.92, "content": "L/\\mathbb{Q}", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [239, 145, 487, 158], "score": 1.0, "content": " be a non-CM totally complex dihedral extension not con-", "type": "text"}], "index": 2}, {"bbox": [126, 157, 487, 170], "spans": [{"bbox": [126, 157, 160, 170], "score": 1.0, "content": "taining ", "type": "text"}, {"bbox": [161, 157, 182, 168], "score": 0.91, "content": "\\sqrt{-1}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [183, 157, 267, 170], "score": 1.0, "content": ", and assume that ", "type": "text"}, {"bbox": [268, 158, 293, 169], "score": 0.93, "content": "L/K_{1}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [293, 157, 317, 170], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [317, 158, 342, 169], "score": 0.93, "content": "L/K_{2}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [343, 157, 487, 170], "score": 1.0, "content": " are essentially ramified. If the", "type": "text"}], "index": 3}, {"bbox": [126, 169, 481, 182], "spans": [{"bbox": [126, 169, 341, 182], "score": 1.0, "content": "fundamental unit of the real quadratic subfield of ", "type": "text"}, {"bbox": [342, 171, 351, 178], "score": 0.87, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [352, 169, 398, 182], "score": 1.0, "content": " has norm ", "type": "text"}, {"bbox": [399, 171, 412, 179], "score": 0.85, "content": "^{-1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [412, 169, 440, 182], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [440, 171, 477, 180], "score": 0.91, "content": "q_{1}q_{2}=2", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [477, 169, 481, 182], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 3, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [126, 145, 487, 182]}, {"type": "text", "bbox": [125, 185, 486, 269], "lines": [{"bbox": [127, 188, 485, 200], "spans": [{"bbox": [127, 188, 227, 200], "score": 1.0, "content": "Proof. Notice first that ", "type": "text"}, {"bbox": [227, 189, 233, 197], "score": 0.9, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [233, 188, 333, 200], "score": 1.0, "content": " cannot be real (in fact,", "type": "text"}, {"bbox": [334, 190, 343, 197], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [343, 188, 485, 200], "score": 1.0, "content": " is not totally real by assumption,", "type": "text"}], "index": 5}, {"bbox": [126, 199, 486, 212], "spans": [{"bbox": [126, 199, 170, 212], "score": 1.0, "content": "and since ", "type": "text"}, {"bbox": [170, 201, 188, 211], "score": 0.93, "content": "L/k", "type": "inline_equation", "height": 10, "width": 18}, {"bbox": [188, 199, 459, 212], "score": 1.0, "content": " is a cyclic quartic extension, no infinite prime can ramify in ", "type": "text"}, {"bbox": [459, 201, 478, 211], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [479, 199, 486, 212], "score": 1.0, "content": ");", "type": "text"}], "index": 6}, {"bbox": [126, 212, 486, 224], "spans": [{"bbox": [126, 212, 213, 224], "score": 1.0, "content": "thus exactly one of ", "type": "text"}, {"bbox": [213, 213, 223, 222], "score": 0.9, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [223, 212, 228, 224], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [229, 213, 239, 222], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [239, 212, 486, 224], "score": 1.0, "content": " is real, and the other is complex. Multiplying the class", "type": "text"}], "index": 7}, {"bbox": [125, 224, 486, 236], "spans": [{"bbox": [125, 224, 289, 236], "score": 1.0, "content": "number formulas, Proposition 3, for ", "type": "text"}, {"bbox": [289, 225, 311, 235], "score": 0.94, "content": "L/k_{1}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [311, 224, 334, 236], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [334, 225, 356, 235], "score": 0.94, "content": "L/k_{2}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [357, 224, 408, 236], "score": 1.0, "content": " (note that ", "type": "text"}, {"bbox": [409, 226, 435, 233], "score": 0.91, "content": "\\upsilon=0", "type": "inline_equation", "height": 7, "width": 26}, {"bbox": [436, 224, 486, 236], "score": 1.0, "content": " since both", "type": "text"}], "index": 8}, {"bbox": [126, 236, 487, 248], "spans": [{"bbox": [126, 237, 151, 247], "score": 0.94, "content": "L/K_{1}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [151, 236, 173, 248], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [173, 237, 198, 247], "score": 0.93, "content": "L/K_{2}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [198, 236, 365, 248], "score": 1.0, "content": " are essentially ramified) we find that ", "type": "text"}, {"bbox": [365, 238, 388, 247], "score": 0.91, "content": "2q_{1}q_{2}", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [388, 236, 487, 248], "score": 1.0, "content": " is a square. If we can", "type": "text"}], "index": 9}, {"bbox": [125, 248, 486, 260], "spans": [{"bbox": [125, 248, 174, 260], "score": 1.0, "content": "prove that", "type": "text"}, {"bbox": [175, 250, 217, 259], "score": 0.92, "content": "q_{1},q_{2}\\leq2", "type": "inline_equation", "height": 9, "width": 42}, {"bbox": [218, 248, 246, 260], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [247, 250, 270, 259], "score": 0.92, "content": "2q_{1}q_{2}", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [270, 248, 486, 260], "score": 1.0, "content": " is a square between 2 and 8, which implies that", "type": "text"}], "index": 10}, {"bbox": [125, 259, 340, 272], "spans": [{"bbox": [125, 259, 188, 272], "score": 1.0, "content": "we must have ", "type": "text"}, {"bbox": [189, 262, 230, 271], "score": 0.93, "content": "2q_{1}q_{2}=4", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [230, 259, 252, 272], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [252, 262, 288, 270], "score": 0.93, "content": "q_{1}q_{2}=2", "type": "inline_equation", "height": 8, "width": 36}, {"bbox": [289, 259, 340, 272], "score": 1.0, "content": " as claimed.", "type": "text"}], "index": 11}], "index": 8, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [125, 188, 487, 272]}, {"type": "text", "bbox": [125, 270, 486, 306], "lines": [{"bbox": [137, 271, 485, 284], "spans": [{"bbox": [137, 271, 266, 284], "score": 1.0, "content": "We start by remarking that if ", "type": "text"}, {"bbox": [266, 273, 276, 282], "score": 0.92, "content": "\\zeta\\eta", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [277, 271, 366, 284], "score": 1.0, "content": " becomes a square in ", "type": "text"}, {"bbox": [366, 273, 374, 280], "score": 0.88, "content": "L", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [374, 271, 406, 284], "score": 1.0, "content": ", where", "type": "text"}, {"bbox": [407, 273, 412, 282], "score": 0.9, "content": "\\zeta", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [412, 271, 485, 284], "score": 1.0, "content": " is a root of unity", "type": "text"}], "index": 12}, {"bbox": [126, 284, 486, 295], "spans": [{"bbox": [126, 284, 137, 295], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [138, 285, 145, 293], "score": 0.89, "content": "L", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [145, 284, 239, 295], "score": 1.0, "content": ", then so does one of", "type": "text"}, {"bbox": [239, 286, 253, 294], "score": 0.9, "content": "\\pm\\eta", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [253, 284, 486, 295], "score": 1.0, "content": ". This follows from the fact that the only non-trivial", "type": "text"}], "index": 13}, {"bbox": [125, 295, 485, 307], "spans": [{"bbox": [125, 295, 251, 307], "score": 1.0, "content": "roots of unity that can be in ", "type": "text"}, {"bbox": [251, 297, 258, 304], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [258, 295, 380, 307], "score": 1.0, "content": " are the sixth roots of unity", "type": "text"}, {"bbox": [380, 297, 397, 307], "score": 0.93, "content": "\\langle\\zeta_{6}\\rangle", "type": "inline_equation", "height": 10, "width": 17}, {"bbox": [397, 295, 442, 307], "score": 1.0, "content": ", and here ", "type": "text"}, {"bbox": [442, 296, 482, 307], "score": 0.93, "content": "\\zeta_{6}=-\\zeta_{3}^{2}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [482, 295, 485, 307], "score": 1.0, "content": ".", "type": "text"}], "index": 14}], "index": 13, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [125, 271, 486, 307]}, {"type": "text", "bbox": [125, 306, 486, 390], "lines": [{"bbox": [137, 307, 486, 319], "spans": [{"bbox": [137, 307, 222, 319], "score": 1.0, "content": "Now we prove that ", "type": "text"}, {"bbox": [222, 309, 249, 318], "score": 0.92, "content": "q_{1}\\leq2", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [250, 307, 439, 319], "score": 1.0, "content": " under the assumptions we made; the claim ", "type": "text"}, {"bbox": [439, 309, 466, 318], "score": 0.92, "content": "q_{2}\\leq2", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [467, 307, 486, 319], "score": 1.0, "content": " will", "type": "text"}], "index": 15}, {"bbox": [125, 320, 487, 330], "spans": [{"bbox": [125, 320, 318, 330], "score": 1.0, "content": "then follow by symmetry. Assume first that ", "type": "text"}, {"bbox": [318, 321, 327, 330], "score": 0.92, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [328, 320, 392, 330], "score": 1.0, "content": " is real and let ", "type": "text"}, {"bbox": [393, 324, 397, 328], "score": 0.88, "content": "\\varepsilon", "type": "inline_equation", "height": 4, "width": 4}, {"bbox": [398, 320, 487, 330], "score": 1.0, "content": " be the fundamental", "type": "text"}], "index": 16}, {"bbox": [126, 331, 487, 344], "spans": [{"bbox": [126, 331, 159, 344], "score": 1.0, "content": "unit of ", "type": "text"}, {"bbox": [159, 333, 169, 342], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [169, 331, 244, 344], "score": 1.0, "content": ". We claim that ", "type": "text"}, {"bbox": [244, 331, 285, 342], "score": 0.93, "content": "\\sqrt{\\pm\\varepsilon}\\notin L", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [286, 331, 403, 344], "score": 1.0, "content": ". Suppose otherwise; then ", "type": "text"}, {"bbox": [404, 331, 444, 343], "score": 0.93, "content": "k_{1}(\\sqrt{\\pm\\varepsilon})", "type": "inline_equation", "height": 12, "width": 40}, {"bbox": [444, 331, 487, 344], "score": 1.0, "content": " is one of", "type": "text"}], "index": 17}, {"bbox": [126, 343, 487, 356], "spans": [{"bbox": [126, 346, 139, 354], "score": 0.89, "content": "K_{1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [139, 343, 145, 356], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [145, 345, 158, 356], "score": 0.91, "content": "K_{1}^{\\prime}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [158, 343, 174, 356], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [174, 346, 183, 353], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [184, 343, 201, 356], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [201, 344, 268, 356], "score": 0.94, "content": "k_{1}(\\sqrt{\\pm\\varepsilon}\\,)=K_{1}", "type": "inline_equation", "height": 12, "width": 67}, {"bbox": [268, 343, 297, 356], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [297, 344, 367, 356], "score": 0.94, "content": "K_{1}^{\\prime}=k_{1}(\\sqrt{\\pm\\varepsilon^{\\prime}}\\,)", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [367, 343, 389, 356], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [390, 344, 453, 356], "score": 0.93, "content": "K=k_{1}\\big(\\sqrt{\\varepsilon\\varepsilon^{\\prime}}\\big)", "type": "inline_equation", "height": 12, "width": 63}, {"bbox": [453, 343, 487, 356], "score": 1.0, "content": ". (Here", "type": "text"}], "index": 18}, {"bbox": [125, 355, 485, 369], "spans": [{"bbox": [125, 355, 175, 369], "score": 1.0, "content": "and below ", "type": "text"}, {"bbox": [175, 357, 212, 365], "score": 0.91, "content": "x^{\\prime}\\,=\\,x^{\\sigma}", "type": "inline_equation", "height": 8, "width": 37}, {"bbox": [212, 355, 443, 369], "score": 1.0, "content": ".) This however cannot occur since by assumption ", "type": "text"}, {"bbox": [443, 357, 485, 366], "score": 0.92, "content": "\\varepsilon\\varepsilon^{\\prime}\\,=\\,-1", "type": "inline_equation", "height": 9, "width": 42}], "index": 19}, {"bbox": [125, 368, 487, 380], "spans": [{"bbox": [125, 368, 189, 380], "score": 1.0, "content": "implying that ", "type": "text"}, {"bbox": [190, 368, 232, 379], "score": 0.92, "content": "\\sqrt{-1}\\in L", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [232, 368, 367, 380], "score": 1.0, "content": ", a contradiction. Similarly, if ", "type": "text"}, {"bbox": [367, 369, 432, 380], "score": 0.94, "content": "k_{1}({\\sqrt{\\pm\\varepsilon}}\\,)\\,=\\,K", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [432, 368, 487, 380], "score": 1.0, "content": ", then again", "type": "text"}], "index": 20}, {"bbox": [126, 380, 170, 392], "spans": [{"bbox": [126, 380, 166, 391], "score": 0.92, "content": "\\sqrt{-1}\\in L", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [167, 380, 170, 392], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 18, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [125, 307, 487, 392]}, {"type": "text", "bbox": [125, 390, 486, 450], "lines": [{"bbox": [137, 390, 486, 405], "spans": [{"bbox": [137, 390, 164, 405], "score": 1.0, "content": "Thus ", "type": "text"}, {"bbox": [164, 392, 210, 403], "score": 0.94, "content": "\\sqrt{\\pm\\varepsilon}~\\notin~L", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [210, 390, 238, 405], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [239, 393, 308, 403], "score": 0.94, "content": "E_{1}~=~\\langle-1,\\varepsilon,\\eta\\rangle", "type": "inline_equation", "height": 10, "width": 69}, {"bbox": [309, 390, 379, 405], "score": 1.0, "content": "for some unit ", "type": "text"}, {"bbox": [379, 394, 414, 403], "score": 0.93, "content": "\\eta\\ \\in\\ E_{1}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [414, 390, 486, 405], "score": 1.0, "content": ". Suppose that", "type": "text"}], "index": 22}, {"bbox": [126, 404, 485, 417], "spans": [{"bbox": [126, 406, 164, 416], "score": 0.93, "content": "\\sqrt{u\\eta}\\in L", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [165, 404, 228, 417], "score": 1.0, "content": " for some unit ", "type": "text"}, {"bbox": [228, 406, 256, 415], "score": 0.92, "content": "u\\in k_{1}", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [257, 404, 289, 417], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [289, 405, 351, 416], "score": 0.94, "content": "L=K_{1}(\\sqrt{u\\eta}\\,)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [352, 404, 485, 417], "score": 1.0, "content": ", contradicting our assumption", "type": "text"}], "index": 23}, {"bbox": [126, 417, 486, 429], "spans": [{"bbox": [126, 417, 147, 429], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [147, 418, 172, 428], "score": 0.94, "content": "L/K_{1}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [172, 417, 413, 429], "score": 1.0, "content": " is essentially ramified. The same argument shows that ", "type": "text"}, {"bbox": [414, 417, 455, 428], "score": 0.94, "content": "\\sqrt{u\\eta^{\\prime}}\\notin L", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [455, 417, 486, 429], "score": 1.0, "content": ", hence", "type": "text"}], "index": 24}, {"bbox": [126, 428, 484, 442], "spans": [{"bbox": [126, 428, 154, 442], "score": 1.0, "content": "either ", "type": "text"}, {"bbox": [155, 430, 227, 440], "score": 0.94, "content": "E_{L}\\,=\\,\\langle\\zeta,\\varepsilon,\\eta,\\eta^{\\prime}\\rangle", "type": "inline_equation", "height": 10, "width": 72}, {"bbox": [227, 428, 250, 442], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [251, 431, 279, 439], "score": 0.93, "content": "q_{1}=1", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [280, 428, 295, 442], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [296, 429, 389, 440], "score": 0.93, "content": "E_{L}=\\langle\\zeta,\\varepsilon,\\eta,\\sqrt{u\\eta\\eta^{\\prime}}\\rangle", "type": "inline_equation", "height": 11, "width": 93}, {"bbox": [390, 428, 455, 442], "score": 1.0, "content": "for some unit ", "type": "text"}, {"bbox": [455, 430, 484, 439], "score": 0.92, "content": "u\\in k_{1}", "type": "inline_equation", "height": 9, "width": 29}], "index": 25}, {"bbox": [126, 441, 479, 452], "spans": [{"bbox": [126, 441, 145, 452], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [145, 443, 173, 451], "score": 0.92, "content": "q_{1}=2", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [173, 441, 203, 452], "score": 1.0, "content": ". Here ", "type": "text"}, {"bbox": [203, 442, 208, 451], "score": 0.88, "content": "\\zeta", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [209, 441, 431, 452], "score": 1.0, "content": " is a root of unity generating the torsion subgroup ", "type": "text"}, {"bbox": [431, 442, 446, 451], "score": 0.92, "content": "W_{L}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [447, 441, 461, 452], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [461, 442, 475, 451], "score": 0.91, "content": "E_{L}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [475, 441, 479, 452], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 24, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [126, 390, 486, 452]}, {"type": "text", "bbox": [125, 451, 486, 474], "lines": [{"bbox": [137, 451, 486, 465], "spans": [{"bbox": [137, 451, 270, 465], "score": 1.0, "content": "Next consider the case where ", "type": "text"}, {"bbox": [271, 454, 280, 463], "score": 0.92, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [280, 451, 371, 465], "score": 1.0, "content": " is complex, and let ", "type": "text"}, {"bbox": [371, 457, 376, 461], "score": 0.87, "content": "\\varepsilon", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [377, 451, 486, 465], "score": 1.0, "content": " denote the fundamental", "type": "text"}], "index": 27}, {"bbox": [126, 464, 425, 476], "spans": [{"bbox": [126, 464, 158, 476], "score": 1.0, "content": "unit of ", "type": "text"}, {"bbox": [158, 466, 168, 475], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [168, 464, 201, 476], "score": 1.0, "content": ". Then", "type": "text"}, {"bbox": [201, 466, 214, 474], "score": 0.86, "content": "\\pm\\varepsilon", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [214, 464, 311, 476], "score": 1.0, "content": " stays fundamental in ", "type": "text"}, {"bbox": [311, 466, 318, 473], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [318, 464, 425, 476], "score": 1.0, "content": " by the argument above.", "type": "text"}], "index": 28}], "index": 27.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [126, 451, 486, 476]}, {"type": "text", "bbox": [125, 475, 486, 522], "lines": [{"bbox": [137, 477, 485, 488], "spans": [{"bbox": [137, 477, 156, 488], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [156, 481, 162, 488], "score": 0.9, "content": "\\eta", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [162, 477, 280, 488], "score": 1.0, "content": " be a fundamental unit in ", "type": "text"}, {"bbox": [280, 478, 294, 487], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [294, 477, 313, 488], "score": 1.0, "content": ". If", "type": "text"}, {"bbox": [313, 479, 326, 488], "score": 0.89, "content": "\\pm\\eta", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [327, 477, 419, 488], "score": 1.0, "content": " became a square in ", "type": "text"}, {"bbox": [419, 478, 426, 486], "score": 0.89, "content": "L", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [426, 477, 485, 488], "score": 1.0, "content": ", then clearly", "type": "text"}], "index": 29}, {"bbox": [126, 488, 487, 501], "spans": [{"bbox": [126, 489, 151, 500], "score": 0.93, "content": "L/K_{1}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [151, 488, 375, 501], "score": 1.0, "content": " could not be essentially ramified. Thus if we have ", "type": "text"}, {"bbox": [376, 491, 403, 499], "score": 0.91, "content": "q_{1}\\geq4", "type": "inline_equation", "height": 8, "width": 27}, {"bbox": [404, 488, 432, 501], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [432, 489, 474, 500], "score": 0.93, "content": "\\pm\\varepsilon\\eta=\\alpha^{2}", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [475, 488, 487, 501], "score": 1.0, "content": " is", "type": "text"}], "index": 30}, {"bbox": [125, 500, 487, 513], "spans": [{"bbox": [125, 500, 177, 513], "score": 1.0, "content": "a square in ", "type": "text"}, {"bbox": [178, 502, 185, 509], "score": 0.89, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [185, 500, 235, 513], "score": 1.0, "content": ". Applying ", "type": "text"}, {"bbox": [236, 505, 241, 509], "score": 0.87, "content": "\\tau", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [242, 500, 370, 513], "score": 1.0, "content": " to this relation we find that ", "type": "text"}, {"bbox": [371, 501, 409, 510], "score": 0.9, "content": "-1=\\varepsilon\\varepsilon^{\\prime}", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [410, 500, 475, 513], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [475, 502, 482, 509], "score": 0.87, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [483, 500, 487, 513], "score": 1.0, "content": ",", "type": "text"}], "index": 31}, {"bbox": [125, 512, 486, 524], "spans": [{"bbox": [125, 512, 277, 524], "score": 1.0, "content": "contradicting the assumption that", "type": "text"}, {"bbox": [278, 514, 285, 521], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [286, 512, 363, 524], "score": 1.0, "content": " does not contain ", "type": "text"}, {"bbox": [363, 513, 385, 523], "score": 0.92, "content": "\\sqrt{-1}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [385, 512, 388, 524], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [476, 513, 486, 522], "score": 0.9836314916610718, "content": "\u53e3", "type": "text"}], "index": 32}], "index": 30.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [125, 477, 487, 524]}, {"type": "text", "bbox": [125, 530, 486, 567], "lines": [{"bbox": [126, 532, 487, 545], "spans": [{"bbox": [126, 532, 261, 545], "score": 1.0, "content": "Proposition 7. Suppose that ", "type": "text"}, {"bbox": [261, 534, 291, 543], "score": 0.92, "content": "q_{2}=1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [292, 532, 327, 545], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [327, 533, 355, 544], "score": 0.94, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [355, 532, 487, 545], "score": 1.0, "content": " is essentially ramified if and", "type": "text"}], "index": 33}, {"bbox": [127, 543, 486, 558], "spans": [{"bbox": [127, 543, 158, 558], "score": 1.0, "content": "only if ", "type": "text"}, {"bbox": [158, 546, 189, 555], "score": 0.84, "content": "\\kappa_{2}=1", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [190, 543, 205, 558], "score": 1.0, "content": "; if ", "type": "text"}, {"bbox": [206, 545, 234, 556], "score": 0.9, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [234, 543, 380, 558], "score": 1.0, "content": " is not essentially ramified, then ", "type": "text"}, {"bbox": [380, 545, 424, 556], "score": 0.93, "content": "\\kappa_{2}=\\langle[6]\\rangle", "type": "inline_equation", "height": 11, "width": 44}, {"bbox": [425, 543, 460, 558], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [460, 545, 486, 555], "score": 0.87, "content": "K_{2}=", "type": "inline_equation", "height": 10, "width": 26}], "index": 34}, {"bbox": [126, 556, 223, 568], "spans": [{"bbox": [126, 556, 160, 568], "score": 0.92, "content": "k_{2}(\\sqrt{\\beta}\\,)", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [160, 556, 182, 568], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [182, 556, 219, 568], "score": 0.92, "content": "(\\beta)=\\mathfrak{b}^{2}", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [220, 556, 223, 568], "score": 1.0, "content": ".", "type": "text"}], "index": 35}], "index": 34, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [126, 532, 487, 568]}, {"type": "text", "bbox": [125, 573, 486, 632], "lines": [{"bbox": [126, 574, 485, 587], "spans": [{"bbox": [126, 574, 243, 587], "score": 1.0, "content": "Proof. First notice that if ", "type": "text"}, {"bbox": [243, 576, 271, 586], "score": 0.93, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [271, 574, 415, 587], "score": 1.0, "content": " is not essentially ramified, then ", "type": "text"}, {"bbox": [416, 577, 446, 586], "score": 0.92, "content": "\\kappa_{2}\\neq1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [446, 574, 485, 587], "score": 1.0, "content": ": in fact,", "type": "text"}], "index": 36}, {"bbox": [124, 586, 486, 599], "spans": [{"bbox": [124, 586, 221, 599], "score": 1.0, "content": "in this case we have ", "type": "text"}, {"bbox": [221, 587, 261, 598], "score": 0.93, "content": "(\\beta)\\;=\\;6^{2}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [261, 586, 334, 599], "score": 1.0, "content": ", and if we had ", "type": "text"}, {"bbox": [335, 589, 367, 597], "score": 0.91, "content": "\\,\\kappa_{2}\\,=\\,1", "type": "inline_equation", "height": 8, "width": 32}, {"bbox": [367, 586, 397, 599], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [398, 588, 403, 596], "score": 0.83, "content": "\\mathfrak{b}", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [403, 586, 486, 599], "score": 1.0, "content": " would have to be", "type": "text"}], "index": 37}, {"bbox": [126, 599, 486, 611], "spans": [{"bbox": [126, 599, 188, 611], "score": 1.0, "content": "principal, say ", "type": "text"}, {"bbox": [189, 600, 222, 610], "score": 0.93, "content": "{\\mathfrak{b}}=(\\gamma)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [222, 599, 309, 611], "score": 1.0, "content": ". This implies that", "type": "text"}, {"bbox": [310, 600, 346, 610], "score": 0.93, "content": "\\beta\\,=\\,\\varepsilon\\gamma^{2}", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [347, 599, 412, 611], "score": 1.0, "content": " for some unit ", "type": "text"}, {"bbox": [412, 601, 441, 609], "score": 0.92, "content": "\\varepsilon\\in k_{2}", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [441, 599, 486, 611], "score": 1.0, "content": ", which in", "type": "text"}], "index": 38}, {"bbox": [125, 611, 487, 623], "spans": [{"bbox": [125, 611, 159, 623], "score": 1.0, "content": "view of ", "type": "text"}, {"bbox": [160, 613, 187, 622], "score": 0.92, "content": "q_{2}=1", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [187, 611, 244, 623], "score": 1.0, "content": " implies that", "type": "text"}, {"bbox": [245, 615, 249, 620], "score": 0.89, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 4}, {"bbox": [250, 611, 375, 623], "score": 1.0, "content": " must be a square. But then ", "type": "text"}, {"bbox": [375, 613, 381, 622], "score": 0.91, "content": "\\beta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [382, 611, 487, 623], "score": 1.0, "content": " would be a square, and", "type": "text"}], "index": 39}, {"bbox": [125, 623, 204, 635], "spans": [{"bbox": [125, 623, 204, 635], "score": 1.0, "content": "this is impossible.", "type": "text"}], "index": 40}], "index": 38, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [124, 574, 487, 635]}, {"type": "text", "bbox": [124, 633, 486, 700], "lines": [{"bbox": [138, 635, 487, 647], "spans": [{"bbox": [138, 635, 226, 647], "score": 1.0, "content": "Conversely, suppose", "type": "text"}, {"bbox": [227, 636, 255, 646], "score": 0.92, "content": "\\kappa_{2}\\neq1", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [256, 635, 280, 647], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [280, 639, 285, 644], "score": 0.87, "content": "\\mathfrak{a}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [285, 635, 399, 647], "score": 1.0, "content": " be a nonprincipal ideal in", "type": "text"}, {"bbox": [400, 636, 409, 645], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [410, 635, 487, 647], "score": 1.0, "content": " of absolute norm", "type": "text"}], "index": 41}, {"bbox": [126, 646, 486, 660], "spans": [{"bbox": [126, 652, 132, 657], "score": 0.85, "content": "a", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [132, 647, 216, 660], "score": 1.0, "content": ", and assume that ", "type": "text"}, {"bbox": [216, 649, 250, 659], "score": 0.94, "content": "{\\mathfrak{a}}=(\\alpha)", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [251, 647, 266, 660], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [267, 649, 280, 658], "score": 0.92, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [280, 647, 315, 660], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [316, 646, 363, 659], "score": 0.95, "content": "\\alpha^{1-\\sigma^{2}}=\\eta", "type": "inline_equation", "height": 13, "width": 47}, {"bbox": [363, 647, 429, 660], "score": 1.0, "content": " for some unit ", "type": "text"}, {"bbox": [430, 649, 462, 659], "score": 0.93, "content": "\\eta\\in E_{2}", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [462, 647, 486, 660], "score": 1.0, "content": ", and", "type": "text"}], "index": 42}, {"bbox": [123, 658, 485, 676], "spans": [{"bbox": [123, 658, 167, 676], "score": 1.0, "content": "similarly ", "type": "text"}, {"bbox": [168, 661, 219, 673], "score": 0.93, "content": "\\alpha^{\\sigma-\\sigma^{3}}\\,=\\,\\eta^{\\prime}", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [220, 658, 256, 676], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [256, 663, 264, 673], "score": 0.9, "content": "\\eta^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [265, 658, 324, 676], "score": 1.0, "content": " is a unit in ", "type": "text"}, {"bbox": [324, 663, 336, 673], "score": 0.92, "content": "E_{2}^{\\prime}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [336, 658, 390, 676], "score": 1.0, "content": ". But then ", "type": "text"}, {"bbox": [390, 660, 485, 673], "score": 0.9, "content": "\\eta\\eta^{\\prime}\\,=\\,\\alpha^{1+\\sigma-\\sigma^{2}-\\sigma^{3}}\\,\\stackrel{\\cdot2}{=}", "type": "inline_equation", "height": 13, "width": 95}], "index": 43}, {"bbox": [125, 674, 487, 690], "spans": [{"bbox": [125, 675, 269, 689], "score": 0.89, "content": "N_{L/k}\\alpha\\,=\\,\\pm N_{L/k}\\mathfrak{a}\\,=\\,\\pm a^{2}\\,\\stackrel{?}{=}\\,\\pm1", "type": "inline_equation", "height": 14, "width": 144}, {"bbox": [270, 674, 285, 690], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [285, 677, 299, 686], "score": 0.89, "content": "L^{\\times}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [299, 674, 335, 690], "score": 1.0, "content": ", where", "type": "text"}, {"bbox": [336, 675, 344, 686], "score": 0.71, "content": "\\underline{{\\underline{{2}}}}", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [344, 674, 487, 690], "score": 1.0, "content": " means equal up to a square in", "type": "text"}], "index": 44}, {"bbox": [126, 689, 484, 702], "spans": [{"bbox": [126, 690, 140, 698], "score": 0.89, "content": "L^{\\times}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [140, 689, 172, 702], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [172, 690, 194, 700], "score": 0.92, "content": "\\pm\\eta\\eta^{\\prime}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [194, 689, 258, 702], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [259, 691, 266, 698], "score": 0.88, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [266, 689, 376, 702], "score": 1.0, "content": ", so our assumption that ", "type": "text"}, {"bbox": [376, 691, 405, 700], "score": 0.93, "content": "q_{2}=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [405, 689, 463, 702], "score": 1.0, "content": " implies that ", "type": "text"}, {"bbox": [464, 690, 484, 700], "score": 0.91, "content": "\\pm\\eta\\eta^{\\prime}", "type": "inline_equation", "height": 10, "width": 20}], "index": 45}, {"bbox": [124, 114, 486, 127], "spans": [{"bbox": [124, 114, 218, 127], "score": 1.0, "content": "must be a square in ", "type": "text", "cross_page": true}, {"bbox": [219, 116, 229, 125], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10, "cross_page": true}, {"bbox": [229, 114, 377, 127], "score": 1.0, "content": ". The same argument show that ", "type": "text", "cross_page": true}, {"bbox": [378, 115, 404, 126], "score": 0.94, "content": "\\pm\\eta/\\eta^{\\prime}", "type": "inline_equation", "height": 11, "width": 26, "cross_page": true}, {"bbox": [405, 114, 472, 127], "score": 1.0, "content": " is a square in ", "type": "text", "cross_page": true}, {"bbox": [472, 116, 482, 125], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10, "cross_page": true}, {"bbox": [482, 114, 486, 127], "score": 1.0, "content": ",", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [124, 126, 487, 140], "spans": [{"bbox": [124, 126, 189, 140], "score": 1.0, "content": "hence we find ", "type": "text", "cross_page": true}, {"bbox": [189, 129, 217, 139], "score": 0.93, "content": "\\eta\\in k_{2}", "type": "inline_equation", "height": 10, "width": 28, "cross_page": true}, {"bbox": [218, 126, 250, 140], "score": 1.0, "content": ". Thus ", "type": "text", "cross_page": true}, {"bbox": [251, 126, 276, 137], "score": 0.93, "content": "\\alpha^{1-\\sigma^{2}}", "type": "inline_equation", "height": 11, "width": 25, "cross_page": true}, {"bbox": [277, 126, 329, 140], "score": 1.0, "content": " is fixed by ", "type": "text", "cross_page": true}, {"bbox": [329, 128, 340, 137], "score": 0.91, "content": "\\sigma^{2}", "type": "inline_equation", "height": 9, "width": 11, "cross_page": true}, {"bbox": [340, 126, 375, 140], "score": 1.0, "content": " and so ", "type": "text", "cross_page": true}, {"bbox": [375, 128, 433, 139], "score": 0.93, "content": "\\beta:=\\alpha^{2}\\in k_{2}", "type": "inline_equation", "height": 11, "width": 58, "cross_page": true}, {"bbox": [433, 126, 487, 140], "score": 1.0, "content": ". This gives", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [126, 139, 487, 151], "spans": [{"bbox": [126, 140, 186, 151], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\beta}\\,)", "type": "inline_equation", "height": 11, "width": 60, "cross_page": true}, {"bbox": [186, 139, 219, 151], "score": 1.0, "content": ", hence ", "type": "text", "cross_page": true}, {"bbox": [219, 141, 247, 151], "score": 0.94, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 10, "width": 28, "cross_page": true}, {"bbox": [248, 139, 432, 151], "score": 1.0, "content": " is not essentially ramified, and moreover, ", "type": "text", "cross_page": true}, {"bbox": [433, 141, 456, 149], "score": 0.91, "content": "a\\sim{\\mathfrak{b}}", "type": "inline_equation", "height": 8, "width": 23, "cross_page": true}, {"bbox": [456, 139, 462, 151], "score": 1.0, "content": ".", "type": "text", "cross_page": true}, {"bbox": [465, 140, 487, 150], "score": 0.9668625593185425, "content": "\u518f\u53e3", "type": "text", "cross_page": true}], "index": 2}], "index": 43, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [123, 635, 487, 702]}]}	{"layout_dets": [{"category_id": 1, "poly": [349, 516, 1351, 516, 1351, 749, 349, 749], "score": 0.977}, {"category_id": 1, "poly": [348, 1321, 1352, 1321, 1352, 1452, 348, 1452], "score": 0.969}, {"category_id": 1, "poly": [347, 1760, 1352, 1760, 1352, 1946, 347, 1946], "score": 0.969}, {"category_id": 1, "poly": [349, 1592, 1352, 1592, 1352, 1758, 349, 1758], "score": 0.966}, {"category_id": 1, "poly": [348, 851, 1352, 851, 1352, 1085, 348, 1085], "score": 0.963}, {"category_id": 1, "poly": [348, 1086, 1351, 1086, 1351, 1251, 348, 1251], "score": 0.957}, {"category_id": 1, "poly": [349, 397, 1352, 397, 1352, 499, 349, 499], "score": 0.95}, {"category_id": 1, "poly": [348, 751, 1351, 751, 1351, 850, 348, 850], "score": 0.947}, {"category_id": 1, "poly": [349, 1473, 1352, 1473, 1352, 1575, 349, 1575], "score": 0.946}, 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This definition is independent of the choice of . Proposition 6. Let be a non-CM totally complex dihedral extension not con- taining , and assume that and are essentially ramified. If the fundamental unit of the real quadratic subfield of has norm , then . Proof. Notice first that cannot be real (in fact, is not totally real by assumption, and since is a cyclic quartic extension, no infinite prime can ramify in ); thus exactly one of , is real, and the other is complex. Multiplying the class number formulas, Proposition 3, for and (note that since both and are essentially ramified) we find that is a square. If we can prove that , then is a square between 2 and 8, which implies that we must have and as claimed. We start by remarking that if becomes a square in , where is a root of unity in , then so does one of . This follows from the fact that the only non-trivial roots of unity that can be in are the sixth roots of unity , and here . Now we prove that under the assumptions we made; the claim will then follow by symmetry. Assume first that is real and let be the fundamental unit of . We claim that . Suppose otherwise; then is one of , or . If , then and . (Here and below .) This however cannot occur since by assumption implying that , a contradiction. Similarly, if , then again . Thus , and for some unit . Suppose that for some unit . Then , contradicting our assumption that is essentially ramified. The same argument shows that , hence either and or for some unit and . Here is a root of unity generating the torsion subgroup of . Next consider the case where is complex, and let denote the fundamental unit of . Then stays fundamental in by the argument above. Let be a fundamental unit in . If became a square in , then clearly could not be essentially ramified. Thus if we have , then is a square in . Applying to this relation we find that is a square in , contradicting the assumption that does not contain . 口 Proposition 7. Suppose that . Then is essentially ramified if and only if ; if is not essentially ramified, then , where and . Proof. First notice that if is not essentially ramified, then : in fact, in this case we have , and if we had , then would have to be principal, say . This implies that for some unit , which in view of implies that must be a square. But then would be a square, and this is impossible. Conversely, suppose . Let be a nonprincipal ideal in of absolute norm , and assume that in . Then for some unit , and similarly , where is a unit in . But then in , where means equal up to a square in . Thus is a square in , so our assumption that implies that must be a square in . The same argument show that is a square in , hence we find . Thus is fixed by and so . This gives , hence is not essentially ramified, and moreover, . 冏口	<div class="pdf-page"> <p>Recall that a quadratic extension is called essentially ramified if is not an ideal square. This definition is independent of the choice of .</p> <p>Proposition 6. Let be a non-CM totally complex dihedral extension not con- taining , and assume that and are essentially ramified. If the fundamental unit of the real quadratic subfield of has norm , then .</p> <p>Proof. Notice first that cannot be real (in fact, is not totally real by assumption, and since is a cyclic quartic extension, no infinite prime can ramify in ); thus exactly one of , is real, and the other is complex. Multiplying the class number formulas, Proposition 3, for and (note that since both and are essentially ramified) we find that is a square. If we can prove that , then is a square between 2 and 8, which implies that we must have and as claimed.</p> <p>We start by remarking that if becomes a square in , where is a root of unity in , then so does one of . This follows from the fact that the only non-trivial roots of unity that can be in are the sixth roots of unity , and here .</p> <p>Now we prove that under the assumptions we made; the claim will then follow by symmetry. Assume first that is real and let be the fundamental unit of . We claim that . Suppose otherwise; then is one of , or . If , then and . (Here and below .) This however cannot occur since by assumption implying that , a contradiction. Similarly, if , then again .</p> <p>Thus , and for some unit . Suppose that for some unit . Then , contradicting our assumption that is essentially ramified. The same argument shows that , hence either and or for some unit and . Here is a root of unity generating the torsion subgroup of .</p> <p>Next consider the case where is complex, and let denote the fundamental unit of . Then stays fundamental in by the argument above.</p> <p>Let be a fundamental unit in . If became a square in , then clearly could not be essentially ramified. Thus if we have , then is a square in . Applying to this relation we find that is a square in , contradicting the assumption that does not contain . 口</p> <p>Proposition 7. Suppose that . Then is essentially ramified if and only if ; if is not essentially ramified, then , where and .</p> <p>Proof. First notice that if is not essentially ramified, then : in fact, in this case we have , and if we had , then would have to be principal, say . This implies that for some unit , which in view of implies that must be a square. But then would be a square, and this is impossible.</p> <p>Conversely, suppose . Let be a nonprincipal ideal in of absolute norm , and assume that in . Then for some unit , and similarly , where is a unit in . But then in , where means equal up to a square in . Thus is a square in , so our assumption that implies that must be a square in . The same argument show that is a square in , hence we find . Thus is fixed by and so . This gives , hence is not essentially ramified, and moreover, . 冏口</p> </div>	<div class="pdf-page"> <div class="pdf-discarded" data-x="209" data-y="117" data-width="10" data-height="11" style="opacity: 0.5;">8</div> <p class="pdf-text" data-x="209" data-y="143" data-width="604" data-height="32">Recall that a quadratic extension is called essentially ramified if is not an ideal square. This definition is independent of the choice of .</p> <p class="pdf-text" data-x="209" data-y="183" data-width="604" data-height="48">Proposition 6. Let be a non-CM totally complex dihedral extension not con- taining , and assume that and are essentially ramified. If the fundamental unit of the real quadratic subfield of has norm , then .</p> <p class="pdf-text" data-x="209" data-y="239" data-width="604" data-height="108">Proof. Notice first that cannot be real (in fact, is not totally real by assumption, and since is a cyclic quartic extension, no infinite prime can ramify in ); thus exactly one of , is real, and the other is complex. Multiplying the class number formulas, Proposition 3, for and (note that since both and are essentially ramified) we find that is a square. If we can prove that , then is a square between 2 and 8, which implies that we must have and as claimed.</p> <p class="pdf-text" data-x="209" data-y="349" data-width="604" data-height="46">We start by remarking that if becomes a square in , where is a root of unity in , then so does one of . This follows from the fact that the only non-trivial roots of unity that can be in are the sixth roots of unity , and here .</p> <p class="pdf-text" data-x="209" data-y="395" data-width="604" data-height="109">Now we prove that under the assumptions we made; the claim will then follow by symmetry. Assume first that is real and let be the fundamental unit of . We claim that . Suppose otherwise; then is one of , or . If , then and . (Here and below .) This however cannot occur since by assumption implying that , a contradiction. Similarly, if , then again .</p> <p class="pdf-text" data-x="209" data-y="504" data-width="604" data-height="77">Thus , and for some unit . Suppose that for some unit . Then , contradicting our assumption that is essentially ramified. The same argument shows that , hence either and or for some unit and . Here is a root of unity generating the torsion subgroup of .</p> <p class="pdf-text" data-x="209" data-y="583" data-width="604" data-height="29">Next consider the case where is complex, and let denote the fundamental unit of . Then stays fundamental in by the argument above.</p> <p class="pdf-text" data-x="209" data-y="614" data-width="604" data-height="60">Let be a fundamental unit in . If became a square in , then clearly could not be essentially ramified. Thus if we have , then is a square in . Applying to this relation we find that is a square in , contradicting the assumption that does not contain . 口</p> <p class="pdf-text" data-x="209" data-y="685" data-width="604" data-height="48">Proposition 7. Suppose that . Then is essentially ramified if and only if ; if is not essentially ramified, then , where and .</p> <p class="pdf-text" data-x="209" data-y="740" data-width="604" data-height="77">Proof. First notice that if is not essentially ramified, then : in fact, in this case we have , and if we had , then would have to be principal, say . This implies that for some unit , which in view of implies that must be a square. But then would be a square, and this is impossible.</p> <p class="pdf-text" data-x="207" data-y="818" data-width="606" data-height="87">Conversely, suppose . Let be a nonprincipal ideal in of absolute norm , and assume that in . Then for some unit , and similarly , where is a unit in . But then in , where means equal up to a square in . Thus is a square in , so our assumption that implies that must be a square in . The same argument show that is a square in , hence we find . Thus is fixed by and so . 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This definition is independent of the choice of \\alpha .", "Proposition 6. Let L/\\mathbb{Q} be a non-CM totally complex dihedral extension not con-", "taining \\sqrt{-1} , and assume that L/K_{1} and L/K_{2} are essentially ramified. If the", "fundamental unit of the real quadratic subfield of K has norm ^{-1} , then q_{1}q_{2}=2 .", "Proof. Notice first that k cannot be real (in fact, K is not totally real by assumption,", "and since L/k is a cyclic quartic extension, no infinite prime can ramify in K/k );", "thus exactly one of k_{1} , k_{2} is real, and the other is complex. Multiplying the class", "number formulas, Proposition 3, for L/k_{1} and L/k_{2} (note that \\upsilon=0 since both", "L/K_{1} and L/K_{2} are essentially ramified) we find that 2q_{1}q_{2} is a square. If we can", "prove that q_{1},q_{2}\\leq2 , then 2q_{1}q_{2} is a square between 2 and 8, which implies that", "we must have 2q_{1}q_{2}=4 and q_{1}q_{2}=2 as claimed.", "We start by remarking that if \\zeta\\eta becomes a square in L , where \\zeta is a root of unity", "in L , then so does one of \\pm\\eta . This follows from the fact that the only non-trivial", "roots of unity that can be in L are the sixth roots of unity \\langle\\zeta_{6}\\rangle , and here \\zeta_{6}=-\\zeta_{3}^{2} .", "Now we prove that q_{1}\\leq2 under the assumptions we made; the claim q_{2}\\leq2 will", "then follow by symmetry. Assume first that k_{1} is real and let \\varepsilon be the fundamental", "unit of k_{1} . We claim that \\sqrt{\\pm\\varepsilon}\\notin L . Suppose otherwise; then k_{1}(\\sqrt{\\pm\\varepsilon}) is one of", "K_{1} , K_{1}^{\\prime} or K . If k_{1}(\\sqrt{\\pm\\varepsilon}\\,)=K_{1} , then K_{1}^{\\prime}=k_{1}(\\sqrt{\\pm\\varepsilon^{\\prime}}\\,) and K=k_{1}\\big(\\sqrt{\\varepsilon\\varepsilon^{\\prime}}\\big) . (Here", "and below x^{\\prime}\\,=\\,x^{\\sigma} .) This however cannot occur since by assumption \\varepsilon\\varepsilon^{\\prime}\\,=\\,-1", "implying that \\sqrt{-1}\\in L , a contradiction. Similarly, if k_{1}({\\sqrt{\\pm\\varepsilon}}\\,)\\,=\\,K , then again", "\\sqrt{-1}\\in L .", "Thus \\sqrt{\\pm\\varepsilon}~\\notin~L , and E_{1}~=~\\langle-1,\\varepsilon,\\eta\\rangle for some unit \\eta\\ \\in\\ E_{1} . Suppose that", "\\sqrt{u\\eta}\\in L for some unit u\\in k_{1} . Then L=K_{1}(\\sqrt{u\\eta}\\,) , contradicting our assumption", "that L/K_{1} is essentially ramified. The same argument shows that \\sqrt{u\\eta^{\\prime}}\\notin L , hence", "either E_{L}\\,=\\,\\langle\\zeta,\\varepsilon,\\eta,\\eta^{\\prime}\\rangle and q_{1}=1 or E_{L}=\\langle\\zeta,\\varepsilon,\\eta,\\sqrt{u\\eta\\eta^{\\prime}}\\rangle for some unit u\\in k_{1}", "and q_{1}=2 . Here \\zeta is a root of unity generating the torsion subgroup W_{L} of E_{L} .", "Next consider the case where k_{1} is complex, and let \\varepsilon denote the fundamental", "unit of k_{2} . Then \\pm\\varepsilon stays fundamental in L by the argument above.", "Let \\eta be a fundamental unit in K_{1} . If \\pm\\eta became a square in L , then clearly", "L/K_{1} could not be essentially ramified. Thus if we have q_{1}\\geq4 , then \\pm\\varepsilon\\eta=\\alpha^{2} is", "a square in L . Applying \\tau to this relation we find that -1=\\varepsilon\\varepsilon^{\\prime} is a square in L ,", "contradicting the assumption that L does not contain \\sqrt{-1} . 口", "Proposition 7. Suppose that q_{2}=1 . Then K_{2}/k_{2} is essentially ramified if and", "only if \\kappa_{2}=1 ; if K_{2}/k_{2} is not essentially ramified, then \\kappa_{2}=\\langle[6]\\rangle , where K_{2}=", "k_{2}(\\sqrt{\\beta}\\,) and (\\beta)=\\mathfrak{b}^{2} .", "Proof. First notice that if K_{2}/k_{2} is not essentially ramified, then \\kappa_{2}\\neq1 : in fact,", "in this case we have (\\beta)\\;=\\;6^{2} , and if we had \\,\\kappa_{2}\\,=\\,1 , then \\mathfrak{b} would have to be", "principal, say {\\mathfrak{b}}=(\\gamma) . This implies that \\beta\\,=\\,\\varepsilon\\gamma^{2} for some unit \\varepsilon\\in k_{2} , which in", "view of q_{2}=1 implies that \\varepsilon must be a square. But then \\beta would be a square, and", "this is impossible.", "Conversely, suppose \\kappa_{2}\\neq1 . Let \\mathfrak{a} be a nonprincipal ideal in k_{2} of absolute norm", "a , and assume that {\\mathfrak{a}}=(\\alpha) in K_{2} . Then \\alpha^{1-\\sigma^{2}}=\\eta for some unit \\eta\\in E_{2} , and", "similarly \\alpha^{\\sigma-\\sigma^{3}}\\,=\\,\\eta^{\\prime} , where \\eta^{\\prime} is a unit in E_{2}^{\\prime} . But then \\eta\\eta^{\\prime}\\,=\\,\\alpha^{1+\\sigma-\\sigma^{2}-\\sigma^{3}}\\,\\stackrel{\\cdot2}{=}", "N_{L/k}\\alpha\\,=\\,\\pm N_{L/k}\\mathfrak{a}\\,=\\,\\pm a^{2}\\,\\stackrel{?}{=}\\,\\pm1 in L^{\\times} , where \\underline{{\\underline{{2}}}} means equal up to a square in", "L^{\\times} . Thus \\pm\\eta\\eta^{\\prime} is a square in L , so our assumption that q_{2}=1 implies that \\pm\\eta\\eta^{\\prime}", "must be a square in k_{2} . The same argument show that \\pm\\eta/\\eta^{\\prime} is a square in k_{2} ,", "hence we find \\eta\\in k_{2} . Thus \\alpha^{1-\\sigma^{2}} is fixed by \\sigma^{2} and so \\beta:=\\alpha^{2}\\in k_{2} . This gives", "K_{2}=k_{2}(\\sqrt{\\beta}\\,) , hence K_{2}/k_{2} is not essentially ramified, and moreover, a\\sim{\\mathfrak{b}} . 冏口" ], "index": [ 0, 1, 2, 3, 4, 7, 8, 9, 10, 11, 12, 13, 19, 20, 21, 34, 35, 36, 37, 38, 39, 40, 56, 57, 58, 59, 60, 83, 84, 112, 113, 114, 115, 145, 146, 147, 181, 182, 183, 184, 185, 222, 223, 224, 225, 226, 227, 228, 229 ] }	{ "coordinates": [], "index": [], "caption": [], "caption_coordinates": [] }	{ "type": [], "coordinates": [], "content": [], "index": [] }	{ "type": [], "coordinates": [], "content": [], "index": [] }	{"document_info": {"total_pages": 14, "parse_type": "txt", "version": "1.0.1", "total_content_list_items": 155}, "content_statistics": {"total_blocks": 182, "total_lines": 526, "total_images": 4, "total_equations": 28, "total_tables": 11, "total_characters": 29658, "total_words": 5331, "block_type_distribution": {"title": 7, "text": 126, "interline_equation": 14, "discarded": 24, "table_caption": 3, "table_body": 4, "image_body": 2, "list": 2}}, "model_statistics": {"total_layout_detections": 2826, "avg_confidence": 0.9470955414012739, "min_confidence": 0.26, "max_confidence": 1.0}, "page_statistics": [{"page_number": 0, "page_size": [612.0, 792.0], "blocks": 11, "lines": 38, "images": 0, "equations": 2, "tables": 0, "characters": 2362, "words": 398, "discarded_blocks": 1}, {"page_number": 1, "page_size": [612.0, 792.0], "blocks": 10, "lines": 30, "images": 0, "equations": 0, "tables": 3, "characters": 1688, "words": 305, "discarded_blocks": 1}, {"page_number": 2, "page_size": [612.0, 792.0], "blocks": 15, "lines": 44, "images": 0, "equations": 4, "tables": 0, "characters": 2417, "words": 416, "discarded_blocks": 2}, {"page_number": 3, "page_size": [612.0, 792.0], "blocks": 7, "lines": 27, "images": 0, "equations": 0, "tables": 3, "characters": 1591, "words": 291, "discarded_blocks": 1}, {"page_number": 4, "page_size": [612.0, 792.0], "blocks": 23, "lines": 38, "images": 0, "equations": 6, "tables": 0, "characters": 1604, "words": 303, "discarded_blocks": 4}, {"page_number": 5, "page_size": [612.0, 792.0], "blocks": 17, "lines": 40, "images": 0, "equations": 6, "tables": 0, "characters": 2131, "words": 393, "discarded_blocks": 1}, {"page_number": 6, "page_size": [612.0, 792.0], "blocks": 15, "lines": 36, "images": 2, "equations": 4, "tables": 0, "characters": 1791, "words": 324, "discarded_blocks": 2}, {"page_number": 7, "page_size": [612.0, 792.0], "blocks": 12, "lines": 49, "images": 0, "equations": 0, "tables": 0, "characters": 3049, "words": 571, "discarded_blocks": 1}, {"page_number": 8, "page_size": [612.0, 792.0], "blocks": 11, "lines": 42, "images": 0, "equations": 0, "tables": 0, "characters": 2636, "words": 479, "discarded_blocks": 2}, {"page_number": 9, "page_size": [612.0, 792.0], "blocks": 14, "lines": 35, "images": 2, "equations": 0, "tables": 0, "characters": 2036, "words": 375, "discarded_blocks": 1}, {"page_number": 10, "page_size": [612.0, 792.0], "blocks": 17, "lines": 35, "images": 0, "equations": 6, "tables": 0, "characters": 1789, "words": 327, "discarded_blocks": 2}, {"page_number": 11, "page_size": [612.0, 792.0], "blocks": 14, "lines": 40, "images": 0, "equations": 0, "tables": 2, "characters": 2094, "words": 399, "discarded_blocks": 2}, {"page_number": 12, "page_size": [612.0, 792.0], "blocks": 10, "lines": 44, "images": 0, "equations": 0, "tables": 3, "characters": 2572, "words": 487, "discarded_blocks": 2}, {"page_number": 13, "page_size": [612.0, 792.0], "blocks": 6, "lines": 28, "images": 0, "equations": 0, "tables": 0, "characters": 1898, "words": 263, "discarded_blocks": 2}]}
0003244v1	8	[ 612, 792 ]	{ "type": [ "text", "text", "text", "text", "text", "text", "text", "text", "text", "discarded", "discarded" ], "coordinates": [ [ 209, 144, 813, 193 ], [ 209, 202, 813, 234 ], [ 207, 276, 813, 497 ], [ 209, 502, 813, 552 ], [ 209, 559, 813, 624 ], [ 209, 632, 813, 663 ], [ 210, 671, 813, 733 ], [ 209, 734, 813, 827 ], [ 209, 827, 813, 905 ], [ 398, 116, 624, 128 ], [ 801, 117, 813, 128 ] ], "content": [ "", "From now on assume that is one of the imaginary quadratic fields of type A) or ) as explained in the Introduction. Let", "Then there exist two unramified cyclic quartic extensions of which are over (see Proposition 2). Let us say a few words about their construction. Consider e.g. case B); by R´edei’s theory (see [12]), the -factorization implies that unramified cyclic quartic extensions of are constructed by choosing a “primitive” solution of and putting with (primitive here means that should not be divisible by rational integers); the other unramified cyclic quartic extension is then . If we put , then it is an elementary exerc i se to show that is a square in , hence we also have etc. If , then it is easy to see that we may choose as the fundamental unit of ; if , then genus theory says that a) the class number of is twice an odd number ; and b) the prime ideal above in is in the principal genus, so is principal. Again it can be checked that for a suitable choice of the sign.", "Example. Consider the case ; here , and the positive sign is correct since mod 4 is primary. The minimal polynomial of is : compare Table 1.", "The fields and will play a dominant role in the proof below; they are both contai n ed in for , and it is the ambiguous class group that contains the information we are interested in.", "Lemma 6. The field has odd class number (even in the strict sense), and we have 2. In particular, is cyclic (though possibly trivial).", "Proof. The class group in the strict sense of is cyclic of order 2 by R´edei’s theory [12] (since in case A) and in case B)). Since is the Hilbert class field of in the strict sense, its class number in the strict sense is odd.", "Next we apply the ambiguous class number formula. In case A), is complex, and exactly the two primes above ramify in . Note that with primary of norm ; there are four primes above in , and exactly two of them divide to an odd power, so by the decomposition law in quadratic Kummer extensions. By Proposition 4 and the remarks following it, , and is cyclic.", "In case B), however, is real; since has norm , it has mixed signature, hence there are exactly two infinite primes that ramify in . As in case A), there are two finite primes above that ramify in , so we get . Since has odd class number in the strict sense, has units of independent signs. This implies that the group of units that are positive at the two ramified infinite primes has -rank 2, i.e. by consideration of the infinite primes alone. In particular, in case B). 口", "IMAGINARY QUADRATIC FIELDS", "9" ], "index": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ] }	[{"type": "text", "text": "", "page_idx": 8}, {"type": "text", "text": "From now on assume that $k$ is one of the imaginary quadratic fields of type A) or $\\mathrm{B}$ ) as explained in the Introduction. Let ", "page_idx": 8}, {"type": "text", "text": "Then there exist two unramified cyclic quartic extensions of $k$ which are $D_{4}$ over $\\mathbb{Q}$ (see Proposition 2). Let us say a few words about their construction. Consider e.g. case B); by R\u00b4edei\u2019s theory (see [12]), the $C_{4}$ -factorization $d=d_{1}d_{2}\\cdot d_{3}$ implies that unramified cyclic quartic extensions of $k\\,=\\,\\mathbb{Q}({\\sqrt{d}}\\,)$ are constructed by choosing a \u201cprimitive\u201d solution $\\left(x,y,z\\right)$ of $d_{1}d_{2}X^{2}+d_{3}Y^{2}\\,=\\,Z^{2}$ and putting $L=k(\\sqrt{d_{1}d_{2}},\\sqrt{\\alpha}\\,)$ with $\\alpha=z+x\\sqrt{d_{1}d_{2}}$ (primitive here means that $\\alpha$ should not be divisible by rational integers); the other unramified cyclic quartic extension is then $\\widetilde{L}=k(\\sqrt{d_{1}d_{2}},\\sqrt{d_{1}\\alpha}\\,)$ . If we put $\\beta\\,=\\,{\\textstyle\\frac{1}{2}}(z+y\\sqrt{d_{3}}\\,)$ , then it is an elementary exerc i se to show that $\\alpha\\beta$ is a square in $L$ , hence we also have $L=k(\\sqrt{d_{3}},\\sqrt{\\beta}\\,)$ etc. If $d_{3}=-4$ , then it is easy to see that we may choose $\\beta$ as the fundamental unit of $k_{2}$ ; if $d_{3}\\neq-4$ , then genus theory says that a) the class number $h$ of $k_{2}$ is twice an odd number $u$ ; and b) the prime ideal ${\\mathfrak{p}}_{3}$ above $d_{3}$ in $k_{2}$ is in the principal genus, so ${\\mathfrak{p}}_{3}^{u}=(\\pi_{3})$ is principal. Again it can be checked that $\\beta=\\pm\\pi_{3}$ for a suitable choice of the sign. ", "page_idx": 8}, {"type": "text", "text": "Example. Consider the case $d\\,=\\,-31\\cdot5\\cdot8$ ; here $\\pi_{3}\\,=\\,\\pm(3+2{\\sqrt{10}}\\,)$ , and the positive sign is correct since $3\\,{+}\\,2{\\sqrt{10}}\\equiv(1\\,{+}\\,{\\sqrt{10}}\\,)^{2}$ mod 4 is primary. The minimal polynomial of $\\sqrt{\\pi_{3}}$ is $f(x)=x^{4}-6x^{2}-31$ : compare Table 1. ", "page_idx": 8}, {"type": "text", "text": "The fields $K_{2}=k_{2}(\\sqrt{\\alpha}\\,)$ and $\\tilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)$ will play a dominant role in the proof below; they are both contai n ed in $M=F({\\sqrt{\\alpha}}\\,)$ for $F=k_{2}(\\sqrt{d_{2}}\\,)$ , and it is the ambiguous class group $\\mathrm{Am}(M/F)$ that contains the information we are interested in. ", "page_idx": 8}, {"type": "text", "text": "Lemma 6. The field $F$ has odd class number (even in the strict sense), and we have $\\#\\operatorname{Am}(M/F)\\mid$ 2. In particular, $\\mathrm{Cl_{2}}(M)$ is cyclic (though possibly trivial). ", "page_idx": 8}, {"type": "text", "text": "Proof. The class group in the strict sense of $k_{2}$ is cyclic of order 2 by R\u00b4edei\u2019s theory [12] (since $(d_{2}/p_{3})=(d_{3}/p_{2})=-1$ in case A) and $(d_{1}/p_{2})=(d_{2}/p_{1})=-1$ in case B)). Since $F$ is the Hilbert class field of $k_{2}$ in the strict sense, its class number in the strict sense is odd. ", "page_idx": 8}, {"type": "text", "text": "Next we apply the ambiguous class number formula. In case A), $F$ is complex, and exactly the two primes above $d_{3}$ ramify in $M/F$ . Note that $M\\,=\\,F({\\sqrt{\\alpha}}\\,)$ with $\\alpha$ primary of norm $d_{3}y^{2}$ ; there are four primes above $d_{3}$ in $F$ , and exactly two of them divide $\\alpha$ to an odd power, so $t\\ =\\ 2$ by the decomposition law in quadratic Kummer extensions. By Proposition 4 and the remarks following it, $\\#\\operatorname{Am}_{2}(M/F)=2/(E:H)\\leq2$ , and $\\mathrm{Cl_{2}}(M)$ is cyclic. ", "page_idx": 8}, {"type": "text", "text": "In case B), however, $F$ is real; since $\\alpha\\,\\in\\,k_{2}$ has norm $d_{3}y^{2}\\,<\\,0$ , it has mixed signature, hence there are exactly two infinite primes that ramify in $M/F$ . As in case A), there are two finite primes above $d_{3}$ that ramify in $M/F$ , so we get $\\#\\operatorname{Am}_{2}(M/F)=8/(E:H)$ . Since $F$ has odd class number in the strict sense, $F$ has units of independent signs. This implies that the group of units that are positive at the two ramified infinite primes has $\\mathbb{Z}$ -rank 2, i.e. $(E:H)\\geq4$ by consideration of the infinite primes alone. In particular, $\\#\\operatorname{Am}_{2}(M/F)\\leq2$ in case B). \u53e3 ", "page_idx": 8}]	{"preproc_blocks": [{"type": "text", "bbox": [125, 112, 486, 150], "lines": [{"bbox": [124, 114, 486, 127], "spans": [{"bbox": [124, 114, 218, 127], "score": 1.0, "content": "must be a square in ", "type": "text"}, {"bbox": [219, 116, 229, 125], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [229, 114, 377, 127], "score": 1.0, "content": ". The same argument show that ", "type": "text"}, {"bbox": [378, 115, 404, 126], "score": 0.94, "content": "\\pm\\eta/\\eta^{\\prime}", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [405, 114, 472, 127], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [472, 116, 482, 125], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [482, 114, 486, 127], "score": 1.0, "content": ",", "type": "text"}], "index": 0}, {"bbox": [124, 126, 487, 140], "spans": [{"bbox": [124, 126, 189, 140], "score": 1.0, "content": "hence we find ", "type": "text"}, {"bbox": [189, 129, 217, 139], "score": 0.93, "content": "\\eta\\in k_{2}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [218, 126, 250, 140], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [251, 126, 276, 137], "score": 0.93, "content": "\\alpha^{1-\\sigma^{2}}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [277, 126, 329, 140], "score": 1.0, "content": " is fixed by ", "type": "text"}, {"bbox": [329, 128, 340, 137], "score": 0.91, "content": "\\sigma^{2}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [340, 126, 375, 140], "score": 1.0, "content": " and so ", "type": "text"}, {"bbox": [375, 128, 433, 139], "score": 0.93, "content": "\\beta:=\\alpha^{2}\\in k_{2}", "type": "inline_equation", "height": 11, "width": 58}, {"bbox": [433, 126, 487, 140], "score": 1.0, "content": ". This gives", "type": "text"}], "index": 1}, {"bbox": [126, 139, 487, 151], "spans": [{"bbox": [126, 140, 186, 151], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\beta}\\,)", "type": "inline_equation", "height": 11, "width": 60}, {"bbox": [186, 139, 219, 151], "score": 1.0, "content": ", hence ", "type": "text"}, {"bbox": [219, 141, 247, 151], "score": 0.94, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [248, 139, 432, 151], "score": 1.0, "content": " is not essentially ramified, and moreover, ", "type": "text"}, {"bbox": [433, 141, 456, 149], "score": 0.91, "content": "a\\sim{\\mathfrak{b}}", "type": "inline_equation", "height": 8, "width": 23}, {"bbox": [456, 139, 462, 151], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [465, 140, 487, 150], "score": 0.9668625593185425, "content": "\u518f\u53e3", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [125, 157, 486, 181], "lines": [{"bbox": [137, 159, 484, 172], "spans": [{"bbox": [137, 159, 255, 172], "score": 1.0, "content": "From now on assume that ", "type": "text"}, {"bbox": [255, 162, 261, 169], "score": 0.89, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [261, 159, 484, 172], "score": 1.0, "content": " is one of the imaginary quadratic fields of type A)", "type": "text"}], "index": 3}, {"bbox": [126, 172, 316, 183], "spans": [{"bbox": [126, 172, 137, 183], "score": 1.0, "content": "or", "type": "text"}, {"bbox": [138, 173, 145, 181], "score": 0.43, "content": "\\mathrm{B}", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [146, 172, 316, 183], "score": 1.0, "content": ") as explained in the Introduction. Let", "type": "text"}], "index": 4}], "index": 3.5}, {"type": "text", "bbox": [124, 214, 486, 385], "lines": [{"bbox": [138, 217, 484, 228], "spans": [{"bbox": [138, 217, 414, 228], "score": 1.0, "content": "Then there exist two unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [414, 218, 420, 225], "score": 0.89, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [420, 217, 471, 228], "score": 1.0, "content": " which are ", "type": "text"}, {"bbox": [472, 218, 484, 227], "score": 0.92, "content": "D_{4}", "type": "inline_equation", "height": 9, "width": 12}], "index": 5}, {"bbox": [126, 229, 486, 240], "spans": [{"bbox": [126, 229, 148, 240], "score": 1.0, "content": "over ", "type": "text"}, {"bbox": [149, 230, 157, 239], "score": 0.9, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [157, 229, 486, 240], "score": 1.0, "content": " (see Proposition 2). Let us say a few words about their construction.", "type": "text"}], "index": 6}, {"bbox": [126, 240, 484, 252], "spans": [{"bbox": [126, 240, 360, 252], "score": 1.0, "content": "Consider e.g. case B); by R\u00b4edei\u2019s theory (see [12]), the", "type": "text"}, {"bbox": [361, 242, 372, 250], "score": 0.91, "content": "C_{4}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [373, 240, 433, 252], "score": 1.0, "content": "-factorization ", "type": "text"}, {"bbox": [433, 242, 484, 250], "score": 0.93, "content": "d=d_{1}d_{2}\\cdot d_{3}", "type": "inline_equation", "height": 8, "width": 51}], "index": 7}, {"bbox": [125, 252, 486, 264], "spans": [{"bbox": [125, 253, 359, 264], "score": 1.0, "content": "implies that unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [359, 252, 412, 264], "score": 0.94, "content": "k\\,=\\,\\mathbb{Q}({\\sqrt{d}}\\,)", "type": "inline_equation", "height": 12, "width": 53}, {"bbox": [413, 253, 486, 264], "score": 1.0, "content": " are constructed", "type": "text"}], "index": 8}, {"bbox": [124, 263, 487, 278], "spans": [{"bbox": [124, 263, 282, 278], "score": 1.0, "content": "by choosing a \u201cprimitive\u201d solution ", "type": "text"}, {"bbox": [283, 266, 316, 276], "score": 0.93, "content": "\\left(x,y,z\\right)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [316, 263, 331, 278], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [332, 265, 428, 275], "score": 0.92, "content": "d_{1}d_{2}X^{2}+d_{3}Y^{2}\\,=\\,Z^{2}", "type": "inline_equation", "height": 10, "width": 96}, {"bbox": [429, 263, 487, 278], "score": 1.0, "content": " and putting", "type": "text"}], "index": 9}, {"bbox": [126, 276, 487, 289], "spans": [{"bbox": [126, 277, 208, 288], "score": 0.94, "content": "L=k(\\sqrt{d_{1}d_{2}},\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 82}, {"bbox": [209, 276, 234, 289], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [234, 277, 305, 288], "score": 0.93, "content": "\\alpha=z+x\\sqrt{d_{1}d_{2}}", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [305, 276, 428, 289], "score": 1.0, "content": " (primitive here means that ", "type": "text"}, {"bbox": [429, 281, 435, 286], "score": 0.89, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [436, 276, 487, 289], "score": 1.0, "content": " should not", "type": "text"}], "index": 10}, {"bbox": [125, 288, 487, 301], "spans": [{"bbox": [125, 288, 487, 301], "score": 1.0, "content": "be divisible by rational integers); the other unramified cyclic quartic extension is", "type": "text"}], "index": 11}, {"bbox": [126, 300, 486, 315], "spans": [{"bbox": [126, 300, 149, 315], "score": 1.0, "content": "then", "type": "text"}, {"bbox": [149, 301, 241, 313], "score": 0.92, "content": "\\widetilde{L}=k(\\sqrt{d_{1}d_{2}},\\sqrt{d_{1}\\alpha}\\,)", "type": "inline_equation", "height": 12, "width": 92}, {"bbox": [242, 300, 294, 315], "score": 1.0, "content": ". If we put ", "type": "text"}, {"bbox": [294, 302, 372, 314], "score": 0.96, "content": "\\beta\\,=\\,{\\textstyle\\frac{1}{2}}(z+y\\sqrt{d_{3}}\\,)", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [372, 300, 486, 315], "score": 1.0, "content": ", then it is an elementary", "type": "text"}], "index": 12}, {"bbox": [126, 315, 486, 326], "spans": [{"bbox": [126, 315, 220, 326], "score": 1.0, "content": "exerc i se to show that ", "type": "text"}, {"bbox": [221, 316, 233, 326], "score": 0.92, "content": "\\alpha\\beta", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [234, 315, 296, 326], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [297, 317, 304, 324], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [304, 315, 394, 326], "score": 1.0, "content": ", hence we also have ", "type": "text"}, {"bbox": [394, 315, 466, 326], "score": 0.94, "content": "L=k(\\sqrt{d_{3}},\\sqrt{\\beta}\\,)", "type": "inline_equation", "height": 11, "width": 72}, {"bbox": [466, 315, 486, 326], "score": 1.0, "content": " etc.", "type": "text"}], "index": 13}, {"bbox": [125, 326, 487, 339], "spans": [{"bbox": [125, 326, 136, 339], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [136, 329, 172, 337], "score": 0.94, "content": "d_{3}=-4", "type": "inline_equation", "height": 8, "width": 36}, {"bbox": [172, 326, 359, 339], "score": 1.0, "content": ", then it is easy to see that we may choose ", "type": "text"}, {"bbox": [360, 329, 366, 338], "score": 0.89, "content": "\\beta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [366, 326, 487, 339], "score": 1.0, "content": " as the fundamental unit of", "type": "text"}], "index": 14}, {"bbox": [126, 338, 487, 351], "spans": [{"bbox": [126, 340, 136, 349], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 338, 150, 351], "score": 1.0, "content": "; if ", "type": "text"}, {"bbox": [151, 340, 187, 349], "score": 0.93, "content": "d_{3}\\neq-4", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [187, 338, 405, 351], "score": 1.0, "content": ", then genus theory says that a) the class number ", "type": "text"}, {"bbox": [405, 340, 411, 348], "score": 0.91, "content": "h", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [411, 338, 425, 351], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [425, 340, 435, 349], "score": 0.92, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [435, 338, 487, 351], "score": 1.0, "content": " is twice an", "type": "text"}], "index": 15}, {"bbox": [126, 352, 486, 363], "spans": [{"bbox": [126, 352, 181, 363], "score": 1.0, "content": "odd number ", "type": "text"}, {"bbox": [181, 355, 187, 360], "score": 0.88, "content": "u", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [187, 352, 291, 363], "score": 1.0, "content": "; and b) the prime ideal ", "type": "text"}, {"bbox": [291, 354, 301, 362], "score": 0.89, "content": "{\\mathfrak{p}}_{3}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [301, 352, 331, 363], "score": 1.0, "content": " above ", "type": "text"}, {"bbox": [331, 353, 341, 361], "score": 0.91, "content": "d_{3}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [341, 352, 354, 363], "score": 1.0, "content": " in", "type": "text"}, {"bbox": [355, 353, 365, 361], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [365, 352, 486, 363], "score": 1.0, "content": " is in the principal genus, so", "type": "text"}], "index": 16}, {"bbox": [126, 363, 486, 375], "spans": [{"bbox": [126, 364, 168, 374], "score": 0.93, "content": "{\\mathfrak{p}}_{3}^{u}=(\\pi_{3})", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [168, 363, 356, 375], "score": 1.0, "content": " is principal. Again it can be checked that ", "type": "text"}, {"bbox": [356, 364, 394, 374], "score": 0.93, "content": "\\beta=\\pm\\pi_{3}", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [394, 363, 486, 375], "score": 1.0, "content": " for a suitable choice", "type": "text"}], "index": 17}, {"bbox": [125, 374, 175, 388], "spans": [{"bbox": [125, 374, 175, 388], "score": 1.0, "content": "of the sign.", "type": "text"}], "index": 18}], "index": 11.5}, {"type": "text", "bbox": [125, 389, 486, 427], "lines": [{"bbox": [126, 392, 486, 404], "spans": [{"bbox": [126, 392, 262, 404], "score": 1.0, "content": "Example. Consider the case ", "type": "text"}, {"bbox": [262, 394, 329, 402], "score": 0.9, "content": "d\\,=\\,-31\\cdot5\\cdot8", "type": "inline_equation", "height": 8, "width": 67}, {"bbox": [329, 392, 357, 404], "score": 1.0, "content": "; here ", "type": "text"}, {"bbox": [358, 392, 443, 404], "score": 0.92, "content": "\\pi_{3}\\,=\\,\\pm(3+2{\\sqrt{10}}\\,)", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [444, 392, 486, 404], "score": 1.0, "content": ", and the", "type": "text"}], "index": 19}, {"bbox": [126, 404, 486, 417], "spans": [{"bbox": [126, 404, 248, 417], "score": 1.0, "content": "positive sign is correct since ", "type": "text"}, {"bbox": [248, 405, 346, 416], "score": 0.94, "content": "3\\,{+}\\,2{\\sqrt{10}}\\equiv(1\\,{+}\\,{\\sqrt{10}}\\,)^{2}", "type": "inline_equation", "height": 11, "width": 98}, {"bbox": [346, 404, 486, 417], "score": 1.0, "content": " mod 4 is primary. The minimal", "type": "text"}], "index": 20}, {"bbox": [126, 416, 397, 429], "spans": [{"bbox": [126, 416, 189, 429], "score": 1.0, "content": "polynomial of ", "type": "text"}, {"bbox": [189, 418, 208, 429], "score": 0.93, "content": "\\sqrt{\\pi_{3}}", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [208, 416, 220, 429], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [221, 417, 313, 428], "score": 0.92, "content": "f(x)=x^{4}-6x^{2}-31", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [313, 416, 397, 429], "score": 1.0, "content": ": compare Table 1.", "type": "text"}], "index": 21}], "index": 20}, {"type": "text", "bbox": [125, 433, 486, 483], "lines": [{"bbox": [137, 435, 486, 449], "spans": [{"bbox": [137, 435, 184, 449], "score": 1.0, "content": "The fields ", "type": "text"}, {"bbox": [185, 437, 246, 448], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [246, 435, 269, 449], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [269, 435, 340, 448], "score": 0.92, "content": "\\tilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [340, 435, 486, 449], "score": 1.0, "content": " will play a dominant role in the", "type": "text"}], "index": 22}, {"bbox": [125, 448, 486, 460], "spans": [{"bbox": [125, 449, 297, 460], "score": 1.0, "content": "proof below; they are both contai n ed in ", "type": "text"}, {"bbox": [297, 449, 353, 460], "score": 0.94, "content": "M=F({\\sqrt{\\alpha}}\\,)", "type": "inline_equation", "height": 11, "width": 56}, {"bbox": [353, 449, 370, 460], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [370, 448, 428, 460], "score": 0.94, "content": "F=k_{2}(\\sqrt{d_{2}}\\,)", "type": "inline_equation", "height": 12, "width": 58}, {"bbox": [429, 449, 486, 460], "score": 1.0, "content": ", and it is the", "type": "text"}], "index": 23}, {"bbox": [126, 460, 487, 474], "spans": [{"bbox": [126, 460, 227, 474], "score": 1.0, "content": "ambiguous class group ", "type": "text"}, {"bbox": [228, 461, 275, 472], "score": 0.91, "content": "\\mathrm{Am}(M/F)", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [275, 460, 487, 474], "score": 1.0, "content": " that contains the information we are interested", "type": "text"}], "index": 24}, {"bbox": [126, 473, 138, 484], "spans": [{"bbox": [126, 473, 138, 484], "score": 1.0, "content": "in.", "type": "text"}], "index": 25}], "index": 23.5}, {"type": "text", "bbox": [125, 489, 486, 513], "lines": [{"bbox": [124, 491, 487, 503], "spans": [{"bbox": [124, 491, 223, 503], "score": 1.0, "content": "Lemma 6. The field ", "type": "text"}, {"bbox": [223, 493, 232, 500], "score": 0.87, "content": "F", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [232, 491, 487, 503], "score": 1.0, "content": " has odd class number (even in the strict sense), and we", "type": "text"}], "index": 26}, {"bbox": [125, 502, 469, 515], "spans": [{"bbox": [125, 502, 149, 515], "score": 1.0, "content": "have ", "type": "text"}, {"bbox": [149, 504, 213, 514], "score": 0.84, "content": "\\#\\operatorname{Am}(M/F)\\mid", "type": "inline_equation", "height": 10, "width": 64}, {"bbox": [213, 502, 287, 515], "score": 1.0, "content": "2. In particular, ", "type": "text"}, {"bbox": [288, 504, 321, 514], "score": 0.92, "content": "\\mathrm{Cl_{2}}(M)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [321, 502, 469, 515], "score": 1.0, "content": " is cyclic (though possibly trivial).", "type": "text"}], "index": 27}], "index": 26.5}, {"type": "text", "bbox": [126, 519, 486, 567], "lines": [{"bbox": [125, 521, 486, 535], "spans": [{"bbox": [125, 521, 316, 535], "score": 1.0, "content": "Proof. The class group in the strict sense of ", "type": "text"}, {"bbox": [317, 524, 326, 532], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [326, 521, 486, 535], "score": 1.0, "content": " is cyclic of order 2 by R\u00b4edei\u2019s theory", "type": "text"}], "index": 28}, {"bbox": [126, 533, 486, 547], "spans": [{"bbox": [126, 533, 173, 547], "score": 1.0, "content": "[12] (since ", "type": "text"}, {"bbox": [173, 535, 277, 545], "score": 0.92, "content": "(d_{2}/p_{3})=(d_{3}/p_{2})=-1", "type": "inline_equation", "height": 10, "width": 104}, {"bbox": [278, 533, 347, 547], "score": 1.0, "content": " in case A) and ", "type": "text"}, {"bbox": [348, 535, 452, 545], "score": 0.92, "content": "(d_{1}/p_{2})=(d_{2}/p_{1})=-1", "type": "inline_equation", "height": 10, "width": 104}, {"bbox": [452, 533, 486, 547], "score": 1.0, "content": " in case", "type": "text"}], "index": 29}, {"bbox": [125, 546, 486, 558], "spans": [{"bbox": [125, 546, 173, 558], "score": 1.0, "content": "B)). Since ", "type": "text"}, {"bbox": [173, 547, 181, 555], "score": 0.9, "content": "F", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [182, 546, 304, 558], "score": 1.0, "content": " is the Hilbert class field of ", "type": "text"}, {"bbox": [304, 547, 314, 556], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [314, 546, 486, 558], "score": 1.0, "content": " in the strict sense, its class number in", "type": "text"}], "index": 30}, {"bbox": [125, 558, 225, 570], "spans": [{"bbox": [125, 558, 225, 570], "score": 1.0, "content": "the strict sense is odd.", "type": "text"}], "index": 31}], "index": 29.5}, {"type": "text", "bbox": [125, 568, 486, 640], "lines": [{"bbox": [137, 569, 484, 581], "spans": [{"bbox": [137, 569, 424, 581], "score": 1.0, "content": "Next we apply the ambiguous class number formula. In case A), ", "type": "text"}, {"bbox": [425, 571, 433, 578], "score": 0.89, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [433, 569, 484, 581], "score": 1.0, "content": " is complex,", "type": "text"}], "index": 32}, {"bbox": [126, 581, 485, 594], "spans": [{"bbox": [126, 581, 283, 594], "score": 1.0, "content": "and exactly the two primes above ", "type": "text"}, {"bbox": [283, 583, 293, 592], "score": 0.91, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [293, 581, 342, 594], "score": 1.0, "content": " ramify in ", "type": "text"}, {"bbox": [343, 583, 366, 593], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [366, 581, 424, 594], "score": 1.0, "content": ". Note that ", "type": "text"}, {"bbox": [425, 582, 485, 593], "score": 0.94, "content": "M\\,=\\,F({\\sqrt{\\alpha}}\\,)", "type": "inline_equation", "height": 11, "width": 60}], "index": 33}, {"bbox": [126, 594, 486, 606], "spans": [{"bbox": [126, 594, 149, 606], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [150, 598, 156, 603], "score": 0.88, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [156, 594, 237, 606], "score": 1.0, "content": " primary of norm ", "type": "text"}, {"bbox": [238, 594, 257, 604], "score": 0.93, "content": "d_{3}y^{2}", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [258, 594, 392, 606], "score": 1.0, "content": "; there are four primes above ", "type": "text"}, {"bbox": [392, 595, 402, 604], "score": 0.91, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [402, 594, 418, 606], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [418, 595, 426, 603], "score": 0.9, "content": "F", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [427, 594, 486, 606], "score": 1.0, "content": ", and exactly", "type": "text"}], "index": 34}, {"bbox": [126, 606, 487, 618], "spans": [{"bbox": [126, 606, 216, 618], "score": 1.0, "content": "two of them divide ", "type": "text"}, {"bbox": [217, 610, 223, 614], "score": 0.89, "content": "\\alpha", "type": "inline_equation", "height": 4, "width": 6}, {"bbox": [224, 606, 324, 618], "score": 1.0, "content": " to an odd power, so ", "type": "text"}, {"bbox": [325, 608, 352, 614], "score": 0.91, "content": "t\\ =\\ 2", "type": "inline_equation", "height": 6, "width": 27}, {"bbox": [352, 606, 487, 618], "score": 1.0, "content": " by the decomposition law in", "type": "text"}], "index": 35}, {"bbox": [125, 617, 487, 630], "spans": [{"bbox": [125, 617, 487, 630], "score": 1.0, "content": "quadratic Kummer extensions. By Proposition 4 and the remarks following it,", "type": "text"}], "index": 36}, {"bbox": [125, 629, 360, 641], "spans": [{"bbox": [125, 630, 262, 641], "score": 0.93, "content": "\\#\\operatorname{Am}_{2}(M/F)=2/(E:H)\\leq2", "type": "inline_equation", "height": 11, "width": 137}, {"bbox": [262, 629, 286, 641], "score": 1.0, "content": ", and", "type": "text"}, {"bbox": [287, 630, 320, 641], "score": 0.86, "content": "\\mathrm{Cl_{2}}(M)", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [320, 629, 360, 641], "score": 1.0, "content": " is cyclic.", "type": "text"}], "index": 37}], "index": 34.5}, {"type": "text", "bbox": [125, 640, 486, 700], "lines": [{"bbox": [137, 641, 486, 653], "spans": [{"bbox": [137, 641, 232, 653], "score": 1.0, "content": "In case B), however, ", "type": "text"}, {"bbox": [232, 643, 240, 650], "score": 0.89, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [241, 641, 303, 653], "score": 1.0, "content": " is real; since ", "type": "text"}, {"bbox": [303, 643, 333, 652], "score": 0.93, "content": "\\alpha\\,\\in\\,k_{2}", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [334, 641, 382, 653], "score": 1.0, "content": " has norm ", "type": "text"}, {"bbox": [383, 642, 423, 652], "score": 0.94, "content": "d_{3}y^{2}\\,<\\,0", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [423, 641, 486, 653], "score": 1.0, "content": ", it has mixed", "type": "text"}], "index": 38}, {"bbox": [125, 653, 486, 665], "spans": [{"bbox": [125, 653, 439, 665], "score": 1.0, "content": "signature, hence there are exactly two infinite primes that ramify in ", "type": "text"}, {"bbox": [439, 654, 462, 665], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [463, 653, 486, 665], "score": 1.0, "content": ". As", "type": "text"}], "index": 39}, {"bbox": [125, 664, 486, 678], "spans": [{"bbox": [125, 664, 331, 678], "score": 1.0, "content": "in case A), there are two finite primes above ", "type": "text"}, {"bbox": [331, 667, 340, 676], "score": 0.92, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [341, 664, 412, 678], "score": 1.0, "content": " that ramify in ", "type": "text"}, {"bbox": [412, 666, 435, 677], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [436, 664, 486, 678], "score": 1.0, "content": ", so we get", "type": "text"}], "index": 40}, {"bbox": [126, 678, 485, 689], "spans": [{"bbox": [126, 678, 246, 689], "score": 0.91, "content": "\\#\\operatorname{Am}_{2}(M/F)=8/(E:H)", "type": "inline_equation", "height": 11, "width": 120}, {"bbox": [247, 678, 281, 689], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [281, 679, 289, 686], "score": 0.91, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [290, 678, 476, 689], "score": 1.0, "content": " has odd class number in the strict sense, ", "type": "text"}, {"bbox": [477, 679, 485, 686], "score": 0.9, "content": "F", "type": "inline_equation", "height": 7, "width": 8}], "index": 41}, {"bbox": [126, 690, 486, 702], "spans": [{"bbox": [126, 690, 486, 702], "score": 1.0, "content": "has units of independent signs. This implies that the group of units that are positive", "type": "text"}], "index": 42}], "index": 40}], "layout_bboxes": [], "page_idx": 8, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [238, 90, 373, 99], "lines": [{"bbox": [239, 93, 372, 100], "spans": [{"bbox": [239, 93, 372, 100], "score": 1.0, "content": "IMAGINARY QUADRATIC FIELDS", "type": "text"}]}]}, {"type": "discarded", "bbox": [479, 91, 486, 99], "lines": [{"bbox": [480, 93, 486, 101], "spans": [{"bbox": [480, 93, 486, 101], "score": 1.0, "content": "9", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 112, 486, 150], "lines": [], "index": 1, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [124, 114, 487, 151], "lines_deleted": true}, {"type": "text", "bbox": [125, 157, 486, 181], "lines": [{"bbox": [137, 159, 484, 172], "spans": [{"bbox": [137, 159, 255, 172], "score": 1.0, "content": "From now on assume that ", "type": "text"}, {"bbox": [255, 162, 261, 169], "score": 0.89, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [261, 159, 484, 172], "score": 1.0, "content": " is one of the imaginary quadratic fields of type A)", "type": "text"}], "index": 3}, {"bbox": [126, 172, 316, 183], "spans": [{"bbox": [126, 172, 137, 183], "score": 1.0, "content": "or", "type": "text"}, {"bbox": [138, 173, 145, 181], "score": 0.43, "content": "\\mathrm{B}", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [146, 172, 316, 183], "score": 1.0, "content": ") as explained in the Introduction. Let", "type": "text"}], "index": 4}], "index": 3.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [126, 159, 484, 183]}, {"type": "text", "bbox": [124, 214, 486, 385], "lines": [{"bbox": [138, 217, 484, 228], "spans": [{"bbox": [138, 217, 414, 228], "score": 1.0, "content": "Then there exist two unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [414, 218, 420, 225], "score": 0.89, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [420, 217, 471, 228], "score": 1.0, "content": " which are ", "type": "text"}, {"bbox": [472, 218, 484, 227], "score": 0.92, "content": "D_{4}", "type": "inline_equation", "height": 9, "width": 12}], "index": 5}, {"bbox": [126, 229, 486, 240], "spans": [{"bbox": [126, 229, 148, 240], "score": 1.0, "content": "over ", "type": "text"}, {"bbox": [149, 230, 157, 239], "score": 0.9, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [157, 229, 486, 240], "score": 1.0, "content": " (see Proposition 2). Let us say a few words about their construction.", "type": "text"}], "index": 6}, {"bbox": [126, 240, 484, 252], "spans": [{"bbox": [126, 240, 360, 252], "score": 1.0, "content": "Consider e.g. case B); by R\u00b4edei\u2019s theory (see [12]), the", "type": "text"}, {"bbox": [361, 242, 372, 250], "score": 0.91, "content": "C_{4}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [373, 240, 433, 252], "score": 1.0, "content": "-factorization ", "type": "text"}, {"bbox": [433, 242, 484, 250], "score": 0.93, "content": "d=d_{1}d_{2}\\cdot d_{3}", "type": "inline_equation", "height": 8, "width": 51}], "index": 7}, {"bbox": [125, 252, 486, 264], "spans": [{"bbox": [125, 253, 359, 264], "score": 1.0, "content": "implies that unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [359, 252, 412, 264], "score": 0.94, "content": "k\\,=\\,\\mathbb{Q}({\\sqrt{d}}\\,)", "type": "inline_equation", "height": 12, "width": 53}, {"bbox": [413, 253, 486, 264], "score": 1.0, "content": " are constructed", "type": "text"}], "index": 8}, {"bbox": [124, 263, 487, 278], "spans": [{"bbox": [124, 263, 282, 278], "score": 1.0, "content": "by choosing a \u201cprimitive\u201d solution ", "type": "text"}, {"bbox": [283, 266, 316, 276], "score": 0.93, "content": "\\left(x,y,z\\right)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [316, 263, 331, 278], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [332, 265, 428, 275], "score": 0.92, "content": "d_{1}d_{2}X^{2}+d_{3}Y^{2}\\,=\\,Z^{2}", "type": "inline_equation", "height": 10, "width": 96}, {"bbox": [429, 263, 487, 278], "score": 1.0, "content": " and putting", "type": "text"}], "index": 9}, {"bbox": [126, 276, 487, 289], "spans": [{"bbox": [126, 277, 208, 288], "score": 0.94, "content": "L=k(\\sqrt{d_{1}d_{2}},\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 82}, {"bbox": [209, 276, 234, 289], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [234, 277, 305, 288], "score": 0.93, "content": "\\alpha=z+x\\sqrt{d_{1}d_{2}}", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [305, 276, 428, 289], "score": 1.0, "content": " (primitive here means that ", "type": "text"}, {"bbox": [429, 281, 435, 286], "score": 0.89, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [436, 276, 487, 289], "score": 1.0, "content": " should not", "type": "text"}], "index": 10}, {"bbox": [125, 288, 487, 301], "spans": [{"bbox": [125, 288, 487, 301], "score": 1.0, "content": "be divisible by rational integers); the other unramified cyclic quartic extension is", "type": "text"}], "index": 11}, {"bbox": [126, 300, 486, 315], "spans": [{"bbox": [126, 300, 149, 315], "score": 1.0, "content": "then", "type": "text"}, {"bbox": [149, 301, 241, 313], "score": 0.92, "content": "\\widetilde{L}=k(\\sqrt{d_{1}d_{2}},\\sqrt{d_{1}\\alpha}\\,)", "type": "inline_equation", "height": 12, "width": 92}, {"bbox": [242, 300, 294, 315], "score": 1.0, "content": ". If we put ", "type": "text"}, {"bbox": [294, 302, 372, 314], "score": 0.96, "content": "\\beta\\,=\\,{\\textstyle\\frac{1}{2}}(z+y\\sqrt{d_{3}}\\,)", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [372, 300, 486, 315], "score": 1.0, "content": ", then it is an elementary", "type": "text"}], "index": 12}, {"bbox": [126, 315, 486, 326], "spans": [{"bbox": [126, 315, 220, 326], "score": 1.0, "content": "exerc i se to show that ", "type": "text"}, {"bbox": [221, 316, 233, 326], "score": 0.92, "content": "\\alpha\\beta", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [234, 315, 296, 326], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [297, 317, 304, 324], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [304, 315, 394, 326], "score": 1.0, "content": ", hence we also have ", "type": "text"}, {"bbox": [394, 315, 466, 326], "score": 0.94, "content": "L=k(\\sqrt{d_{3}},\\sqrt{\\beta}\\,)", "type": "inline_equation", "height": 11, "width": 72}, {"bbox": [466, 315, 486, 326], "score": 1.0, "content": " etc.", "type": "text"}], "index": 13}, {"bbox": [125, 326, 487, 339], "spans": [{"bbox": [125, 326, 136, 339], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [136, 329, 172, 337], "score": 0.94, "content": "d_{3}=-4", "type": "inline_equation", "height": 8, "width": 36}, {"bbox": [172, 326, 359, 339], "score": 1.0, "content": ", then it is easy to see that we may choose ", "type": "text"}, {"bbox": [360, 329, 366, 338], "score": 0.89, "content": "\\beta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [366, 326, 487, 339], "score": 1.0, "content": " as the fundamental unit of", "type": "text"}], "index": 14}, {"bbox": [126, 338, 487, 351], "spans": [{"bbox": [126, 340, 136, 349], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 338, 150, 351], "score": 1.0, "content": "; if ", "type": "text"}, {"bbox": [151, 340, 187, 349], "score": 0.93, "content": "d_{3}\\neq-4", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [187, 338, 405, 351], "score": 1.0, "content": ", then genus theory says that a) the class number ", "type": "text"}, {"bbox": [405, 340, 411, 348], "score": 0.91, "content": "h", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [411, 338, 425, 351], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [425, 340, 435, 349], "score": 0.92, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [435, 338, 487, 351], "score": 1.0, "content": " is twice an", "type": "text"}], "index": 15}, {"bbox": [126, 352, 486, 363], "spans": [{"bbox": [126, 352, 181, 363], "score": 1.0, "content": "odd number ", "type": "text"}, {"bbox": [181, 355, 187, 360], "score": 0.88, "content": "u", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [187, 352, 291, 363], "score": 1.0, "content": "; and b) the prime ideal ", "type": "text"}, {"bbox": [291, 354, 301, 362], "score": 0.89, "content": "{\\mathfrak{p}}_{3}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [301, 352, 331, 363], "score": 1.0, "content": " above ", "type": "text"}, {"bbox": [331, 353, 341, 361], "score": 0.91, "content": "d_{3}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [341, 352, 354, 363], "score": 1.0, "content": " in", "type": "text"}, {"bbox": [355, 353, 365, 361], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [365, 352, 486, 363], "score": 1.0, "content": " is in the principal genus, so", "type": "text"}], "index": 16}, {"bbox": [126, 363, 486, 375], "spans": [{"bbox": [126, 364, 168, 374], "score": 0.93, "content": "{\\mathfrak{p}}_{3}^{u}=(\\pi_{3})", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [168, 363, 356, 375], "score": 1.0, "content": " is principal. Again it can be checked that ", "type": "text"}, {"bbox": [356, 364, 394, 374], "score": 0.93, "content": "\\beta=\\pm\\pi_{3}", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [394, 363, 486, 375], "score": 1.0, "content": " for a suitable choice", "type": "text"}], "index": 17}, {"bbox": [125, 374, 175, 388], "spans": [{"bbox": [125, 374, 175, 388], "score": 1.0, "content": "of the sign.", "type": "text"}], "index": 18}], "index": 11.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [124, 217, 487, 388]}, {"type": "text", "bbox": [125, 389, 486, 427], "lines": [{"bbox": [126, 392, 486, 404], "spans": [{"bbox": [126, 392, 262, 404], "score": 1.0, "content": "Example. Consider the case ", "type": "text"}, {"bbox": [262, 394, 329, 402], "score": 0.9, "content": "d\\,=\\,-31\\cdot5\\cdot8", "type": "inline_equation", "height": 8, "width": 67}, {"bbox": [329, 392, 357, 404], "score": 1.0, "content": "; here ", "type": "text"}, {"bbox": [358, 392, 443, 404], "score": 0.92, "content": "\\pi_{3}\\,=\\,\\pm(3+2{\\sqrt{10}}\\,)", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [444, 392, 486, 404], "score": 1.0, "content": ", and the", "type": "text"}], "index": 19}, {"bbox": [126, 404, 486, 417], "spans": [{"bbox": [126, 404, 248, 417], "score": 1.0, "content": "positive sign is correct since ", "type": "text"}, {"bbox": [248, 405, 346, 416], "score": 0.94, "content": "3\\,{+}\\,2{\\sqrt{10}}\\equiv(1\\,{+}\\,{\\sqrt{10}}\\,)^{2}", "type": "inline_equation", "height": 11, "width": 98}, {"bbox": [346, 404, 486, 417], "score": 1.0, "content": " mod 4 is primary. The minimal", "type": "text"}], "index": 20}, {"bbox": [126, 416, 397, 429], "spans": [{"bbox": [126, 416, 189, 429], "score": 1.0, "content": "polynomial of ", "type": "text"}, {"bbox": [189, 418, 208, 429], "score": 0.93, "content": "\\sqrt{\\pi_{3}}", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [208, 416, 220, 429], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [221, 417, 313, 428], "score": 0.92, "content": "f(x)=x^{4}-6x^{2}-31", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [313, 416, 397, 429], "score": 1.0, "content": ": compare Table 1.", "type": "text"}], "index": 21}], "index": 20, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [126, 392, 486, 429]}, {"type": "text", "bbox": [125, 433, 486, 483], "lines": [{"bbox": [137, 435, 486, 449], "spans": [{"bbox": [137, 435, 184, 449], "score": 1.0, "content": "The fields ", "type": "text"}, {"bbox": [185, 437, 246, 448], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [246, 435, 269, 449], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [269, 435, 340, 448], "score": 0.92, "content": "\\tilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [340, 435, 486, 449], "score": 1.0, "content": " will play a dominant role in the", "type": "text"}], "index": 22}, {"bbox": [125, 448, 486, 460], "spans": [{"bbox": [125, 449, 297, 460], "score": 1.0, "content": "proof below; they are both contai n ed in ", "type": "text"}, {"bbox": [297, 449, 353, 460], "score": 0.94, "content": "M=F({\\sqrt{\\alpha}}\\,)", "type": "inline_equation", "height": 11, "width": 56}, {"bbox": [353, 449, 370, 460], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [370, 448, 428, 460], "score": 0.94, "content": "F=k_{2}(\\sqrt{d_{2}}\\,)", "type": "inline_equation", "height": 12, "width": 58}, {"bbox": [429, 449, 486, 460], "score": 1.0, "content": ", and it is the", "type": "text"}], "index": 23}, {"bbox": [126, 460, 487, 474], "spans": [{"bbox": [126, 460, 227, 474], "score": 1.0, "content": "ambiguous class group ", "type": "text"}, {"bbox": [228, 461, 275, 472], "score": 0.91, "content": "\\mathrm{Am}(M/F)", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [275, 460, 487, 474], "score": 1.0, "content": " that contains the information we are interested", "type": "text"}], "index": 24}, {"bbox": [126, 473, 138, 484], "spans": [{"bbox": [126, 473, 138, 484], "score": 1.0, "content": "in.", "type": "text"}], "index": 25}], "index": 23.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [125, 435, 487, 484]}, {"type": "text", "bbox": [125, 489, 486, 513], "lines": [{"bbox": [124, 491, 487, 503], "spans": [{"bbox": [124, 491, 223, 503], "score": 1.0, "content": "Lemma 6. The field ", "type": "text"}, {"bbox": [223, 493, 232, 500], "score": 0.87, "content": "F", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [232, 491, 487, 503], "score": 1.0, "content": " has odd class number (even in the strict sense), and we", "type": "text"}], "index": 26}, {"bbox": [125, 502, 469, 515], "spans": [{"bbox": [125, 502, 149, 515], "score": 1.0, "content": "have ", "type": "text"}, {"bbox": [149, 504, 213, 514], "score": 0.84, "content": "\\#\\operatorname{Am}(M/F)\\mid", "type": "inline_equation", "height": 10, "width": 64}, {"bbox": [213, 502, 287, 515], "score": 1.0, "content": "2. In particular, ", "type": "text"}, {"bbox": [288, 504, 321, 514], "score": 0.92, "content": "\\mathrm{Cl_{2}}(M)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [321, 502, 469, 515], "score": 1.0, "content": " is cyclic (though possibly trivial).", "type": "text"}], "index": 27}], "index": 26.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [124, 491, 487, 515]}, {"type": "text", "bbox": [126, 519, 486, 567], "lines": [{"bbox": [125, 521, 486, 535], "spans": [{"bbox": [125, 521, 316, 535], "score": 1.0, "content": "Proof. The class group in the strict sense of ", "type": "text"}, {"bbox": [317, 524, 326, 532], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [326, 521, 486, 535], "score": 1.0, "content": " is cyclic of order 2 by R\u00b4edei\u2019s theory", "type": "text"}], "index": 28}, {"bbox": [126, 533, 486, 547], "spans": [{"bbox": [126, 533, 173, 547], "score": 1.0, "content": "[12] (since ", "type": "text"}, {"bbox": [173, 535, 277, 545], "score": 0.92, "content": "(d_{2}/p_{3})=(d_{3}/p_{2})=-1", "type": "inline_equation", "height": 10, "width": 104}, {"bbox": [278, 533, 347, 547], "score": 1.0, "content": " in case A) and ", "type": "text"}, {"bbox": [348, 535, 452, 545], "score": 0.92, "content": "(d_{1}/p_{2})=(d_{2}/p_{1})=-1", "type": "inline_equation", "height": 10, "width": 104}, {"bbox": [452, 533, 486, 547], "score": 1.0, "content": " in case", "type": "text"}], "index": 29}, {"bbox": [125, 546, 486, 558], "spans": [{"bbox": [125, 546, 173, 558], "score": 1.0, "content": "B)). Since ", "type": "text"}, {"bbox": [173, 547, 181, 555], "score": 0.9, "content": "F", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [182, 546, 304, 558], "score": 1.0, "content": " is the Hilbert class field of ", "type": "text"}, {"bbox": [304, 547, 314, 556], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [314, 546, 486, 558], "score": 1.0, "content": " in the strict sense, its class number in", "type": "text"}], "index": 30}, {"bbox": [125, 558, 225, 570], "spans": [{"bbox": [125, 558, 225, 570], "score": 1.0, "content": "the strict sense is odd.", "type": "text"}], "index": 31}], "index": 29.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [125, 521, 486, 570]}, {"type": "text", "bbox": [125, 568, 486, 640], "lines": [{"bbox": [137, 569, 484, 581], "spans": [{"bbox": [137, 569, 424, 581], "score": 1.0, "content": "Next we apply the ambiguous class number formula. In case A), ", "type": "text"}, {"bbox": [425, 571, 433, 578], "score": 0.89, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [433, 569, 484, 581], "score": 1.0, "content": " is complex,", "type": "text"}], "index": 32}, {"bbox": [126, 581, 485, 594], "spans": [{"bbox": [126, 581, 283, 594], "score": 1.0, "content": "and exactly the two primes above ", "type": "text"}, {"bbox": [283, 583, 293, 592], "score": 0.91, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [293, 581, 342, 594], "score": 1.0, "content": " ramify in ", "type": "text"}, {"bbox": [343, 583, 366, 593], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [366, 581, 424, 594], "score": 1.0, "content": ". Note that ", "type": "text"}, {"bbox": [425, 582, 485, 593], "score": 0.94, "content": "M\\,=\\,F({\\sqrt{\\alpha}}\\,)", "type": "inline_equation", "height": 11, "width": 60}], "index": 33}, {"bbox": [126, 594, 486, 606], "spans": [{"bbox": [126, 594, 149, 606], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [150, 598, 156, 603], "score": 0.88, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [156, 594, 237, 606], "score": 1.0, "content": " primary of norm ", "type": "text"}, {"bbox": [238, 594, 257, 604], "score": 0.93, "content": "d_{3}y^{2}", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [258, 594, 392, 606], "score": 1.0, "content": "; there are four primes above ", "type": "text"}, {"bbox": [392, 595, 402, 604], "score": 0.91, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [402, 594, 418, 606], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [418, 595, 426, 603], "score": 0.9, "content": "F", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [427, 594, 486, 606], "score": 1.0, "content": ", and exactly", "type": "text"}], "index": 34}, {"bbox": [126, 606, 487, 618], "spans": [{"bbox": [126, 606, 216, 618], "score": 1.0, "content": "two of them divide ", "type": "text"}, {"bbox": [217, 610, 223, 614], "score": 0.89, "content": "\\alpha", "type": "inline_equation", "height": 4, "width": 6}, {"bbox": [224, 606, 324, 618], "score": 1.0, "content": " to an odd power, so ", "type": "text"}, {"bbox": [325, 608, 352, 614], "score": 0.91, "content": "t\\ =\\ 2", "type": "inline_equation", "height": 6, "width": 27}, {"bbox": [352, 606, 487, 618], "score": 1.0, "content": " by the decomposition law in", "type": "text"}], "index": 35}, {"bbox": [125, 617, 487, 630], "spans": [{"bbox": [125, 617, 487, 630], "score": 1.0, "content": "quadratic Kummer extensions. By Proposition 4 and the remarks following it,", "type": "text"}], "index": 36}, {"bbox": [125, 629, 360, 641], "spans": [{"bbox": [125, 630, 262, 641], "score": 0.93, "content": "\\#\\operatorname{Am}_{2}(M/F)=2/(E:H)\\leq2", "type": "inline_equation", "height": 11, "width": 137}, {"bbox": [262, 629, 286, 641], "score": 1.0, "content": ", and", "type": "text"}, {"bbox": [287, 630, 320, 641], "score": 0.86, "content": "\\mathrm{Cl_{2}}(M)", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [320, 629, 360, 641], "score": 1.0, "content": " is cyclic.", "type": "text"}], "index": 37}], "index": 34.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [125, 569, 487, 641]}, {"type": "text", "bbox": [125, 640, 486, 700], "lines": [{"bbox": [137, 641, 486, 653], "spans": [{"bbox": [137, 641, 232, 653], "score": 1.0, "content": "In case B), however, ", "type": "text"}, {"bbox": [232, 643, 240, 650], "score": 0.89, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [241, 641, 303, 653], "score": 1.0, "content": " is real; since ", "type": "text"}, {"bbox": [303, 643, 333, 652], "score": 0.93, "content": "\\alpha\\,\\in\\,k_{2}", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [334, 641, 382, 653], "score": 1.0, "content": " has norm ", "type": "text"}, {"bbox": [383, 642, 423, 652], "score": 0.94, "content": "d_{3}y^{2}\\,<\\,0", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [423, 641, 486, 653], "score": 1.0, "content": ", it has mixed", "type": "text"}], "index": 38}, {"bbox": [125, 653, 486, 665], "spans": [{"bbox": [125, 653, 439, 665], "score": 1.0, "content": "signature, hence there are exactly two infinite primes that ramify in ", "type": "text"}, {"bbox": [439, 654, 462, 665], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [463, 653, 486, 665], "score": 1.0, "content": ". As", "type": "text"}], "index": 39}, {"bbox": [125, 664, 486, 678], "spans": [{"bbox": [125, 664, 331, 678], "score": 1.0, "content": "in case A), there are two finite primes above ", "type": "text"}, {"bbox": [331, 667, 340, 676], "score": 0.92, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [341, 664, 412, 678], "score": 1.0, "content": " that ramify in ", "type": "text"}, {"bbox": [412, 666, 435, 677], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [436, 664, 486, 678], "score": 1.0, "content": ", so we get", "type": "text"}], "index": 40}, {"bbox": [126, 678, 485, 689], "spans": [{"bbox": [126, 678, 246, 689], "score": 0.91, "content": "\\#\\operatorname{Am}_{2}(M/F)=8/(E:H)", "type": "inline_equation", "height": 11, "width": 120}, {"bbox": [247, 678, 281, 689], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [281, 679, 289, 686], "score": 0.91, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [290, 678, 476, 689], "score": 1.0, "content": " has odd class number in the strict sense, ", "type": "text"}, {"bbox": [477, 679, 485, 686], "score": 0.9, "content": "F", "type": "inline_equation", "height": 7, "width": 8}], "index": 41}, {"bbox": [126, 690, 486, 702], "spans": [{"bbox": [126, 690, 486, 702], "score": 1.0, "content": "has units of independent signs. This implies that the group of units that are positive", "type": "text"}], "index": 42}, {"bbox": [125, 114, 486, 126], "spans": [{"bbox": [125, 114, 297, 126], "score": 1.0, "content": "at the two ramified infinite primes has ", "type": "text", "cross_page": true}, {"bbox": [297, 116, 304, 123], "score": 0.9, "content": "\\mathbb{Z}", "type": "inline_equation", "height": 7, "width": 7, "cross_page": true}, {"bbox": [304, 114, 358, 126], "score": 1.0, "content": "-rank 2, i.e. ", "type": "text", "cross_page": true}, {"bbox": [358, 115, 410, 126], "score": 0.92, "content": "(E:H)\\geq4", "type": "inline_equation", "height": 11, "width": 52, "cross_page": true}, {"bbox": [410, 114, 486, 126], "score": 1.0, "content": " by consideration", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [125, 127, 486, 138], "spans": [{"bbox": [125, 127, 311, 138], "score": 1.0, "content": "of the infinite primes alone. In particular, ", "type": "text", "cross_page": true}, {"bbox": [311, 127, 391, 138], "score": 0.92, "content": "\\#\\operatorname{Am}_{2}(M/F)\\leq2", "type": "inline_equation", "height": 11, "width": 80, "cross_page": true}, {"bbox": [391, 127, 441, 138], "score": 1.0, "content": " in case B).", "type": "text", "cross_page": true}, {"bbox": [476, 127, 486, 137], "score": 0.9776784181594849, "content": "\u53e3", "type": "text", "cross_page": true}], "index": 1}], "index": 40, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [125, 641, 486, 702]}]}	{"layout_dets": [{"category_id": 1, "poly": [347, 597, 1352, 597, 1352, 1070, 347, 1070], "score": 0.982}, {"category_id": 1, "poly": [348, 1578, 1351, 1578, 1351, 1778, 348, 1778], "score": 0.978}, {"category_id": 1, "poly": [349, 1780, 1352, 1780, 1352, 1945, 349, 1945], "score": 0.967}, {"category_id": 1, "poly": [349, 1203, 1351, 1203, 1351, 1342, 349, 1342], "score": 0.967}, {"category_id": 1, "poly": [350, 1443, 1350, 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982, 1248, 982, 1279, 827, 1279], "score": 0.94, "latex": "M=F({\\sqrt{\\alpha}}\\,)"}, {"category_id": 13, "poly": [611, 392, 688, 392, 688, 421, 611, 421], "score": 0.94, "latex": "K_{2}/k_{2}"}, {"category_id": 13, "poly": [514, 1215, 685, 1215, 685, 1245, 514, 1245], "score": 0.94, "latex": "K_{2}=k_{2}(\\sqrt{\\alpha}\\,)"}, {"category_id": 13, "poly": [609, 520, 797, 520, 797, 551, 609, 551], "score": 0.94, "latex": "k_{2}=\\mathbb{Q}({\\sqrt{d_{2}d_{3}}}\\,)"}, {"category_id": 13, "poly": [999, 702, 1147, 702, 1147, 734, 999, 734], "score": 0.94, "latex": "k\\,=\\,\\mathbb{Q}({\\sqrt{d}}\\,)"}, {"category_id": 13, "poly": [351, 390, 518, 390, 518, 421, 351, 421], "score": 0.94, "latex": "K_{2}=k_{2}(\\sqrt{\\beta}\\,)"}, {"category_id": 13, "poly": [691, 1125, 962, 1125, 962, 1157, 691, 1157], "score": 0.94, "latex": "3\\,{+}\\,2{\\sqrt{10}}\\equiv(1\\,{+}\\,{\\sqrt{10}}\\,)^{2}"}, {"category_id": 13, "poly": [379, 914, 478, 914, 478, 938, 379, 938], "score": 0.94, "latex": "d_{3}=-4"}, {"category_id": 13, "poly": [1096, 876, 1296, 876, 1296, 908, 1096, 908], "score": 0.94, "latex": "L=k(\\sqrt{d_{3}},\\sqrt{\\beta}\\,)"}, {"category_id": 13, "poly": [1181, 1618, 1348, 1618, 1348, 1649, 1181, 1649], "score": 0.94, "latex": "M\\,=\\,F({\\sqrt{\\alpha}}\\,)"}, {"category_id": 13, "poly": [1050, 322, 1124, 322, 1124, 351, 1050, 351], "score": 0.94, "latex": "\\pm\\eta/\\eta^{\\prime}"}, {"category_id": 13, "poly": [1064, 1784, 1175, 1784, 1175, 1813, 1064, 1813], "score": 0.94, "latex": "d_{3}y^{2}\\,<\\,0"}, {"category_id": 13, "poly": [351, 770, 580, 770, 580, 802, 351, 802], "score": 0.94, "latex": "L=k(\\sqrt{d_{1}d_{2}},\\sqrt{\\alpha}\\,)"}, {"category_id": 13, "poly": [609, 561, 796, 561, 796, 592, 609, 592], "score": 0.93, "latex": "k_{2}=\\mathbb{Q}({\\sqrt{d_{1}d_{2}}}\\,)"}, {"category_id": 13, "poly": [384, 520, 546, 520, 546, 551, 384, 551], "score": 0.93, "latex": "k_{1}=\\mathbb{Q}(\\sqrt{d_{1}}\\,)"}, {"category_id": 13, "poly": [384, 560, 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"\\mathrm{Cl_{2}}(M)"}, {"category_id": 13, "poly": [1183, 947, 1210, 947, 1210, 971, 1183, 971], "score": 0.92, "latex": "k_{2}"}, {"category_id": 13, "poly": [923, 738, 1191, 738, 1191, 766, 923, 766], "score": 0.92, "latex": "d_{1}d_{2}X^{2}+d_{3}Y^{2}\\,=\\,Z^{2}"}, {"category_id": 13, "poly": [1312, 607, 1347, 607, 1347, 631, 1312, 631], "score": 0.92, "latex": "D_{4}"}, {"category_id": 13, "poly": [921, 1854, 947, 1854, 947, 1878, 921, 1878], "score": 0.92, "latex": "d_{3}"}, {"category_id": 13, "poly": [614, 1160, 870, 1160, 870, 1191, 614, 1191], "score": 0.92, "latex": "f(x)=x^{4}-6x^{2}-31"}, {"category_id": 13, "poly": [922, 981, 948, 981, 948, 1005, 922, 1005], "score": 0.91, "latex": "d_{3}"}, {"category_id": 13, "poly": [1091, 1655, 1117, 1655, 1117, 1679, 1091, 1679], "score": 0.91, "latex": "d_{3}"}, {"category_id": 13, "poly": [847, 1522, 873, 1522, 873, 1546, 847, 1546], "score": 0.91, "latex": "k_{2}"}, {"category_id": 13, "poly": [634, 1283, 764, 1283, 764, 1312, 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the imaginary quadratic fields of type A) or ) as explained in the Introduction. Let Then there exist two unramified cyclic quartic extensions of which are over (see Proposition 2). Let us say a few words about their construction. Consider e.g. case B); by R´edei’s theory (see [12]), the -factorization implies that unramified cyclic quartic extensions of are constructed by choosing a “primitive” solution of and putting with (primitive here means that should not be divisible by rational integers); the other unramified cyclic quartic extension is then . If we put , then it is an elementary exerc i se to show that is a square in , hence we also have etc. If , then it is easy to see that we may choose as the fundamental unit of ; if , then genus theory says that a) the class number of is twice an odd number ; and b) the prime ideal above in is in the principal genus, so is principal. Again it can be checked that for a suitable choice of the sign. Example. Consider the case ; here , and the positive sign is correct since mod 4 is primary. The minimal polynomial of is : compare Table 1. The fields and will play a dominant role in the proof below; they are both contai n ed in for , and it is the ambiguous class group that contains the information we are interested in. Lemma 6. The field has odd class number (even in the strict sense), and we have 2. In particular, is cyclic (though possibly trivial). Proof. The class group in the strict sense of is cyclic of order 2 by R´edei’s theory [12] (since in case A) and in case B)). Since is the Hilbert class field of in the strict sense, its class number in the strict sense is odd. Next we apply the ambiguous class number formula. In case A), is complex, and exactly the two primes above ramify in . Note that with primary of norm ; there are four primes above in , and exactly two of them divide to an odd power, so by the decomposition law in quadratic Kummer extensions. By Proposition 4 and the remarks following it, , and is cyclic. In case B), however, is real; since has norm , it has mixed signature, hence there are exactly two infinite primes that ramify in . As in case A), there are two finite primes above that ramify in , so we get . Since has odd class number in the strict sense, has units of independent signs. This implies that the group of units that are positive at the two ramified infinite primes has -rank 2, i.e. by consideration of the infinite primes alone. In particular, in case B). 口	<div class="pdf-page"> <p>From now on assume that is one of the imaginary quadratic fields of type A) or ) as explained in the Introduction. Let</p> <p>Then there exist two unramified cyclic quartic extensions of which are over (see Proposition 2). Let us say a few words about their construction. Consider e.g. case B); by R´edei’s theory (see [12]), the -factorization implies that unramified cyclic quartic extensions of are constructed by choosing a “primitive” solution of and putting with (primitive here means that should not be divisible by rational integers); the other unramified cyclic quartic extension is then . If we put , then it is an elementary exerc i se to show that is a square in , hence we also have etc. If , then it is easy to see that we may choose as the fundamental unit of ; if , then genus theory says that a) the class number of is twice an odd number ; and b) the prime ideal above in is in the principal genus, so is principal. Again it can be checked that for a suitable choice of the sign.</p> <p>Example. Consider the case ; here , and the positive sign is correct since mod 4 is primary. The minimal polynomial of is : compare Table 1.</p> <p>The fields and will play a dominant role in the proof below; they are both contai n ed in for , and it is the ambiguous class group that contains the information we are interested in.</p> <p>Lemma 6. The field has odd class number (even in the strict sense), and we have 2. In particular, is cyclic (though possibly trivial).</p> <p>Proof. The class group in the strict sense of is cyclic of order 2 by R´edei’s theory [12] (since in case A) and in case B)). Since is the Hilbert class field of in the strict sense, its class number in the strict sense is odd.</p> <p>Next we apply the ambiguous class number formula. In case A), is complex, and exactly the two primes above ramify in . Note that with primary of norm ; there are four primes above in , and exactly two of them divide to an odd power, so by the decomposition law in quadratic Kummer extensions. By Proposition 4 and the remarks following it, , and is cyclic.</p> <p>In case B), however, is real; since has norm , it has mixed signature, hence there are exactly two infinite primes that ramify in . As in case A), there are two finite primes above that ramify in , so we get . Since has odd class number in the strict sense, has units of independent signs. This implies that the group of units that are positive at the two ramified infinite primes has -rank 2, i.e. by consideration of the infinite primes alone. In particular, in case B). 口</p> </div>	<div class="pdf-page"> <div class="pdf-discarded" data-x="398" data-y="116" data-width="226" data-height="12" style="opacity: 0.5;">IMAGINARY QUADRATIC FIELDS</div> <div class="pdf-discarded" data-x="801" data-y="117" data-width="12" data-height="11" style="opacity: 0.5;">9</div> <p class="pdf-text" data-x="209" data-y="202" data-width="604" data-height="32">From now on assume that is one of the imaginary quadratic fields of type A) or ) as explained in the Introduction. Let</p> <p class="pdf-text" data-x="207" data-y="276" data-width="606" data-height="221">Then there exist two unramified cyclic quartic extensions of which are over (see Proposition 2). Let us say a few words about their construction. Consider e.g. case B); by R´edei’s theory (see [12]), the -factorization implies that unramified cyclic quartic extensions of are constructed by choosing a “primitive” solution of and putting with (primitive here means that should not be divisible by rational integers); the other unramified cyclic quartic extension is then . If we put , then it is an elementary exerc i se to show that is a square in , hence we also have etc. If , then it is easy to see that we may choose as the fundamental unit of ; if , then genus theory says that a) the class number of is twice an odd number ; and b) the prime ideal above in is in the principal genus, so is principal. Again it can be checked that for a suitable choice of the sign.</p> <p class="pdf-text" data-x="209" data-y="502" data-width="604" data-height="50">Example. Consider the case ; here , and the positive sign is correct since mod 4 is primary. The minimal polynomial of is : compare Table 1.</p> <p class="pdf-text" data-x="209" data-y="559" data-width="604" data-height="65">The fields and will play a dominant role in the proof below; they are both contai n ed in for , and it is the ambiguous class group that contains the information we are interested in.</p> <p class="pdf-text" data-x="209" data-y="632" data-width="604" data-height="31">Lemma 6. The field has odd class number (even in the strict sense), and we have 2. In particular, is cyclic (though possibly trivial).</p> <p class="pdf-text" data-x="210" data-y="671" data-width="603" data-height="62">Proof. The class group in the strict sense of is cyclic of order 2 by R´edei’s theory [12] (since in case A) and in case B)). Since is the Hilbert class field of in the strict sense, its class number in the strict sense is odd.</p> <p class="pdf-text" data-x="209" data-y="734" data-width="604" data-height="93">Next we apply the ambiguous class number formula. In case A), is complex, and exactly the two primes above ramify in . Note that with primary of norm ; there are four primes above in , and exactly two of them divide to an odd power, so by the decomposition law in quadratic Kummer extensions. By Proposition 4 and the remarks following it, , and is cyclic.</p> <p class="pdf-text" data-x="209" data-y="827" data-width="604" data-height="78">In case B), however, is real; since has norm , it has mixed signature, hence there are exactly two infinite primes that ramify in . As in case A), there are two finite primes above that ramify in , so we get . Since has odd class number in the strict sense, has units of independent signs. This implies that the group of units that are positive at the two ramified infinite primes has -rank 2, i.e. by consideration of the infinite primes alone. In particular, in case B). 口</p> </div>	{ "type": [ "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "inline_equation", "inline_equation" ], "coordinates": [ [ 229, 205, 809, 222 ], [ 210, 222, 528, 236 ], [ 230, 280, 809, 294 ], [ 210, 296, 813, 310 ], [ 210, 310, 809, 325 ], [ 209, 325, 813, 341 ], [ 207, 340, 814, 359 ], [ 210, 356, 814, 373 ], [ 209, 372, 814, 389 ], [ 210, 387, 813, 407 ], [ 210, 407, 813, 421 ], [ 209, 421, 814, 438 ], [ 210, 437, 814, 453 ], [ 210, 455, 813, 469 ], [ 210, 469, 813, 484 ], [ 209, 483, 292, 501 ], [ 210, 506, 813, 522 ], [ 210, 522, 813, 539 ], [ 210, 537, 664, 554 ], [ 229, 562, 813, 580 ], [ 209, 579, 813, 594 ], [ 210, 594, 814, 612 ], [ 210, 611, 230, 625 ], [ 207, 634, 814, 650 ], [ 209, 649, 784, 665 ], [ 209, 673, 813, 691 ], [ 210, 689, 813, 707 ], [ 209, 705, 813, 721 ], [ 209, 721, 376, 736 ], [ 229, 735, 809, 751 ], [ 210, 751, 811, 768 ], [ 210, 768, 813, 783 ], [ 210, 783, 814, 799 ], [ 209, 797, 814, 814 ], [ 209, 813, 602, 828 ], [ 229, 828, 813, 844 ], [ 209, 844, 813, 859 ], [ 209, 858, 813, 876 ], [ 210, 876, 811, 890 ], [ 210, 892, 813, 907 ], [ 209, 147, 813, 162 ], [ 209, 164, 813, 178 ] ], "content": [ "From now on assume that k is one of the imaginary quadratic fields of type A)", "or \\mathrm{B} ) as explained in the Introduction. Let", "Then there exist two unramified cyclic quartic extensions of k which are D_{4}", "over \\mathbb{Q} (see Proposition 2). Let us say a few words about their construction.", "Consider e.g. case B); by R´edei’s theory (see [12]), the C_{4} -factorization d=d_{1}d_{2}\\cdot d_{3}", "implies that unramified cyclic quartic extensions of k\\,=\\,\\mathbb{Q}({\\sqrt{d}}\\,) are constructed", "by choosing a “primitive” solution \\left(x,y,z\\right) of d_{1}d_{2}X^{2}+d_{3}Y^{2}\\,=\\,Z^{2} and putting", "L=k(\\sqrt{d_{1}d_{2}},\\sqrt{\\alpha}\\,) with \\alpha=z+x\\sqrt{d_{1}d_{2}} (primitive here means that \\alpha should not", "be divisible by rational integers); the other unramified cyclic quartic extension is", "then \\widetilde{L}=k(\\sqrt{d_{1}d_{2}},\\sqrt{d_{1}\\alpha}\\,) . If we put \\beta\\,=\\,{\\textstyle\\frac{1}{2}}(z+y\\sqrt{d_{3}}\\,) , then it is an elementary", "exerc i se to show that \\alpha\\beta is a square in L , hence we also have L=k(\\sqrt{d_{3}},\\sqrt{\\beta}\\,) etc.", "If d_{3}=-4 , then it is easy to see that we may choose \\beta as the fundamental unit of", "k_{2} ; if d_{3}\\neq-4 , then genus theory says that a) the class number h of k_{2} is twice an", "odd number u ; and b) the prime ideal {\\mathfrak{p}}_{3} above d_{3} in k_{2} is in the principal genus, so", "{\\mathfrak{p}}_{3}^{u}=(\\pi_{3}) is principal. Again it can be checked that \\beta=\\pm\\pi_{3} for a suitable choice", "of the sign.", "Example. Consider the case d\\,=\\,-31\\cdot5\\cdot8 ; here \\pi_{3}\\,=\\,\\pm(3+2{\\sqrt{10}}\\,) , and the", "positive sign is correct since 3\\,{+}\\,2{\\sqrt{10}}\\equiv(1\\,{+}\\,{\\sqrt{10}}\\,)^{2} mod 4 is primary. The minimal", "polynomial of \\sqrt{\\pi_{3}} is f(x)=x^{4}-6x^{2}-31 : compare Table 1.", "The fields K_{2}=k_{2}(\\sqrt{\\alpha}\\,) and \\tilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big) will play a dominant role in the", "proof below; they are both contai n ed in M=F({\\sqrt{\\alpha}}\\,) for F=k_{2}(\\sqrt{d_{2}}\\,) , and it is the", "ambiguous class group \\mathrm{Am}(M/F) that contains the information we are interested", "in.", "Lemma 6. The field F has odd class number (even in the strict sense), and we", "have \\#\\operatorname{Am}(M/F)\\mid 2. In particular, \\mathrm{Cl_{2}}(M) is cyclic (though possibly trivial).", "Proof. The class group in the strict sense of k_{2} is cyclic of order 2 by R´edei’s theory", "[12] (since (d_{2}/p_{3})=(d_{3}/p_{2})=-1 in case A) and (d_{1}/p_{2})=(d_{2}/p_{1})=-1 in case", "B)). Since F is the Hilbert class field of k_{2} in the strict sense, its class number in", "the strict sense is odd.", "Next we apply the ambiguous class number formula. In case A), F is complex,", "and exactly the two primes above d_{3} ramify in M/F . Note that M\\,=\\,F({\\sqrt{\\alpha}}\\,)", "with \\alpha primary of norm d_{3}y^{2} ; there are four primes above d_{3} in F , and exactly", "two of them divide \\alpha to an odd power, so t\\ =\\ 2 by the decomposition law in", "quadratic Kummer extensions. By Proposition 4 and the remarks following it,", "\\#\\operatorname{Am}_{2}(M/F)=2/(E:H)\\leq2 , and \\mathrm{Cl_{2}}(M) is cyclic.", "In case B), however, F is real; since \\alpha\\,\\in\\,k_{2} has norm d_{3}y^{2}\\,<\\,0 , it has mixed", "signature, hence there are exactly two infinite primes that ramify in M/F . As", "in case A), there are two finite primes above d_{3} that ramify in M/F , so we get", "\\#\\operatorname{Am}_{2}(M/F)=8/(E:H) . Since F has odd class number in the strict sense, F", "has units of independent signs. This implies that the group of units that are positive", "at the two ramified infinite primes has \\mathbb{Z} -rank 2, i.e. (E:H)\\geq4 by consideration", "of the infinite primes alone. In particular, \\#\\operatorname{Am}_{2}(M/F)\\leq2 in case B). 口" ], "index": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 37, 38, 39, 40, 60, 61, 85, 86, 87, 88, 114, 115, 116, 117, 118, 119, 149, 150, 151, 152, 153, 154, 155 ] }	{ "coordinates": [], "index": [], "caption": [], "caption_coordinates": [] }	{ "type": [], "coordinates": [], "content": [], "index": [] }	{ "type": [], "coordinates": [], "content": [], "index": [] }	{"document_info": {"total_pages": 14, "parse_type": "txt", "version": "1.0.1", "total_content_list_items": 155}, "content_statistics": {"total_blocks": 182, "total_lines": 526, "total_images": 4, "total_equations": 28, "total_tables": 11, "total_characters": 29658, "total_words": 5331, "block_type_distribution": {"title": 7, "text": 126, "interline_equation": 14, "discarded": 24, "table_caption": 3, "table_body": 4, "image_body": 2, "list": 2}}, "model_statistics": {"total_layout_detections": 2826, "avg_confidence": 0.9470955414012739, "min_confidence": 0.26, "max_confidence": 1.0}, "page_statistics": [{"page_number": 0, "page_size": [612.0, 792.0], "blocks": 11, "lines": 38, "images": 0, "equations": 2, "tables": 0, "characters": 2362, "words": 398, "discarded_blocks": 1}, {"page_number": 1, "page_size": [612.0, 792.0], "blocks": 10, "lines": 30, "images": 0, "equations": 0, "tables": 3, "characters": 1688, "words": 305, "discarded_blocks": 1}, {"page_number": 2, "page_size": [612.0, 792.0], "blocks": 15, "lines": 44, "images": 0, "equations": 4, "tables": 0, "characters": 2417, "words": 416, "discarded_blocks": 2}, {"page_number": 3, "page_size": [612.0, 792.0], "blocks": 7, "lines": 27, "images": 0, "equations": 0, "tables": 3, "characters": 1591, "words": 291, "discarded_blocks": 1}, {"page_number": 4, "page_size": [612.0, 792.0], "blocks": 23, "lines": 38, "images": 0, "equations": 6, "tables": 0, "characters": 1604, "words": 303, "discarded_blocks": 4}, {"page_number": 5, "page_size": [612.0, 792.0], "blocks": 17, "lines": 40, "images": 0, "equations": 6, "tables": 0, "characters": 2131, "words": 393, "discarded_blocks": 1}, {"page_number": 6, "page_size": [612.0, 792.0], "blocks": 15, "lines": 36, "images": 2, "equations": 4, "tables": 0, "characters": 1791, "words": 324, "discarded_blocks": 2}, {"page_number": 7, "page_size": [612.0, 792.0], "blocks": 12, "lines": 49, "images": 0, "equations": 0, "tables": 0, "characters": 3049, "words": 571, "discarded_blocks": 1}, {"page_number": 8, "page_size": [612.0, 792.0], "blocks": 11, "lines": 42, "images": 0, "equations": 0, "tables": 0, "characters": 2636, "words": 479, "discarded_blocks": 2}, {"page_number": 9, "page_size": [612.0, 792.0], "blocks": 14, "lines": 35, "images": 2, "equations": 0, "tables": 0, "characters": 2036, "words": 375, "discarded_blocks": 1}, {"page_number": 10, "page_size": [612.0, 792.0], "blocks": 17, "lines": 35, "images": 0, "equations": 6, "tables": 0, "characters": 1789, "words": 327, "discarded_blocks": 2}, {"page_number": 11, "page_size": [612.0, 792.0], "blocks": 14, "lines": 40, "images": 0, "equations": 0, "tables": 2, "characters": 2094, "words": 399, "discarded_blocks": 2}, {"page_number": 12, "page_size": [612.0, 792.0], "blocks": 10, "lines": 44, "images": 0, "equations": 0, "tables": 3, "characters": 2572, "words": 487, "discarded_blocks": 2}, {"page_number": 13, "page_size": [612.0, 792.0], "blocks": 6, "lines": 28, "images": 0, "equations": 0, "tables": 0, "characters": 1898, "words": 263, "discarded_blocks": 2}]}
0003244v1	9	[ 612, 792 ]	{ "type": [ "text", "text", "text", "list", "text", "text", "text", "text", "text", "text", "image_body", "text", "text", "discarded" ], "coordinates": [ [ 207, 143, 814, 177 ], [ 209, 187, 813, 218 ], [ 207, 224, 813, 274 ], [ 222, 276, 814, 327 ], [ 209, 334, 813, 367 ], [ 209, 368, 813, 402 ], [ 207, 402, 607, 416 ], [ 207, 431, 814, 462 ], [ 207, 462, 814, 561 ], [ 220, 561, 811, 577 ], [ 423, 594, 629, 704 ], [ 207, 716, 814, 857 ], [ 207, 857, 813, 905 ], [ 210, 117, 225, 128 ] ], "content": [ "", "Next we derive some relations between the class groups of and ; these relations will allow us to use each of them as our field in Theorem 1.", "Proposition 8. Let and be the two unramified cyclic quartic extensions of , and let and be two qu a dratic extensions of in and , respectively, which are not normal o ver .", "a) We have 4 if and only if ; b) , then one of or has type , whereas the other is cyclic of order .", "Proof. Notice that the prime dividing splits in . Throughout this proof, let be one of the primes of dividing .", "If we write for some , then . In fact, and are the only extensions of with the p roperties", "1. is a quadratic extension unramified outside ;", "Therefore it suffices to observe that if has these properties, then so does . But this is elementary.", "In particular, the compositum is an extension of type over with subextensions , and . Clearly is the unramified quadratic extension of , so bo t h and are unramified. If had 2-class number 2, then would have odd class numb e r, and would also be the 2-class field of . Thus implies that . This proves part a) of the proposition.", "Before we go on, we give a Hasse diagram for the fields occurring in this proof:", "", "Now assume that . Since is cyclic by Lemma 6, there is a unique quadratic unramified extension , and the uniqueness implies at once that is normal. Hence is a group of order 8 containing a subgroup of type : in fact, if were cyclic, then the primes ramifying in would also ramify in contradicting the fact that is unramified. There are three groups satisfying these conditions: , and . We claim that is non-abelian; once we have proved this, it follows that exactly one of the groups and is cyclic, and that the other is not, which is what we want to prove.", "So assume that is abelian. Then is ramified at two finite primes and of dividing (in ); if and denote the quadratic subextensions of different from then and must be ramified at a finite prime (since", "10" ], "index": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 ] }	[{"type": "text", "text": "", "page_idx": 9}, {"type": "text", "text": "Next we derive some relations between the class groups of $K_{2}$ and ${\\tilde{K}}_{2}$ ; these relations will allow us to use each of them as our field $K$ in Theorem 1. ", "page_idx": 9}, {"type": "text", "text": "Proposition 8. Let $L$ and $\\widetilde{L}$ be the two unramified cyclic quartic extensions of $k$ , and let $K_{2}$ and ${\\widetilde{K}}_{2}$ be two qu a dratic extensions of $k_{2}$ in $L$ and $\\widetilde{L}$ , respectively, which are not normal o ver $\\mathbb{Q}$ . ", "page_idx": 9}, {"type": "text", "text": "a) We have 4 $\|\\mathit{\\Omega}_{h}(K_{2})$ if and only if $4\\mid h(\\widetilde{K}_{2})$ ; \nb) $I f\\,4\\mid h(K_{2})$ , then one of $\\mathrm{Cl}_{2}(K_{2})$ or $\\mathrm{Cl}_{2}(\\widetilde{K}_{2})$ has type $(2,2)$ , whereas the other is cyclic of order $\\geq4$ . ", "page_idx": 9}, {"type": "text", "text": "Proof. Notice that the prime dividing $\\mathrm{disc}(k_{1})$ splits in $k_{2}$ . Throughout this proof, let $\\mathfrak{p}$ be one of the primes of $k_{2}$ dividing $\\mathrm{disc}(k_{1})$ . ", "page_idx": 9}, {"type": "text", "text": "If we write $K_{2}=k_{2}(\\sqrt{\\alpha}\\,)$ for some $\\alpha\\in k_{2}$ , then $\\widetilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)$ . In fact, $K_{2}$ and ${\\tilde{K}}_{2}$ are the only extensions $F/k_{2}$ of $k_{2}$ with the p roperties ", "page_idx": 9}, {"type": "text", "text": "1. $F/k_{2}$ is a quadratic extension unramified outside $\\mathfrak{p}$ ; ", "page_idx": 9}, {"type": "text", "text": "Therefore it suffices to observe that if $k_{2}(\\sqrt{\\alpha}\\,)$ has these properties, then so does $k_{2}(\\sqrt{d_{2}\\alpha}\\,)$ . But this is elementary. ", "page_idx": 9}, {"type": "text", "text": "In particular, the compositum $M\\,=\\,K_{2}\\tilde{K}_{2}\\,=\\,k_{2}(\\sqrt{d_{2}},\\sqrt{\\alpha}\\,)$ is an extension of type $(2,2)$ over $k_{2}$ with subextensions $K_{2}$ , $\\widetilde{K}_{2}$ and $F=k_{2}(\\sqrt{d_{2}}\\,)$ . Clearly $F$ is the unramified quadratic extension of $k_{2}$ , so bo t h $M/K_{2}$ and $M/\\widetilde{K}_{2}$ are unramified. If $K_{2}$ had 2-class number 2, then $M$ would have odd class numb e r, and $M$ would also be the 2-class field of ${\\tilde{K}}_{2}$ . Thus $2\\parallel h(K_{2})$ implies that $2\\parallel h(\\widetilde{K}_{2})$ . This proves part a) of the proposition. ", "page_idx": 9}, {"type": "text", "text": "Before we go on, we give a Hasse diagram for the fields occurring in this proof: ", "page_idx": 9}, {"type": "image", "img_path": "images/d372d949890fcb556f1018db84ad61f8f2a0dc1ea6a962a89ead2aeec6283a9b.jpg", "img_caption": [], "img_footnote": [], "page_idx": 9}, {"type": "text", "text": "Now assume that $4\\,\\mid\\,h(K_{2})$ . Since $\\mathrm{Cl_{2}}(M)$ is cyclic by Lemma 6, there is a unique quadratic unramified extension $N/M$ , and the uniqueness implies at once that $N/k_{2}$ is normal. Hence $G\\,=\\,\\operatorname{Gal}(N/k_{2})$ is a group of order 8 containing a subgroup of type $(2,2)\\,\\simeq\\,\\mathrm{Gal}(N/F)$ : in fact, if $\\operatorname{Gal}(N/F)$ were cyclic, then the primes ramifying in $M/F$ would also ramify in $N/M$ contradicting the fact that $N/M$ is unramified. There are three groups satisfying these conditions: $G=(2,4)$ , $G=(2,2,2)$ and $G=D_{4}$ . We claim that $G$ is non-abelian; once we have proved this, it follows that exactly one of the groups $\\mathrm{Gal}(N/K_{2})$ and $\\mathrm{Gal}(N/\\widetilde{K}_{2})$ is cyclic, and that the other is not, which is what we want to prove. ", "page_idx": 9}, {"type": "text", "text": "So assume that $G$ is abelian. Then $M/F$ is ramified at two finite primes $\\mathfrak{q}$ and ${\\mathfrak{q}}^{\\prime}$ of $F$ dividing $\\mathfrak{p}$ (in $k_{2}$ ); if $F_{1}$ and $F_{2}$ denote the quadratic subextensions of $N/F$ different from $M$ then $F_{1}/F$ and $F_{2}/F$ must be ramified at a finite prime (since ", "page_idx": 9}]	{"preproc_blocks": [{"type": "text", "bbox": [124, 111, 487, 137], "lines": [{"bbox": [125, 114, 486, 126], "spans": [{"bbox": [125, 114, 297, 126], "score": 1.0, "content": "at the two ramified infinite primes has ", "type": "text"}, {"bbox": [297, 116, 304, 123], "score": 0.9, "content": "\\mathbb{Z}", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [304, 114, 358, 126], "score": 1.0, "content": "-rank 2, i.e. ", "type": "text"}, {"bbox": [358, 115, 410, 126], "score": 0.92, "content": "(E:H)\\geq4", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [410, 114, 486, 126], "score": 1.0, "content": " by consideration", "type": "text"}], "index": 0}, {"bbox": [125, 127, 486, 138], "spans": [{"bbox": [125, 127, 311, 138], "score": 1.0, "content": "of the infinite primes alone. In particular, ", "type": "text"}, {"bbox": [311, 127, 391, 138], "score": 0.92, "content": "\\#\\operatorname{Am}_{2}(M/F)\\leq2", "type": "inline_equation", "height": 11, "width": 80}, {"bbox": [391, 127, 441, 138], "score": 1.0, "content": " in case B).", "type": "text"}, {"bbox": [476, 127, 486, 137], "score": 0.9776784181594849, "content": "\u53e3", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [125, 145, 486, 169], "lines": [{"bbox": [136, 146, 487, 159], "spans": [{"bbox": [136, 147, 404, 159], "score": 1.0, "content": "Next we derive some relations between the class groups of ", "type": "text"}, {"bbox": [404, 149, 417, 158], "score": 0.92, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [417, 147, 441, 159], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [442, 146, 455, 158], "score": 0.92, "content": "{\\tilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [455, 147, 487, 159], "score": 1.0, "content": "; these", "type": "text"}], "index": 2}, {"bbox": [125, 159, 439, 170], "spans": [{"bbox": [125, 159, 363, 170], "score": 1.0, "content": "relations will allow us to use each of them as our field", "type": "text"}, {"bbox": [364, 161, 373, 168], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [374, 159, 439, 170], "score": 1.0, "content": " in Theorem 1.", "type": "text"}], "index": 3}], "index": 2.5}, {"type": "text", "bbox": [124, 174, 486, 212], "lines": [{"bbox": [126, 177, 485, 190], "spans": [{"bbox": [126, 177, 218, 190], "score": 1.0, "content": "Proposition 8. Let ", "type": "text"}, {"bbox": [218, 180, 226, 187], "score": 0.76, "content": "L", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [226, 177, 248, 190], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [248, 177, 255, 187], "score": 0.86, "content": "\\widetilde{L}", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [256, 177, 476, 190], "score": 1.0, "content": " be the two unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [476, 180, 482, 187], "score": 0.83, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [482, 177, 485, 190], "score": 1.0, "content": ",", "type": "text"}], "index": 4}, {"bbox": [126, 189, 486, 203], "spans": [{"bbox": [126, 190, 158, 203], "score": 1.0, "content": "and let ", "type": "text"}, {"bbox": [158, 192, 171, 201], "score": 0.89, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [172, 190, 192, 203], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [193, 189, 206, 201], "score": 0.92, "content": "{\\widetilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [206, 190, 339, 203], "score": 1.0, "content": " be two qu a dratic extensions of", "type": "text"}, {"bbox": [340, 192, 349, 201], "score": 0.81, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [350, 190, 363, 203], "score": 1.0, "content": " in", "type": "text"}, {"bbox": [364, 191, 371, 200], "score": 0.62, "content": "L", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [372, 190, 392, 203], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [393, 189, 400, 200], "score": 0.86, "content": "\\widetilde{L}", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [400, 190, 486, 203], "score": 1.0, "content": ", respectively, which", "type": "text"}], "index": 5}, {"bbox": [126, 203, 228, 214], "spans": [{"bbox": [126, 203, 216, 214], "score": 1.0, "content": "are not normal o ver ", "type": "text"}, {"bbox": [217, 204, 225, 213], "score": 0.89, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [225, 203, 228, 214], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5}, {"type": "text", "bbox": [133, 214, 487, 253], "lines": [{"bbox": [136, 216, 340, 229], "spans": [{"bbox": [136, 217, 199, 229], "score": 1.0, "content": "a) We have 4", "type": "text"}, {"bbox": [199, 219, 232, 229], "score": 0.85, "content": "\|\\mathit{\\Omega}_{h}(K_{2})", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [232, 217, 294, 229], "score": 1.0, "content": " if and only if", "type": "text"}, {"bbox": [295, 216, 336, 229], "score": 0.89, "content": "4\\mid h(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [336, 217, 340, 229], "score": 1.0, "content": ";", "type": "text"}], "index": 7}, {"bbox": [135, 230, 486, 243], "spans": [{"bbox": [135, 231, 153, 243], "score": 1.0, "content": "b) ", "type": "text"}, {"bbox": [154, 232, 200, 243], "score": 0.67, "content": "I f\\,4\\mid h(K_{2})", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [201, 231, 256, 243], "score": 1.0, "content": ", then one of", "type": "text"}, {"bbox": [257, 232, 292, 243], "score": 0.9, "content": "\\mathrm{Cl}_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [292, 231, 306, 243], "score": 1.0, "content": " or", "type": "text"}, {"bbox": [307, 230, 342, 243], "score": 0.92, "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [343, 231, 382, 243], "score": 1.0, "content": " has type ", "type": "text"}, {"bbox": [382, 232, 405, 243], "score": 0.84, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [405, 231, 486, 243], "score": 1.0, "content": ", whereas the other", "type": "text"}], "index": 8}, {"bbox": [150, 243, 246, 254], "spans": [{"bbox": [150, 243, 226, 254], "score": 1.0, "content": "is cyclic of order", "type": "text"}, {"bbox": [227, 245, 243, 253], "score": 0.83, "content": "\\geq4", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [243, 243, 246, 254], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 8}, {"type": "text", "bbox": [125, 259, 486, 284], "lines": [{"bbox": [126, 262, 486, 274], "spans": [{"bbox": [126, 262, 293, 274], "score": 1.0, "content": "Proof. Notice that the prime dividing ", "type": "text"}, {"bbox": [293, 262, 327, 273], "score": 0.82, "content": "\\mathrm{disc}(k_{1})", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [327, 262, 367, 274], "score": 1.0, "content": " splits in ", "type": "text"}, {"bbox": [368, 263, 378, 272], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [378, 262, 486, 274], "score": 1.0, "content": ". Throughout this proof,", "type": "text"}], "index": 10}, {"bbox": [125, 273, 342, 286], "spans": [{"bbox": [125, 273, 140, 286], "score": 1.0, "content": "let ", "type": "text"}, {"bbox": [140, 277, 146, 285], "score": 0.85, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [146, 273, 252, 286], "score": 1.0, "content": " be one of the primes of ", "type": "text"}, {"bbox": [252, 275, 262, 284], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [262, 273, 304, 286], "score": 1.0, "content": " dividing ", "type": "text"}, {"bbox": [304, 275, 338, 285], "score": 0.85, "content": "\\mathrm{disc}(k_{1})", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [339, 273, 342, 286], "score": 1.0, "content": ".", "type": "text"}], "index": 11}], "index": 10.5}, {"type": "text", "bbox": [125, 285, 486, 311], "lines": [{"bbox": [136, 285, 484, 299], "spans": [{"bbox": [136, 286, 189, 299], "score": 1.0, "content": "If we write ", "type": "text"}, {"bbox": [189, 287, 251, 298], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [252, 286, 296, 299], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [296, 288, 326, 297], "score": 0.92, "content": "\\alpha\\in k_{2}", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [326, 286, 355, 299], "score": 1.0, "content": ", then", "type": "text"}, {"bbox": [356, 285, 427, 298], "score": 0.93, "content": "\\widetilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [427, 286, 471, 299], "score": 1.0, "content": ". In fact, ", "type": "text"}, {"bbox": [471, 289, 484, 297], "score": 0.91, "content": "K_{2}", "type": "inline_equation", "height": 8, "width": 13}], "index": 12}, {"bbox": [126, 299, 399, 312], "spans": [{"bbox": [126, 300, 145, 312], "score": 1.0, "content": "and", "type": "text"}, {"bbox": [145, 299, 158, 311], "score": 0.93, "content": "{\\tilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [159, 300, 265, 312], "score": 1.0, "content": " are the only extensions ", "type": "text"}, {"bbox": [265, 301, 287, 312], "score": 0.93, "content": "F/k_{2}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [287, 300, 301, 312], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [302, 302, 311, 311], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [312, 300, 399, 312], "score": 1.0, "content": " with the p roperties", "type": "text"}], "index": 13}], "index": 12.5}, {"type": "text", "bbox": [124, 311, 363, 322], "lines": [{"bbox": [126, 311, 363, 325], "spans": [{"bbox": [126, 311, 137, 325], "score": 1.0, "content": "1. ", "type": "text"}, {"bbox": [138, 313, 160, 324], "score": 0.91, "content": "F/k_{2}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [160, 311, 354, 325], "score": 1.0, "content": " is a quadratic extension unramified outside ", "type": "text"}, {"bbox": [354, 316, 360, 323], "score": 0.84, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [360, 311, 363, 325], "score": 1.0, "content": ";", "type": "text"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [124, 334, 487, 358], "lines": [{"bbox": [125, 335, 486, 348], "spans": [{"bbox": [125, 335, 297, 348], "score": 1.0, "content": "Therefore it suffices to observe that if ", "type": "text"}, {"bbox": [297, 336, 331, 347], "score": 0.94, "content": "k_{2}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [331, 335, 486, 348], "score": 1.0, "content": " has these properties, then so does", "type": "text"}], "index": 15}, {"bbox": [126, 347, 277, 360], "spans": [{"bbox": [126, 348, 169, 360], "score": 0.94, "content": "k_{2}(\\sqrt{d_{2}\\alpha}\\,)", "type": "inline_equation", "height": 12, "width": 43}, {"bbox": [170, 347, 277, 360], "score": 1.0, "content": ". But this is elementary.", "type": "text"}], "index": 16}], "index": 15.5}, {"type": "text", "bbox": [124, 358, 487, 434], "lines": [{"bbox": [137, 360, 488, 374], "spans": [{"bbox": [137, 360, 276, 374], "score": 1.0, "content": "In particular, the compositum ", "type": "text"}, {"bbox": [276, 360, 402, 372], "score": 0.94, "content": "M\\,=\\,K_{2}\\tilde{K}_{2}\\,=\\,k_{2}(\\sqrt{d_{2}},\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 12, "width": 126}, {"bbox": [402, 360, 488, 374], "score": 1.0, "content": " is an extension of", "type": "text"}], "index": 17}, {"bbox": [125, 373, 486, 387], "spans": [{"bbox": [125, 374, 148, 387], "score": 1.0, "content": "type ", "type": "text"}, {"bbox": [149, 375, 171, 386], "score": 0.93, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [171, 374, 195, 387], "score": 1.0, "content": " over ", "type": "text"}, {"bbox": [195, 376, 205, 385], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [205, 374, 294, 387], "score": 1.0, "content": " with subextensions ", "type": "text"}, {"bbox": [294, 376, 307, 385], "score": 0.89, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [308, 374, 313, 387], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [313, 373, 326, 385], "score": 0.92, "content": "\\widetilde{K}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [327, 374, 348, 387], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [349, 374, 408, 385], "score": 0.92, "content": "F=k_{2}(\\sqrt{d_{2}}\\,)", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [408, 374, 449, 387], "score": 1.0, "content": ". Clearly ", "type": "text"}, {"bbox": [450, 376, 457, 383], "score": 0.9, "content": "F", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [458, 374, 486, 387], "score": 1.0, "content": " is the", "type": "text"}], "index": 18}, {"bbox": [126, 387, 487, 400], "spans": [{"bbox": [126, 388, 275, 400], "score": 1.0, "content": "unramified quadratic extension of ", "type": "text"}, {"bbox": [275, 389, 285, 398], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [285, 388, 326, 400], "score": 1.0, "content": ", so bo t h ", "type": "text"}, {"bbox": [326, 389, 354, 399], "score": 0.93, "content": "M/K_{2}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [354, 388, 376, 400], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [376, 387, 405, 399], "score": 0.94, "content": "M/\\widetilde{K}_{2}", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [405, 388, 487, 400], "score": 1.0, "content": " are unramified. If", "type": "text"}], "index": 19}, {"bbox": [126, 399, 486, 411], "spans": [{"bbox": [126, 401, 139, 410], "score": 0.92, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [139, 399, 261, 411], "score": 1.0, "content": " had 2-class number 2, then ", "type": "text"}, {"bbox": [261, 401, 272, 408], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [272, 399, 425, 411], "score": 1.0, "content": " would have odd class numb e r, and ", "type": "text"}, {"bbox": [426, 401, 437, 408], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [437, 399, 486, 411], "score": 1.0, "content": " would also", "type": "text"}], "index": 20}, {"bbox": [124, 411, 486, 425], "spans": [{"bbox": [124, 411, 219, 425], "score": 1.0, "content": "be the 2-class field of", "type": "text"}, {"bbox": [220, 411, 233, 423], "score": 0.92, "content": "{\\tilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [233, 411, 264, 425], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [265, 413, 307, 424], "score": 0.95, "content": "2\\parallel h(K_{2})", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [307, 411, 364, 425], "score": 1.0, "content": " implies that ", "type": "text"}, {"bbox": [364, 411, 406, 424], "score": 0.94, "content": "2\\parallel h(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [407, 411, 486, 425], "score": 1.0, "content": ". This proves part", "type": "text"}], "index": 21}, {"bbox": [125, 424, 220, 436], "spans": [{"bbox": [125, 424, 220, 436], "score": 1.0, "content": "a) of the proposition.", "type": "text"}], "index": 22}], "index": 19.5}, {"type": "text", "bbox": [132, 434, 485, 447], "lines": [{"bbox": [137, 436, 484, 449], "spans": [{"bbox": [137, 436, 484, 449], "score": 1.0, "content": "Before we go on, we give a Hasse diagram for the fields occurring in this proof:", "type": "text"}], "index": 23}], "index": 23}, {"type": "image", "bbox": [253, 460, 376, 545], "blocks": [{"type": "image_body", "bbox": [253, 460, 376, 545], "group_id": 0, "lines": [{"bbox": [253, 460, 376, 545], "spans": [{"bbox": [253, 460, 376, 545], "score": 0.944, "type": "image", "image_path": "d372d949890fcb556f1018db84ad61f8f2a0dc1ea6a962a89ead2aeec6283a9b.jpg"}]}], "index": 24.5, "virtual_lines": [{"bbox": [253, 460, 376, 502.5], "spans": [], "index": 24}, {"bbox": [253, 502.5, 376, 545.0], "spans": [], "index": 25}]}], "index": 24.5}, {"type": "text", "bbox": [124, 554, 487, 663], "lines": [{"bbox": [137, 556, 487, 569], "spans": [{"bbox": [137, 556, 220, 569], "score": 1.0, "content": "Now assume that ", "type": "text"}, {"bbox": [220, 558, 263, 568], "score": 0.93, "content": "4\\,\\mid\\,h(K_{2})", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [264, 556, 300, 569], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [301, 558, 334, 568], "score": 0.91, "content": "\\mathrm{Cl_{2}}(M)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [334, 556, 487, 569], "score": 1.0, "content": " is cyclic by Lemma 6, there is a", "type": "text"}], "index": 26}, {"bbox": [126, 569, 486, 580], "spans": [{"bbox": [126, 569, 299, 580], "score": 1.0, "content": "unique quadratic unramified extension ", "type": "text"}, {"bbox": [299, 569, 323, 580], "score": 0.91, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [323, 569, 486, 580], "score": 1.0, "content": ", and the uniqueness implies at once", "type": "text"}], "index": 27}, {"bbox": [125, 579, 487, 593], "spans": [{"bbox": [125, 579, 148, 593], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 581, 171, 592], "score": 0.93, "content": "N/k_{2}", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [172, 579, 257, 593], "score": 1.0, "content": " is normal. Hence ", "type": "text"}, {"bbox": [258, 581, 329, 592], "score": 0.93, "content": "G\\,=\\,\\operatorname{Gal}(N/k_{2})", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [329, 579, 487, 593], "score": 1.0, "content": " is a group of order 8 containing a", "type": "text"}], "index": 28}, {"bbox": [126, 592, 486, 604], "spans": [{"bbox": [126, 592, 205, 604], "score": 1.0, "content": "subgroup of type ", "type": "text"}, {"bbox": [206, 593, 288, 604], "score": 0.93, "content": "(2,2)\\,\\simeq\\,\\mathrm{Gal}(N/F)", "type": "inline_equation", "height": 11, "width": 82}, {"bbox": [288, 592, 343, 604], "score": 1.0, "content": ": in fact, if ", "type": "text"}, {"bbox": [343, 593, 388, 604], "score": 0.92, "content": "\\operatorname{Gal}(N/F)", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [388, 592, 486, 604], "score": 1.0, "content": " were cyclic, then the", "type": "text"}], "index": 29}, {"bbox": [126, 605, 486, 616], "spans": [{"bbox": [126, 605, 217, 616], "score": 1.0, "content": "primes ramifying in ", "type": "text"}, {"bbox": [217, 605, 240, 616], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [240, 605, 339, 616], "score": 1.0, "content": " would also ramify in ", "type": "text"}, {"bbox": [339, 605, 363, 616], "score": 0.94, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [363, 605, 486, 616], "score": 1.0, "content": " contradicting the fact that", "type": "text"}], "index": 30}, {"bbox": [126, 616, 485, 628], "spans": [{"bbox": [126, 617, 150, 628], "score": 0.94, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [150, 616, 438, 628], "score": 1.0, "content": " is unramified. There are three groups satisfying these conditions: ", "type": "text"}, {"bbox": [439, 617, 482, 628], "score": 0.94, "content": "G=(2,4)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [483, 616, 485, 628], "score": 1.0, "content": ",", "type": "text"}], "index": 31}, {"bbox": [126, 628, 486, 640], "spans": [{"bbox": [126, 629, 180, 640], "score": 0.94, "content": "G=(2,2,2)", "type": "inline_equation", "height": 11, "width": 54}, {"bbox": [181, 628, 203, 640], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [204, 630, 239, 639], "score": 0.93, "content": "G=D_{4}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [239, 628, 314, 640], "score": 1.0, "content": ". We claim that ", "type": "text"}, {"bbox": [314, 630, 322, 637], "score": 0.91, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [322, 628, 486, 640], "score": 1.0, "content": " is non-abelian; once we have proved", "type": "text"}], "index": 32}, {"bbox": [125, 640, 485, 654], "spans": [{"bbox": [125, 641, 324, 654], "score": 1.0, "content": "this, it follows that exactly one of the groups ", "type": "text"}, {"bbox": [324, 642, 373, 653], "score": 0.91, "content": "\\mathrm{Gal}(N/K_{2})", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [374, 641, 395, 654], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [396, 640, 445, 653], "score": 0.92, "content": "\\mathrm{Gal}(N/\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 49}, {"bbox": [445, 641, 485, 654], "score": 1.0, "content": " is cyclic,", "type": "text"}], "index": 33}, {"bbox": [125, 653, 383, 665], "spans": [{"bbox": [125, 653, 383, 665], "score": 1.0, "content": "and that the other is not, which is what we want to prove.", "type": "text"}], "index": 34}], "index": 30}, {"type": "text", "bbox": [124, 663, 486, 700], "lines": [{"bbox": [137, 665, 487, 678], "spans": [{"bbox": [137, 665, 208, 678], "score": 1.0, "content": "So assume that ", "type": "text"}, {"bbox": [208, 667, 216, 674], "score": 0.91, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [217, 665, 295, 678], "score": 1.0, "content": " is abelian. Then ", "type": "text"}, {"bbox": [295, 666, 318, 677], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [318, 665, 460, 678], "score": 1.0, "content": " is ramified at two finite primes ", "type": "text"}, {"bbox": [460, 669, 465, 676], "score": 0.87, "content": "\\mathfrak{q}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [466, 665, 487, 678], "score": 1.0, "content": " and", "type": "text"}], "index": 35}, {"bbox": [126, 677, 485, 690], "spans": [{"bbox": [126, 678, 134, 688], "score": 0.9, "content": "{\\mathfrak{q}}^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [134, 677, 147, 690], "score": 1.0, "content": " of", "type": "text"}, {"bbox": [148, 679, 156, 686], "score": 0.91, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [156, 677, 196, 690], "score": 1.0, "content": " dividing", "type": "text"}, {"bbox": [197, 681, 202, 688], "score": 0.85, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [203, 677, 220, 690], "score": 1.0, "content": " (in ", "type": "text"}, {"bbox": [221, 679, 230, 687], "score": 0.86, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [231, 677, 248, 690], "score": 1.0, "content": "); if", "type": "text"}, {"bbox": [249, 679, 259, 687], "score": 0.92, "content": "F_{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [260, 677, 282, 690], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [282, 679, 293, 687], "score": 0.92, "content": "F_{2}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [293, 677, 463, 690], "score": 1.0, "content": " denote the quadratic subextensions of ", "type": "text"}, {"bbox": [464, 678, 485, 689], "score": 0.94, "content": "N/F", "type": "inline_equation", "height": 11, "width": 21}], "index": 36}, {"bbox": [125, 689, 486, 702], "spans": [{"bbox": [125, 689, 190, 702], "score": 1.0, "content": "different from ", "type": "text"}, {"bbox": [190, 691, 201, 698], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [201, 689, 228, 702], "score": 1.0, "content": " then ", "type": "text"}, {"bbox": [228, 690, 252, 701], "score": 0.94, "content": "F_{1}/F", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [253, 689, 275, 702], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [276, 690, 300, 701], "score": 0.94, "content": "F_{2}/F", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [300, 689, 486, 702], "score": 1.0, "content": " must be ramified at a finite prime (since", "type": "text"}], "index": 37}], "index": 36}], "layout_bboxes": [], "page_idx": 9, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [{"type": "image", "bbox": [253, 460, 376, 545], "blocks": [{"type": "image_body", "bbox": [253, 460, 376, 545], "group_id": 0, "lines": [{"bbox": [253, 460, 376, 545], "spans": [{"bbox": [253, 460, 376, 545], "score": 0.944, "type": "image", "image_path": "d372d949890fcb556f1018db84ad61f8f2a0dc1ea6a962a89ead2aeec6283a9b.jpg"}]}], "index": 24.5, "virtual_lines": [{"bbox": [253, 460, 376, 502.5], "spans": [], "index": 24}, {"bbox": [253, 502.5, 376, 545.0], "spans": [], "index": 25}]}], "index": 24.5}], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [126, 91, 135, 99], "lines": [{"bbox": [125, 92, 136, 102], "spans": [{"bbox": [125, 92, 136, 102], "score": 1.0, "content": "10", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [124, 111, 487, 137], "lines": [], "index": 0.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [125, 114, 486, 138], "lines_deleted": true}, {"type": "text", "bbox": [125, 145, 486, 169], "lines": [{"bbox": [136, 146, 487, 159], "spans": [{"bbox": [136, 147, 404, 159], "score": 1.0, "content": "Next we derive some relations between the class groups of ", "type": "text"}, {"bbox": [404, 149, 417, 158], "score": 0.92, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [417, 147, 441, 159], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [442, 146, 455, 158], "score": 0.92, "content": "{\\tilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [455, 147, 487, 159], "score": 1.0, "content": "; these", "type": "text"}], "index": 2}, {"bbox": [125, 159, 439, 170], "spans": [{"bbox": [125, 159, 363, 170], "score": 1.0, "content": "relations will allow us to use each of them as our field", "type": "text"}, {"bbox": [364, 161, 373, 168], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [374, 159, 439, 170], "score": 1.0, "content": " in Theorem 1.", "type": "text"}], "index": 3}], "index": 2.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [125, 146, 487, 170]}, {"type": "text", "bbox": [124, 174, 486, 212], "lines": [{"bbox": [126, 177, 485, 190], "spans": [{"bbox": [126, 177, 218, 190], "score": 1.0, "content": "Proposition 8. Let ", "type": "text"}, {"bbox": [218, 180, 226, 187], "score": 0.76, "content": "L", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [226, 177, 248, 190], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [248, 177, 255, 187], "score": 0.86, "content": "\\widetilde{L}", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [256, 177, 476, 190], "score": 1.0, "content": " be the two unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [476, 180, 482, 187], "score": 0.83, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [482, 177, 485, 190], "score": 1.0, "content": ",", "type": "text"}], "index": 4}, {"bbox": [126, 189, 486, 203], "spans": [{"bbox": [126, 190, 158, 203], "score": 1.0, "content": "and let ", "type": "text"}, {"bbox": [158, 192, 171, 201], "score": 0.89, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [172, 190, 192, 203], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [193, 189, 206, 201], "score": 0.92, "content": "{\\widetilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [206, 190, 339, 203], "score": 1.0, "content": " be two qu a dratic extensions of", "type": "text"}, {"bbox": [340, 192, 349, 201], "score": 0.81, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [350, 190, 363, 203], "score": 1.0, "content": " in", "type": "text"}, {"bbox": [364, 191, 371, 200], "score": 0.62, "content": "L", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [372, 190, 392, 203], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [393, 189, 400, 200], "score": 0.86, "content": "\\widetilde{L}", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [400, 190, 486, 203], "score": 1.0, "content": ", respectively, which", "type": "text"}], "index": 5}, {"bbox": [126, 203, 228, 214], "spans": [{"bbox": [126, 203, 216, 214], "score": 1.0, "content": "are not normal o ver ", "type": "text"}, {"bbox": [217, 204, 225, 213], "score": 0.89, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [225, 203, 228, 214], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [126, 177, 486, 214]}, {"type": "list", "bbox": [133, 214, 487, 253], "lines": [{"bbox": [136, 216, 340, 229], "spans": [{"bbox": [136, 217, 199, 229], "score": 1.0, "content": "a) We have 4", "type": "text"}, {"bbox": [199, 219, 232, 229], "score": 0.85, "content": "\|\\mathit{\\Omega}_{h}(K_{2})", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [232, 217, 294, 229], "score": 1.0, "content": " if and only if", "type": "text"}, {"bbox": [295, 216, 336, 229], "score": 0.89, "content": "4\\mid h(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [336, 217, 340, 229], "score": 1.0, "content": ";", "type": "text"}], "index": 7, "is_list_start_line": true, "is_list_end_line": true}, {"bbox": [135, 230, 486, 243], "spans": [{"bbox": [135, 231, 153, 243], "score": 1.0, "content": "b) ", "type": "text"}, {"bbox": [154, 232, 200, 243], "score": 0.67, "content": "I f\\,4\\mid h(K_{2})", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [201, 231, 256, 243], "score": 1.0, "content": ", then one of", "type": "text"}, {"bbox": [257, 232, 292, 243], "score": 0.9, "content": "\\mathrm{Cl}_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [292, 231, 306, 243], "score": 1.0, "content": " or", "type": "text"}, {"bbox": [307, 230, 342, 243], "score": 0.92, "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [343, 231, 382, 243], "score": 1.0, "content": " has type ", "type": "text"}, {"bbox": [382, 232, 405, 243], "score": 0.84, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [405, 231, 486, 243], "score": 1.0, "content": ", whereas the other", "type": "text"}], "index": 8, "is_list_start_line": true}, {"bbox": [150, 243, 246, 254], "spans": [{"bbox": [150, 243, 226, 254], "score": 1.0, "content": "is cyclic of order", "type": "text"}, {"bbox": [227, 245, 243, 253], "score": 0.83, "content": "\\geq4", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [243, 243, 246, 254], "score": 1.0, "content": ".", "type": "text"}], "index": 9, "is_list_end_line": true}], "index": 8, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [135, 216, 486, 254]}, {"type": "text", "bbox": [125, 259, 486, 284], "lines": [{"bbox": [126, 262, 486, 274], "spans": [{"bbox": [126, 262, 293, 274], "score": 1.0, "content": "Proof. Notice that the prime dividing ", "type": "text"}, {"bbox": [293, 262, 327, 273], "score": 0.82, "content": "\\mathrm{disc}(k_{1})", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [327, 262, 367, 274], "score": 1.0, "content": " splits in ", "type": "text"}, {"bbox": [368, 263, 378, 272], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [378, 262, 486, 274], "score": 1.0, "content": ". Throughout this proof,", "type": "text"}], "index": 10}, {"bbox": [125, 273, 342, 286], "spans": [{"bbox": [125, 273, 140, 286], "score": 1.0, "content": "let ", "type": "text"}, {"bbox": [140, 277, 146, 285], "score": 0.85, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [146, 273, 252, 286], "score": 1.0, "content": " be one of the primes of ", "type": "text"}, {"bbox": [252, 275, 262, 284], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [262, 273, 304, 286], "score": 1.0, "content": " dividing ", "type": "text"}, {"bbox": [304, 275, 338, 285], "score": 0.85, "content": "\\mathrm{disc}(k_{1})", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [339, 273, 342, 286], "score": 1.0, "content": ".", "type": "text"}], "index": 11}], "index": 10.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [125, 262, 486, 286]}, {"type": "text", "bbox": [125, 285, 486, 311], "lines": [{"bbox": [136, 285, 484, 299], "spans": [{"bbox": [136, 286, 189, 299], "score": 1.0, "content": "If we write ", "type": "text"}, {"bbox": [189, 287, 251, 298], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [252, 286, 296, 299], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [296, 288, 326, 297], "score": 0.92, "content": "\\alpha\\in k_{2}", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [326, 286, 355, 299], "score": 1.0, "content": ", then", "type": "text"}, {"bbox": [356, 285, 427, 298], "score": 0.93, "content": "\\widetilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [427, 286, 471, 299], "score": 1.0, "content": ". In fact, ", "type": "text"}, {"bbox": [471, 289, 484, 297], "score": 0.91, "content": "K_{2}", "type": "inline_equation", "height": 8, "width": 13}], "index": 12}, {"bbox": [126, 299, 399, 312], "spans": [{"bbox": [126, 300, 145, 312], "score": 1.0, "content": "and", "type": "text"}, {"bbox": [145, 299, 158, 311], "score": 0.93, "content": "{\\tilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [159, 300, 265, 312], "score": 1.0, "content": " are the only extensions ", "type": "text"}, {"bbox": [265, 301, 287, 312], "score": 0.93, "content": "F/k_{2}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [287, 300, 301, 312], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [302, 302, 311, 311], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [312, 300, 399, 312], "score": 1.0, "content": " with the p roperties", "type": "text"}], "index": 13}], "index": 12.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [126, 285, 484, 312]}, {"type": "text", "bbox": [124, 311, 363, 322], "lines": [{"bbox": [126, 311, 363, 325], "spans": [{"bbox": [126, 311, 137, 325], "score": 1.0, "content": "1. ", "type": "text"}, {"bbox": [138, 313, 160, 324], "score": 0.91, "content": "F/k_{2}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [160, 311, 354, 325], "score": 1.0, "content": " is a quadratic extension unramified outside ", "type": "text"}, {"bbox": [354, 316, 360, 323], "score": 0.84, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [360, 311, 363, 325], "score": 1.0, "content": ";", "type": "text"}], "index": 14}], "index": 14, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [126, 311, 363, 325]}, {"type": "text", "bbox": [124, 334, 487, 358], "lines": [{"bbox": [125, 335, 486, 348], "spans": [{"bbox": [125, 335, 297, 348], "score": 1.0, "content": "Therefore it suffices to observe that if ", "type": "text"}, {"bbox": [297, 336, 331, 347], "score": 0.94, "content": "k_{2}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [331, 335, 486, 348], "score": 1.0, "content": " has these properties, then so does", "type": "text"}], "index": 15}, {"bbox": [126, 347, 277, 360], "spans": [{"bbox": [126, 348, 169, 360], "score": 0.94, "content": "k_{2}(\\sqrt{d_{2}\\alpha}\\,)", "type": "inline_equation", "height": 12, "width": 43}, {"bbox": [170, 347, 277, 360], "score": 1.0, "content": ". But this is elementary.", "type": "text"}], "index": 16}], "index": 15.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [125, 335, 486, 360]}, {"type": "text", "bbox": [124, 358, 487, 434], "lines": [{"bbox": [137, 360, 488, 374], "spans": [{"bbox": [137, 360, 276, 374], "score": 1.0, "content": "In particular, the compositum ", "type": "text"}, {"bbox": [276, 360, 402, 372], "score": 0.94, "content": "M\\,=\\,K_{2}\\tilde{K}_{2}\\,=\\,k_{2}(\\sqrt{d_{2}},\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 12, "width": 126}, {"bbox": [402, 360, 488, 374], "score": 1.0, "content": " is an extension of", "type": "text"}], "index": 17}, {"bbox": [125, 373, 486, 387], "spans": [{"bbox": [125, 374, 148, 387], "score": 1.0, "content": "type ", "type": "text"}, {"bbox": [149, 375, 171, 386], "score": 0.93, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [171, 374, 195, 387], "score": 1.0, "content": " over ", "type": "text"}, {"bbox": [195, 376, 205, 385], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [205, 374, 294, 387], "score": 1.0, "content": " with subextensions ", "type": "text"}, {"bbox": [294, 376, 307, 385], "score": 0.89, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [308, 374, 313, 387], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [313, 373, 326, 385], "score": 0.92, "content": "\\widetilde{K}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [327, 374, 348, 387], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [349, 374, 408, 385], "score": 0.92, "content": "F=k_{2}(\\sqrt{d_{2}}\\,)", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [408, 374, 449, 387], "score": 1.0, "content": ". Clearly ", "type": "text"}, {"bbox": [450, 376, 457, 383], "score": 0.9, "content": "F", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [458, 374, 486, 387], "score": 1.0, "content": " is the", "type": "text"}], "index": 18}, {"bbox": [126, 387, 487, 400], "spans": [{"bbox": [126, 388, 275, 400], "score": 1.0, "content": "unramified quadratic extension of ", "type": "text"}, {"bbox": [275, 389, 285, 398], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [285, 388, 326, 400], "score": 1.0, "content": ", so bo t h ", "type": "text"}, {"bbox": [326, 389, 354, 399], "score": 0.93, "content": "M/K_{2}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [354, 388, 376, 400], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [376, 387, 405, 399], "score": 0.94, "content": "M/\\widetilde{K}_{2}", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [405, 388, 487, 400], "score": 1.0, "content": " are unramified. If", "type": "text"}], "index": 19}, {"bbox": [126, 399, 486, 411], "spans": [{"bbox": [126, 401, 139, 410], "score": 0.92, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [139, 399, 261, 411], "score": 1.0, "content": " had 2-class number 2, then ", "type": "text"}, {"bbox": [261, 401, 272, 408], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [272, 399, 425, 411], "score": 1.0, "content": " would have odd class numb e r, and ", "type": "text"}, {"bbox": [426, 401, 437, 408], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [437, 399, 486, 411], "score": 1.0, "content": " would also", "type": "text"}], "index": 20}, {"bbox": [124, 411, 486, 425], "spans": [{"bbox": [124, 411, 219, 425], "score": 1.0, "content": "be the 2-class field of", "type": "text"}, {"bbox": [220, 411, 233, 423], "score": 0.92, "content": "{\\tilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [233, 411, 264, 425], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [265, 413, 307, 424], "score": 0.95, "content": "2\\parallel h(K_{2})", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [307, 411, 364, 425], "score": 1.0, "content": " implies that ", "type": "text"}, {"bbox": [364, 411, 406, 424], "score": 0.94, "content": "2\\parallel h(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [407, 411, 486, 425], "score": 1.0, "content": ". This proves part", "type": "text"}], "index": 21}, {"bbox": [125, 424, 220, 436], "spans": [{"bbox": [125, 424, 220, 436], "score": 1.0, "content": "a) of the proposition.", "type": "text"}], "index": 22}], "index": 19.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [124, 360, 488, 436]}, {"type": "text", "bbox": [132, 434, 485, 447], "lines": [{"bbox": [137, 436, 484, 449], "spans": [{"bbox": [137, 436, 484, 449], "score": 1.0, "content": "Before we go on, we give a Hasse diagram for the fields occurring in this proof:", "type": "text"}], "index": 23}], "index": 23, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [137, 436, 484, 449]}, {"type": "image", "bbox": [253, 460, 376, 545], "blocks": [{"type": "image_body", "bbox": [253, 460, 376, 545], "group_id": 0, "lines": [{"bbox": [253, 460, 376, 545], "spans": [{"bbox": [253, 460, 376, 545], "score": 0.944, "type": "image", "image_path": "d372d949890fcb556f1018db84ad61f8f2a0dc1ea6a962a89ead2aeec6283a9b.jpg"}]}], "index": 24.5, "virtual_lines": [{"bbox": [253, 460, 376, 502.5], "spans": [], "index": 24}, {"bbox": [253, 502.5, 376, 545.0], "spans": [], "index": 25}]}], "index": 24.5, "page_num": "page_9", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 554, 487, 663], "lines": [{"bbox": [137, 556, 487, 569], "spans": [{"bbox": [137, 556, 220, 569], "score": 1.0, "content": "Now assume that ", "type": "text"}, {"bbox": [220, 558, 263, 568], "score": 0.93, "content": "4\\,\\mid\\,h(K_{2})", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [264, 556, 300, 569], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [301, 558, 334, 568], "score": 0.91, "content": "\\mathrm{Cl_{2}}(M)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [334, 556, 487, 569], "score": 1.0, "content": " is cyclic by Lemma 6, there is a", "type": "text"}], "index": 26}, {"bbox": [126, 569, 486, 580], "spans": [{"bbox": [126, 569, 299, 580], "score": 1.0, "content": "unique quadratic unramified extension ", "type": "text"}, {"bbox": [299, 569, 323, 580], "score": 0.91, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [323, 569, 486, 580], "score": 1.0, "content": ", and the uniqueness implies at once", "type": "text"}], "index": 27}, {"bbox": [125, 579, 487, 593], "spans": [{"bbox": [125, 579, 148, 593], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 581, 171, 592], "score": 0.93, "content": "N/k_{2}", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [172, 579, 257, 593], "score": 1.0, "content": " is normal. Hence ", "type": "text"}, {"bbox": [258, 581, 329, 592], "score": 0.93, "content": "G\\,=\\,\\operatorname{Gal}(N/k_{2})", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [329, 579, 487, 593], "score": 1.0, "content": " is a group of order 8 containing a", "type": "text"}], "index": 28}, {"bbox": [126, 592, 486, 604], "spans": [{"bbox": [126, 592, 205, 604], "score": 1.0, "content": "subgroup of type ", "type": "text"}, {"bbox": [206, 593, 288, 604], "score": 0.93, "content": "(2,2)\\,\\simeq\\,\\mathrm{Gal}(N/F)", "type": "inline_equation", "height": 11, "width": 82}, {"bbox": [288, 592, 343, 604], "score": 1.0, "content": ": in fact, if ", "type": "text"}, {"bbox": [343, 593, 388, 604], "score": 0.92, "content": "\\operatorname{Gal}(N/F)", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [388, 592, 486, 604], "score": 1.0, "content": " were cyclic, then the", "type": "text"}], "index": 29}, {"bbox": [126, 605, 486, 616], "spans": [{"bbox": [126, 605, 217, 616], "score": 1.0, "content": "primes ramifying in ", "type": "text"}, {"bbox": [217, 605, 240, 616], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [240, 605, 339, 616], "score": 1.0, "content": " would also ramify in ", "type": "text"}, {"bbox": [339, 605, 363, 616], "score": 0.94, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [363, 605, 486, 616], "score": 1.0, "content": " contradicting the fact that", "type": "text"}], "index": 30}, {"bbox": [126, 616, 485, 628], "spans": [{"bbox": [126, 617, 150, 628], "score": 0.94, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [150, 616, 438, 628], "score": 1.0, "content": " is unramified. There are three groups satisfying these conditions: ", "type": "text"}, {"bbox": [439, 617, 482, 628], "score": 0.94, "content": "G=(2,4)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [483, 616, 485, 628], "score": 1.0, "content": ",", "type": "text"}], "index": 31}, {"bbox": [126, 628, 486, 640], "spans": [{"bbox": [126, 629, 180, 640], "score": 0.94, "content": "G=(2,2,2)", "type": "inline_equation", "height": 11, "width": 54}, {"bbox": [181, 628, 203, 640], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [204, 630, 239, 639], "score": 0.93, "content": "G=D_{4}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [239, 628, 314, 640], "score": 1.0, "content": ". We claim that ", "type": "text"}, {"bbox": [314, 630, 322, 637], "score": 0.91, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [322, 628, 486, 640], "score": 1.0, "content": " is non-abelian; once we have proved", "type": "text"}], "index": 32}, {"bbox": [125, 640, 485, 654], "spans": [{"bbox": [125, 641, 324, 654], "score": 1.0, "content": "this, it follows that exactly one of the groups ", "type": "text"}, {"bbox": [324, 642, 373, 653], "score": 0.91, "content": "\\mathrm{Gal}(N/K_{2})", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [374, 641, 395, 654], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [396, 640, 445, 653], "score": 0.92, "content": "\\mathrm{Gal}(N/\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 49}, {"bbox": [445, 641, 485, 654], "score": 1.0, "content": " is cyclic,", "type": "text"}], "index": 33}, {"bbox": [125, 653, 383, 665], "spans": [{"bbox": [125, 653, 383, 665], "score": 1.0, "content": "and that the other is not, which is what we want to prove.", "type": "text"}], "index": 34}], "index": 30, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [125, 556, 487, 665]}, {"type": "text", "bbox": [124, 663, 486, 700], "lines": [{"bbox": [137, 665, 487, 678], "spans": [{"bbox": [137, 665, 208, 678], "score": 1.0, "content": "So assume that ", "type": "text"}, {"bbox": [208, 667, 216, 674], "score": 0.91, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [217, 665, 295, 678], "score": 1.0, "content": " is abelian. Then ", "type": "text"}, {"bbox": [295, 666, 318, 677], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [318, 665, 460, 678], "score": 1.0, "content": " is ramified at two finite primes ", "type": "text"}, {"bbox": [460, 669, 465, 676], "score": 0.87, "content": "\\mathfrak{q}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [466, 665, 487, 678], "score": 1.0, "content": " and", "type": "text"}], "index": 35}, {"bbox": [126, 677, 485, 690], "spans": [{"bbox": [126, 678, 134, 688], "score": 0.9, "content": "{\\mathfrak{q}}^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [134, 677, 147, 690], "score": 1.0, "content": " of", "type": "text"}, {"bbox": [148, 679, 156, 686], "score": 0.91, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [156, 677, 196, 690], "score": 1.0, "content": " dividing", "type": "text"}, {"bbox": [197, 681, 202, 688], "score": 0.85, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [203, 677, 220, 690], "score": 1.0, "content": " (in ", "type": "text"}, {"bbox": [221, 679, 230, 687], "score": 0.86, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [231, 677, 248, 690], "score": 1.0, "content": "); if", "type": "text"}, {"bbox": [249, 679, 259, 687], "score": 0.92, "content": "F_{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [260, 677, 282, 690], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [282, 679, 293, 687], "score": 0.92, "content": "F_{2}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [293, 677, 463, 690], "score": 1.0, "content": " denote the quadratic subextensions of ", "type": "text"}, {"bbox": [464, 678, 485, 689], "score": 0.94, "content": "N/F", "type": "inline_equation", "height": 11, "width": 21}], "index": 36}, {"bbox": [125, 689, 486, 702], "spans": [{"bbox": [125, 689, 190, 702], "score": 1.0, "content": "different from ", "type": "text"}, {"bbox": [190, 691, 201, 698], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [201, 689, 228, 702], "score": 1.0, "content": " then ", "type": "text"}, {"bbox": [228, 690, 252, 701], "score": 0.94, "content": "F_{1}/F", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [253, 689, 275, 702], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [276, 690, 300, 701], "score": 0.94, "content": "F_{2}/F", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [300, 689, 486, 702], "score": 1.0, "content": " must be ramified at a finite prime (since", "type": "text"}], "index": 37}], "index": 36, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [125, 665, 487, 702]}]}	{"layout_dets": [{"category_id": 1, "poly": [347, 1539, 1353, 1539, 1353, 1843, 347, 1843], "score": 0.981}, {"category_id": 1, "poly": [347, 997, 1354, 997, 1354, 1207, 347, 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each of them as our field in Theorem 1. Proposition 8. Let and be the two unramified cyclic quartic extensions of , and let and be two qu a dratic extensions of in and , respectively, which are not normal o ver . - a) We have 4 if and only if ; b) , then one of or has type , whereas the other is cyclic of order . Proof. Notice that the prime dividing splits in . Throughout this proof, let be one of the primes of dividing . If we write for some , then . In fact, and are the only extensions of with the p roperties 1. is a quadratic extension unramified outside ; Therefore it suffices to observe that if has these properties, then so does . But this is elementary. In particular, the compositum is an extension of type over with subextensions , and . Clearly is the unramified quadratic extension of , so bo t h and are unramified. If had 2-class number 2, then would have odd class numb e r, and would also be the 2-class field of . Thus implies that . This proves part a) of the proposition. Before we go on, we give a Hasse diagram for the fields occurring in this proof: Now assume that . Since is cyclic by Lemma 6, there is a unique quadratic unramified extension , and the uniqueness implies at once that is normal. Hence is a group of order 8 containing a subgroup of type : in fact, if were cyclic, then the primes ramifying in would also ramify in contradicting the fact that is unramified. There are three groups satisfying these conditions: , and . We claim that is non-abelian; once we have proved this, it follows that exactly one of the groups and is cyclic, and that the other is not, which is what we want to prove. So assume that is abelian. Then is ramified at two finite primes and of dividing (in ); if and denote the quadratic subextensions of different from then and must be ramified at a finite prime (since	<div class="pdf-page"> <p>Next we derive some relations between the class groups of and ; these relations will allow us to use each of them as our field in Theorem 1.</p> <p>Proposition 8. Let and be the two unramified cyclic quartic extensions of , and let and be two qu a dratic extensions of in and , respectively, which are not normal o ver .</p> <ul> <li>a) We have 4 if and only if ; b) , then one of or has type , whereas the other is cyclic of order .</li> </ul> <p>Proof. Notice that the prime dividing splits in . Throughout this proof, let be one of the primes of dividing .</p> <p>If we write for some , then . In fact, and are the only extensions of with the p roperties</p> <p>1. is a quadratic extension unramified outside ;</p> <p>Therefore it suffices to observe that if has these properties, then so does . But this is elementary.</p> <p>In particular, the compositum is an extension of type over with subextensions , and . Clearly is the unramified quadratic extension of , so bo t h and are unramified. If had 2-class number 2, then would have odd class numb e r, and would also be the 2-class field of . Thus implies that . This proves part a) of the proposition.</p> <p>Before we go on, we give a Hasse diagram for the fields occurring in this proof:</p> <p>Now assume that . Since is cyclic by Lemma 6, there is a unique quadratic unramified extension , and the uniqueness implies at once that is normal. Hence is a group of order 8 containing a subgroup of type : in fact, if were cyclic, then the primes ramifying in would also ramify in contradicting the fact that is unramified. There are three groups satisfying these conditions: , and . We claim that is non-abelian; once we have proved this, it follows that exactly one of the groups and is cyclic, and that the other is not, which is what we want to prove.</p> <p>So assume that is abelian. Then is ramified at two finite primes and of dividing (in ); if and denote the quadratic subextensions of different from then and must be ramified at a finite prime (since</p> </div>	<div class="pdf-page"> <div class="pdf-discarded" data-x="210" data-y="117" data-width="15" data-height="11" style="opacity: 0.5;">10</div> <p class="pdf-text" data-x="209" data-y="187" data-width="604" data-height="31">Next we derive some relations between the class groups of and ; these relations will allow us to use each of them as our field in Theorem 1.</p> <p class="pdf-text" data-x="207" data-y="224" data-width="606" data-height="50">Proposition 8. Let and be the two unramified cyclic quartic extensions of , and let and be two qu a dratic extensions of in and , respectively, which are not normal o ver .</p> <ul class="pdf-list" data-x="222" data-y="276" data-width="592" data-height="51"> <li>a) We have 4 if and only if ; b) , then one of or has type , whereas the other is cyclic of order .</li> </ul> <p class="pdf-text" data-x="209" data-y="334" data-width="604" data-height="33">Proof. Notice that the prime dividing splits in . Throughout this proof, let be one of the primes of dividing .</p> <p class="pdf-text" data-x="209" data-y="368" data-width="604" data-height="34">If we write for some , then . In fact, and are the only extensions of with the p roperties</p> <p class="pdf-text" data-x="207" data-y="402" data-width="400" data-height="14">1. is a quadratic extension unramified outside ;</p> <p class="pdf-text" data-x="207" data-y="431" data-width="607" data-height="31">Therefore it suffices to observe that if has these properties, then so does . But this is elementary.</p> <p class="pdf-text" data-x="207" data-y="462" data-width="607" data-height="99">In particular, the compositum is an extension of type over with subextensions , and . Clearly is the unramified quadratic extension of , so bo t h and are unramified. If had 2-class number 2, then would have odd class numb e r, and would also be the 2-class field of . Thus implies that . This proves part a) of the proposition.</p> <p class="pdf-text" data-x="220" data-y="561" data-width="591" data-height="16">Before we go on, we give a Hasse diagram for the fields occurring in this proof:</p> <p class="pdf-text" data-x="207" data-y="716" data-width="607" data-height="141">Now assume that . Since is cyclic by Lemma 6, there is a unique quadratic unramified extension , and the uniqueness implies at once that is normal. Hence is a group of order 8 containing a subgroup of type : in fact, if were cyclic, then the primes ramifying in would also ramify in contradicting the fact that is unramified. There are three groups satisfying these conditions: , and . We claim that is non-abelian; once we have proved this, it follows that exactly one of the groups and is cyclic, and that the other is not, which is what we want to prove.</p> <p class="pdf-text" data-x="207" data-y="857" data-width="606" data-height="48">So assume that is abelian. Then is ramified at two finite primes and of dividing (in ); if and denote the quadratic subextensions of different from then and must be ramified at a finite prime (since</p> </div>	{ "type": [ "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "text", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "inline_equation", "inline_equation", "inline_equation" ], "coordinates": [ [ 227, 188, 814, 205 ], [ 209, 205, 734, 219 ], [ 210, 228, 811, 245 ], [ 210, 244, 813, 262 ], [ 210, 262, 381, 276 ], [ 227, 279, 568, 296 ], [ 225, 297, 813, 314 ], [ 250, 314, 411, 328 ], [ 210, 338, 813, 354 ], [ 209, 352, 572, 369 ], [ 227, 368, 809, 386 ], [ 210, 386, 667, 403 ], [ 210, 402, 607, 420 ], [ 209, 433, 813, 449 ], [ 210, 448, 463, 465 ], [ 229, 465, 816, 483 ], [ 209, 482, 813, 500 ], [ 210, 500, 814, 517 ], [ 210, 515, 813, 531 ], [ 207, 531, 813, 549 ], [ 209, 548, 368, 563 ], [ 229, 563, 809, 580 ], [ 423, 594, 629, 704 ], [ 229, 718, 814, 735 ], [ 210, 735, 813, 749 ], [ 209, 748, 814, 766 ], [ 210, 765, 813, 780 ], [ 210, 782, 813, 796 ], [ 210, 796, 811, 811 ], [ 210, 811, 813, 827 ], [ 209, 827, 811, 845 ], [ 209, 844, 640, 859 ], [ 229, 859, 814, 876 ], [ 210, 875, 811, 892 ], [ 209, 890, 813, 907 ] ], "content": [ "Next we derive some relations between the class groups of K_{2} and {\\tilde{K}}_{2} ; these", "relations will allow us to use each of them as our field K in Theorem 1.", "Proposition 8. Let L and \\widetilde{L} be the two unramified cyclic quartic extensions of k ,", "and let K_{2} and {\\widetilde{K}}_{2} be two qu a dratic extensions of k_{2} in L and \\widetilde{L} , respectively, which", "are not normal o ver \\mathbb{Q} .", "a) We have 4 \|\\mathit{\\Omega}_{h}(K_{2}) if and only if 4\\mid h(\\widetilde{K}_{2}) ;", "b) I f\\,4\\mid h(K_{2}) , then one of \\mathrm{Cl}_{2}(K_{2}) or \\mathrm{Cl}_{2}(\\widetilde{K}_{2}) has type (2,2) , whereas the other", "is cyclic of order \\geq4 .", "Proof. Notice that the prime dividing \\mathrm{disc}(k_{1}) splits in k_{2} . Throughout this proof,", "let \\mathfrak{p} be one of the primes of k_{2} dividing \\mathrm{disc}(k_{1}) .", "If we write K_{2}=k_{2}(\\sqrt{\\alpha}\\,) for some \\alpha\\in k_{2} , then \\widetilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big) . In fact, K_{2}", "and {\\tilde{K}}_{2} are the only extensions F/k_{2} of k_{2} with the p roperties", "1. F/k_{2} is a quadratic extension unramified outside \\mathfrak{p} ;", "Therefore it suffices to observe that if k_{2}(\\sqrt{\\alpha}\\,) has these properties, then so does", "k_{2}(\\sqrt{d_{2}\\alpha}\\,) . But this is elementary.", "In particular, the compositum M\\,=\\,K_{2}\\tilde{K}_{2}\\,=\\,k_{2}(\\sqrt{d_{2}},\\sqrt{\\alpha}\\,) is an extension of", "type (2,2) over k_{2} with subextensions K_{2} , \\widetilde{K}_{2} and F=k_{2}(\\sqrt{d_{2}}\\,) . Clearly F is the", "unramified quadratic extension of k_{2} , so bo t h M/K_{2} and M/\\widetilde{K}_{2} are unramified. If", "K_{2} had 2-class number 2, then M would have odd class numb e r, and M would also", "be the 2-class field of {\\tilde{K}}_{2} . Thus 2\\parallel h(K_{2}) implies that 2\\parallel h(\\widetilde{K}_{2}) . This proves part", "a) of the proposition.", "Before we go on, we give a Hasse diagram for the fields occurring in this proof:", "", "Now assume that 4\\,\\mid\\,h(K_{2}) . Since \\mathrm{Cl_{2}}(M) is cyclic by Lemma 6, there is a", "unique quadratic unramified extension N/M , and the uniqueness implies at once", "that N/k_{2} is normal. Hence G\\,=\\,\\operatorname{Gal}(N/k_{2}) is a group of order 8 containing a", "subgroup of type (2,2)\\,\\simeq\\,\\mathrm{Gal}(N/F) : in fact, if \\operatorname{Gal}(N/F) were cyclic, then the", "primes ramifying in M/F would also ramify in N/M contradicting the fact that", "N/M is unramified. There are three groups satisfying these conditions: G=(2,4) ,", "G=(2,2,2) and G=D_{4} . We claim that G is non-abelian; once we have proved", "this, it follows that exactly one of the groups \\mathrm{Gal}(N/K_{2}) and \\mathrm{Gal}(N/\\widetilde{K}_{2}) is cyclic,", "and that the other is not, which is what we want to prove.", "So assume that G is abelian. Then M/F is ramified at two finite primes \\mathfrak{q} and", "{\\mathfrak{q}}^{\\prime} of F dividing \\mathfrak{p} (in k_{2} ); if F_{1} and F_{2} denote the quadratic subextensions of N/F", "different from M then F_{1}/F and F_{2}/F must be ramified at a finite prime (since" ], "index": [ 0, 1, 2, 3, 4, 7, 8, 9, 15, 16, 25, 26, 37, 50, 51, 65, 66, 67, 68, 69, 70, 86, 108, 131, 132, 133, 134, 135, 136, 137, 138, 139, 163, 164, 165 ] }	{ "coordinates": [ [ 423, 594, 629, 704 ], [ 423, 594, 629, 704 ] ], "index": [ 0, 1 ], "caption": [ "", "" ], "caption_coordinates": [ [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ] ] }	{ "type": [], "coordinates": [], "content": [], "index": [] }	{ "type": [], "coordinates": [], "content": [], "index": [] }	{"document_info": {"total_pages": 14, "parse_type": "txt", "version": "1.0.1", "total_content_list_items": 155}, "content_statistics": {"total_blocks": 182, "total_lines": 526, "total_images": 4, "total_equations": 28, "total_tables": 11, "total_characters": 29658, "total_words": 5331, "block_type_distribution": {"title": 7, "text": 126, "interline_equation": 14, "discarded": 24, "table_caption": 3, "table_body": 4, "image_body": 2, "list": 2}}, "model_statistics": {"total_layout_detections": 2826, "avg_confidence": 0.9470955414012739, "min_confidence": 0.26, "max_confidence": 1.0}, "page_statistics": [{"page_number": 0, "page_size": [612.0, 792.0], "blocks": 11, "lines": 38, "images": 0, "equations": 2, "tables": 0, "characters": 2362, "words": 398, "discarded_blocks": 1}, {"page_number": 1, "page_size": [612.0, 792.0], "blocks": 10, "lines": 30, "images": 0, "equations": 0, "tables": 3, "characters": 1688, "words": 305, "discarded_blocks": 1}, {"page_number": 2, "page_size": [612.0, 792.0], "blocks": 15, "lines": 44, "images": 0, "equations": 4, "tables": 0, "characters": 2417, "words": 416, "discarded_blocks": 2}, {"page_number": 3, "page_size": [612.0, 792.0], "blocks": 7, "lines": 27, "images": 0, "equations": 0, "tables": 3, "characters": 1591, "words": 291, "discarded_blocks": 1}, {"page_number": 4, "page_size": [612.0, 792.0], "blocks": 23, "lines": 38, "images": 0, "equations": 6, "tables": 0, "characters": 1604, "words": 303, "discarded_blocks": 4}, {"page_number": 5, "page_size": [612.0, 792.0], "blocks": 17, "lines": 40, "images": 0, "equations": 6, "tables": 0, "characters": 2131, "words": 393, "discarded_blocks": 1}, {"page_number": 6, "page_size": [612.0, 792.0], "blocks": 15, "lines": 36, "images": 2, "equations": 4, "tables": 0, "characters": 1791, "words": 324, "discarded_blocks": 2}, {"page_number": 7, "page_size": [612.0, 792.0], "blocks": 12, "lines": 49, "images": 0, "equations": 0, "tables": 0, "characters": 3049, "words": 571, "discarded_blocks": 1}, {"page_number": 8, "page_size": [612.0, 792.0], "blocks": 11, "lines": 42, "images": 0, "equations": 0, "tables": 0, "characters": 2636, "words": 479, "discarded_blocks": 2}, {"page_number": 9, "page_size": [612.0, 792.0], "blocks": 14, "lines": 35, "images": 2, "equations": 0, "tables": 0, "characters": 2036, "words": 375, "discarded_blocks": 1}, {"page_number": 10, "page_size": [612.0, 792.0], "blocks": 17, "lines": 35, "images": 0, "equations": 6, "tables": 0, "characters": 1789, "words": 327, "discarded_blocks": 2}, {"page_number": 11, "page_size": [612.0, 792.0], "blocks": 14, "lines": 40, "images": 0, "equations": 0, "tables": 2, "characters": 2094, "words": 399, "discarded_blocks": 2}, {"page_number": 12, "page_size": [612.0, 792.0], "blocks": 10, "lines": 44, "images": 0, "equations": 0, "tables": 3, "characters": 2572, "words": 487, "discarded_blocks": 2}, {"page_number": 13, "page_size": [612.0, 792.0], "blocks": 6, "lines": 28, "images": 0, "equations": 0, "tables": 0, "characters": 1898, "words": 263, "discarded_blocks": 2}]}
0003244v1	10	[ 612, 792 ]	{ "type": [ "text", "text", "text", "text", "text", "text", "interline_equation", "text", "interline_equation", "text", "interline_equation", "text", "title", "text", "text", "discarded", "discarded" ], "coordinates": [ [ 209, 143, 813, 222 ], [ 209, 223, 813, 355 ], [ 209, 356, 813, 389 ], [ 209, 400, 814, 447 ], [ 210, 491, 778, 509 ], [ 209, 518, 813, 550 ], [ 332, 559, 689, 593 ], [ 207, 598, 319, 614 ], [ 332, 625, 689, 658 ], [ 209, 664, 813, 695 ], [ 366, 700, 655, 734 ], [ 207, 736, 814, 769 ], [ 436, 788, 583, 805 ], [ 209, 813, 813, 844 ], [ 207, 855, 813, 905 ], [ 398, 116, 624, 128 ], [ 796, 117, 811, 128 ] ], "content": [ "has odd class number in the strict sense: see Lemma 6); since both and are normal (even abelian) over , ramification at implies ramification at the conjugated ideal . Hence both and ramify in and , and since they also ramify in , they must ramify completely in , again contradicting the fact that is unramified.", "We have proved that and contain subgroups of type (4) and , respectively. Now we wish to apply P roposition 5. But we have to compute . Since the class number of is even, it is sufficient to show that . In case A), there is e xactly one ramified prime (it divides ), hen c e . In case B), there are two ramified primes (one is infin i te, the other divides ), hence ; but is not a norm residue at the ramified infinite prime, h e nce and as claimed.", "Now P roposition 5 implies that is cyclic of order , and that . This concludes our proof. 口", "Proposition 9. Assume that is one of the imaginary quadratic fields of type or ) as explained in the Introduction. Then there exist two unramified cyclic quartic extensions of . Let be one of them, and write", "Then unless possibly when in case ).", "Proof. Observe that in case A) and B); Kuroda’s class number formulas for and gives", "", "in case A) and", "", "in case B). Multiplying them together and plugging in the class number formula for yields", "", "Now , and (by Proposition 6), and taking the square root we find as claimed. 口", "5. Classification", "In this section we apply the results obtained in the last few sections to give a proof for Theorem 1.", "Proof of Theorem 1. Let be one of the two cyclic quartic unramified extensions of , and let be the subgroup of fixing . Then satisfies the assumptions of Proposition 1, thus there are only the following possibilities:", "IMAGINARY QUADRATIC FIELDS", "11" ], "index": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 ] }	[{"type": "text", "text": "$F$ has odd class number in the strict sense: see Lemma 6); since both $F_{1}$ and $F_{2}$ are normal (even abelian) over $k_{2}$ , ramification at $\\mathfrak{q}$ implies ramification at the conjugated ideal ${\\mathfrak{q}}^{\\prime}$ . Hence both $\\mathfrak{q}$ and ${\\mathfrak{q}}^{\\prime}$ ramify in $F_{1}/F$ and $F_{2}/F$ , and since they also ramify in $M/F$ , they must ramify completely in $N/F$ , again contradicting the fact that $N/M$ is unramified. ", "page_idx": 10}, {"type": "text", "text": "We have proved that $\\mathrm{Cl_{2}}(K_{2})$ and $\\mathrm{Cl}_{2}(\\widetilde{K}_{2})$ contain subgroups of type (4) and $(2,2)$ , respectively. Now we wish to apply P roposition 5. But we have to compute $\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})$ . Since the class number of ${\\widetilde{K}}_{2}$ is even, it is sufficient to show that $\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})\\,\\leq\\,2$ . In case A), there is e xactly one ramified prime (it divides $d_{1}$ ), hen c e $\\#\\operatorname{Am}_{2}({\\widetilde{K}}_{2}/k_{2})\\,=\\,2/(E:H)\\,\\le\\,2$ . In case B), there are two ramified primes (one is infin i te, the other divides $d_{3}$ ), hence $\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})=4/(E:H)$ ; but $^{-1}$ is not a norm residue at the ramified infinite prime, h e nce $(E:H)\\ge2$ and $\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})\\leq2$ as claimed. ", "page_idx": 10}, {"type": "text", "text": "Now P roposition 5 implies that $\\mathrm{Cl}_{2}(K_{2})$ is cyclic of order $\\geq4$ , and that $\\mathrm{Cl}_{2}(\\widetilde{K}_{2})\\simeq$ $(2,2)$ . This concludes our proof. \u53e3 ", "page_idx": 10}, {"type": "text", "text": "Proposition 9. Assume that $k$ is one of the imaginary quadratic fields of type $A)$ or $B$ ) as explained in the Introduction. Then there exist two unramified cyclic quartic extensions of $k$ . Let $L$ be one of them, and write ", "page_idx": 10}, {"type": "text", "text": "Then $\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}$ unless possibly when $d_{3}=-4$ in case $B$ ). ", "page_idx": 10}, {"type": "text", "text": "Proof. Observe that $\\upsilon=0$ in case A) and B); Kuroda\u2019s class number formulas for $L/k_{1}$ and $L/k_{2}$ gives ", "page_idx": 10}, {"type": "equation", "text": "$$\nh_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{2h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{4h_{2}(k_{2})^{2}}\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "in case A) and ", "page_idx": 10}, {"type": "equation", "text": "$$\nh_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{4h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{2h_{2}(k_{2})^{2}}\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "in case B). Multiplying them together and plugging in the class number formula for $K/\\mathbb{Q}$ yields ", "page_idx": 10}, {"type": "equation", "text": "$$\nh_{2}(L)^{2}=\\frac{q_{1}\\,q_{2}}{8}\\frac{h_{2}(K_{1})^{2}\\,h_{2}(K_{2})^{2}\\,h_{2}(k)^{2}}{h_{2}(k_{1})^{2}\\,h_{2}(k_{2})^{2}}.\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "Now $h_{2}(k_{1})=1$ , $h_{2}(k_{2})=2$ and $q_{1}q_{2}=2$ (by Proposition 6), and taking the square root we find $\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}$ as claimed. \u53e3 ", "page_idx": 10}, {"type": "text", "text": "5. Classification ", "text_level": 1, "page_idx": 10}, {"type": "text", "text": "In this section we apply the results obtained in the last few sections to give a proof for Theorem 1. ", "page_idx": 10}, {"type": "text", "text": "Proof of Theorem 1. Let $L$ be one of the two cyclic quartic unramified extensions of $k$ , and let $N$ be the subgroup of $\\operatorname{Gal}(k^{2}/k)$ fixing $L$ . Then $N$ satisfies the assumptions of Proposition 1, thus there are only the following possibilities: ", "page_idx": 10}]	{"preproc_blocks": [{"type": "text", "bbox": [125, 111, 486, 172], "lines": [{"bbox": [126, 114, 484, 127], "spans": [{"bbox": [126, 116, 134, 123], "score": 0.9, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [134, 114, 439, 127], "score": 1.0, "content": " has odd class number in the strict sense: see Lemma 6); since both ", "type": "text"}, {"bbox": [439, 116, 450, 125], "score": 0.92, "content": "F_{1}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [450, 114, 473, 127], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [474, 116, 484, 125], "score": 0.92, "content": "F_{2}", "type": "inline_equation", "height": 9, "width": 10}], "index": 0}, {"bbox": [126, 127, 486, 139], "spans": [{"bbox": [126, 127, 268, 139], "score": 1.0, "content": "are normal (even abelian) over ", "type": "text"}, {"bbox": [269, 128, 279, 137], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [279, 127, 356, 139], "score": 1.0, "content": ", ramification at ", "type": "text"}, {"bbox": [356, 130, 361, 137], "score": 0.87, "content": "\\mathfrak{q}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [362, 127, 486, 139], "score": 1.0, "content": " implies ramification at the", "type": "text"}], "index": 1}, {"bbox": [126, 138, 485, 150], "spans": [{"bbox": [126, 138, 199, 150], "score": 1.0, "content": "conjugated ideal", "type": "text"}, {"bbox": [200, 139, 208, 149], "score": 0.91, "content": "{\\mathfrak{q}}^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [208, 138, 267, 150], "score": 1.0, "content": ". Hence both ", "type": "text"}, {"bbox": [267, 142, 272, 149], "score": 0.88, "content": "\\mathfrak{q}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [272, 138, 294, 150], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [294, 139, 302, 149], "score": 0.91, "content": "{\\mathfrak{q}}^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [302, 138, 347, 150], "score": 1.0, "content": " ramify in ", "type": "text"}, {"bbox": [347, 139, 371, 149], "score": 0.94, "content": "F_{1}/F", "type": "inline_equation", "height": 10, "width": 24}, {"bbox": [371, 138, 393, 150], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [393, 139, 417, 150], "score": 0.93, "content": "F_{2}/F", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [417, 138, 485, 150], "score": 1.0, "content": ", and since they", "type": "text"}], "index": 2}, {"bbox": [125, 150, 486, 163], "spans": [{"bbox": [125, 150, 188, 163], "score": 1.0, "content": "also ramify in ", "type": "text"}, {"bbox": [189, 151, 212, 162], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [212, 150, 357, 163], "score": 1.0, "content": ", they must ramify completely in ", "type": "text"}, {"bbox": [357, 151, 378, 162], "score": 0.93, "content": "N/F", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [379, 150, 486, 163], "score": 1.0, "content": ", again contradicting the", "type": "text"}], "index": 3}, {"bbox": [126, 162, 255, 174], "spans": [{"bbox": [126, 162, 167, 174], "score": 1.0, "content": "fact that ", "type": "text"}, {"bbox": [167, 163, 191, 174], "score": 0.93, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [191, 162, 255, 174], "score": 1.0, "content": " is unramified.", "type": "text"}], "index": 4}], "index": 2}, {"type": "text", "bbox": [125, 173, 486, 275], "lines": [{"bbox": [137, 174, 487, 188], "spans": [{"bbox": [137, 174, 234, 188], "score": 1.0, "content": "We have proved that ", "type": "text"}, {"bbox": [235, 176, 270, 187], "score": 0.94, "content": "\\mathrm{Cl_{2}}(K_{2})", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [270, 174, 294, 188], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [294, 174, 329, 187], "score": 0.93, "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [330, 174, 487, 188], "score": 1.0, "content": " contain subgroups of type (4) and", "type": "text"}], "index": 5}, {"bbox": [126, 187, 486, 200], "spans": [{"bbox": [126, 188, 148, 199], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 187, 486, 200], "score": 1.0, "content": ", respectively. Now we wish to apply P roposition 5. But we have to compute", "type": "text"}], "index": 6}, {"bbox": [126, 199, 486, 212], "spans": [{"bbox": [126, 199, 192, 212], "score": 0.92, "content": "\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})", "type": "inline_equation", "height": 13, "width": 66}, {"bbox": [192, 200, 314, 212], "score": 1.0, "content": ". Since the class number of", "type": "text"}, {"bbox": [314, 199, 327, 211], "score": 0.93, "content": "{\\widetilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [327, 200, 486, 212], "score": 1.0, "content": " is even, it is sufficient to show that", "type": "text"}], "index": 7}, {"bbox": [126, 212, 487, 227], "spans": [{"bbox": [126, 212, 214, 225], "score": 0.9, "content": "\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})\\,\\leq\\,2", "type": "inline_equation", "height": 13, "width": 88}, {"bbox": [214, 212, 487, 227], "score": 1.0, "content": ". In case A), there is e xactly one ramified prime (it divides", "type": "text"}], "index": 8}, {"bbox": [126, 226, 486, 238], "spans": [{"bbox": [126, 228, 136, 237], "score": 0.47, "content": "d_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 226, 174, 238], "score": 1.0, "content": "), hen c e ", "type": "text"}, {"bbox": [175, 226, 324, 238], "score": 0.94, "content": "\\#\\operatorname{Am}_{2}({\\widetilde{K}}_{2}/k_{2})\\,=\\,2/(E:H)\\,\\le\\,2", "type": "inline_equation", "height": 12, "width": 149}, {"bbox": [324, 226, 486, 238], "score": 1.0, "content": ". In case B), there are two ramified", "type": "text"}], "index": 9}, {"bbox": [126, 239, 486, 252], "spans": [{"bbox": [126, 240, 307, 252], "score": 1.0, "content": "primes (one is infin i te, the other divides ", "type": "text"}, {"bbox": [308, 242, 317, 251], "score": 0.89, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [318, 240, 356, 252], "score": 1.0, "content": "), hence ", "type": "text"}, {"bbox": [356, 239, 483, 252], "score": 0.94, "content": "\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})=4/(E:H)", "type": "inline_equation", "height": 13, "width": 127}, {"bbox": [483, 240, 486, 252], "score": 1.0, "content": ";", "type": "text"}], "index": 10}, {"bbox": [125, 251, 487, 264], "spans": [{"bbox": [125, 251, 144, 264], "score": 1.0, "content": "but", "type": "text"}, {"bbox": [144, 254, 157, 262], "score": 0.89, "content": "^{-1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [157, 251, 414, 264], "score": 1.0, "content": " is not a norm residue at the ramified infinite prime, h e nce ", "type": "text"}, {"bbox": [414, 253, 466, 263], "score": 0.92, "content": "(E:H)\\ge2", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [466, 251, 487, 264], "score": 1.0, "content": " and", "type": "text"}], "index": 11}, {"bbox": [126, 264, 262, 277], "spans": [{"bbox": [126, 264, 210, 277], "score": 0.93, "content": "\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})\\leq2", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [210, 265, 262, 277], "score": 1.0, "content": " as claimed.", "type": "text"}], "index": 12}], "index": 8.5}, {"type": "text", "bbox": [125, 276, 486, 301], "lines": [{"bbox": [137, 277, 487, 290], "spans": [{"bbox": [137, 278, 272, 290], "score": 1.0, "content": "Now P roposition 5 implies that ", "type": "text"}, {"bbox": [272, 280, 308, 290], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K_{2})", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [308, 278, 380, 290], "score": 1.0, "content": " is cyclic of order", "type": "text"}, {"bbox": [380, 281, 396, 289], "score": 0.7, "content": "\\geq4", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [396, 278, 439, 290], "score": 1.0, "content": ", and that", "type": "text"}, {"bbox": [439, 277, 487, 290], "score": 0.91, "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})\\simeq", "type": "inline_equation", "height": 13, "width": 48}], "index": 13}, {"bbox": [126, 290, 487, 302], "spans": [{"bbox": [126, 291, 148, 302], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 290, 267, 302], "score": 1.0, "content": ". This concludes our proof.", "type": "text"}, {"bbox": [476, 291, 487, 301], "score": 0.9864116907119751, "content": "\u53e3", "type": "text"}], "index": 14}], "index": 13.5}, {"type": "text", "bbox": [125, 310, 487, 346], "lines": [{"bbox": [125, 311, 486, 326], "spans": [{"bbox": [125, 311, 261, 326], "score": 1.0, "content": "Proposition 9. Assume that ", "type": "text"}, {"bbox": [261, 314, 268, 322], "score": 0.75, "content": "k", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [268, 311, 486, 326], "score": 1.0, "content": " is one of the imaginary quadratic fields of type", "type": "text"}], "index": 15}, {"bbox": [126, 324, 486, 337], "spans": [{"bbox": [126, 325, 138, 336], "score": 0.28, "content": "A)", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [139, 324, 154, 337], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [154, 326, 162, 334], "score": 0.77, "content": "B", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [163, 324, 486, 337], "score": 1.0, "content": ") as explained in the Introduction. Then there exist two unramified cyclic", "type": "text"}], "index": 16}, {"bbox": [126, 337, 374, 348], "spans": [{"bbox": [126, 337, 219, 348], "score": 1.0, "content": "quartic extensions of ", "type": "text"}, {"bbox": [219, 338, 225, 345], "score": 0.81, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [226, 337, 249, 348], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [250, 338, 257, 345], "score": 0.8, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [257, 337, 374, 348], "score": 1.0, "content": " be one of them, and write", "type": "text"}], "index": 17}], "index": 16}, {"type": "text", "bbox": [126, 380, 465, 394], "lines": [{"bbox": [127, 382, 464, 396], "spans": [{"bbox": [127, 382, 151, 396], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [152, 383, 282, 396], "score": 0.91, "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}", "type": "inline_equation", "height": 13, "width": 130}, {"bbox": [282, 382, 376, 396], "score": 1.0, "content": " unless possibly when ", "type": "text"}, {"bbox": [377, 385, 413, 394], "score": 0.9, "content": "d_{3}=-4", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [413, 382, 450, 396], "score": 1.0, "content": " in case ", "type": "text"}, {"bbox": [450, 385, 457, 392], "score": 0.77, "content": "B", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [458, 382, 464, 396], "score": 1.0, "content": ").", "type": "text"}], "index": 18}], "index": 18}, {"type": "text", "bbox": [125, 401, 486, 426], "lines": [{"bbox": [127, 403, 486, 415], "spans": [{"bbox": [127, 403, 217, 415], "score": 1.0, "content": "Proof. Observe that ", "type": "text"}, {"bbox": [217, 405, 242, 412], "score": 0.88, "content": "\\upsilon=0", "type": "inline_equation", "height": 7, "width": 25}, {"bbox": [242, 403, 486, 415], "score": 1.0, "content": " in case A) and B); Kuroda\u2019s class number formulas for", "type": "text"}], "index": 19}, {"bbox": [126, 415, 218, 428], "spans": [{"bbox": [126, 416, 147, 426], "score": 0.92, "content": "L/k_{1}", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [148, 415, 169, 428], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [170, 416, 192, 426], "score": 0.93, "content": "L/k_{2}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [192, 415, 218, 428], "score": 1.0, "content": " gives", "type": "text"}], "index": 20}], "index": 19.5}, {"type": "interline_equation", "bbox": [199, 433, 412, 459], "lines": [{"bbox": [199, 433, 412, 459], "spans": [{"bbox": [199, 433, 412, 459], "score": 0.91, "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{2h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{4h_{2}(k_{2})^{2}}", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "text", "bbox": [124, 463, 191, 475], "lines": [{"bbox": [124, 465, 191, 477], "spans": [{"bbox": [124, 465, 191, 477], "score": 1.0, "content": "in case A) and", "type": "text"}], "index": 22}], "index": 22}, {"type": "interline_equation", "bbox": [199, 484, 412, 509], "lines": [{"bbox": [199, 484, 412, 509], "spans": [{"bbox": [199, 484, 412, 509], "score": 0.9, "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{4h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{2h_{2}(k_{2})^{2}}", "type": "interline_equation"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [125, 514, 486, 538], "lines": [{"bbox": [125, 516, 486, 529], "spans": [{"bbox": [125, 516, 486, 529], "score": 1.0, "content": "in case B). Multiplying them together and plugging in the class number formula", "type": "text"}], "index": 24}, {"bbox": [126, 528, 192, 541], "spans": [{"bbox": [126, 528, 141, 541], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [141, 529, 163, 540], "score": 0.93, "content": "K/\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [163, 528, 192, 541], "score": 1.0, "content": " yields", "type": "text"}], "index": 25}], "index": 24.5}, {"type": "interline_equation", "bbox": [219, 542, 392, 568], "lines": [{"bbox": [219, 542, 392, 568], "spans": [{"bbox": [219, 542, 392, 568], "score": 0.94, "content": "h_{2}(L)^{2}=\\frac{q_{1}\\,q_{2}}{8}\\frac{h_{2}(K_{1})^{2}\\,h_{2}(K_{2})^{2}\\,h_{2}(k)^{2}}{h_{2}(k_{1})^{2}\\,h_{2}(k_{2})^{2}}.", "type": "interline_equation"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [124, 570, 487, 595], "lines": [{"bbox": [125, 572, 487, 586], "spans": [{"bbox": [125, 572, 147, 586], "score": 1.0, "content": "Now", "type": "text"}, {"bbox": [148, 573, 194, 584], "score": 0.93, "content": "h_{2}(k_{1})=1", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [195, 572, 199, 586], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [199, 573, 246, 584], "score": 0.92, "content": "h_{2}(k_{2})=2", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [246, 572, 267, 586], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [267, 574, 303, 583], "score": 0.92, "content": "q_{1}q_{2}=2", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [304, 572, 487, 586], "score": 1.0, "content": " (by Proposition 6), and taking the square", "type": "text"}], "index": 27}, {"bbox": [126, 585, 486, 596], "spans": [{"bbox": [126, 585, 182, 596], "score": 1.0, "content": "root we find ", "type": "text"}, {"bbox": [182, 585, 312, 596], "score": 0.91, "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}", "type": "inline_equation", "height": 11, "width": 130}, {"bbox": [312, 585, 364, 596], "score": 1.0, "content": " as claimed.", "type": "text"}, {"bbox": [475, 585, 486, 595], "score": 0.9940622448921204, "content": "\u53e3", "type": "text"}], "index": 28}], "index": 27.5}, {"type": "title", "bbox": [261, 610, 349, 623], "lines": [{"bbox": [262, 612, 349, 625], "spans": [{"bbox": [262, 612, 349, 625], "score": 1.0, "content": "5. Classification", "type": "text"}], "index": 29}], "index": 29}, {"type": "text", "bbox": [125, 629, 486, 653], "lines": [{"bbox": [136, 631, 487, 643], "spans": [{"bbox": [136, 631, 487, 643], "score": 1.0, "content": "In this section we apply the results obtained in the last few sections to give a", "type": "text"}], "index": 30}, {"bbox": [124, 644, 218, 654], "spans": [{"bbox": [124, 644, 218, 654], "score": 1.0, "content": "proof for Theorem 1.", "type": "text"}], "index": 31}], "index": 30.5}, {"type": "text", "bbox": [124, 662, 486, 700], "lines": [{"bbox": [126, 665, 487, 677], "spans": [{"bbox": [126, 665, 236, 677], "score": 1.0, "content": "Proof of Theorem 1. Let ", "type": "text"}, {"bbox": [237, 667, 244, 674], "score": 0.91, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [244, 665, 487, 677], "score": 1.0, "content": " be one of the two cyclic quartic unramified extensions", "type": "text"}], "index": 32}, {"bbox": [126, 677, 486, 689], "spans": [{"bbox": [126, 677, 138, 689], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 679, 145, 686], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [145, 677, 189, 689], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [189, 679, 199, 686], "score": 0.9, "content": "N", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [199, 677, 294, 689], "score": 1.0, "content": " be the subgroup of ", "type": "text"}, {"bbox": [295, 677, 339, 689], "score": 0.93, "content": "\\operatorname{Gal}(k^{2}/k)", "type": "inline_equation", "height": 12, "width": 44}, {"bbox": [339, 677, 372, 689], "score": 1.0, "content": " fixing ", "type": "text"}, {"bbox": [372, 679, 379, 686], "score": 0.89, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [380, 677, 418, 689], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [419, 679, 428, 686], "score": 0.91, "content": "N", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [428, 677, 486, 689], "score": 1.0, "content": " satisfies the", "type": "text"}], "index": 33}, {"bbox": [126, 690, 458, 702], "spans": [{"bbox": [126, 690, 458, 702], "score": 1.0, "content": "assumptions of Proposition 1, thus there are only the following possibilities:", "type": "text"}], "index": 34}], "index": 33}], "layout_bboxes": [], "page_idx": 10, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [199, 433, 412, 459], "lines": [{"bbox": [199, 433, 412, 459], "spans": [{"bbox": [199, 433, 412, 459], "score": 0.91, "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{2h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{4h_{2}(k_{2})^{2}}", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "interline_equation", "bbox": [199, 484, 412, 509], "lines": [{"bbox": [199, 484, 412, 509], "spans": [{"bbox": [199, 484, 412, 509], "score": 0.9, "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{4h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{2h_{2}(k_{2})^{2}}", "type": "interline_equation"}], "index": 23}], "index": 23}, {"type": "interline_equation", "bbox": [219, 542, 392, 568], "lines": [{"bbox": [219, 542, 392, 568], "spans": [{"bbox": [219, 542, 392, 568], "score": 0.94, "content": "h_{2}(L)^{2}=\\frac{q_{1}\\,q_{2}}{8}\\frac{h_{2}(K_{1})^{2}\\,h_{2}(K_{2})^{2}\\,h_{2}(k)^{2}}{h_{2}(k_{1})^{2}\\,h_{2}(k_{2})^{2}}.", "type": "interline_equation"}], "index": 26}], "index": 26}], "discarded_blocks": [{"type": "discarded", "bbox": [238, 90, 373, 99], "lines": [{"bbox": [239, 93, 372, 100], "spans": [{"bbox": [239, 93, 372, 100], "score": 1.0, "content": "IMAGINARY QUADRATIC FIELDS", "type": "text"}]}]}, {"type": "discarded", "bbox": [476, 91, 485, 99], "lines": [{"bbox": [476, 93, 486, 101], "spans": [{"bbox": [476, 93, 486, 101], "score": 1.0, "content": "11", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 111, 486, 172], "lines": [{"bbox": [126, 114, 484, 127], "spans": [{"bbox": [126, 116, 134, 123], "score": 0.9, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [134, 114, 439, 127], "score": 1.0, "content": " has odd class number in the strict sense: see Lemma 6); since both ", "type": "text"}, {"bbox": [439, 116, 450, 125], "score": 0.92, "content": "F_{1}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [450, 114, 473, 127], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [474, 116, 484, 125], "score": 0.92, "content": "F_{2}", "type": "inline_equation", "height": 9, "width": 10}], "index": 0}, {"bbox": [126, 127, 486, 139], "spans": [{"bbox": [126, 127, 268, 139], "score": 1.0, "content": "are normal (even abelian) over ", "type": "text"}, {"bbox": [269, 128, 279, 137], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [279, 127, 356, 139], "score": 1.0, "content": ", ramification at ", "type": "text"}, {"bbox": [356, 130, 361, 137], "score": 0.87, "content": "\\mathfrak{q}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [362, 127, 486, 139], "score": 1.0, "content": " implies ramification at the", "type": "text"}], "index": 1}, {"bbox": [126, 138, 485, 150], "spans": [{"bbox": [126, 138, 199, 150], "score": 1.0, "content": "conjugated ideal", "type": "text"}, {"bbox": [200, 139, 208, 149], "score": 0.91, "content": "{\\mathfrak{q}}^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [208, 138, 267, 150], "score": 1.0, "content": ". Hence both ", "type": "text"}, {"bbox": [267, 142, 272, 149], "score": 0.88, "content": "\\mathfrak{q}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [272, 138, 294, 150], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [294, 139, 302, 149], "score": 0.91, "content": "{\\mathfrak{q}}^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [302, 138, 347, 150], "score": 1.0, "content": " ramify in ", "type": "text"}, {"bbox": [347, 139, 371, 149], "score": 0.94, "content": "F_{1}/F", "type": "inline_equation", "height": 10, "width": 24}, {"bbox": [371, 138, 393, 150], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [393, 139, 417, 150], "score": 0.93, "content": "F_{2}/F", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [417, 138, 485, 150], "score": 1.0, "content": ", and since they", "type": "text"}], "index": 2}, {"bbox": [125, 150, 486, 163], "spans": [{"bbox": [125, 150, 188, 163], "score": 1.0, "content": "also ramify in ", "type": "text"}, {"bbox": [189, 151, 212, 162], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [212, 150, 357, 163], "score": 1.0, "content": ", they must ramify completely in ", "type": "text"}, {"bbox": [357, 151, 378, 162], "score": 0.93, "content": "N/F", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [379, 150, 486, 163], "score": 1.0, "content": ", again contradicting the", "type": "text"}], "index": 3}, {"bbox": [126, 162, 255, 174], "spans": [{"bbox": [126, 162, 167, 174], "score": 1.0, "content": "fact that ", "type": "text"}, {"bbox": [167, 163, 191, 174], "score": 0.93, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [191, 162, 255, 174], "score": 1.0, "content": " is unramified.", "type": "text"}], "index": 4}], "index": 2, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [125, 114, 486, 174]}, {"type": "text", "bbox": [125, 173, 486, 275], "lines": [{"bbox": [137, 174, 487, 188], "spans": [{"bbox": [137, 174, 234, 188], "score": 1.0, "content": "We have proved that ", "type": "text"}, {"bbox": [235, 176, 270, 187], "score": 0.94, "content": "\\mathrm{Cl_{2}}(K_{2})", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [270, 174, 294, 188], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [294, 174, 329, 187], "score": 0.93, "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [330, 174, 487, 188], "score": 1.0, "content": " contain subgroups of type (4) and", "type": "text"}], "index": 5}, {"bbox": [126, 187, 486, 200], "spans": [{"bbox": [126, 188, 148, 199], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 187, 486, 200], "score": 1.0, "content": ", respectively. Now we wish to apply P roposition 5. But we have to compute", "type": "text"}], "index": 6}, {"bbox": [126, 199, 486, 212], "spans": [{"bbox": [126, 199, 192, 212], "score": 0.92, "content": "\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})", "type": "inline_equation", "height": 13, "width": 66}, {"bbox": [192, 200, 314, 212], "score": 1.0, "content": ". Since the class number of", "type": "text"}, {"bbox": [314, 199, 327, 211], "score": 0.93, "content": "{\\widetilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [327, 200, 486, 212], "score": 1.0, "content": " is even, it is sufficient to show that", "type": "text"}], "index": 7}, {"bbox": [126, 212, 487, 227], "spans": [{"bbox": [126, 212, 214, 225], "score": 0.9, "content": "\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})\\,\\leq\\,2", "type": "inline_equation", "height": 13, "width": 88}, {"bbox": [214, 212, 487, 227], "score": 1.0, "content": ". In case A), there is e xactly one ramified prime (it divides", "type": "text"}], "index": 8}, {"bbox": [126, 226, 486, 238], "spans": [{"bbox": [126, 228, 136, 237], "score": 0.47, "content": "d_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 226, 174, 238], "score": 1.0, "content": "), hen c e ", "type": "text"}, {"bbox": [175, 226, 324, 238], "score": 0.94, "content": "\\#\\operatorname{Am}_{2}({\\widetilde{K}}_{2}/k_{2})\\,=\\,2/(E:H)\\,\\le\\,2", "type": "inline_equation", "height": 12, "width": 149}, {"bbox": [324, 226, 486, 238], "score": 1.0, "content": ". In case B), there are two ramified", "type": "text"}], "index": 9}, {"bbox": [126, 239, 486, 252], "spans": [{"bbox": [126, 240, 307, 252], "score": 1.0, "content": "primes (one is infin i te, the other divides ", "type": "text"}, {"bbox": [308, 242, 317, 251], "score": 0.89, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [318, 240, 356, 252], "score": 1.0, "content": "), hence ", "type": "text"}, {"bbox": [356, 239, 483, 252], "score": 0.94, "content": "\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})=4/(E:H)", "type": "inline_equation", "height": 13, "width": 127}, {"bbox": [483, 240, 486, 252], "score": 1.0, "content": ";", "type": "text"}], "index": 10}, {"bbox": [125, 251, 487, 264], "spans": [{"bbox": [125, 251, 144, 264], "score": 1.0, "content": "but", "type": "text"}, {"bbox": [144, 254, 157, 262], "score": 0.89, "content": "^{-1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [157, 251, 414, 264], "score": 1.0, "content": " is not a norm residue at the ramified infinite prime, h e nce ", "type": "text"}, {"bbox": [414, 253, 466, 263], "score": 0.92, "content": "(E:H)\\ge2", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [466, 251, 487, 264], "score": 1.0, "content": " and", "type": "text"}], "index": 11}, {"bbox": [126, 264, 262, 277], "spans": [{"bbox": [126, 264, 210, 277], "score": 0.93, "content": "\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})\\leq2", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [210, 265, 262, 277], "score": 1.0, "content": " as claimed.", "type": "text"}], "index": 12}], "index": 8.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [125, 174, 487, 277]}, {"type": "text", "bbox": [125, 276, 486, 301], "lines": [{"bbox": [137, 277, 487, 290], "spans": [{"bbox": [137, 278, 272, 290], "score": 1.0, "content": "Now P roposition 5 implies that ", "type": "text"}, {"bbox": [272, 280, 308, 290], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K_{2})", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [308, 278, 380, 290], "score": 1.0, "content": " is cyclic of order", "type": "text"}, {"bbox": [380, 281, 396, 289], "score": 0.7, "content": "\\geq4", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [396, 278, 439, 290], "score": 1.0, "content": ", and that", "type": "text"}, {"bbox": [439, 277, 487, 290], "score": 0.91, "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})\\simeq", "type": "inline_equation", "height": 13, "width": 48}], "index": 13}, {"bbox": [126, 290, 487, 302], "spans": [{"bbox": [126, 291, 148, 302], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 290, 267, 302], "score": 1.0, "content": ". This concludes our proof.", "type": "text"}, {"bbox": [476, 291, 487, 301], "score": 0.9864116907119751, "content": "\u53e3", "type": "text"}], "index": 14}], "index": 13.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [126, 277, 487, 302]}, {"type": "text", "bbox": [125, 310, 487, 346], "lines": [{"bbox": [125, 311, 486, 326], "spans": [{"bbox": [125, 311, 261, 326], "score": 1.0, "content": "Proposition 9. Assume that ", "type": "text"}, {"bbox": [261, 314, 268, 322], "score": 0.75, "content": "k", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [268, 311, 486, 326], "score": 1.0, "content": " is one of the imaginary quadratic fields of type", "type": "text"}], "index": 15}, {"bbox": [126, 324, 486, 337], "spans": [{"bbox": [126, 325, 138, 336], "score": 0.28, "content": "A)", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [139, 324, 154, 337], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [154, 326, 162, 334], "score": 0.77, "content": "B", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [163, 324, 486, 337], "score": 1.0, "content": ") as explained in the Introduction. Then there exist two unramified cyclic", "type": "text"}], "index": 16}, {"bbox": [126, 337, 374, 348], "spans": [{"bbox": [126, 337, 219, 348], "score": 1.0, "content": "quartic extensions of ", "type": "text"}, {"bbox": [219, 338, 225, 345], "score": 0.81, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [226, 337, 249, 348], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [250, 338, 257, 345], "score": 0.8, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [257, 337, 374, 348], "score": 1.0, "content": " be one of them, and write", "type": "text"}], "index": 17}], "index": 16, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [125, 311, 486, 348]}, {"type": "text", "bbox": [126, 380, 465, 394], "lines": [{"bbox": [127, 382, 464, 396], "spans": [{"bbox": [127, 382, 151, 396], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [152, 383, 282, 396], "score": 0.91, "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}", "type": "inline_equation", "height": 13, "width": 130}, {"bbox": [282, 382, 376, 396], "score": 1.0, "content": " unless possibly when ", "type": "text"}, {"bbox": [377, 385, 413, 394], "score": 0.9, "content": "d_{3}=-4", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [413, 382, 450, 396], "score": 1.0, "content": " in case ", "type": "text"}, {"bbox": [450, 385, 457, 392], "score": 0.77, "content": "B", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [458, 382, 464, 396], "score": 1.0, "content": ").", "type": "text"}], "index": 18}], "index": 18, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [127, 382, 464, 396]}, {"type": "text", "bbox": [125, 401, 486, 426], "lines": [{"bbox": [127, 403, 486, 415], "spans": [{"bbox": [127, 403, 217, 415], "score": 1.0, "content": "Proof. Observe that ", "type": "text"}, {"bbox": [217, 405, 242, 412], "score": 0.88, "content": "\\upsilon=0", "type": "inline_equation", "height": 7, "width": 25}, {"bbox": [242, 403, 486, 415], "score": 1.0, "content": " in case A) and B); Kuroda\u2019s class number formulas for", "type": "text"}], "index": 19}, {"bbox": [126, 415, 218, 428], "spans": [{"bbox": [126, 416, 147, 426], "score": 0.92, "content": "L/k_{1}", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [148, 415, 169, 428], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [170, 416, 192, 426], "score": 0.93, "content": "L/k_{2}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [192, 415, 218, 428], "score": 1.0, "content": " gives", "type": "text"}], "index": 20}], "index": 19.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [126, 403, 486, 428]}, {"type": "interline_equation", "bbox": [199, 433, 412, 459], "lines": [{"bbox": [199, 433, 412, 459], "spans": [{"bbox": [199, 433, 412, 459], "score": 0.91, "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{2h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{4h_{2}(k_{2})^{2}}", "type": "interline_equation"}], "index": 21}], "index": 21, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 463, 191, 475], "lines": [{"bbox": [124, 465, 191, 477], "spans": [{"bbox": [124, 465, 191, 477], "score": 1.0, "content": "in case A) and", "type": "text"}], "index": 22}], "index": 22, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [124, 465, 191, 477]}, {"type": "interline_equation", "bbox": [199, 484, 412, 509], "lines": [{"bbox": [199, 484, 412, 509], "spans": [{"bbox": [199, 484, 412, 509], "score": 0.9, "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{4h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{2h_{2}(k_{2})^{2}}", "type": "interline_equation"}], "index": 23}], "index": 23, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 514, 486, 538], "lines": [{"bbox": [125, 516, 486, 529], "spans": [{"bbox": [125, 516, 486, 529], "score": 1.0, "content": "in case B). Multiplying them together and plugging in the class number formula", "type": "text"}], "index": 24}, {"bbox": [126, 528, 192, 541], "spans": [{"bbox": [126, 528, 141, 541], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [141, 529, 163, 540], "score": 0.93, "content": "K/\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [163, 528, 192, 541], "score": 1.0, "content": " yields", "type": "text"}], "index": 25}], "index": 24.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [125, 516, 486, 541]}, {"type": "interline_equation", "bbox": [219, 542, 392, 568], "lines": [{"bbox": [219, 542, 392, 568], "spans": [{"bbox": [219, 542, 392, 568], "score": 0.94, "content": "h_{2}(L)^{2}=\\frac{q_{1}\\,q_{2}}{8}\\frac{h_{2}(K_{1})^{2}\\,h_{2}(K_{2})^{2}\\,h_{2}(k)^{2}}{h_{2}(k_{1})^{2}\\,h_{2}(k_{2})^{2}}.", "type": "interline_equation"}], "index": 26}], "index": 26, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 570, 487, 595], "lines": [{"bbox": [125, 572, 487, 586], "spans": [{"bbox": [125, 572, 147, 586], "score": 1.0, "content": "Now", "type": "text"}, {"bbox": [148, 573, 194, 584], "score": 0.93, "content": "h_{2}(k_{1})=1", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [195, 572, 199, 586], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [199, 573, 246, 584], "score": 0.92, "content": "h_{2}(k_{2})=2", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [246, 572, 267, 586], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [267, 574, 303, 583], "score": 0.92, "content": "q_{1}q_{2}=2", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [304, 572, 487, 586], "score": 1.0, "content": " (by Proposition 6), and taking the square", "type": "text"}], "index": 27}, {"bbox": [126, 585, 486, 596], "spans": [{"bbox": [126, 585, 182, 596], "score": 1.0, "content": "root we find ", "type": "text"}, {"bbox": [182, 585, 312, 596], "score": 0.91, "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}", "type": "inline_equation", "height": 11, "width": 130}, {"bbox": [312, 585, 364, 596], "score": 1.0, "content": " as claimed.", "type": "text"}, {"bbox": [475, 585, 486, 595], "score": 0.9940622448921204, "content": "\u53e3", "type": "text"}], "index": 28}], "index": 27.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [125, 572, 487, 596]}, {"type": "title", "bbox": [261, 610, 349, 623], "lines": [{"bbox": [262, 612, 349, 625], "spans": [{"bbox": [262, 612, 349, 625], "score": 1.0, "content": "5. Classification", "type": "text"}], "index": 29}], "index": 29, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 629, 486, 653], "lines": [{"bbox": [136, 631, 487, 643], "spans": [{"bbox": [136, 631, 487, 643], "score": 1.0, "content": "In this section we apply the results obtained in the last few sections to give a", "type": "text"}], "index": 30}, {"bbox": [124, 644, 218, 654], "spans": [{"bbox": [124, 644, 218, 654], "score": 1.0, "content": "proof for Theorem 1.", "type": "text"}], "index": 31}], "index": 30.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [124, 631, 487, 654]}, {"type": "text", "bbox": [124, 662, 486, 700], "lines": [{"bbox": [126, 665, 487, 677], "spans": [{"bbox": [126, 665, 236, 677], "score": 1.0, "content": "Proof of Theorem 1. 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"score": 1.0, "text": ""}, {"category_id": 15, "poly": [785.0, 1063.0, 1047.0, 1063.0, 1047.0, 1102.0, 785.0, 1102.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1149.0, 1063.0, 1250.0, 1063.0, 1250.0, 1102.0, 1149.0, 1102.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1273.0, 1063.0, 1291.0, 1063.0, 1291.0, 1102.0, 1273.0, 1102.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [728.0, 1702.0, 971.0, 1702.0, 971.0, 1737.0, 728.0, 1737.0], "score": 1.0, "text": ""}], "page_info": {"page_no": 10, "height": 2200, "width": 1700}}	IMAGINARY QUADRATIC FIELDS 11 has odd class number in the strict sense: see Lemma 6); since both and are normal (even abelian) over , ramification at implies ramification at the conjugated ideal . Hence both and ramify in and , and since they also ramify in , they must ramify completely in , again contradicting the fact that is unramified. We have proved that and contain subgroups of type (4) and , respectively. Now we wish to apply P roposition 5. But we have to compute . Since the class number of is even, it is sufficient to show that . In case A), there is e xactly one ramified prime (it divides ), hen c e . In case B), there are two ramified primes (one is infin i te, the other divides ), hence ; but is not a norm residue at the ramified infinite prime, h e nce and as claimed. Now P roposition 5 implies that is cyclic of order , and that . This concludes our proof. 口 Proposition 9. Assume that is one of the imaginary quadratic fields of type or ) as explained in the Introduction. Then there exist two unramified cyclic quartic extensions of . Let be one of them, and write Then unless possibly when in case ). Proof. Observe that in case A) and B); Kuroda’s class number formulas for and gives $$ h_{2}(L)\ =\ \frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{2h_{2}(k_{1})^{2}}\ =\ \frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{4h_{2}(k_{2})^{2}} $$ in case A) and $$ h_{2}(L)\ =\ \frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{4h_{2}(k_{1})^{2}}\ =\ \frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{2h_{2}(k_{2})^{2}} $$ in case B). Multiplying them together and plugging in the class number formula for yields $$ h_{2}(L)^{2}=\frac{q_{1}\,q_{2}}{8}\frac{h_{2}(K_{1})^{2}\,h_{2}(K_{2})^{2}\,h_{2}(k)^{2}}{h_{2}(k_{1})^{2}\,h_{2}(k_{2})^{2}}. $$ Now , and (by Proposition 6), and taking the square root we find as claimed. 口 # 5. Classification In this section we apply the results obtained in the last few sections to give a proof for Theorem 1. Proof of Theorem 1. Let be one of the two cyclic quartic unramified extensions of , and let be the subgroup of fixing . Then satisfies the assumptions of Proposition 1, thus there are only the following possibilities:	<div class="pdf-page"> <p>has odd class number in the strict sense: see Lemma 6); since both and are normal (even abelian) over , ramification at implies ramification at the conjugated ideal . Hence both and ramify in and , and since they also ramify in , they must ramify completely in , again contradicting the fact that is unramified.</p> <p>We have proved that and contain subgroups of type (4) and , respectively. Now we wish to apply P roposition 5. But we have to compute . Since the class number of is even, it is sufficient to show that . In case A), there is e xactly one ramified prime (it divides ), hen c e . In case B), there are two ramified primes (one is infin i te, the other divides ), hence ; but is not a norm residue at the ramified infinite prime, h e nce and as claimed.</p> <p>Now P roposition 5 implies that is cyclic of order , and that . This concludes our proof. 口</p> <p>Proposition 9. Assume that is one of the imaginary quadratic fields of type or ) as explained in the Introduction. Then there exist two unramified cyclic quartic extensions of . Let be one of them, and write</p> <p>Then unless possibly when in case ).</p> <p>Proof. Observe that in case A) and B); Kuroda’s class number formulas for and gives</p> <p>in case A) and</p> <p>in case B). Multiplying them together and plugging in the class number formula for yields</p> <p>Now , and (by Proposition 6), and taking the square root we find as claimed. 口</p> <h1>5. Classification</h1> <p>In this section we apply the results obtained in the last few sections to give a proof for Theorem 1.</p> <p>Proof of Theorem 1. Let be one of the two cyclic quartic unramified extensions of , and let be the subgroup of fixing . Then satisfies the assumptions of Proposition 1, thus there are only the following possibilities:</p> </div>	<div class="pdf-page"> <div class="pdf-discarded" data-x="398" data-y="116" data-width="226" data-height="12" style="opacity: 0.5;">IMAGINARY QUADRATIC FIELDS</div> <div class="pdf-discarded" data-x="796" data-y="117" data-width="15" data-height="11" style="opacity: 0.5;">11</div> <p class="pdf-text" data-x="209" data-y="143" data-width="604" data-height="79">has odd class number in the strict sense: see Lemma 6); since both and are normal (even abelian) over , ramification at implies ramification at the conjugated ideal . Hence both and ramify in and , and since they also ramify in , they must ramify completely in , again contradicting the fact that is unramified.</p> <p class="pdf-text" data-x="209" data-y="223" data-width="604" data-height="132">We have proved that and contain subgroups of type (4) and , respectively. Now we wish to apply P roposition 5. But we have to compute . Since the class number of is even, it is sufficient to show that . In case A), there is e xactly one ramified prime (it divides ), hen c e . In case B), there are two ramified primes (one is infin i te, the other divides ), hence ; but is not a norm residue at the ramified infinite prime, h e nce and as claimed.</p> <p class="pdf-text" data-x="209" data-y="356" data-width="604" data-height="33">Now P roposition 5 implies that is cyclic of order , and that . This concludes our proof. 口</p> <p class="pdf-text" data-x="209" data-y="400" data-width="605" data-height="47">Proposition 9. Assume that is one of the imaginary quadratic fields of type or ) as explained in the Introduction. Then there exist two unramified cyclic quartic extensions of . Let be one of them, and write</p> <p class="pdf-text" data-x="210" data-y="491" data-width="568" data-height="18">Then unless possibly when in case ).</p> <p class="pdf-text" data-x="209" data-y="518" data-width="604" data-height="32">Proof. Observe that in case A) and B); Kuroda’s class number formulas for and gives</p> <p class="pdf-text" data-x="207" data-y="598" data-width="112" data-height="16">in case A) and</p> <p class="pdf-text" data-x="209" data-y="664" data-width="604" data-height="31">in case B). Multiplying them together and plugging in the class number formula for yields</p> <p class="pdf-text" data-x="207" data-y="736" data-width="607" data-height="33">Now , and (by Proposition 6), and taking the square root we find as claimed. 口</p> <h1 class="pdf-title" data-x="436" data-y="788" data-width="147" data-height="17">5. Classification</h1> <p class="pdf-text" data-x="209" data-y="813" data-width="604" data-height="31">In this section we apply the results obtained in the last few sections to give a proof for Theorem 1.</p> <p class="pdf-text" data-x="207" data-y="855" data-width="606" data-height="50">Proof of Theorem 1. Let be one of the two cyclic quartic unramified extensions of , and let be the subgroup of fixing . Then satisfies the assumptions of Proposition 1, thus there are only the following possibilities:</p> </div>	{ "type": [ "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "interline_equation", "text", "interline_equation", "text", "inline_equation", "interline_equation", "inline_equation", "inline_equation", "text", "text", "text", "inline_equation", "inline_equation", "text" ], "coordinates": [ [ 210, 147, 809, 164 ], [ 210, 164, 813, 179 ], [ 210, 178, 811, 193 ], [ 209, 193, 813, 210 ], [ 210, 209, 426, 224 ], [ 229, 224, 814, 243 ], [ 210, 241, 813, 258 ], [ 210, 257, 813, 274 ], [ 210, 274, 814, 293 ], [ 210, 292, 813, 307 ], [ 210, 309, 813, 325 ], [ 209, 324, 814, 341 ], [ 210, 341, 438, 358 ], [ 229, 358, 814, 374 ], [ 210, 374, 814, 390 ], [ 209, 402, 813, 421 ], [ 210, 418, 813, 435 ], [ 210, 435, 625, 449 ], [ 212, 493, 776, 512 ], [ 212, 521, 813, 536 ], [ 210, 536, 364, 553 ], [ 332, 559, 689, 593 ], [ 207, 601, 319, 616 ], [ 332, 625, 689, 658 ], [ 209, 667, 813, 683 ], [ 210, 682, 321, 699 ], [ 366, 700, 655, 734 ], [ 209, 739, 814, 757 ], [ 210, 756, 813, 770 ], [ 438, 791, 583, 808 ], [ 227, 815, 814, 831 ], [ 207, 832, 364, 845 ], [ 210, 859, 814, 875 ], [ 210, 875, 813, 890 ], [ 210, 892, 766, 907 ] ], "content": [ "F has odd class number in the strict sense: see Lemma 6); since both F_{1} and F_{2}", "are normal (even abelian) over k_{2} , ramification at \\mathfrak{q} implies ramification at the", "conjugated ideal {\\mathfrak{q}}^{\\prime} . Hence both \\mathfrak{q} and {\\mathfrak{q}}^{\\prime} ramify in F_{1}/F and F_{2}/F , and since they", "also ramify in M/F , they must ramify completely in N/F , again contradicting the", "fact that N/M is unramified.", "We have proved that \\mathrm{Cl_{2}}(K_{2}) and \\mathrm{Cl}_{2}(\\widetilde{K}_{2}) contain subgroups of type (4) and", "(2,2) , respectively. Now we wish to apply P roposition 5. But we have to compute", "\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2}) . Since the class number of {\\widetilde{K}}_{2} is even, it is sufficient to show that", "\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})\\,\\leq\\,2 . In case A), there is e xactly one ramified prime (it divides", "d_{1} ), hen c e \\#\\operatorname{Am}_{2}({\\widetilde{K}}_{2}/k_{2})\\,=\\,2/(E:H)\\,\\le\\,2 . In case B), there are two ramified", "primes (one is infin i te, the other divides d_{3} ), hence \\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})=4/(E:H) ;", "but ^{-1} is not a norm residue at the ramified infinite prime, h e nce (E:H)\\ge2 and", "\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})\\leq2 as claimed.", "Now P roposition 5 implies that \\mathrm{Cl}_{2}(K_{2}) is cyclic of order \\geq4 , and that \\mathrm{Cl}_{2}(\\widetilde{K}_{2})\\simeq", "(2,2) . This concludes our proof. 口", "Proposition 9. Assume that k is one of the imaginary quadratic fields of type", "A) or B ) as explained in the Introduction. Then there exist two unramified cyclic", "quartic extensions of k . Let L be one of them, and write", "Then \\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array} unless possibly when d_{3}=-4 in case B ).", "Proof. Observe that \\upsilon=0 in case A) and B); Kuroda’s class number formulas for", "L/k_{1} and L/k_{2} gives", "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{2h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{4h_{2}(k_{2})^{2}}", "in case A) and", "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{4h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{2h_{2}(k_{2})^{2}}", "in case B). Multiplying them together and plugging in the class number formula", "for K/\\mathbb{Q} yields", "h_{2}(L)^{2}=\\frac{q_{1}\\,q_{2}}{8}\\frac{h_{2}(K_{1})^{2}\\,h_{2}(K_{2})^{2}\\,h_{2}(k)^{2}}{h_{2}(k_{1})^{2}\\,h_{2}(k_{2})^{2}}.", "Now h_{2}(k_{1})=1 , h_{2}(k_{2})=2 and q_{1}q_{2}=2 (by Proposition 6), and taking the square", "root we find \\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array} as claimed. 口", "5. Classification", "In this section we apply the results obtained in the last few sections to give a", "proof for Theorem 1.", "Proof of Theorem 1. Let L be one of the two cyclic quartic unramified extensions", "of k , and let N be the subgroup of \\operatorname{Gal}(k^{2}/k) fixing L . Then N satisfies the", "assumptions of Proposition 1, thus there are only the following possibilities:" ], "index": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 18, 19, 33, 34, 35, 51, 70, 71, 91, 113, 136, 160, 161, 186, 213, 214, 242, 272, 273, 304, 305, 306 ] }	{ "coordinates": [], "index": [], "caption": [], "caption_coordinates": [] }	{ "type": [ "interline_equation", "interline_equation", "interline_equation", "interline_equation", "interline_equation", "interline_equation" ], "coordinates": [ [ 332, 559, 689, 593 ], [ 332, 625, 689, 658 ], [ 366, 700, 655, 734 ], [ 332, 559, 689, 593 ], [ 332, 625, 689, 658 ], [ 366, 700, 655, 734 ] ], "content": [ "", "", "", "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{2h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{4h_{2}(k_{2})^{2}}", "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{4h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{2h_{2}(k_{2})^{2}}", "h_{2}(L)^{2}=\\frac{q_{1}\\,q_{2}}{8}\\frac{h_{2}(K_{1})^{2}\\,h_{2}(K_{2})^{2}\\,h_{2}(k)^{2}}{h_{2}(k_{1})^{2}\\,h_{2}(k_{2})^{2}}." ], "index": [ 0, 1, 2, 3, 4, 5 ] }	{ "type": [], "coordinates": [], "content": [], "index": [] }	{"document_info": {"total_pages": 14, "parse_type": "txt", "version": "1.0.1", "total_content_list_items": 155}, "content_statistics": {"total_blocks": 182, "total_lines": 526, "total_images": 4, "total_equations": 28, "total_tables": 11, "total_characters": 29658, "total_words": 5331, "block_type_distribution": {"title": 7, "text": 126, "interline_equation": 14, "discarded": 24, "table_caption": 3, "table_body": 4, "image_body": 2, "list": 2}}, "model_statistics": {"total_layout_detections": 2826, "avg_confidence": 0.9470955414012739, "min_confidence": 0.26, "max_confidence": 1.0}, "page_statistics": [{"page_number": 0, "page_size": [612.0, 792.0], "blocks": 11, "lines": 38, "images": 0, "equations": 2, "tables": 0, "characters": 2362, "words": 398, "discarded_blocks": 1}, {"page_number": 1, "page_size": [612.0, 792.0], "blocks": 10, "lines": 30, "images": 0, "equations": 0, "tables": 3, "characters": 1688, "words": 305, "discarded_blocks": 1}, {"page_number": 2, "page_size": [612.0, 792.0], "blocks": 15, "lines": 44, "images": 0, "equations": 4, "tables": 0, "characters": 2417, "words": 416, "discarded_blocks": 2}, {"page_number": 3, "page_size": [612.0, 792.0], "blocks": 7, "lines": 27, "images": 0, "equations": 0, "tables": 3, "characters": 1591, "words": 291, "discarded_blocks": 1}, {"page_number": 4, "page_size": [612.0, 792.0], "blocks": 23, "lines": 38, "images": 0, "equations": 6, "tables": 0, "characters": 1604, "words": 303, "discarded_blocks": 4}, {"page_number": 5, "page_size": [612.0, 792.0], "blocks": 17, "lines": 40, "images": 0, "equations": 6, "tables": 0, "characters": 2131, "words": 393, "discarded_blocks": 1}, {"page_number": 6, "page_size": [612.0, 792.0], "blocks": 15, "lines": 36, "images": 2, "equations": 4, "tables": 0, "characters": 1791, "words": 324, "discarded_blocks": 2}, {"page_number": 7, "page_size": [612.0, 792.0], "blocks": 12, "lines": 49, "images": 0, "equations": 0, "tables": 0, "characters": 3049, "words": 571, "discarded_blocks": 1}, {"page_number": 8, "page_size": [612.0, 792.0], "blocks": 11, "lines": 42, "images": 0, "equations": 0, "tables": 0, "characters": 2636, "words": 479, "discarded_blocks": 2}, {"page_number": 9, "page_size": [612.0, 792.0], "blocks": 14, "lines": 35, "images": 2, "equations": 0, "tables": 0, "characters": 2036, "words": 375, "discarded_blocks": 1}, {"page_number": 10, "page_size": [612.0, 792.0], "blocks": 17, "lines": 35, "images": 0, "equations": 6, "tables": 0, "characters": 1789, "words": 327, "discarded_blocks": 2}, {"page_number": 11, "page_size": [612.0, 792.0], "blocks": 14, "lines": 40, "images": 0, "equations": 0, "tables": 2, "characters": 2094, "words": 399, "discarded_blocks": 2}, {"page_number": 12, "page_size": [612.0, 792.0], "blocks": 10, "lines": 44, "images": 0, "equations": 0, "tables": 3, "characters": 2572, "words": 487, "discarded_blocks": 2}, {"page_number": 13, "page_size": [612.0, 792.0], "blocks": 6, "lines": 28, "images": 0, "equations": 0, "tables": 0, "characters": 1898, "words": 263, "discarded_blocks": 2}]}
0003244v1	11	[ 612, 792 ]	{ "type": [ "table_body", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "discarded", "discarded" ], "coordinates": [ [ 381, 142, 640, 215 ], [ 207, 241, 813, 320 ], [ 207, 340, 811, 385 ], [ 207, 386, 813, 431 ], [ 209, 433, 813, 493 ], [ 209, 493, 811, 540 ], [ 207, 540, 813, 571 ], [ 212, 590, 813, 621 ], [ 207, 625, 814, 714 ], [ 209, 729, 813, 857 ], [ 209, 857, 811, 888 ], [ 229, 889, 416, 903 ], [ 796, 888, 813, 902 ], [ 209, 117, 225, 128 ] ], "content": [ "", "Here, the first two columns follow from Proposition 1, the last (which we do not claim to hold if in case B)) is a consequence of the class number formula of Proposition 9. In particular, we have if one of the class numbers or is at least 8. Therefore it suffices to examine the cases and (recall from above that is always even).", "We start by considering case A); it is sufficient to show that . We now apply Proposition 5; notice that we may do so by the proof of Proposition 8.", "a) If , then by Proposition 5, hence by Proposition 7 and then by Proposition 6. The class number formulas in the proof of Proposition 9 now give and .", "It can be shown using the ambiguous class number formula that is trivial if and only if is a quadratic nonresidue modulo the prime ideal over in ; by Scholz’s reciprocity law, this is equivalent to , and this agrees with the criterion given in [1].", "b) If , we may assume that from Proposition 8.b). Then by Proposition 5, by Proposition 7 and by Proposition 6. Using the class number formula we get and .", "Thus in both cases we have , and by the table at the beginning of this proof this implies that rank in case A).", "Next we consider case B); here we have to distinguish between (case ) and (case ).", "Let us start with case ). a) If , then , and as above. The class number formula gives and . b) If (which we may assume without loss of generality by Proposition 8.b)) then , and , again exactly as above. This implies and .", "Here we apply Kuroda’s class number formula (see [10]) to , and since and , we get . From (for a suitable choice of ; the other possibility is , where is the fundamental unit of , we deduce that the uni t , which still is fundamental in , becomes a square in , and this implies that . Moreover, we have , where mod 4 are prime factors of and in , respectively. This shows that has even class number, because is easily seen to be unramified.", "Thus , and so we find that is divisible by . In particular, we always have in this case.", "This concludes the proof.", "口", "12" ], "index": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 ] }	[{"type": "table", "img_path": "images/295f4e323014a6c679e39dff1a665cb1d71c61c9d44701d37ba45e8e808d410a.jpg", "table_caption": [], "table_footnote": [], "table_body": "\n\n<html><body><table><tr><td>d(G')</td><td>h2(L)</td><td>h2(K1)h2(K2)</td></tr><tr><td>1</td><td>2m</td><td>2</td></tr><tr><td>2</td><td>2m+1</td><td>4</td></tr><tr><td>>3</td><td>\u22652m+2</td><td>M8</td></tr></table></body></html>\n\n", "page_idx": 11}, {"type": "text", "text": "Here, the first two columns follow from Proposition 1, the last (which we do not claim to hold if $d_{3}=-4$ in case B)) is a consequence of the class number formula of Proposition 9. In particular, we have $d(G^{\\prime})\\geq3$ if one of the class numbers $h_{2}(K_{1})$ or $h_{2}(K_{2})$ is at least 8. Therefore it suffices to examine the cases $h_{2}(K_{2})=2$ and $h_{2}(K_{2})=4$ (recall from above that $h_{2}(K_{2})$ is always even). ", "page_idx": 11}, {"type": "text", "text": "We start by considering case A); it is sufficient to show that $h_{2}(K_{1})h_{2}(K_{2})\\neq4$ . We now apply Proposition 5; notice that we may do so by the proof of Proposition 8. ", "page_idx": 11}, {"type": "text", "text": "a) If $h_{2}(K_{2})\\,=\\,2$ , then $\\#\\kappa_{2}\\,=\\,2$ by Proposition 5, hence $q_{2}\\,=\\,2$ by Proposition 7 and then $q_{1}=1$ by Proposition 6. The class number formulas in the proof of Proposition 9 now give $h_{2}(K_{1})=1$ and $h_{2}(L)=2^{m}$ . ", "page_idx": 11}, {"type": "text", "text": "It can be shown using the ambiguous class number formula that $\\mathrm{Cl}_{2}(K_{1})$ is trivial if and only if $\\varepsilon_{1}$ is a quadratic nonresidue modulo the prime ideal over $d_{2}$ in $k_{1}$ ; by Scholz\u2019s reciprocity law, this is equivalent to $(d_{1}/d_{2})_{4}(d_{2}/d_{1})_{4}=1$ , and this agrees with the criterion given in [1]. ", "page_idx": 11}, {"type": "text", "text": "b) If $h_{2}(K_{2})=4$ , we may assume that $\\mathrm{Cl}_{2}(K_{2})=(4)$ from Proposition 8.b). Then $\\#\\kappa_{2}=2$ by Proposition 5, $q_{2}=2$ by Proposition 7 and $q_{1}=1$ by Proposition 6. Using the class number formula we get $h_{2}(K_{1})=2$ and $h_{2}(L)=2^{m+2}$ . ", "page_idx": 11}, {"type": "text", "text": "Thus in both cases we have $h_{2}(K_{1})h_{2}(K_{2})\\neq4$ , and by the table at the beginning of this proof this implies that rank $\\mathrm{Cl}_{2}(k^{1})\\neq2$ in case A). ", "page_idx": 11}, {"type": "text", "text": "Next we consider case B); here we have to distinguish between $d_{3}\\neq-4$ (case $B_{1}$ ) and $d_{3}=-4$ (case $B_{2}$ ). ", "page_idx": 11}, {"type": "text", "text": "Let us start with case $B_{1}$ ). a) If $h_{2}(K_{2})=2$ , then $\\#\\kappa_{2}=2$ , $q_{2}=2$ and $q_{1}=1$ as above. The class number formula gives $h_{2}(K_{1})=2$ and $h_{2}(L)=2^{m+1}$ . b) If $\\mathrm{Cl}_{2}(K_{2})=(4)$ (which we may assume without loss of generality by Proposition 8.b)) then $\\#\\kappa_{2}\\,=\\,2$ , $q_{2}\\,=\\,2$ and $q_{1}\\,=\\,1$ , again exactly as above. This implies $h_{2}(K_{1})=4$ and $h_{2}(L)=2^{m+3}$ . ", "page_idx": 11}, {"type": "text", "text": "Here we apply Kuroda\u2019s class number formula (see [10]) to $L/k_{1}$ , and since $h_{2}(k_{1})=$ $^{1}$ and $h_{2}(K_{1})=h_{2}(K_{1}^{\\prime})$ , we get $\\begin{array}{r}{h_{2}(L)=\\frac{1}{2}q_{1}h_{2}(K_{1})^{2}h_{2}(k)=2^{m}q_{1}h_{2}(K_{1})^{2}}\\end{array}$ . From $K_{2}=k_{2}(\\sqrt{\\varepsilon}\\,)$ (for a suitable choice of $L$ ; the other possibility is ${\\tilde{K}}_{2}=k_{2}({\\sqrt{d_{2}\\varepsilon}}\\,))$ , where $\\varepsilon$ is the fundamental unit of $k_{2}$ , we deduce that the uni t $\\varepsilon$ , which still is fundamental in $k$ , becomes a square in $L$ , and this implies that $q_{1}\\geq2$ . Moreover, we have $K_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\lambda})$ , where $\\pi,\\lambda\\,\\equiv\\,1$ mod 4 are prime factors of $d_{1}$ and $d_{2}$ in $k_{1}\\,=\\,\\mathbb{Q}(i)$ , respectively. This shows that $K_{1}$ has even class number, because $K_{1}(\\sqrt{\\pi}\\,)/K_{1}$ is easily seen to be unramified. ", "page_idx": 11}, {"type": "text", "text": "Thus $2\\mid q_{1},\\,2\\mid h_{2}(K_{1})$ , and so we find that $h_{2}(L)$ is divisible by $2^{m}\\cdot2\\cdot4=2^{m+3}$ . In particular, we always have $d(G^{\\prime})\\geq3$ in this case. ", "page_idx": 11}, {"type": "text", "text": "This concludes the proof. ", "page_idx": 11}]	{"preproc_blocks": [{"type": "table", "bbox": [228, 110, 383, 167], "blocks": [{"type": "table_body", "bbox": [228, 110, 383, 167], "group_id": 0, "lines": [{"bbox": [228, 110, 383, 167], "spans": [{"bbox": [228, 110, 383, 167], "score": 0.961, "html": "<html><body><table><tr><td>d(G')</td><td>h2(L)</td><td>h2(K1)h2(K2)</td></tr><tr><td>1</td><td>2m</td><td>2</td></tr><tr><td>2</td><td>2m+1</td><td>4</td></tr><tr><td>>3</td><td>\u22652m+2</td><td>M8</td></tr></table></body></html>", "type": "table", "image_path": "295f4e323014a6c679e39dff1a665cb1d71c61c9d44701d37ba45e8e808d410a.jpg"}]}], "index": 2, "virtual_lines": [{"bbox": [228, 110, 383, 123], "spans": [], "index": 0}, {"bbox": [228, 123, 383, 136], "spans": [], "index": 1}, {"bbox": [228, 136, 383, 149], "spans": [], "index": 2}, {"bbox": [228, 149, 383, 162], "spans": [], "index": 3}, {"bbox": [228, 162, 383, 175], "spans": [], "index": 4}]}], "index": 2}, {"type": "text", "bbox": [124, 187, 486, 248], "lines": [{"bbox": [125, 189, 486, 201], "spans": [{"bbox": [125, 189, 486, 201], "score": 1.0, "content": "Here, the first two columns follow from Proposition 1, the last (which we do not", "type": "text"}], "index": 5}, {"bbox": [125, 201, 487, 213], "spans": [{"bbox": [125, 201, 194, 213], "score": 1.0, "content": "claim to hold if ", "type": "text"}, {"bbox": [194, 203, 230, 212], "score": 0.92, "content": "d_{3}=-4", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [230, 201, 487, 213], "score": 1.0, "content": " in case B)) is a consequence of the class number formula of", "type": "text"}], "index": 6}, {"bbox": [126, 213, 485, 225], "spans": [{"bbox": [126, 213, 290, 225], "score": 1.0, "content": "Proposition 9. In particular, we have ", "type": "text"}, {"bbox": [290, 214, 332, 225], "score": 0.94, "content": "d(G^{\\prime})\\geq3", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [332, 213, 453, 225], "score": 1.0, "content": " if one of the class numbers ", "type": "text"}, {"bbox": [454, 214, 485, 225], "score": 0.94, "content": "h_{2}(K_{1})", "type": "inline_equation", "height": 11, "width": 31}], "index": 7}, {"bbox": [126, 225, 486, 237], "spans": [{"bbox": [126, 225, 138, 237], "score": 1.0, "content": "or ", "type": "text"}, {"bbox": [138, 226, 169, 237], "score": 0.94, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [170, 225, 415, 237], "score": 1.0, "content": " is at least 8. Therefore it suffices to examine the cases ", "type": "text"}, {"bbox": [415, 226, 465, 237], "score": 0.94, "content": "h_{2}(K_{2})=2", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [466, 225, 486, 237], "score": 1.0, "content": " and", "type": "text"}], "index": 8}, {"bbox": [126, 237, 385, 250], "spans": [{"bbox": [126, 238, 175, 249], "score": 0.93, "content": "h_{2}(K_{2})=4", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [176, 237, 282, 250], "score": 1.0, "content": " (recall from above that ", "type": "text"}, {"bbox": [282, 238, 313, 249], "score": 0.94, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [313, 237, 385, 250], "score": 1.0, "content": " is always even).", "type": "text"}], "index": 9}], "index": 7}, {"type": "text", "bbox": [124, 263, 485, 298], "lines": [{"bbox": [137, 263, 486, 278], "spans": [{"bbox": [137, 263, 401, 278], "score": 1.0, "content": "We start by considering case A); it is sufficient to show that ", "type": "text"}, {"bbox": [402, 266, 482, 276], "score": 0.94, "content": "h_{2}(K_{1})h_{2}(K_{2})\\neq4", "type": "inline_equation", "height": 10, "width": 80}, {"bbox": [483, 263, 486, 278], "score": 1.0, "content": ".", "type": "text"}], "index": 10}, {"bbox": [125, 275, 486, 290], "spans": [{"bbox": [125, 275, 486, 290], "score": 1.0, "content": "We now apply Proposition 5; notice that we may do so by the proof of Proposition", "type": "text"}], "index": 11}, {"bbox": [126, 290, 135, 299], "spans": [{"bbox": [126, 290, 135, 299], "score": 1.0, "content": "8.", "type": "text"}], "index": 12}], "index": 11}, {"type": "text", "bbox": [124, 299, 486, 334], "lines": [{"bbox": [125, 300, 487, 313], "spans": [{"bbox": [125, 300, 149, 313], "score": 1.0, "content": "a) If ", "type": "text"}, {"bbox": [150, 302, 202, 312], "score": 0.92, "content": "h_{2}(K_{2})\\,=\\,2", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [202, 300, 232, 313], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [232, 302, 272, 311], "score": 0.92, "content": "\\#\\kappa_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 40}, {"bbox": [272, 300, 385, 313], "score": 1.0, "content": " by Proposition 5, hence ", "type": "text"}, {"bbox": [385, 303, 416, 312], "score": 0.92, "content": "q_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [416, 300, 487, 313], "score": 1.0, "content": " by Proposition", "type": "text"}], "index": 13}, {"bbox": [124, 311, 487, 326], "spans": [{"bbox": [124, 311, 178, 326], "score": 1.0, "content": "7 and then ", "type": "text"}, {"bbox": [178, 315, 208, 323], "score": 0.92, "content": "q_{1}=1", "type": "inline_equation", "height": 8, "width": 30}, {"bbox": [209, 311, 487, 326], "score": 1.0, "content": " by Proposition 6. The class number formulas in the proof of", "type": "text"}], "index": 14}, {"bbox": [125, 324, 356, 336], "spans": [{"bbox": [125, 324, 228, 336], "score": 1.0, "content": "Proposition 9 now give", "type": "text"}, {"bbox": [229, 325, 279, 336], "score": 0.94, "content": "h_{2}(K_{1})=1", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [279, 324, 300, 336], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [301, 325, 352, 336], "score": 0.94, "content": "h_{2}(L)=2^{m}", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [352, 324, 356, 336], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 14}, {"type": "text", "bbox": [125, 335, 486, 382], "lines": [{"bbox": [137, 336, 486, 348], "spans": [{"bbox": [137, 336, 411, 348], "score": 1.0, "content": "It can be shown using the ambiguous class number formula that", "type": "text"}, {"bbox": [412, 338, 447, 348], "score": 0.92, "content": "\\mathrm{Cl}_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [447, 336, 486, 348], "score": 1.0, "content": " is trivial", "type": "text"}], "index": 16}, {"bbox": [125, 348, 485, 361], "spans": [{"bbox": [125, 348, 184, 361], "score": 1.0, "content": "if and only if ", "type": "text"}, {"bbox": [185, 353, 194, 359], "score": 0.89, "content": "\\varepsilon_{1}", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [194, 348, 434, 361], "score": 1.0, "content": " is a quadratic nonresidue modulo the prime ideal over ", "type": "text"}, {"bbox": [434, 350, 444, 359], "score": 0.92, "content": "d_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [444, 348, 458, 361], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [459, 350, 468, 359], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [469, 348, 485, 361], "score": 1.0, "content": "; by", "type": "text"}], "index": 17}, {"bbox": [126, 361, 486, 373], "spans": [{"bbox": [126, 361, 322, 373], "score": 1.0, "content": "Scholz\u2019s reciprocity law, this is equivalent to ", "type": "text"}, {"bbox": [322, 361, 414, 372], "score": 0.93, "content": "(d_{1}/d_{2})_{4}(d_{2}/d_{1})_{4}=1", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [414, 361, 486, 373], "score": 1.0, "content": ", and this agrees", "type": "text"}], "index": 18}, {"bbox": [126, 372, 258, 384], "spans": [{"bbox": [126, 372, 258, 384], "score": 1.0, "content": "with the criterion given in [1].", "type": "text"}], "index": 19}], "index": 17.5}, {"type": "text", "bbox": [125, 382, 485, 418], "lines": [{"bbox": [125, 383, 486, 397], "spans": [{"bbox": [125, 383, 148, 397], "score": 1.0, "content": "b) If ", "type": "text"}, {"bbox": [149, 385, 198, 396], "score": 0.93, "content": "h_{2}(K_{2})=4", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [199, 383, 295, 397], "score": 1.0, "content": ", we may assume that", "type": "text"}, {"bbox": [296, 385, 358, 396], "score": 0.94, "content": "\\mathrm{Cl}_{2}(K_{2})=(4)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [358, 383, 486, 397], "score": 1.0, "content": " from Proposition 8.b). Then", "type": "text"}], "index": 20}, {"bbox": [126, 397, 486, 409], "spans": [{"bbox": [126, 398, 164, 407], "score": 0.92, "content": "\\#\\kappa_{2}=2", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [164, 397, 247, 409], "score": 1.0, "content": " by Proposition 5, ", "type": "text"}, {"bbox": [247, 398, 276, 407], "score": 0.93, "content": "q_{2}=2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [276, 397, 376, 409], "score": 1.0, "content": " by Proposition 7 and ", "type": "text"}, {"bbox": [376, 398, 405, 407], "score": 0.92, "content": "q_{1}=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [406, 397, 486, 409], "score": 1.0, "content": " by Proposition 6.", "type": "text"}], "index": 21}, {"bbox": [124, 407, 435, 421], "spans": [{"bbox": [124, 407, 298, 421], "score": 1.0, "content": "Using the class number formula we get ", "type": "text"}, {"bbox": [299, 409, 348, 420], "score": 0.94, "content": "h_{2}(K_{1})=2", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [348, 407, 370, 421], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [370, 408, 431, 420], "score": 0.93, "content": "h_{2}(L)=2^{m+2}", "type": "inline_equation", "height": 12, "width": 61}, {"bbox": [432, 407, 435, 421], "score": 1.0, "content": ".", "type": "text"}], "index": 22}], "index": 21}, {"type": "text", "bbox": [124, 418, 486, 442], "lines": [{"bbox": [137, 419, 486, 433], "spans": [{"bbox": [137, 419, 256, 433], "score": 1.0, "content": "Thus in both cases we have", "type": "text"}, {"bbox": [257, 421, 337, 432], "score": 0.93, "content": "h_{2}(K_{1})h_{2}(K_{2})\\neq4", "type": "inline_equation", "height": 11, "width": 80}, {"bbox": [337, 419, 486, 433], "score": 1.0, "content": ", and by the table at the beginning", "type": "text"}], "index": 23}, {"bbox": [126, 432, 380, 444], "spans": [{"bbox": [126, 432, 278, 444], "score": 1.0, "content": "of this proof this implies that rank", "type": "text"}, {"bbox": [279, 433, 329, 443], "score": 0.89, "content": "\\mathrm{Cl}_{2}(k^{1})\\neq2", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [330, 432, 380, 444], "score": 1.0, "content": " in case A).", "type": "text"}], "index": 24}], "index": 23.5}, {"type": "text", "bbox": [127, 457, 486, 481], "lines": [{"bbox": [136, 459, 487, 472], "spans": [{"bbox": [136, 459, 421, 472], "score": 1.0, "content": "Next we consider case B); here we have to distinguish between ", "type": "text"}, {"bbox": [421, 461, 460, 471], "score": 0.93, "content": "d_{3}\\neq-4", "type": "inline_equation", "height": 10, "width": 39}, {"bbox": [460, 459, 487, 472], "score": 1.0, "content": " (case", "type": "text"}], "index": 25}, {"bbox": [126, 471, 248, 484], "spans": [{"bbox": [126, 473, 138, 482], "score": 0.82, "content": "B_{1}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [138, 471, 164, 484], "score": 1.0, "content": ") and ", "type": "text"}, {"bbox": [164, 473, 201, 482], "score": 0.92, "content": "d_{3}=-4", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [201, 471, 228, 484], "score": 1.0, "content": " (case ", "type": "text"}, {"bbox": [228, 473, 241, 482], "score": 0.88, "content": "B_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [241, 471, 248, 484], "score": 1.0, "content": ").", "type": "text"}], "index": 26}], "index": 25.5}, {"type": "text", "bbox": [124, 484, 487, 553], "lines": [{"bbox": [137, 483, 256, 496], "spans": [{"bbox": [137, 483, 236, 496], "score": 1.0, "content": "Let us start with case ", "type": "text"}, {"bbox": [236, 485, 248, 494], "score": 0.9, "content": "B_{1}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [249, 483, 256, 496], "score": 1.0, "content": ").", "type": "text"}], "index": 27}, {"bbox": [126, 495, 486, 509], "spans": [{"bbox": [126, 495, 149, 509], "score": 1.0, "content": "a) If ", "type": "text"}, {"bbox": [149, 496, 200, 507], "score": 0.93, "content": "h_{2}(K_{2})=2", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [200, 495, 230, 509], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [230, 497, 268, 506], "score": 0.91, "content": "\\#\\kappa_{2}=2", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [269, 495, 275, 509], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [275, 497, 304, 506], "score": 0.91, "content": "q_{2}=2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [304, 495, 327, 509], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [328, 497, 357, 506], "score": 0.93, "content": "q_{1}=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [357, 495, 486, 509], "score": 1.0, "content": " as above. The class number", "type": "text"}], "index": 28}, {"bbox": [123, 507, 324, 520], "spans": [{"bbox": [123, 507, 187, 520], "score": 1.0, "content": "formula gives ", "type": "text"}, {"bbox": [187, 509, 236, 519], "score": 0.94, "content": "h_{2}(K_{1})=2", "type": "inline_equation", "height": 10, "width": 49}, {"bbox": [236, 507, 258, 520], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [259, 508, 320, 519], "score": 0.93, "content": "h_{2}(L)=2^{m+1}", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [320, 507, 324, 520], "score": 1.0, "content": ".", "type": "text"}], "index": 29}, {"bbox": [125, 518, 486, 533], "spans": [{"bbox": [125, 518, 147, 533], "score": 1.0, "content": "b) If ", "type": "text"}, {"bbox": [147, 520, 208, 531], "score": 0.92, "content": "\\mathrm{Cl}_{2}(K_{2})=(4)", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [209, 518, 486, 533], "score": 1.0, "content": " (which we may assume without loss of generality by Proposition", "type": "text"}], "index": 30}, {"bbox": [125, 531, 485, 544], "spans": [{"bbox": [125, 531, 175, 544], "score": 1.0, "content": "8.b)) then ", "type": "text"}, {"bbox": [175, 533, 216, 542], "score": 0.91, "content": "\\#\\kappa_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [217, 531, 223, 544], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [223, 533, 254, 542], "score": 0.91, "content": "q_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [255, 531, 279, 544], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [280, 533, 311, 542], "score": 0.93, "content": "q_{1}\\,=\\,1", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [311, 531, 485, 544], "score": 1.0, "content": ", again exactly as above. This implies", "type": "text"}], "index": 31}, {"bbox": [126, 542, 263, 556], "spans": [{"bbox": [126, 544, 175, 555], "score": 0.94, "content": "h_{2}(K_{1})=4", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [176, 542, 198, 556], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [198, 544, 259, 555], "score": 0.92, "content": "h_{2}(L)=2^{m+3}", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [259, 542, 263, 556], "score": 1.0, "content": ".", "type": "text"}], "index": 32}], "index": 29.5}, {"type": "text", "bbox": [125, 564, 486, 663], "lines": [{"bbox": [125, 566, 487, 580], "spans": [{"bbox": [125, 566, 377, 580], "score": 1.0, "content": "Here we apply Kuroda\u2019s class number formula (see [10]) to ", "type": "text"}, {"bbox": [378, 568, 399, 579], "score": 0.92, "content": "L/k_{1}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [399, 566, 446, 580], "score": 1.0, "content": ", and since ", "type": "text"}, {"bbox": [447, 568, 487, 579], "score": 0.91, "content": "h_{2}(k_{1})=", "type": "inline_equation", "height": 11, "width": 40}], "index": 33}, {"bbox": [126, 579, 486, 592], "spans": [{"bbox": [126, 581, 131, 588], "score": 0.43, "content": "^{1}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [132, 579, 154, 592], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [154, 580, 231, 591], "score": 0.93, "content": "h_{2}(K_{1})=h_{2}(K_{1}^{\\prime})", "type": "inline_equation", "height": 11, "width": 77}, {"bbox": [231, 579, 268, 592], "score": 1.0, "content": ", we get ", "type": "text"}, {"bbox": [269, 579, 453, 591], "score": 0.93, "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{2}q_{1}h_{2}(K_{1})^{2}h_{2}(k)=2^{m}q_{1}h_{2}(K_{1})^{2}}\\end{array}", "type": "inline_equation", "height": 12, "width": 184}, {"bbox": [454, 579, 486, 592], "score": 1.0, "content": ". From", "type": "text"}], "index": 34}, {"bbox": [126, 592, 485, 605], "spans": [{"bbox": [126, 594, 185, 605], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\varepsilon}\\,)", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [185, 594, 294, 605], "score": 1.0, "content": " (for a suitable choice of ", "type": "text"}, {"bbox": [295, 595, 302, 602], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [302, 594, 409, 605], "score": 1.0, "content": "; the other possibility is", "type": "text"}, {"bbox": [409, 592, 482, 605], "score": 0.92, "content": "{\\tilde{K}}_{2}=k_{2}({\\sqrt{d_{2}\\varepsilon}}\\,))", "type": "inline_equation", "height": 13, "width": 73}, {"bbox": [482, 594, 485, 605], "score": 1.0, "content": ",", "type": "text"}], "index": 35}, {"bbox": [126, 604, 487, 617], "spans": [{"bbox": [126, 604, 155, 617], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [155, 609, 160, 614], "score": 0.88, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [161, 604, 286, 617], "score": 1.0, "content": " is the fundamental unit of ", "type": "text"}, {"bbox": [286, 607, 297, 615], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [297, 604, 416, 617], "score": 1.0, "content": ", we deduce that the uni t ", "type": "text"}, {"bbox": [416, 609, 421, 614], "score": 0.88, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [421, 604, 487, 617], "score": 1.0, "content": ", which still is", "type": "text"}], "index": 36}, {"bbox": [124, 617, 487, 630], "spans": [{"bbox": [124, 617, 195, 630], "score": 1.0, "content": "fundamental in ", "type": "text"}, {"bbox": [195, 619, 201, 627], "score": 0.88, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [201, 617, 298, 630], "score": 1.0, "content": ", becomes a square in ", "type": "text"}, {"bbox": [299, 619, 306, 626], "score": 0.88, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [306, 617, 406, 630], "score": 1.0, "content": ", and this implies that ", "type": "text"}, {"bbox": [406, 619, 434, 628], "score": 0.91, "content": "q_{1}\\geq2", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [434, 617, 487, 630], "score": 1.0, "content": ". Moreover,", "type": "text"}], "index": 37}, {"bbox": [125, 628, 484, 642], "spans": [{"bbox": [125, 628, 165, 642], "score": 1.0, "content": "we have ", "type": "text"}, {"bbox": [166, 629, 236, 641], "score": 0.94, "content": "K_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\lambda})", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [236, 628, 273, 642], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [273, 631, 312, 640], "score": 0.73, "content": "\\pi,\\lambda\\,\\equiv\\,1", "type": "inline_equation", "height": 9, "width": 39}, {"bbox": [313, 628, 440, 642], "score": 1.0, "content": " mod 4 are prime factors of ", "type": "text"}, {"bbox": [440, 631, 450, 640], "score": 0.91, "content": "d_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [450, 628, 474, 642], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [475, 631, 484, 640], "score": 0.91, "content": "d_{2}", "type": "inline_equation", "height": 9, "width": 9}], "index": 38}, {"bbox": [126, 642, 486, 653], "spans": [{"bbox": [126, 642, 138, 653], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [138, 642, 184, 653], "score": 0.92, "content": "k_{1}\\,=\\,\\mathbb{Q}(i)", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [184, 642, 326, 653], "score": 1.0, "content": ", respectively. This shows that ", "type": "text"}, {"bbox": [327, 643, 340, 652], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [340, 642, 486, 653], "score": 1.0, "content": " has even class number, because", "type": "text"}], "index": 39}, {"bbox": [126, 653, 319, 665], "spans": [{"bbox": [126, 654, 181, 665], "score": 0.93, "content": "K_{1}(\\sqrt{\\pi}\\,)/K_{1}", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [181, 653, 319, 665], "score": 1.0, "content": " is easily seen to be unramified.", "type": "text"}], "index": 40}], "index": 36.5}, {"type": "text", "bbox": [125, 663, 485, 687], "lines": [{"bbox": [136, 662, 486, 679], "spans": [{"bbox": [136, 662, 162, 679], "score": 1.0, "content": "Thus ", "type": "text"}, {"bbox": [163, 666, 235, 677], "score": 0.91, "content": "2\\mid q_{1},\\,2\\mid h_{2}(K_{1})", "type": "inline_equation", "height": 11, "width": 72}, {"bbox": [235, 662, 325, 679], "score": 1.0, "content": ", and so we find that ", "type": "text"}, {"bbox": [325, 666, 350, 677], "score": 0.94, "content": "h_{2}(L)", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [350, 662, 413, 679], "score": 1.0, "content": " is divisible by", "type": "text"}, {"bbox": [414, 666, 482, 674], "score": 0.92, "content": "2^{m}\\cdot2\\cdot4=2^{m+3}", "type": "inline_equation", "height": 8, "width": 68}, {"bbox": [482, 662, 486, 679], "score": 1.0, "content": ".", "type": "text"}], "index": 41}, {"bbox": [126, 678, 354, 689], "spans": [{"bbox": [126, 678, 256, 689], "score": 1.0, "content": "In particular, we always have ", "type": "text"}, {"bbox": [257, 678, 299, 689], "score": 0.94, "content": "d(G^{\\prime})\\geq3", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [299, 678, 354, 689], "score": 1.0, "content": " in this case.", "type": "text"}], "index": 42}], "index": 41.5}, {"type": "text", "bbox": [137, 688, 249, 699], "lines": [{"bbox": [137, 689, 249, 700], "spans": [{"bbox": [137, 689, 249, 700], "score": 1.0, "content": "This concludes the proof.", "type": "text"}], "index": 43}], "index": 43}], "layout_bboxes": [], "page_idx": 11, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [{"type": "table", "bbox": [228, 110, 383, 167], "blocks": [{"type": "table_body", "bbox": [228, 110, 383, 167], "group_id": 0, "lines": [{"bbox": [228, 110, 383, 167], "spans": [{"bbox": [228, 110, 383, 167], "score": 0.961, "html": "<html><body><table><tr><td>d(G')</td><td>h2(L)</td><td>h2(K1)h2(K2)</td></tr><tr><td>1</td><td>2m</td><td>2</td></tr><tr><td>2</td><td>2m+1</td><td>4</td></tr><tr><td>>3</td><td>\u22652m+2</td><td>M8</td></tr></table></body></html>", "type": "table", "image_path": "295f4e323014a6c679e39dff1a665cb1d71c61c9d44701d37ba45e8e808d410a.jpg"}]}], "index": 2, "virtual_lines": [{"bbox": [228, 110, 383, 123], "spans": [], "index": 0}, {"bbox": [228, 123, 383, 136], "spans": [], "index": 1}, {"bbox": [228, 136, 383, 149], "spans": [], "index": 2}, {"bbox": [228, 149, 383, 162], "spans": [], "index": 3}, {"bbox": [228, 162, 383, 175], "spans": [], "index": 4}]}], "index": 2}], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [476, 687, 486, 698], "lines": [{"bbox": [476, 690, 486, 700], "spans": [{"bbox": [476, 690, 486, 700], "score": 0.9837851524353027, "content": "\u53e3", "type": "text"}]}]}, {"type": "discarded", "bbox": [125, 91, 135, 99], "lines": [{"bbox": [125, 92, 136, 102], "spans": [{"bbox": [125, 92, 136, 102], "score": 1.0, "content": "12", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "table", "bbox": [228, 110, 383, 167], "blocks": [{"type": "table_body", "bbox": [228, 110, 383, 167], "group_id": 0, "lines": [{"bbox": [228, 110, 383, 167], "spans": [{"bbox": [228, 110, 383, 167], "score": 0.961, "html": "<html><body><table><tr><td>d(G')</td><td>h2(L)</td><td>h2(K1)h2(K2)</td></tr><tr><td>1</td><td>2m</td><td>2</td></tr><tr><td>2</td><td>2m+1</td><td>4</td></tr><tr><td>>3</td><td>\u22652m+2</td><td>M8</td></tr></table></body></html>", "type": "table", "image_path": "295f4e323014a6c679e39dff1a665cb1d71c61c9d44701d37ba45e8e808d410a.jpg"}]}], "index": 2, "virtual_lines": [{"bbox": [228, 110, 383, 123], "spans": [], "index": 0}, {"bbox": [228, 123, 383, 136], "spans": [], "index": 1}, {"bbox": [228, 136, 383, 149], "spans": [], "index": 2}, {"bbox": [228, 149, 383, 162], "spans": [], "index": 3}, {"bbox": [228, 162, 383, 175], "spans": [], "index": 4}]}], "index": 2, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 187, 486, 248], "lines": [{"bbox": [125, 189, 486, 201], "spans": [{"bbox": [125, 189, 486, 201], "score": 1.0, "content": "Here, the first two columns follow from Proposition 1, the last (which we do not", "type": "text"}], "index": 5}, {"bbox": [125, 201, 487, 213], "spans": [{"bbox": [125, 201, 194, 213], "score": 1.0, "content": "claim to hold if ", "type": "text"}, {"bbox": [194, 203, 230, 212], "score": 0.92, "content": "d_{3}=-4", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [230, 201, 487, 213], "score": 1.0, "content": " in case B)) is a consequence of the class number formula of", "type": "text"}], "index": 6}, {"bbox": [126, 213, 485, 225], "spans": [{"bbox": [126, 213, 290, 225], "score": 1.0, "content": "Proposition 9. In particular, we have ", "type": "text"}, {"bbox": [290, 214, 332, 225], "score": 0.94, "content": "d(G^{\\prime})\\geq3", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [332, 213, 453, 225], "score": 1.0, "content": " if one of the class numbers ", "type": "text"}, {"bbox": [454, 214, 485, 225], "score": 0.94, "content": "h_{2}(K_{1})", "type": "inline_equation", "height": 11, "width": 31}], "index": 7}, {"bbox": [126, 225, 486, 237], "spans": [{"bbox": [126, 225, 138, 237], "score": 1.0, "content": "or ", "type": "text"}, {"bbox": [138, 226, 169, 237], "score": 0.94, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [170, 225, 415, 237], "score": 1.0, "content": " is at least 8. Therefore it suffices to examine the cases ", "type": "text"}, {"bbox": [415, 226, 465, 237], "score": 0.94, "content": "h_{2}(K_{2})=2", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [466, 225, 486, 237], "score": 1.0, "content": " and", "type": "text"}], "index": 8}, {"bbox": [126, 237, 385, 250], "spans": [{"bbox": [126, 238, 175, 249], "score": 0.93, "content": "h_{2}(K_{2})=4", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [176, 237, 282, 250], "score": 1.0, "content": " (recall from above that ", "type": "text"}, {"bbox": [282, 238, 313, 249], "score": 0.94, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [313, 237, 385, 250], "score": 1.0, "content": " is always even).", "type": "text"}], "index": 9}], "index": 7, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [125, 189, 487, 250]}, {"type": "text", "bbox": [124, 263, 485, 298], "lines": [{"bbox": [137, 263, 486, 278], "spans": [{"bbox": [137, 263, 401, 278], "score": 1.0, "content": "We start by considering case A); it is sufficient to show that ", "type": "text"}, {"bbox": [402, 266, 482, 276], "score": 0.94, "content": "h_{2}(K_{1})h_{2}(K_{2})\\neq4", "type": "inline_equation", "height": 10, "width": 80}, {"bbox": [483, 263, 486, 278], "score": 1.0, "content": ".", "type": "text"}], "index": 10}, {"bbox": [125, 275, 486, 290], "spans": [{"bbox": [125, 275, 486, 290], "score": 1.0, "content": "We now apply Proposition 5; notice that we may do so by the proof of Proposition", "type": "text"}], "index": 11}, {"bbox": [126, 290, 135, 299], "spans": [{"bbox": [126, 290, 135, 299], "score": 1.0, "content": "8.", "type": "text"}], "index": 12}], "index": 11, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [125, 263, 486, 299]}, {"type": "text", "bbox": [124, 299, 486, 334], "lines": [{"bbox": [125, 300, 487, 313], "spans": [{"bbox": [125, 300, 149, 313], "score": 1.0, "content": "a) If ", "type": "text"}, {"bbox": [150, 302, 202, 312], "score": 0.92, "content": "h_{2}(K_{2})\\,=\\,2", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [202, 300, 232, 313], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [232, 302, 272, 311], "score": 0.92, "content": "\\#\\kappa_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 40}, {"bbox": [272, 300, 385, 313], "score": 1.0, "content": " by Proposition 5, hence ", "type": "text"}, {"bbox": [385, 303, 416, 312], "score": 0.92, "content": "q_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [416, 300, 487, 313], "score": 1.0, "content": " by Proposition", "type": "text"}], "index": 13}, {"bbox": [124, 311, 487, 326], "spans": [{"bbox": [124, 311, 178, 326], "score": 1.0, "content": "7 and then ", "type": "text"}, {"bbox": [178, 315, 208, 323], "score": 0.92, "content": "q_{1}=1", "type": "inline_equation", "height": 8, "width": 30}, {"bbox": [209, 311, 487, 326], "score": 1.0, "content": " by Proposition 6. The class number formulas in the proof of", "type": "text"}], "index": 14}, {"bbox": [125, 324, 356, 336], "spans": [{"bbox": [125, 324, 228, 336], "score": 1.0, "content": "Proposition 9 now give", "type": "text"}, {"bbox": [229, 325, 279, 336], "score": 0.94, "content": "h_{2}(K_{1})=1", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [279, 324, 300, 336], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [301, 325, 352, 336], "score": 0.94, "content": "h_{2}(L)=2^{m}", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [352, 324, 356, 336], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 14, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [124, 300, 487, 336]}, {"type": "text", "bbox": [125, 335, 486, 382], "lines": [{"bbox": [137, 336, 486, 348], "spans": [{"bbox": [137, 336, 411, 348], "score": 1.0, "content": "It can be shown using the ambiguous class number formula that", "type": "text"}, {"bbox": [412, 338, 447, 348], "score": 0.92, "content": "\\mathrm{Cl}_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [447, 336, 486, 348], "score": 1.0, "content": " is trivial", "type": "text"}], "index": 16}, {"bbox": [125, 348, 485, 361], "spans": [{"bbox": [125, 348, 184, 361], "score": 1.0, "content": "if and only if ", "type": "text"}, {"bbox": [185, 353, 194, 359], "score": 0.89, "content": "\\varepsilon_{1}", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [194, 348, 434, 361], "score": 1.0, "content": " is a quadratic nonresidue modulo the prime ideal over ", "type": "text"}, {"bbox": [434, 350, 444, 359], "score": 0.92, "content": "d_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [444, 348, 458, 361], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [459, 350, 468, 359], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [469, 348, 485, 361], "score": 1.0, "content": "; by", "type": "text"}], "index": 17}, {"bbox": [126, 361, 486, 373], "spans": [{"bbox": [126, 361, 322, 373], "score": 1.0, "content": "Scholz\u2019s reciprocity law, this is equivalent to ", "type": "text"}, {"bbox": [322, 361, 414, 372], "score": 0.93, "content": "(d_{1}/d_{2})_{4}(d_{2}/d_{1})_{4}=1", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [414, 361, 486, 373], "score": 1.0, "content": ", and this agrees", "type": "text"}], "index": 18}, {"bbox": [126, 372, 258, 384], "spans": [{"bbox": [126, 372, 258, 384], "score": 1.0, "content": "with the criterion given in [1].", "type": "text"}], "index": 19}], "index": 17.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [125, 336, 486, 384]}, {"type": "text", "bbox": [125, 382, 485, 418], "lines": [{"bbox": [125, 383, 486, 397], "spans": [{"bbox": [125, 383, 148, 397], "score": 1.0, "content": "b) If ", "type": "text"}, {"bbox": [149, 385, 198, 396], "score": 0.93, "content": "h_{2}(K_{2})=4", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [199, 383, 295, 397], "score": 1.0, "content": ", we may assume that", "type": "text"}, {"bbox": [296, 385, 358, 396], "score": 0.94, "content": "\\mathrm{Cl}_{2}(K_{2})=(4)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [358, 383, 486, 397], "score": 1.0, "content": " from Proposition 8.b). Then", "type": "text"}], "index": 20}, {"bbox": [126, 397, 486, 409], "spans": [{"bbox": [126, 398, 164, 407], "score": 0.92, "content": "\\#\\kappa_{2}=2", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [164, 397, 247, 409], "score": 1.0, "content": " by Proposition 5, ", "type": "text"}, {"bbox": [247, 398, 276, 407], "score": 0.93, "content": "q_{2}=2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [276, 397, 376, 409], "score": 1.0, "content": " by Proposition 7 and ", "type": "text"}, {"bbox": [376, 398, 405, 407], "score": 0.92, "content": "q_{1}=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [406, 397, 486, 409], "score": 1.0, "content": " by Proposition 6.", "type": "text"}], "index": 21}, {"bbox": [124, 407, 435, 421], "spans": [{"bbox": [124, 407, 298, 421], "score": 1.0, "content": "Using the class number formula we get ", "type": "text"}, {"bbox": [299, 409, 348, 420], "score": 0.94, "content": "h_{2}(K_{1})=2", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [348, 407, 370, 421], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [370, 408, 431, 420], "score": 0.93, "content": "h_{2}(L)=2^{m+2}", "type": "inline_equation", "height": 12, "width": 61}, {"bbox": [432, 407, 435, 421], "score": 1.0, "content": ".", "type": "text"}], "index": 22}], "index": 21, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [124, 383, 486, 421]}, {"type": "text", "bbox": [124, 418, 486, 442], "lines": [{"bbox": [137, 419, 486, 433], "spans": [{"bbox": [137, 419, 256, 433], "score": 1.0, "content": "Thus in both cases we have", "type": "text"}, {"bbox": [257, 421, 337, 432], "score": 0.93, "content": "h_{2}(K_{1})h_{2}(K_{2})\\neq4", "type": "inline_equation", "height": 11, "width": 80}, {"bbox": [337, 419, 486, 433], "score": 1.0, "content": ", and by the table at the beginning", "type": "text"}], "index": 23}, {"bbox": [126, 432, 380, 444], "spans": [{"bbox": [126, 432, 278, 444], "score": 1.0, "content": "of this proof this implies that rank", "type": "text"}, {"bbox": [279, 433, 329, 443], "score": 0.89, "content": "\\mathrm{Cl}_{2}(k^{1})\\neq2", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [330, 432, 380, 444], "score": 1.0, "content": " in case A).", "type": "text"}], "index": 24}], "index": 23.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [126, 419, 486, 444]}, {"type": "text", "bbox": [127, 457, 486, 481], "lines": [{"bbox": [136, 459, 487, 472], "spans": [{"bbox": [136, 459, 421, 472], "score": 1.0, "content": "Next we consider case B); here we have to distinguish between ", "type": "text"}, {"bbox": [421, 461, 460, 471], "score": 0.93, "content": "d_{3}\\neq-4", "type": "inline_equation", "height": 10, "width": 39}, {"bbox": [460, 459, 487, 472], "score": 1.0, "content": " (case", "type": "text"}], "index": 25}, {"bbox": [126, 471, 248, 484], "spans": [{"bbox": [126, 473, 138, 482], "score": 0.82, "content": "B_{1}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [138, 471, 164, 484], "score": 1.0, "content": ") and ", "type": "text"}, {"bbox": [164, 473, 201, 482], "score": 0.92, "content": "d_{3}=-4", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [201, 471, 228, 484], "score": 1.0, "content": " (case ", "type": "text"}, {"bbox": [228, 473, 241, 482], "score": 0.88, "content": "B_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [241, 471, 248, 484], "score": 1.0, "content": ").", "type": "text"}], "index": 26}], "index": 25.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [126, 459, 487, 484]}, {"type": "text", "bbox": [124, 484, 487, 553], "lines": [{"bbox": [137, 483, 256, 496], "spans": [{"bbox": [137, 483, 236, 496], "score": 1.0, "content": "Let us start with case ", "type": "text"}, {"bbox": [236, 485, 248, 494], "score": 0.9, "content": "B_{1}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [249, 483, 256, 496], "score": 1.0, "content": ").", "type": "text"}], "index": 27}, {"bbox": [126, 495, 486, 509], "spans": [{"bbox": [126, 495, 149, 509], "score": 1.0, "content": "a) If ", "type": "text"}, {"bbox": [149, 496, 200, 507], "score": 0.93, "content": "h_{2}(K_{2})=2", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [200, 495, 230, 509], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [230, 497, 268, 506], "score": 0.91, "content": "\\#\\kappa_{2}=2", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [269, 495, 275, 509], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [275, 497, 304, 506], "score": 0.91, "content": "q_{2}=2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [304, 495, 327, 509], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [328, 497, 357, 506], "score": 0.93, "content": "q_{1}=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [357, 495, 486, 509], "score": 1.0, "content": " as above. The class number", "type": "text"}], "index": 28}, {"bbox": [123, 507, 324, 520], "spans": [{"bbox": [123, 507, 187, 520], "score": 1.0, "content": "formula gives ", "type": "text"}, {"bbox": [187, 509, 236, 519], "score": 0.94, "content": "h_{2}(K_{1})=2", "type": "inline_equation", "height": 10, "width": 49}, {"bbox": [236, 507, 258, 520], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [259, 508, 320, 519], "score": 0.93, "content": "h_{2}(L)=2^{m+1}", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [320, 507, 324, 520], "score": 1.0, "content": ".", "type": "text"}], "index": 29}, {"bbox": [125, 518, 486, 533], "spans": [{"bbox": [125, 518, 147, 533], "score": 1.0, "content": "b) If ", "type": "text"}, {"bbox": [147, 520, 208, 531], "score": 0.92, "content": "\\mathrm{Cl}_{2}(K_{2})=(4)", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [209, 518, 486, 533], "score": 1.0, "content": " (which we may assume without loss of generality by Proposition", "type": "text"}], "index": 30}, {"bbox": [125, 531, 485, 544], "spans": [{"bbox": [125, 531, 175, 544], "score": 1.0, "content": "8.b)) then ", "type": "text"}, {"bbox": [175, 533, 216, 542], "score": 0.91, "content": "\\#\\kappa_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [217, 531, 223, 544], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [223, 533, 254, 542], "score": 0.91, "content": "q_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [255, 531, 279, 544], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [280, 533, 311, 542], "score": 0.93, "content": "q_{1}\\,=\\,1", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [311, 531, 485, 544], "score": 1.0, "content": ", again exactly as above. This implies", "type": "text"}], "index": 31}, {"bbox": [126, 542, 263, 556], "spans": [{"bbox": [126, 544, 175, 555], "score": 0.94, "content": "h_{2}(K_{1})=4", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [176, 542, 198, 556], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [198, 544, 259, 555], "score": 0.92, "content": "h_{2}(L)=2^{m+3}", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [259, 542, 263, 556], "score": 1.0, "content": ".", "type": "text"}], "index": 32}], "index": 29.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [123, 483, 486, 556]}, {"type": "text", "bbox": [125, 564, 486, 663], "lines": [{"bbox": [125, 566, 487, 580], "spans": [{"bbox": [125, 566, 377, 580], "score": 1.0, "content": "Here we apply Kuroda\u2019s class number formula (see [10]) to ", "type": "text"}, {"bbox": [378, 568, 399, 579], "score": 0.92, "content": "L/k_{1}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [399, 566, 446, 580], "score": 1.0, "content": ", and since ", "type": "text"}, {"bbox": [447, 568, 487, 579], "score": 0.91, "content": "h_{2}(k_{1})=", "type": "inline_equation", "height": 11, "width": 40}], "index": 33}, {"bbox": [126, 579, 486, 592], "spans": [{"bbox": [126, 581, 131, 588], "score": 0.43, "content": "^{1}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [132, 579, 154, 592], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [154, 580, 231, 591], "score": 0.93, "content": "h_{2}(K_{1})=h_{2}(K_{1}^{\\prime})", "type": "inline_equation", "height": 11, "width": 77}, {"bbox": [231, 579, 268, 592], "score": 1.0, "content": ", we get ", "type": "text"}, {"bbox": [269, 579, 453, 591], "score": 0.93, "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{2}q_{1}h_{2}(K_{1})^{2}h_{2}(k)=2^{m}q_{1}h_{2}(K_{1})^{2}}\\end{array}", "type": "inline_equation", "height": 12, "width": 184}, {"bbox": [454, 579, 486, 592], "score": 1.0, "content": ". From", "type": "text"}], "index": 34}, {"bbox": [126, 592, 485, 605], "spans": [{"bbox": [126, 594, 185, 605], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\varepsilon}\\,)", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [185, 594, 294, 605], "score": 1.0, "content": " (for a suitable choice of ", "type": "text"}, {"bbox": [295, 595, 302, 602], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [302, 594, 409, 605], "score": 1.0, "content": "; the other possibility is", "type": "text"}, {"bbox": [409, 592, 482, 605], "score": 0.92, "content": "{\\tilde{K}}_{2}=k_{2}({\\sqrt{d_{2}\\varepsilon}}\\,))", "type": "inline_equation", "height": 13, "width": 73}, {"bbox": [482, 594, 485, 605], "score": 1.0, "content": ",", "type": "text"}], "index": 35}, {"bbox": [126, 604, 487, 617], "spans": [{"bbox": [126, 604, 155, 617], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [155, 609, 160, 614], "score": 0.88, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [161, 604, 286, 617], "score": 1.0, "content": " is the fundamental unit of ", "type": "text"}, {"bbox": [286, 607, 297, 615], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [297, 604, 416, 617], "score": 1.0, "content": ", we deduce that the uni t ", "type": "text"}, {"bbox": [416, 609, 421, 614], "score": 0.88, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [421, 604, 487, 617], "score": 1.0, "content": ", which still is", "type": "text"}], "index": 36}, {"bbox": [124, 617, 487, 630], "spans": [{"bbox": [124, 617, 195, 630], "score": 1.0, "content": "fundamental in ", "type": "text"}, {"bbox": [195, 619, 201, 627], "score": 0.88, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [201, 617, 298, 630], "score": 1.0, "content": ", becomes a square in ", "type": "text"}, {"bbox": [299, 619, 306, 626], "score": 0.88, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [306, 617, 406, 630], "score": 1.0, "content": ", and this implies that ", "type": "text"}, {"bbox": [406, 619, 434, 628], "score": 0.91, "content": "q_{1}\\geq2", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [434, 617, 487, 630], "score": 1.0, "content": ". Moreover,", "type": "text"}], "index": 37}, {"bbox": [125, 628, 484, 642], "spans": [{"bbox": [125, 628, 165, 642], "score": 1.0, "content": "we have ", "type": "text"}, {"bbox": [166, 629, 236, 641], "score": 0.94, "content": "K_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\lambda})", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [236, 628, 273, 642], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [273, 631, 312, 640], "score": 0.73, "content": "\\pi,\\lambda\\,\\equiv\\,1", "type": "inline_equation", "height": 9, "width": 39}, {"bbox": [313, 628, 440, 642], "score": 1.0, "content": " mod 4 are prime factors of ", "type": "text"}, {"bbox": [440, 631, 450, 640], "score": 0.91, "content": "d_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [450, 628, 474, 642], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [475, 631, 484, 640], "score": 0.91, "content": "d_{2}", "type": "inline_equation", "height": 9, "width": 9}], "index": 38}, {"bbox": [126, 642, 486, 653], "spans": [{"bbox": [126, 642, 138, 653], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [138, 642, 184, 653], "score": 0.92, "content": "k_{1}\\,=\\,\\mathbb{Q}(i)", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [184, 642, 326, 653], "score": 1.0, "content": ", respectively. This shows that ", "type": "text"}, {"bbox": [327, 643, 340, 652], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [340, 642, 486, 653], "score": 1.0, "content": " has even class number, because", "type": "text"}], "index": 39}, {"bbox": [126, 653, 319, 665], "spans": [{"bbox": [126, 654, 181, 665], "score": 0.93, "content": "K_{1}(\\sqrt{\\pi}\\,)/K_{1}", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [181, 653, 319, 665], "score": 1.0, "content": " is easily seen to be unramified.", "type": "text"}], "index": 40}], "index": 36.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [124, 566, 487, 665]}, {"type": "text", "bbox": [125, 663, 485, 687], "lines": [{"bbox": [136, 662, 486, 679], "spans": [{"bbox": [136, 662, 162, 679], "score": 1.0, "content": "Thus ", "type": "text"}, {"bbox": [163, 666, 235, 677], "score": 0.91, "content": "2\\mid q_{1},\\,2\\mid h_{2}(K_{1})", "type": "inline_equation", "height": 11, "width": 72}, {"bbox": [235, 662, 325, 679], "score": 1.0, "content": ", and so we find that ", "type": "text"}, {"bbox": [325, 666, 350, 677], "score": 0.94, "content": "h_{2}(L)", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [350, 662, 413, 679], "score": 1.0, "content": " is divisible by", "type": "text"}, {"bbox": [414, 666, 482, 674], "score": 0.92, "content": "2^{m}\\cdot2\\cdot4=2^{m+3}", "type": "inline_equation", "height": 8, "width": 68}, {"bbox": [482, 662, 486, 679], "score": 1.0, "content": ".", "type": "text"}], "index": 41}, {"bbox": [126, 678, 354, 689], "spans": [{"bbox": [126, 678, 256, 689], "score": 1.0, "content": "In particular, we always have ", "type": "text"}, {"bbox": [257, 678, 299, 689], "score": 0.94, "content": "d(G^{\\prime})\\geq3", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [299, 678, 354, 689], "score": 1.0, "content": " in this case.", "type": "text"}], "index": 42}], "index": 41.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [126, 662, 486, 689]}, {"type": "text", "bbox": [137, 688, 249, 699], "lines": [{"bbox": [137, 689, 249, 700], "spans": [{"bbox": [137, 689, 249, 700], "score": 1.0, "content": "This concludes the proof.", "type": "text"}], "index": 43}], "index": 43, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [137, 689, 249, 700]}]}	{"layout_dets": [{"category_id": 1, "poly": [347, 520, 1352, 520, 1352, 689, 347, 689], "score": 0.967}, {"category_id": 5, "poly": [635, 306, 1065, 306, 1065, 464, 635, 464], "score": 0.961, "html": "<html><body><table><tr><td>d(G')</td><td>h2(L)</td><td>h2(K1)h2(K2)</td></tr><tr><td>1</td><td>2m</td><td>2</td></tr><tr><td>2</td><td>2m+1</td><td>4</td></tr><tr><td>>3</td><td>\u22652m+2</td><td>M8</td></tr></table></body></html>"}, {"category_id": 1, "poly": [348, 1569, 1352, 1569, 1352, 1842, 348, 1842], "score": 0.956}, {"category_id": 1, "poly": [347, 1163, 1350, 1163, 1350, 1230, 347, 1230], 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In particular, we have if one of the class numbers or is at least 8. Therefore it suffices to examine the cases and (recall from above that is always even). We start by considering case A); it is sufficient to show that . We now apply Proposition 5; notice that we may do so by the proof of Proposition 8. a) If , then by Proposition 5, hence by Proposition 7 and then by Proposition 6. The class number formulas in the proof of Proposition 9 now give and . It can be shown using the ambiguous class number formula that is trivial if and only if is a quadratic nonresidue modulo the prime ideal over in ; by Scholz’s reciprocity law, this is equivalent to , and this agrees with the criterion given in [1]. b) If , we may assume that from Proposition 8.b). Then by Proposition 5, by Proposition 7 and by Proposition 6. Using the class number formula we get and . Thus in both cases we have , and by the table at the beginning of this proof this implies that rank in case A). Next we consider case B); here we have to distinguish between (case ) and (case ). Let us start with case ). a) If , then , and as above. The class number formula gives and . b) If (which we may assume without loss of generality by Proposition 8.b)) then , and , again exactly as above. This implies and . Here we apply Kuroda’s class number formula (see [10]) to , and since and , we get . From (for a suitable choice of ; the other possibility is , where is the fundamental unit of , we deduce that the uni t , which still is fundamental in , becomes a square in , and this implies that . Moreover, we have , where mod 4 are prime factors of and in , respectively. This shows that has even class number, because is easily seen to be unramified. Thus , and so we find that is divisible by . In particular, we always have in this case. 口 This concludes the proof.	<div class="pdf-page"> <p>Here, the first two columns follow from Proposition 1, the last (which we do not claim to hold if in case B)) is a consequence of the class number formula of Proposition 9. In particular, we have if one of the class numbers or is at least 8. Therefore it suffices to examine the cases and (recall from above that is always even).</p> <p>We start by considering case A); it is sufficient to show that . We now apply Proposition 5; notice that we may do so by the proof of Proposition 8.</p> <p>a) If , then by Proposition 5, hence by Proposition 7 and then by Proposition 6. The class number formulas in the proof of Proposition 9 now give and .</p> <p>It can be shown using the ambiguous class number formula that is trivial if and only if is a quadratic nonresidue modulo the prime ideal over in ; by Scholz’s reciprocity law, this is equivalent to , and this agrees with the criterion given in [1].</p> <p>b) If , we may assume that from Proposition 8.b). Then by Proposition 5, by Proposition 7 and by Proposition 6. Using the class number formula we get and .</p> <p>Thus in both cases we have , and by the table at the beginning of this proof this implies that rank in case A).</p> <p>Next we consider case B); here we have to distinguish between (case ) and (case ).</p> <p>Let us start with case ). a) If , then , and as above. The class number formula gives and . b) If (which we may assume without loss of generality by Proposition 8.b)) then , and , again exactly as above. This implies and .</p> <p>Here we apply Kuroda’s class number formula (see [10]) to , and since and , we get . From (for a suitable choice of ; the other possibility is , where is the fundamental unit of , we deduce that the uni t , which still is fundamental in , becomes a square in , and this implies that . Moreover, we have , where mod 4 are prime factors of and in , respectively. This shows that has even class number, because is easily seen to be unramified.</p> <p>Thus , and so we find that is divisible by . In particular, we always have in this case.</p> <p>This concludes the proof.</p> </div>	<div class="pdf-page"> <div class="pdf-discarded" data-x="209" data-y="117" data-width="16" data-height="11" style="opacity: 0.5;">12</div> <p class="pdf-text" data-x="207" data-y="241" data-width="606" data-height="79">Here, the first two columns follow from Proposition 1, the last (which we do not claim to hold if in case B)) is a consequence of the class number formula of Proposition 9. In particular, we have if one of the class numbers or is at least 8. Therefore it suffices to examine the cases and (recall from above that is always even).</p> <p class="pdf-text" data-x="207" data-y="340" data-width="604" data-height="45">We start by considering case A); it is sufficient to show that . We now apply Proposition 5; notice that we may do so by the proof of Proposition 8.</p> <p class="pdf-text" data-x="207" data-y="386" data-width="606" data-height="45">a) If , then by Proposition 5, hence by Proposition 7 and then by Proposition 6. The class number formulas in the proof of Proposition 9 now give and .</p> <p class="pdf-text" data-x="209" data-y="433" data-width="604" data-height="60">It can be shown using the ambiguous class number formula that is trivial if and only if is a quadratic nonresidue modulo the prime ideal over in ; by Scholz’s reciprocity law, this is equivalent to , and this agrees with the criterion given in [1].</p> <p class="pdf-text" data-x="209" data-y="493" data-width="602" data-height="47">b) If , we may assume that from Proposition 8.b). Then by Proposition 5, by Proposition 7 and by Proposition 6. Using the class number formula we get and .</p> <p class="pdf-text" data-x="207" data-y="540" data-width="606" data-height="31">Thus in both cases we have , and by the table at the beginning of this proof this implies that rank in case A).</p> <p class="pdf-text" data-x="212" data-y="590" data-width="601" data-height="31">Next we consider case B); here we have to distinguish between (case ) and (case ).</p> <p class="pdf-text" data-x="207" data-y="625" data-width="607" data-height="89">Let us start with case ). a) If , then , and as above. The class number formula gives and . b) If (which we may assume without loss of generality by Proposition 8.b)) then , and , again exactly as above. This implies and .</p> <p class="pdf-text" data-x="209" data-y="729" data-width="604" data-height="128">Here we apply Kuroda’s class number formula (see [10]) to , and since and , we get . From (for a suitable choice of ; the other possibility is , where is the fundamental unit of , we deduce that the uni t , which still is fundamental in , becomes a square in , and this implies that . Moreover, we have , where mod 4 are prime factors of and in , respectively. This shows that has even class number, because is easily seen to be unramified.</p> <p class="pdf-text" data-x="209" data-y="857" data-width="602" data-height="31">Thus , and so we find that is divisible by . In particular, we always have in this case.</p> <div class="pdf-discarded" data-x="796" data-y="888" data-width="17" data-height="14" style="opacity: 0.5;">口</div> <p class="pdf-text" data-x="229" data-y="889" data-width="187" data-height="14">This concludes the proof.</p> </div>	{ "type": [ "text", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text" ], "coordinates": [ [ 381, 142, 640, 215 ], [ 209, 244, 813, 259 ], [ 209, 259, 814, 275 ], [ 210, 275, 811, 290 ], [ 210, 290, 813, 306 ], [ 210, 306, 644, 323 ], [ 229, 340, 813, 359 ], [ 209, 355, 813, 374 ], [ 210, 374, 225, 386 ], [ 209, 387, 814, 404 ], [ 207, 402, 814, 421 ], [ 209, 418, 595, 434 ], [ 229, 434, 813, 449 ], [ 209, 449, 811, 466 ], [ 210, 466, 813, 482 ], [ 210, 480, 431, 496 ], [ 209, 495, 813, 513 ], [ 210, 513, 813, 528 ], [ 207, 526, 727, 544 ], [ 229, 541, 813, 559 ], [ 210, 558, 635, 574 ], [ 227, 593, 814, 610 ], [ 210, 608, 414, 625 ], [ 229, 624, 428, 641 ], [ 210, 640, 813, 658 ], [ 205, 655, 542, 672 ], [ 209, 669, 813, 689 ], [ 209, 686, 811, 703 ], [ 210, 700, 440, 718 ], [ 209, 731, 814, 749 ], [ 210, 748, 813, 765 ], [ 210, 765, 811, 782 ], [ 210, 780, 814, 797 ], [ 207, 797, 814, 814 ], [ 209, 811, 809, 830 ], [ 210, 830, 813, 844 ], [ 210, 844, 533, 859 ], [ 227, 855, 813, 877 ], [ 210, 876, 592, 890 ], [ 229, 890, 416, 905 ] ], "content": [ "", "Here, the first two columns follow from Proposition 1, the last (which we do not", "claim to hold if d_{3}=-4 in case B)) is a consequence of the class number formula of", "Proposition 9. In particular, we have d(G^{\\prime})\\geq3 if one of the class numbers h_{2}(K_{1})", "or h_{2}(K_{2}) is at least 8. Therefore it suffices to examine the cases h_{2}(K_{2})=2 and", "h_{2}(K_{2})=4 (recall from above that h_{2}(K_{2}) is always even).", "We start by considering case A); it is sufficient to show that h_{2}(K_{1})h_{2}(K_{2})\\neq4 .", "We now apply Proposition 5; notice that we may do so by the proof of Proposition", "8.", "a) If h_{2}(K_{2})\\,=\\,2 , then \\#\\kappa_{2}\\,=\\,2 by Proposition 5, hence q_{2}\\,=\\,2 by Proposition", "7 and then q_{1}=1 by Proposition 6. The class number formulas in the proof of", "Proposition 9 now give h_{2}(K_{1})=1 and h_{2}(L)=2^{m} .", "It can be shown using the ambiguous class number formula that \\mathrm{Cl}_{2}(K_{1}) is trivial", "if and only if \\varepsilon_{1} is a quadratic nonresidue modulo the prime ideal over d_{2} in k_{1} ; by", "Scholz’s reciprocity law, this is equivalent to (d_{1}/d_{2})_{4}(d_{2}/d_{1})_{4}=1 , and this agrees", "with the criterion given in [1].", "b) If h_{2}(K_{2})=4 , we may assume that \\mathrm{Cl}_{2}(K_{2})=(4) from Proposition 8.b). Then", "\\#\\kappa_{2}=2 by Proposition 5, q_{2}=2 by Proposition 7 and q_{1}=1 by Proposition 6.", "Using the class number formula we get h_{2}(K_{1})=2 and h_{2}(L)=2^{m+2} .", "Thus in both cases we have h_{2}(K_{1})h_{2}(K_{2})\\neq4 , and by the table at the beginning", "of this proof this implies that rank \\mathrm{Cl}_{2}(k^{1})\\neq2 in case A).", "Next we consider case B); here we have to distinguish between d_{3}\\neq-4 (case", "B_{1} ) and d_{3}=-4 (case B_{2} ).", "Let us start with case B_{1} ).", "a) If h_{2}(K_{2})=2 , then \\#\\kappa_{2}=2 , q_{2}=2 and q_{1}=1 as above. The class number", "formula gives h_{2}(K_{1})=2 and h_{2}(L)=2^{m+1} .", "b) If \\mathrm{Cl}_{2}(K_{2})=(4) (which we may assume without loss of generality by Proposition", "8.b)) then \\#\\kappa_{2}\\,=\\,2 , q_{2}\\,=\\,2 and q_{1}\\,=\\,1 , again exactly as above. This implies", "h_{2}(K_{1})=4 and h_{2}(L)=2^{m+3} .", "Here we apply Kuroda’s class number formula (see [10]) to L/k_{1} , and since h_{2}(k_{1})=", "^{1} and h_{2}(K_{1})=h_{2}(K_{1}^{\\prime}) , we get \\begin{array}{r}{h_{2}(L)=\\frac{1}{2}q_{1}h_{2}(K_{1})^{2}h_{2}(k)=2^{m}q_{1}h_{2}(K_{1})^{2}}\\end{array} . From", "K_{2}=k_{2}(\\sqrt{\\varepsilon}\\,) (for a suitable choice of L ; the other possibility is {\\tilde{K}}_{2}=k_{2}({\\sqrt{d_{2}\\varepsilon}}\\,)) ,", "where \\varepsilon is the fundamental unit of k_{2} , we deduce that the uni t \\varepsilon , which still is", "fundamental in k , becomes a square in L , and this implies that q_{1}\\geq2 . Moreover,", "we have K_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\lambda}) , where \\pi,\\lambda\\,\\equiv\\,1 mod 4 are prime factors of d_{1} and d_{2}", "in k_{1}\\,=\\,\\mathbb{Q}(i) , respectively. This shows that K_{1} has even class number, because", "K_{1}(\\sqrt{\\pi}\\,)/K_{1} is easily seen to be unramified.", "Thus 2\\mid q_{1},\\,2\\mid h_{2}(K_{1}) , and so we find that h_{2}(L) is divisible by 2^{m}\\cdot2\\cdot4=2^{m+3} .", "In particular, we always have d(G^{\\prime})\\geq3 in this case.", "This concludes the proof." ], "index": [ 0, 1, 2, 3, 4, 5, 7, 8, 9, 16, 17, 18, 28, 29, 30, 31, 44, 45, 46, 63, 64, 84, 85, 107, 108, 109, 110, 111, 112, 136, 137, 138, 139, 140, 141, 142, 143, 173, 174, 212 ] }	{ "coordinates": [], "index": [], "caption": [], "caption_coordinates": [] }	{ "type": [], "coordinates": [], "content": [], "index": [] }	{ "type": [ "table", "table_body" ], "coordinates": [ [ 381, 142, 640, 215 ], [ 381, 142, 640, 215 ] ], "content": [ "", "" ], "index": [ 0, 1 ] }	{"document_info": {"total_pages": 14, "parse_type": "txt", "version": "1.0.1", "total_content_list_items": 155}, "content_statistics": {"total_blocks": 182, "total_lines": 526, "total_images": 4, "total_equations": 28, "total_tables": 11, "total_characters": 29658, "total_words": 5331, "block_type_distribution": {"title": 7, "text": 126, "interline_equation": 14, "discarded": 24, "table_caption": 3, "table_body": 4, "image_body": 2, "list": 2}}, "model_statistics": {"total_layout_detections": 2826, "avg_confidence": 0.9470955414012739, "min_confidence": 0.26, "max_confidence": 1.0}, "page_statistics": [{"page_number": 0, "page_size": [612.0, 792.0], "blocks": 11, "lines": 38, "images": 0, "equations": 2, "tables": 0, "characters": 2362, "words": 398, "discarded_blocks": 1}, {"page_number": 1, "page_size": [612.0, 792.0], "blocks": 10, "lines": 30, "images": 0, "equations": 0, "tables": 3, "characters": 1688, "words": 305, "discarded_blocks": 1}, {"page_number": 2, "page_size": [612.0, 792.0], "blocks": 15, "lines": 44, "images": 0, "equations": 4, "tables": 0, "characters": 2417, "words": 416, "discarded_blocks": 2}, {"page_number": 3, "page_size": [612.0, 792.0], "blocks": 7, "lines": 27, "images": 0, "equations": 0, "tables": 3, "characters": 1591, "words": 291, "discarded_blocks": 1}, {"page_number": 4, "page_size": [612.0, 792.0], "blocks": 23, "lines": 38, "images": 0, "equations": 6, "tables": 0, "characters": 1604, "words": 303, "discarded_blocks": 4}, {"page_number": 5, "page_size": [612.0, 792.0], "blocks": 17, "lines": 40, "images": 0, "equations": 6, "tables": 0, "characters": 2131, "words": 393, "discarded_blocks": 1}, {"page_number": 6, "page_size": [612.0, 792.0], "blocks": 15, "lines": 36, "images": 2, "equations": 4, "tables": 0, "characters": 1791, "words": 324, "discarded_blocks": 2}, {"page_number": 7, "page_size": [612.0, 792.0], "blocks": 12, "lines": 49, "images": 0, "equations": 0, "tables": 0, "characters": 3049, "words": 571, "discarded_blocks": 1}, {"page_number": 8, "page_size": [612.0, 792.0], "blocks": 11, "lines": 42, "images": 0, "equations": 0, "tables": 0, "characters": 2636, "words": 479, "discarded_blocks": 2}, {"page_number": 9, "page_size": [612.0, 792.0], "blocks": 14, "lines": 35, "images": 2, "equations": 0, "tables": 0, "characters": 2036, "words": 375, "discarded_blocks": 1}, {"page_number": 10, "page_size": [612.0, 792.0], "blocks": 17, "lines": 35, "images": 0, "equations": 6, "tables": 0, "characters": 1789, "words": 327, "discarded_blocks": 2}, {"page_number": 11, "page_size": [612.0, 792.0], "blocks": 14, "lines": 40, "images": 0, "equations": 0, "tables": 2, "characters": 2094, "words": 399, "discarded_blocks": 2}, {"page_number": 12, "page_size": [612.0, 792.0], "blocks": 10, "lines": 44, "images": 0, "equations": 0, "tables": 3, "characters": 2572, "words": 487, "discarded_blocks": 2}, {"page_number": 13, "page_size": [612.0, 792.0], "blocks": 6, "lines": 28, "images": 0, "equations": 0, "tables": 0, "characters": 1898, "words": 263, "discarded_blocks": 2}]}
0003244v1	12	[ 612, 792 ]	{ "type": [ "text", "text", "text", "table_caption", "table_body", "text", "text", "text", "discarded", "discarded" ], "coordinates": [ [ 209, 144, 813, 191 ], [ 207, 192, 813, 393 ], [ 207, 395, 814, 496 ], [ 209, 497, 813, 527 ], [ 316, 532, 706, 605 ], [ 209, 607, 813, 672 ], [ 207, 672, 813, 841 ], [ 209, 841, 813, 903 ], [ 398, 116, 624, 128 ], [ 796, 117, 813, 128 ] ], "content": [ "The referee (whom we’d like to thank for a couple of helpful remarks) asked whether and infinitely often. Let us show how to prove that both possibilities occur with equal density.", "Before we can do this, we have to study the quadratic extensions and of more closely. We assume that and are odd primes in t he following, and then say how to modify the arguments in the case or . The primes and split in as and . Let denote the odd class number of and write and for primary elements and (this is can easily be proved directly, but it is also a very special case of Hilbert’s first supplementary law for quadratic reciprocity in fields with odd class number (see [7]): if for an ideal with odd norm, then can be chosen primary (i.e. congruent to a square mod ) if and only if is primary (i.e. for all units , where denotes the quadratic residue symbol in )). Let denote the quadratic residue symbol in . Then , so we may choose the conjugates in such a way that and .", "Put and ; we claim that . This is equivalent to , where is a quad r atic unramified extension of . Put a n d apply the ambiguous class number formula to an d : since there is only one ramified prime in each of these two ext e nsions, w e fin d ; note that we have used the assumption that in deducing th a t is inert in .", "In our proof of Theorem 1 we have seen that there are th e following possibilities when \| 4:", "", "In order to decide whether or , recall that we have ; thus must be the field with 2 -class nu mber 2, and this implies and . In particular we see that if and only if as long as with .", "The ambiguous class number formula shows that is cyclic, thus 4 \| if and only if , where is the quadratic unramified extension of . Applying the ambiguous class number formula to , where , we see that if and only if . Now is generated by a root of unity (which always is a norm residue at primes dividing ) and a fundamental unit . Therefore if and only if , where and where denotes the quadratic residue symbol in . Since , we have proved that if and only if the prime ideal above splits in the quadratic extension . But if we fix and , this happens for exactly half of the values of satisfying , .", "If and , then , and we have to choose in such a way that is unramified outside . The residue symbols are defined as Kronecker symbols via the splitting of in the quadratic extension . With these modifactions, the above arguments remain valid.", "IMAGINARY QUADRATIC FIELDS", "13" ], "index": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ] }	[{"type": "text", "text": "The referee (whom we\u2019d like to thank for a couple of helpful remarks) asked whether $h_{2}(K)=2$ and $h_{2}(K)>2$ infinitely often. Let us show how to prove that both possibilities occur with equal density. ", "page_idx": 12}, {"type": "text", "text": "Before we can do this, we have to study the quadratic extensions $K_{1}$ and $\\widetilde{K}_{1}$ of $k_{1}$ more closely. We assume that $d_{2}\\,=\\,p$ and $d_{3}~=~r$ are odd primes in t he following, and then say how to modify the arguments in the case $d_{2}=8$ or $d_{3}=-8$ . The primes $p$ and $r$ split in $k_{1}$ as $p\\mathcal{O}_{1}\\,=\\,\\mathfrak{p p}^{\\prime}$ and $r\\mathcal{O}_{1}\\,=\\,\\mathfrak{r r}^{\\prime}$ . Let $h$ denote the odd class number of $k_{1}$ and write ${\\mathfrak{p}}^{h}\\,=\\,(\\pi)$ and $\\mathfrak{r}^{h}\\,=\\,(\\rho)$ for primary elements $\\pi$ and $\\rho$ (this is can easily be proved directly, but it is also a very special case of Hilbert\u2019s first supplementary law for quadratic reciprocity in fields $K$ with odd class number $h$ (see [7]): if ${\\mathfrak{a}}^{h}\\;=\\;\\alpha{\\mathcal{O}}_{K}$ for an ideal $\\mathfrak{p}$ with odd norm, then $\\alpha$ can be chosen primary (i.e. congruent to a square mod $4{\\cal O}_{K}$ ) if and only if $\\mathfrak{a}$ is primary (i.e. $[\\varepsilon/{\\mathfrak a}]=+1$ for all units $\\varepsilon\\in{\\mathcal{O}}_{K}^{\\times}$ , where $[\\,\\cdot\\,/\\,\\cdot\\,]$ denotes the quadratic residue symbol in $K$ )). Let $\\big[\\cdot\\big/\\cdot\\big]$ denote the quadratic residue symbol in $k_{1}$ . Then $[\\pi/\\rho][\\pi^{\\prime}/\\rho]=[p/\\rho]=(p/r)=-1$ , so we may choose the conjugates in such a way that $[\\pi/\\rho]=+1$ and $[\\pi^{\\prime}/\\rho]=[\\pi/\\rho^{\\prime}]=-1$ . ", "page_idx": 12}, {"type": "text", "text": "Put $K_{1}\\;=\\;k_{1}(\\sqrt{\\pi\\rho}\\,)$ and $\\tilde{K}_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\rho^{\\prime}})$ ; we claim that $h_{2}(\\tilde{K}_{1})\\;=\\;2$ . This is equivalent to $h_{2}(\\widetilde{L}_{1})\\,=\\,1$ , where $\\tilde{L}_{1}\\,=\\,k_{1}(\\sqrt{\\pi},\\sqrt{\\rho^{\\prime}})$ is a quad r atic unramified extension of $\\widetilde{K}_{1}$ . Put $\\widetilde{F}_{1}=k_{1}(\\sqrt{\\pi}\\,)$ a n d apply the ambiguous class number formula to $\\widetilde{F}_{1}/k_{1}$ an d $\\widetilde{L}_{1}/\\widetilde{F}_{1}$ : since there is only one ramified prime in each of these two ext e nsions, w e fin d $\\mathrm{Am}(\\widetilde{F}_{1}/k_{1})\\,=\\,\\mathrm{Am}(\\widetilde{L}_{1}/\\widetilde{F}_{1})\\,=\\,1$ ; note that we have used the assumption that $[\\pi/\\rho^{\\prime}]=-1$ in deducing th a t ${\\mathfrak{r}}^{\\prime}$ is inert in $\\widetilde{F}_{1}/k_{1}$ . ", "page_idx": 12}, {"type": "table", "img_path": "images/dfdcf3e7e1dbc3d44e3061767f642a3e9778b62e79babc37d43af68dae85a584.jpg", "table_caption": ["In our proof of Theorem 1 we have seen that there are th e following possibilities when $h_{2}(K_{2})$ \| 4: "], "table_footnote": [], "table_body": "\n\n<html><body><table><tr><td>q2</td><td>Cl2(K2)</td><td>q1</td><td>h2(K1)</td><td>q2</td><td>Cl2(K2)</td><td>h2(L)</td></tr><tr><td>2</td><td>(2)</td><td>1</td><td>2</td><td>2</td><td>(2)</td><td>2m+1</td></tr><tr><td>2</td><td>(4)</td><td>1</td><td>4</td><td>\uff1f</td><td>(2,2)</td><td>2m+3</td></tr></table></body></html>\n\n", "page_idx": 12}, {"type": "text", "text": "In order to decide whether $\\widetilde{q}_{2}=1$ or $\\widetilde{q}_{2}=2$ , recall that we have $h_{2}(K_{1})=4$ ; thus $\\widetilde{K}_{1}$ must be the field with 2 -class nu mber 2, and this implies $h_{2}(\\widetilde{L})\\,=\\,2^{m+2}$ and $\\widetilde{q}_{2}=1$ . In particular we see that $4\\mid h_{2}(K_{2})$ if and only if $4\\mid h_{2}(K_{1})$ as long as $K_{1}=k_{1}(\\sqrt{\\pi\\rho}\\,)$ with $[\\pi/\\rho]=+1$ . ", "page_idx": 12}, {"type": "text", "text": "The ambiguous class number formula shows that $\\mathrm{Cl}_{2}(K_{1})$ is cyclic, thus 4 \| $h_{2}(K_{1})$ if and only if $2\\mid h_{2}(L_{1})$ , where $L_{1}=K_{1}(\\sqrt{\\pi}\\,)$ is the quadratic unramified extension of $K_{1}$ . Applying the ambiguous class number formula to $L_{1}/F_{1}$ , where $F_{1}\\;=\\;k_{1}(\\sqrt{\\pi}\\,)$ , we see that $2\\:\\:\|\\:\\:h_{2}(L_{1})$ if and only if $(E\\,:\\,H)\\;=\\;1$ . Now $E$ is generated by a root of unity (which always is a norm residue at primes dividing $r\\,\\equiv\\,1\\;\\mathrm{mod}\\;4$ ) and a fundamental unit $\\varepsilon$ . Therefore $(E\\,:\\,H)\\;=\\;1$ if and only if $\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\,=\\,\\{\\varepsilon/\\mathfrak{R}_{2}\\}\\,=\\,+1$ , where $\\mathfrak{r}{\\mathcal{O}}_{F_{1}}\\,=\\,\\mathfrak{R}_{1}\\mathfrak{R}_{2}$ and where $\\{\\,\\cdot\\,/\\,\\cdot\\,\\}$ denotes the quadratic residue symbol in $F_{1}$ . Since $\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\{\\varepsilon/\\mathfrak{R}_{2}\\}=[\\varepsilon/\\mathfrak{r}]=+1$ , we have proved that $4\\mid h_{2}(K_{1})$ if and only if the prime ideal $\\Re_{1}$ above $\\mathfrak{r}$ splits in the quadratic extension $F_{1}(\\sqrt{\\varepsilon})$ . But if we fix $p$ and $q$ , this happens for exactly half of the values of $r$ satisfying $(p/r)=-1$ , $(q/r)=+1$ . ", "page_idx": 12}, {"type": "text", "text": "If $d_{2}=8$ and $p=2$ , then $2\\mathcal{O}_{k_{1}}=22^{\\prime}$ , and we have to choose $2^{h}=(\\pi)$ in such a way that $k_{1}(\\sqrt{\\pi}\\,)/k_{1}$ is unramified outside $\\mathfrak{p}$ . The residue symbols $\\left[\\alpha/2\\right]$ are defined as Kronecker symbols via the splitting of $^{2}$ in the quadratic extension $k_{1}(\\sqrt{\\alpha}\\,)/k_{1}$ . With these modifactions, the above arguments remain valid. ", "page_idx": 12}]	{"preproc_blocks": [{"type": "text", "bbox": [125, 112, 486, 148], "lines": [{"bbox": [137, 114, 486, 127], "spans": [{"bbox": [137, 114, 486, 127], "score": 1.0, "content": "The referee (whom we\u2019d like to thank for a couple of helpful remarks) asked", "type": "text"}], "index": 0}, {"bbox": [126, 126, 485, 138], "spans": [{"bbox": [126, 126, 164, 138], "score": 1.0, "content": "whether ", "type": "text"}, {"bbox": [164, 127, 209, 138], "score": 0.94, "content": "h_{2}(K)=2", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [210, 126, 232, 138], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [232, 127, 278, 138], "score": 0.94, "content": "h_{2}(K)>2", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [278, 126, 485, 138], "score": 1.0, "content": " infinitely often. Let us show how to prove that", "type": "text"}], "index": 1}, {"bbox": [126, 138, 313, 150], "spans": [{"bbox": [126, 138, 313, 150], "score": 1.0, "content": "both possibilities occur with equal density.", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [124, 149, 486, 304], "lines": [{"bbox": [137, 150, 484, 163], "spans": [{"bbox": [137, 150, 434, 163], "score": 1.0, "content": "Before we can do this, we have to study the quadratic extensions ", "type": "text"}, {"bbox": [434, 153, 447, 161], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [448, 150, 471, 163], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [471, 150, 484, 161], "score": 0.92, "content": "\\widetilde{K}_{1}", "type": "inline_equation", "height": 11, "width": 13}], "index": 3}, {"bbox": [125, 161, 487, 176], "spans": [{"bbox": [125, 161, 138, 176], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [138, 164, 148, 173], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [149, 161, 293, 176], "score": 1.0, "content": " more closely. We assume that ", "type": "text"}, {"bbox": [294, 164, 325, 174], "score": 0.93, "content": "d_{2}\\,=\\,p", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [326, 161, 350, 176], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [350, 164, 382, 173], "score": 0.93, "content": "d_{3}~=~r", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [382, 161, 487, 176], "score": 1.0, "content": " are odd primes in t he", "type": "text"}], "index": 4}, {"bbox": [125, 174, 487, 187], "spans": [{"bbox": [125, 174, 404, 187], "score": 1.0, "content": "following, and then say how to modify the arguments in the case ", "type": "text"}, {"bbox": [404, 176, 432, 185], "score": 0.92, "content": "d_{2}=8", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [433, 174, 446, 187], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [446, 177, 482, 185], "score": 0.93, "content": "d_{3}=-8", "type": "inline_equation", "height": 8, "width": 36}, {"bbox": [483, 174, 487, 187], "score": 1.0, "content": ".", "type": "text"}], "index": 5}, {"bbox": [126, 187, 487, 199], "spans": [{"bbox": [126, 187, 180, 199], "score": 1.0, "content": "The primes ", "type": "text"}, {"bbox": [181, 191, 186, 198], "score": 0.9, "content": "p", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [186, 187, 210, 199], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [210, 191, 215, 196], "score": 0.88, "content": "r", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [216, 187, 255, 199], "score": 1.0, "content": " split in ", "type": "text"}, {"bbox": [255, 189, 265, 197], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [265, 187, 282, 199], "score": 1.0, "content": " as ", "type": "text"}, {"bbox": [282, 188, 330, 198], "score": 0.92, "content": "p\\mathcal{O}_{1}\\,=\\,\\mathfrak{p p}^{\\prime}", "type": "inline_equation", "height": 10, "width": 48}, {"bbox": [330, 187, 353, 199], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [354, 188, 399, 197], "score": 0.92, "content": "r\\mathcal{O}_{1}\\,=\\,\\mathfrak{r r}^{\\prime}", "type": "inline_equation", "height": 9, "width": 45}, {"bbox": [399, 187, 427, 199], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [427, 189, 433, 196], "score": 0.87, "content": "h", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [434, 187, 487, 199], "score": 1.0, "content": " denote the", "type": "text"}], "index": 6}, {"bbox": [125, 198, 487, 211], "spans": [{"bbox": [125, 198, 221, 211], "score": 1.0, "content": "odd class number of ", "type": "text"}, {"bbox": [221, 200, 231, 209], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [231, 198, 282, 211], "score": 1.0, "content": " and write ", "type": "text"}, {"bbox": [282, 199, 323, 210], "score": 0.92, "content": "{\\mathfrak{p}}^{h}\\,=\\,(\\pi)", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [324, 198, 348, 211], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [348, 199, 387, 210], "score": 0.92, "content": "\\mathfrak{r}^{h}\\,=\\,(\\rho)", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [388, 198, 487, 211], "score": 1.0, "content": " for primary elements", "type": "text"}], "index": 7}, {"bbox": [126, 210, 487, 223], "spans": [{"bbox": [126, 215, 132, 219], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 4, "width": 6}, {"bbox": [132, 210, 156, 223], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [157, 215, 162, 222], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [163, 210, 487, 223], "score": 1.0, "content": " (this is can easily be proved directly, but it is also a very special case", "type": "text"}], "index": 8}, {"bbox": [125, 223, 486, 235], "spans": [{"bbox": [125, 223, 432, 235], "score": 1.0, "content": "of Hilbert\u2019s first supplementary law for quadratic reciprocity in fields ", "type": "text"}, {"bbox": [433, 225, 442, 232], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [443, 223, 486, 235], "score": 1.0, "content": " with odd", "type": "text"}], "index": 9}, {"bbox": [125, 234, 485, 247], "spans": [{"bbox": [125, 234, 188, 247], "score": 1.0, "content": "class number ", "type": "text"}, {"bbox": [188, 236, 194, 244], "score": 0.88, "content": "h", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [195, 234, 255, 247], "score": 1.0, "content": " (see [7]): if ", "type": "text"}, {"bbox": [256, 235, 306, 245], "score": 0.92, "content": "{\\mathfrak{a}}^{h}\\;=\\;\\alpha{\\mathcal{O}}_{K}", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [307, 234, 367, 247], "score": 1.0, "content": " for an ideal ", "type": "text"}, {"bbox": [368, 236, 374, 246], "score": 0.75, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [374, 234, 478, 247], "score": 1.0, "content": " with odd norm, then ", "type": "text"}, {"bbox": [478, 239, 485, 244], "score": 0.87, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 7}], "index": 10}, {"bbox": [126, 247, 487, 259], "spans": [{"bbox": [126, 247, 378, 259], "score": 1.0, "content": "can be chosen primary (i.e. congruent to a square mod ", "type": "text"}, {"bbox": [378, 248, 398, 257], "score": 0.89, "content": "4{\\cal O}_{K}", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [399, 247, 469, 259], "score": 1.0, "content": ") if and only if ", "type": "text"}, {"bbox": [469, 250, 474, 255], "score": 0.84, "content": "\\mathfrak{a}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [474, 247, 487, 259], "score": 1.0, "content": " is", "type": "text"}], "index": 11}, {"bbox": [125, 258, 487, 272], "spans": [{"bbox": [125, 258, 186, 272], "score": 1.0, "content": "primary (i.e. ", "type": "text"}, {"bbox": [187, 259, 234, 270], "score": 0.93, "content": "[\\varepsilon/{\\mathfrak a}]=+1", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [235, 258, 293, 272], "score": 1.0, "content": " for all units ", "type": "text"}, {"bbox": [293, 259, 327, 271], "score": 0.93, "content": "\\varepsilon\\in{\\mathcal{O}}_{K}^{\\times}", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [327, 258, 362, 272], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [363, 259, 386, 270], "score": 0.95, "content": "[\\,\\cdot\\,/\\,\\cdot\\,]", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [386, 258, 487, 272], "score": 1.0, "content": " denotes the quadratic", "type": "text"}], "index": 12}, {"bbox": [126, 271, 486, 283], "spans": [{"bbox": [126, 271, 205, 283], "score": 1.0, "content": "residue symbol in ", "type": "text"}, {"bbox": [205, 272, 215, 280], "score": 0.86, "content": "K", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [216, 271, 246, 283], "score": 1.0, "content": ")). Let ", "type": "text"}, {"bbox": [247, 272, 270, 282], "score": 0.94, "content": "\\big[\\cdot\\big/\\cdot\\big]", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [270, 271, 445, 283], "score": 1.0, "content": " denote the quadratic residue symbol in ", "type": "text"}, {"bbox": [445, 272, 455, 281], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [455, 271, 486, 283], "score": 1.0, "content": ". Then", "type": "text"}], "index": 13}, {"bbox": [126, 282, 485, 295], "spans": [{"bbox": [126, 284, 271, 294], "score": 0.9, "content": "[\\pi/\\rho][\\pi^{\\prime}/\\rho]=[p/\\rho]=(p/r)=-1", "type": "inline_equation", "height": 10, "width": 145}, {"bbox": [271, 282, 485, 295], "score": 1.0, "content": ", so we may choose the conjugates in such a way", "type": "text"}], "index": 14}, {"bbox": [125, 294, 310, 307], "spans": [{"bbox": [125, 294, 147, 307], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [147, 295, 195, 306], "score": 0.93, "content": "[\\pi/\\rho]=+1", "type": "inline_equation", "height": 11, "width": 48}, {"bbox": [196, 294, 218, 307], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [218, 295, 307, 306], "score": 0.91, "content": "[\\pi^{\\prime}/\\rho]=[\\pi/\\rho^{\\prime}]=-1", "type": "inline_equation", "height": 11, "width": 89}, {"bbox": [307, 294, 310, 307], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 9}, {"type": "text", "bbox": [124, 306, 487, 384], "lines": [{"bbox": [136, 307, 486, 320], "spans": [{"bbox": [136, 307, 158, 320], "score": 1.0, "content": "Put ", "type": "text"}, {"bbox": [159, 309, 228, 320], "score": 0.93, "content": "K_{1}\\;=\\;k_{1}(\\sqrt{\\pi\\rho}\\,)", "type": "inline_equation", "height": 11, "width": 69}, {"bbox": [229, 307, 253, 320], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [253, 307, 326, 319], "score": 0.91, "content": "\\tilde{K}_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\rho^{\\prime}})", "type": "inline_equation", "height": 12, "width": 73}, {"bbox": [326, 307, 400, 320], "score": 1.0, "content": "; we claim that ", "type": "text"}, {"bbox": [400, 307, 454, 319], "score": 0.94, "content": "h_{2}(\\tilde{K}_{1})\\;=\\;2", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [455, 307, 486, 320], "score": 1.0, "content": ". This", "type": "text"}], "index": 16}, {"bbox": [125, 321, 486, 334], "spans": [{"bbox": [125, 321, 198, 334], "score": 1.0, "content": "is equivalent to ", "type": "text"}, {"bbox": [198, 321, 249, 333], "score": 0.92, "content": "h_{2}(\\widetilde{L}_{1})\\,=\\,1", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [250, 321, 286, 334], "score": 1.0, "content": ", where", "type": "text"}, {"bbox": [286, 321, 368, 333], "score": 0.93, "content": "\\tilde{L}_{1}\\,=\\,k_{1}(\\sqrt{\\pi},\\sqrt{\\rho^{\\prime}})", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [369, 321, 486, 334], "score": 1.0, "content": " is a quad r atic unramified", "type": "text"}], "index": 17}, {"bbox": [126, 333, 486, 349], "spans": [{"bbox": [126, 333, 180, 349], "score": 1.0, "content": "extension of", "type": "text"}, {"bbox": [180, 334, 194, 345], "score": 0.9, "content": "\\widetilde{K}_{1}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [194, 333, 219, 349], "score": 1.0, "content": ". Put ", "type": "text"}, {"bbox": [219, 334, 277, 347], "score": 0.93, "content": "\\widetilde{F}_{1}=k_{1}(\\sqrt{\\pi}\\,)", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [278, 333, 486, 349], "score": 1.0, "content": " a n d apply the ambiguous class number formula", "type": "text"}], "index": 18}, {"bbox": [125, 347, 486, 361], "spans": [{"bbox": [125, 347, 138, 361], "score": 1.0, "content": "to", "type": "text"}, {"bbox": [138, 347, 164, 360], "score": 0.95, "content": "\\widetilde{F}_{1}/k_{1}", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [165, 347, 188, 361], "score": 1.0, "content": " an d ", "type": "text"}, {"bbox": [188, 347, 216, 360], "score": 0.94, "content": "\\widetilde{L}_{1}/\\widetilde{F}_{1}", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [216, 347, 486, 361], "score": 1.0, "content": ": since there is only one ramified prime in each of these two", "type": "text"}], "index": 19}, {"bbox": [125, 360, 486, 373], "spans": [{"bbox": [125, 361, 214, 373], "score": 1.0, "content": "ext e nsions, w e fin d ", "type": "text"}, {"bbox": [215, 360, 353, 373], "score": 0.94, "content": "\\mathrm{Am}(\\widetilde{F}_{1}/k_{1})\\,=\\,\\mathrm{Am}(\\widetilde{L}_{1}/\\widetilde{F}_{1})\\,=\\,1", "type": "inline_equation", "height": 13, "width": 138}, {"bbox": [354, 361, 486, 373], "score": 1.0, "content": "; note that we have used the", "type": "text"}], "index": 20}, {"bbox": [126, 374, 415, 386], "spans": [{"bbox": [126, 374, 200, 386], "score": 1.0, "content": "assumption that ", "type": "text"}, {"bbox": [200, 375, 251, 386], "score": 0.94, "content": "[\\pi/\\rho^{\\prime}]=-1", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [251, 374, 329, 386], "score": 1.0, "content": " in deducing th a t ", "type": "text"}, {"bbox": [329, 375, 336, 384], "score": 0.88, "content": "{\\mathfrak{r}}^{\\prime}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [336, 374, 384, 386], "score": 1.0, "content": " is inert in", "type": "text"}, {"bbox": [385, 374, 410, 386], "score": 0.94, "content": "\\widetilde{F}_{1}/k_{1}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [411, 374, 415, 386], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 18.5}, {"type": "table", "bbox": [189, 412, 422, 468], "blocks": [{"type": "table_caption", "bbox": [125, 385, 486, 408], "group_id": 0, "lines": [{"bbox": [137, 386, 485, 399], "spans": [{"bbox": [137, 386, 485, 399], "score": 1.0, "content": "In our proof of Theorem 1 we have seen that there are th e following possibilities", "type": "text"}], "index": 22}, {"bbox": [126, 398, 200, 411], "spans": [{"bbox": [126, 398, 151, 411], "score": 1.0, "content": "when", "type": "text"}, {"bbox": [152, 399, 183, 410], "score": 0.76, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [183, 398, 200, 411], "score": 1.0, "content": " \| 4:", "type": "text"}], "index": 23}], "index": 22.5}, {"type": "table_body", "bbox": [189, 412, 422, 468], "group_id": 0, "lines": [{"bbox": [189, 412, 422, 468], "spans": [{"bbox": [189, 412, 422, 468], "score": 0.975, "html": "<html><body><table><tr><td>q2</td><td>Cl2(K2)</td><td>q1</td><td>h2(K1)</td><td>q2</td><td>Cl2(K2)</td><td>h2(L)</td></tr><tr><td>2</td><td>(2)</td><td>1</td><td>2</td><td>2</td><td>(2)</td><td>2m+1</td></tr><tr><td>2</td><td>(4)</td><td>1</td><td>4</td><td>\uff1f</td><td>(2,2)</td><td>2m+3</td></tr></table></body></html>", "type": "table", "image_path": "dfdcf3e7e1dbc3d44e3061767f642a3e9778b62e79babc37d43af68dae85a584.jpg"}]}], "index": 26, "virtual_lines": [{"bbox": [189, 412, 422, 425], "spans": [], "index": 24}, {"bbox": [189, 425, 422, 438], "spans": [], "index": 25}, {"bbox": [189, 438, 422, 451], "spans": [], "index": 26}, {"bbox": [189, 451, 422, 464], "spans": [], "index": 27}, {"bbox": [189, 464, 422, 477], "spans": [], "index": 28}]}], "index": 24.25}, {"type": "text", "bbox": [125, 470, 486, 520], "lines": [{"bbox": [124, 471, 486, 486], "spans": [{"bbox": [124, 471, 245, 486], "score": 1.0, "content": "In order to decide whether", "type": "text"}, {"bbox": [246, 474, 274, 484], "score": 0.93, "content": "\\widetilde{q}_{2}=1", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [274, 471, 289, 486], "score": 1.0, "content": " or", "type": "text"}, {"bbox": [289, 474, 317, 484], "score": 0.93, "content": "\\widetilde{q}_{2}=2", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [318, 471, 410, 486], "score": 1.0, "content": ", recall that we have ", "type": "text"}, {"bbox": [410, 474, 460, 484], "score": 0.93, "content": "h_{2}(K_{1})=4", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [460, 471, 486, 486], "score": 1.0, "content": "; thus", "type": "text"}], "index": 29}, {"bbox": [126, 483, 487, 498], "spans": [{"bbox": [126, 485, 139, 496], "score": 0.92, "content": "\\widetilde{K}_{1}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [139, 483, 401, 498], "score": 1.0, "content": " must be the field with 2 -class nu mber 2, and this implies ", "type": "text"}, {"bbox": [402, 484, 465, 497], "score": 0.92, "content": "h_{2}(\\widetilde{L})\\,=\\,2^{m+2}", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [465, 483, 487, 498], "score": 1.0, "content": " and", "type": "text"}], "index": 30}, {"bbox": [126, 497, 487, 511], "spans": [{"bbox": [126, 499, 155, 509], "score": 0.92, "content": "\\widetilde{q}_{2}=1", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [156, 497, 278, 511], "score": 1.0, "content": ". In particular we see that ", "type": "text"}, {"bbox": [278, 499, 325, 509], "score": 0.91, "content": "4\\mid h_{2}(K_{2})", "type": "inline_equation", "height": 10, "width": 47}, {"bbox": [325, 497, 392, 511], "score": 1.0, "content": " if and only if ", "type": "text"}, {"bbox": [392, 499, 437, 509], "score": 0.86, "content": "4\\mid h_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [437, 497, 487, 511], "score": 1.0, "content": " as long as", "type": "text"}], "index": 31}, {"bbox": [126, 510, 268, 522], "spans": [{"bbox": [126, 511, 191, 522], "score": 0.94, "content": "K_{1}=k_{1}(\\sqrt{\\pi\\rho}\\,)", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [191, 510, 216, 522], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [217, 511, 265, 521], "score": 0.94, "content": "[\\pi/\\rho]=+1", "type": "inline_equation", "height": 10, "width": 48}, {"bbox": [265, 510, 268, 522], "score": 1.0, "content": ".", "type": "text"}], "index": 32}], "index": 30.5}, {"type": "text", "bbox": [124, 520, 486, 651], "lines": [{"bbox": [137, 521, 487, 534], "spans": [{"bbox": [137, 521, 365, 534], "score": 1.0, "content": "The ambiguous class number formula shows that ", "type": "text"}, {"bbox": [365, 523, 400, 533], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [400, 521, 487, 534], "score": 1.0, "content": " is cyclic, thus 4 \|", "type": "text"}], "index": 33}, {"bbox": [126, 534, 486, 545], "spans": [{"bbox": [126, 535, 157, 545], "score": 0.94, "content": "h_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [157, 534, 221, 545], "score": 1.0, "content": " if and only if ", "type": "text"}, {"bbox": [221, 535, 265, 545], "score": 0.93, "content": "2\\mid h_{2}(L_{1})", "type": "inline_equation", "height": 10, "width": 44}, {"bbox": [265, 534, 299, 545], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [300, 534, 362, 545], "score": 0.94, "content": "L_{1}=K_{1}(\\sqrt{\\pi}\\,)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [362, 534, 486, 545], "score": 1.0, "content": " is the quadratic unramified", "type": "text"}], "index": 34}, {"bbox": [125, 545, 487, 558], "spans": [{"bbox": [125, 545, 182, 558], "score": 1.0, "content": "extension of ", "type": "text"}, {"bbox": [182, 547, 195, 556], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [196, 545, 425, 558], "score": 1.0, "content": ". Applying the ambiguous class number formula to ", "type": "text"}, {"bbox": [425, 547, 452, 557], "score": 0.95, "content": "L_{1}/F_{1}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [453, 545, 487, 558], "score": 1.0, "content": ", where", "type": "text"}], "index": 35}, {"bbox": [126, 558, 487, 570], "spans": [{"bbox": [126, 559, 188, 569], "score": 0.95, "content": "F_{1}\\;=\\;k_{1}(\\sqrt{\\pi}\\,)", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [189, 558, 254, 570], "score": 1.0, "content": ", we see that ", "type": "text"}, {"bbox": [254, 559, 300, 569], "score": 0.89, "content": "2\\:\\:\|\\:\\:h_{2}(L_{1})", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [300, 558, 369, 570], "score": 1.0, "content": " if and only if ", "type": "text"}, {"bbox": [369, 559, 430, 569], "score": 0.92, "content": "(E\\,:\\,H)\\;=\\;1", "type": "inline_equation", "height": 10, "width": 61}, {"bbox": [430, 558, 465, 570], "score": 1.0, "content": ". Now ", "type": "text"}, {"bbox": [465, 559, 474, 567], "score": 0.9, "content": "E", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [474, 558, 487, 570], "score": 1.0, "content": " is", "type": "text"}], "index": 36}, {"bbox": [125, 569, 487, 582], "spans": [{"bbox": [125, 569, 487, 582], "score": 1.0, "content": "generated by a root of unity (which always is a norm residue at primes dividing", "type": "text"}], "index": 37}, {"bbox": [126, 582, 485, 593], "spans": [{"bbox": [126, 583, 184, 591], "score": 0.8, "content": "r\\,\\equiv\\,1\\;\\mathrm{mod}\\;4", "type": "inline_equation", "height": 8, "width": 58}, {"bbox": [185, 582, 304, 593], "score": 1.0, "content": ") and a fundamental unit ", "type": "text"}, {"bbox": [304, 586, 310, 591], "score": 0.89, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [310, 582, 368, 593], "score": 1.0, "content": ". Therefore ", "type": "text"}, {"bbox": [368, 583, 430, 593], "score": 0.91, "content": "(E\\,:\\,H)\\;=\\;1", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [430, 582, 485, 593], "score": 1.0, "content": " if and only", "type": "text"}], "index": 38}, {"bbox": [125, 594, 487, 606], "spans": [{"bbox": [125, 594, 136, 606], "score": 1.0, "content": "if ", "type": "text"}, {"bbox": [136, 595, 247, 605], "score": 0.92, "content": "\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\,=\\,\\{\\varepsilon/\\mathfrak{R}_{2}\\}\\,=\\,+1", "type": "inline_equation", "height": 10, "width": 111}, {"bbox": [248, 594, 284, 606], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [284, 595, 348, 605], "score": 0.91, "content": "\\mathfrak{r}{\\mathcal{O}}_{F_{1}}\\,=\\,\\mathfrak{R}_{1}\\mathfrak{R}_{2}", "type": "inline_equation", "height": 10, "width": 64}, {"bbox": [348, 594, 402, 606], "score": 1.0, "content": " and where ", "type": "text"}, {"bbox": [402, 595, 430, 605], "score": 0.93, "content": "\\{\\,\\cdot\\,/\\,\\cdot\\,\\}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [430, 594, 487, 606], "score": 1.0, "content": " denotes the", "type": "text"}], "index": 39}, {"bbox": [125, 606, 487, 617], "spans": [{"bbox": [125, 606, 247, 617], "score": 1.0, "content": "quadratic residue symbol in ", "type": "text"}, {"bbox": [248, 607, 258, 616], "score": 0.91, "content": "F_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [259, 606, 291, 617], "score": 1.0, "content": ". Since", "type": "text"}, {"bbox": [291, 606, 414, 617], "score": 0.92, "content": "\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\{\\varepsilon/\\mathfrak{R}_{2}\\}=[\\varepsilon/\\mathfrak{r}]=+1", "type": "inline_equation", "height": 11, "width": 123}, {"bbox": [415, 606, 487, 617], "score": 1.0, "content": ", we have proved", "type": "text"}], "index": 40}, {"bbox": [126, 617, 487, 630], "spans": [{"bbox": [126, 617, 147, 630], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 618, 195, 629], "score": 0.91, "content": "4\\mid h_{2}(K_{1})", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [195, 617, 332, 630], "score": 1.0, "content": " if and only if the prime ideal ", "type": "text"}, {"bbox": [333, 619, 345, 628], "score": 0.91, "content": "\\Re_{1}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [346, 617, 378, 630], "score": 1.0, "content": " above ", "type": "text"}, {"bbox": [378, 621, 383, 626], "score": 0.65, "content": "\\mathfrak{r}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [383, 617, 487, 630], "score": 1.0, "content": " splits in the quadratic", "type": "text"}], "index": 41}, {"bbox": [126, 630, 486, 641], "spans": [{"bbox": [126, 630, 169, 641], "score": 1.0, "content": "extension ", "type": "text"}, {"bbox": [170, 630, 203, 641], "score": 0.93, "content": "F_{1}(\\sqrt{\\varepsilon})", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [203, 630, 266, 641], "score": 1.0, "content": ". But if we fix ", "type": "text"}, {"bbox": [267, 633, 272, 640], "score": 0.88, "content": "p", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [272, 630, 293, 641], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [294, 633, 299, 640], "score": 0.88, "content": "q", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [299, 630, 486, 641], "score": 1.0, "content": ", this happens for exactly half of the values", "type": "text"}], "index": 42}, {"bbox": [125, 641, 297, 654], "spans": [{"bbox": [125, 641, 137, 654], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [137, 646, 142, 650], "score": 0.89, "content": "r", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [142, 641, 190, 654], "score": 1.0, "content": " satisfying ", "type": "text"}, {"bbox": [190, 642, 239, 653], "score": 0.92, "content": "(p/r)=-1", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [239, 641, 244, 654], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [244, 642, 293, 653], "score": 0.91, "content": "(q/r)=+1", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [293, 641, 297, 654], "score": 1.0, "content": ".", "type": "text"}], "index": 43}], "index": 38}, {"type": "text", "bbox": [125, 651, 486, 699], "lines": [{"bbox": [136, 652, 487, 666], "spans": [{"bbox": [136, 652, 147, 666], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [148, 655, 176, 664], "score": 0.93, "content": "d_{2}=8", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [176, 652, 198, 666], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [198, 655, 222, 664], "score": 0.9, "content": "p=2", "type": "inline_equation", "height": 9, "width": 24}, {"bbox": [222, 652, 250, 666], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [250, 654, 298, 665], "score": 0.92, "content": "2\\mathcal{O}_{k_{1}}=22^{\\prime}", "type": "inline_equation", "height": 11, "width": 48}, {"bbox": [299, 652, 405, 666], "score": 1.0, "content": ", and we have to choose ", "type": "text"}, {"bbox": [405, 653, 442, 665], "score": 0.94, "content": "2^{h}=(\\pi)", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [443, 652, 487, 666], "score": 1.0, "content": " in such a", "type": "text"}], "index": 44}, {"bbox": [127, 665, 486, 677], "spans": [{"bbox": [127, 665, 166, 677], "score": 1.0, "content": "way that ", "type": "text"}, {"bbox": [167, 666, 214, 677], "score": 0.94, "content": "k_{1}(\\sqrt{\\pi}\\,)/k_{1}", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [215, 665, 309, 677], "score": 1.0, "content": " is unramified outside", "type": "text"}, {"bbox": [310, 669, 315, 676], "score": 0.74, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [316, 665, 412, 677], "score": 1.0, "content": ". The residue symbols", "type": "text"}, {"bbox": [413, 666, 435, 677], "score": 0.92, "content": "\\left[\\alpha/2\\right]", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [435, 665, 486, 677], "score": 1.0, "content": " are defined", "type": "text"}], "index": 45}, {"bbox": [125, 677, 486, 689], "spans": [{"bbox": [125, 677, 307, 689], "score": 1.0, "content": "as Kronecker symbols via the splitting of ", "type": "text"}, {"bbox": [308, 681, 313, 686], "score": 0.41, "content": "^{2}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [313, 677, 433, 689], "score": 1.0, "content": " in the quadratic extension ", "type": "text"}, {"bbox": [433, 678, 482, 689], "score": 0.93, "content": "k_{1}(\\sqrt{\\alpha}\\,)/k_{1}", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [482, 677, 486, 689], "score": 1.0, "content": ".", "type": "text"}], "index": 46}, {"bbox": [126, 689, 390, 702], "spans": [{"bbox": [126, 689, 390, 702], "score": 1.0, "content": "With these modifactions, the above arguments remain valid.", "type": "text"}], "index": 47}], "index": 45.5}], "layout_bboxes": [], "page_idx": 12, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [{"type": "table", "bbox": [189, 412, 422, 468], "blocks": [{"type": "table_caption", "bbox": [125, 385, 486, 408], "group_id": 0, "lines": [{"bbox": [137, 386, 485, 399], "spans": [{"bbox": [137, 386, 485, 399], "score": 1.0, "content": "In our proof of Theorem 1 we have seen that there are th e following possibilities", "type": "text"}], "index": 22}, {"bbox": [126, 398, 200, 411], "spans": [{"bbox": [126, 398, 151, 411], "score": 1.0, "content": "when", "type": "text"}, {"bbox": [152, 399, 183, 410], "score": 0.76, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [183, 398, 200, 411], "score": 1.0, "content": " \| 4:", "type": "text"}], "index": 23}], "index": 22.5}, {"type": "table_body", "bbox": [189, 412, 422, 468], "group_id": 0, "lines": [{"bbox": [189, 412, 422, 468], "spans": [{"bbox": [189, 412, 422, 468], "score": 0.975, "html": "<html><body><table><tr><td>q2</td><td>Cl2(K2)</td><td>q1</td><td>h2(K1)</td><td>q2</td><td>Cl2(K2)</td><td>h2(L)</td></tr><tr><td>2</td><td>(2)</td><td>1</td><td>2</td><td>2</td><td>(2)</td><td>2m+1</td></tr><tr><td>2</td><td>(4)</td><td>1</td><td>4</td><td>\uff1f</td><td>(2,2)</td><td>2m+3</td></tr></table></body></html>", "type": "table", "image_path": "dfdcf3e7e1dbc3d44e3061767f642a3e9778b62e79babc37d43af68dae85a584.jpg"}]}], "index": 26, "virtual_lines": [{"bbox": [189, 412, 422, 425], "spans": [], "index": 24}, {"bbox": [189, 425, 422, 438], "spans": [], "index": 25}, {"bbox": [189, 438, 422, 451], "spans": [], "index": 26}, {"bbox": [189, 451, 422, 464], "spans": [], "index": 27}, {"bbox": [189, 464, 422, 477], "spans": [], "index": 28}]}], "index": 24.25}], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [238, 90, 373, 99], "lines": [{"bbox": [239, 93, 372, 100], "spans": [{"bbox": [239, 93, 372, 100], "score": 1.0, "content": "IMAGINARY QUADRATIC FIELDS", "type": "text"}]}]}, {"type": "discarded", "bbox": [476, 91, 486, 99], "lines": [{"bbox": [476, 92, 487, 101], "spans": [{"bbox": [476, 92, 487, 101], "score": 1.0, "content": "13", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 112, 486, 148], "lines": [{"bbox": [137, 114, 486, 127], "spans": [{"bbox": [137, 114, 486, 127], "score": 1.0, "content": "The referee (whom we\u2019d like to thank for a couple of helpful remarks) asked", "type": "text"}], "index": 0}, {"bbox": [126, 126, 485, 138], "spans": [{"bbox": [126, 126, 164, 138], "score": 1.0, "content": "whether ", "type": "text"}, {"bbox": [164, 127, 209, 138], "score": 0.94, "content": "h_{2}(K)=2", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [210, 126, 232, 138], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [232, 127, 278, 138], "score": 0.94, "content": "h_{2}(K)>2", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [278, 126, 485, 138], "score": 1.0, "content": " infinitely often. Let us show how to prove that", "type": "text"}], "index": 1}, {"bbox": [126, 138, 313, 150], "spans": [{"bbox": [126, 138, 313, 150], "score": 1.0, "content": "both possibilities occur with equal density.", "type": "text"}], "index": 2}], "index": 1, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [126, 114, 486, 150]}, {"type": "text", "bbox": [124, 149, 486, 304], "lines": [{"bbox": [137, 150, 484, 163], "spans": [{"bbox": [137, 150, 434, 163], "score": 1.0, "content": "Before we can do this, we have to study the quadratic extensions ", "type": "text"}, {"bbox": [434, 153, 447, 161], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [448, 150, 471, 163], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [471, 150, 484, 161], "score": 0.92, "content": "\\widetilde{K}_{1}", "type": "inline_equation", "height": 11, "width": 13}], "index": 3}, {"bbox": [125, 161, 487, 176], "spans": [{"bbox": [125, 161, 138, 176], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [138, 164, 148, 173], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [149, 161, 293, 176], "score": 1.0, "content": " more closely. We assume that ", "type": "text"}, {"bbox": [294, 164, 325, 174], "score": 0.93, "content": "d_{2}\\,=\\,p", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [326, 161, 350, 176], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [350, 164, 382, 173], "score": 0.93, "content": "d_{3}~=~r", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [382, 161, 487, 176], "score": 1.0, "content": " are odd primes in t he", "type": "text"}], "index": 4}, {"bbox": [125, 174, 487, 187], "spans": [{"bbox": [125, 174, 404, 187], "score": 1.0, "content": "following, and then say how to modify the arguments in the case ", "type": "text"}, {"bbox": [404, 176, 432, 185], "score": 0.92, "content": "d_{2}=8", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [433, 174, 446, 187], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [446, 177, 482, 185], "score": 0.93, "content": "d_{3}=-8", "type": "inline_equation", "height": 8, "width": 36}, {"bbox": [483, 174, 487, 187], "score": 1.0, "content": ".", "type": "text"}], "index": 5}, {"bbox": [126, 187, 487, 199], "spans": [{"bbox": [126, 187, 180, 199], "score": 1.0, "content": "The primes ", "type": "text"}, {"bbox": [181, 191, 186, 198], "score": 0.9, "content": "p", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [186, 187, 210, 199], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [210, 191, 215, 196], "score": 0.88, "content": "r", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [216, 187, 255, 199], "score": 1.0, "content": " split in ", "type": "text"}, {"bbox": [255, 189, 265, 197], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [265, 187, 282, 199], "score": 1.0, "content": " as ", "type": "text"}, {"bbox": [282, 188, 330, 198], "score": 0.92, "content": "p\\mathcal{O}_{1}\\,=\\,\\mathfrak{p p}^{\\prime}", "type": "inline_equation", "height": 10, "width": 48}, {"bbox": [330, 187, 353, 199], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [354, 188, 399, 197], "score": 0.92, "content": "r\\mathcal{O}_{1}\\,=\\,\\mathfrak{r r}^{\\prime}", "type": "inline_equation", "height": 9, "width": 45}, {"bbox": [399, 187, 427, 199], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [427, 189, 433, 196], "score": 0.87, "content": "h", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [434, 187, 487, 199], "score": 1.0, "content": " denote the", "type": "text"}], "index": 6}, {"bbox": [125, 198, 487, 211], "spans": [{"bbox": [125, 198, 221, 211], "score": 1.0, "content": "odd class number of ", "type": "text"}, {"bbox": [221, 200, 231, 209], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [231, 198, 282, 211], "score": 1.0, "content": " and write ", "type": "text"}, {"bbox": [282, 199, 323, 210], "score": 0.92, "content": "{\\mathfrak{p}}^{h}\\,=\\,(\\pi)", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [324, 198, 348, 211], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [348, 199, 387, 210], "score": 0.92, "content": "\\mathfrak{r}^{h}\\,=\\,(\\rho)", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [388, 198, 487, 211], "score": 1.0, "content": " for primary elements", "type": "text"}], "index": 7}, {"bbox": [126, 210, 487, 223], "spans": [{"bbox": [126, 215, 132, 219], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 4, "width": 6}, {"bbox": [132, 210, 156, 223], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [157, 215, 162, 222], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [163, 210, 487, 223], "score": 1.0, "content": " (this is can easily be proved directly, but it is also a very special case", "type": "text"}], "index": 8}, {"bbox": [125, 223, 486, 235], "spans": [{"bbox": [125, 223, 432, 235], "score": 1.0, "content": "of Hilbert\u2019s first supplementary law for quadratic reciprocity in fields ", "type": "text"}, {"bbox": [433, 225, 442, 232], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [443, 223, 486, 235], "score": 1.0, "content": " with odd", "type": "text"}], "index": 9}, {"bbox": [125, 234, 485, 247], "spans": [{"bbox": [125, 234, 188, 247], "score": 1.0, "content": "class number ", "type": "text"}, {"bbox": [188, 236, 194, 244], "score": 0.88, "content": "h", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [195, 234, 255, 247], "score": 1.0, "content": " (see [7]): if ", "type": "text"}, {"bbox": [256, 235, 306, 245], "score": 0.92, "content": "{\\mathfrak{a}}^{h}\\;=\\;\\alpha{\\mathcal{O}}_{K}", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [307, 234, 367, 247], "score": 1.0, "content": " for an ideal ", "type": "text"}, {"bbox": [368, 236, 374, 246], "score": 0.75, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [374, 234, 478, 247], "score": 1.0, "content": " with odd norm, then ", "type": "text"}, {"bbox": [478, 239, 485, 244], "score": 0.87, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 7}], "index": 10}, {"bbox": [126, 247, 487, 259], "spans": [{"bbox": [126, 247, 378, 259], "score": 1.0, "content": "can be chosen primary (i.e. congruent to a square mod ", "type": "text"}, {"bbox": [378, 248, 398, 257], "score": 0.89, "content": "4{\\cal O}_{K}", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [399, 247, 469, 259], "score": 1.0, "content": ") if and only if ", "type": "text"}, {"bbox": [469, 250, 474, 255], "score": 0.84, "content": "\\mathfrak{a}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [474, 247, 487, 259], "score": 1.0, "content": " is", "type": "text"}], "index": 11}, {"bbox": [125, 258, 487, 272], "spans": [{"bbox": [125, 258, 186, 272], "score": 1.0, "content": "primary (i.e. ", "type": "text"}, {"bbox": [187, 259, 234, 270], "score": 0.93, "content": "[\\varepsilon/{\\mathfrak a}]=+1", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [235, 258, 293, 272], "score": 1.0, "content": " for all units ", "type": "text"}, {"bbox": [293, 259, 327, 271], "score": 0.93, "content": "\\varepsilon\\in{\\mathcal{O}}_{K}^{\\times}", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [327, 258, 362, 272], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [363, 259, 386, 270], "score": 0.95, "content": "[\\,\\cdot\\,/\\,\\cdot\\,]", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [386, 258, 487, 272], "score": 1.0, "content": " denotes the quadratic", "type": "text"}], "index": 12}, {"bbox": [126, 271, 486, 283], "spans": [{"bbox": [126, 271, 205, 283], "score": 1.0, "content": "residue symbol in ", "type": "text"}, {"bbox": [205, 272, 215, 280], "score": 0.86, "content": "K", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [216, 271, 246, 283], "score": 1.0, "content": ")). Let ", "type": "text"}, {"bbox": [247, 272, 270, 282], "score": 0.94, "content": "\\big[\\cdot\\big/\\cdot\\big]", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [270, 271, 445, 283], "score": 1.0, "content": " denote the quadratic residue symbol in ", "type": "text"}, {"bbox": [445, 272, 455, 281], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [455, 271, 486, 283], "score": 1.0, "content": ". Then", "type": "text"}], "index": 13}, {"bbox": [126, 282, 485, 295], "spans": [{"bbox": [126, 284, 271, 294], "score": 0.9, "content": "[\\pi/\\rho][\\pi^{\\prime}/\\rho]=[p/\\rho]=(p/r)=-1", "type": "inline_equation", "height": 10, "width": 145}, {"bbox": [271, 282, 485, 295], "score": 1.0, "content": ", so we may choose the conjugates in such a way", "type": "text"}], "index": 14}, {"bbox": [125, 294, 310, 307], "spans": [{"bbox": [125, 294, 147, 307], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [147, 295, 195, 306], "score": 0.93, "content": "[\\pi/\\rho]=+1", "type": "inline_equation", "height": 11, "width": 48}, {"bbox": [196, 294, 218, 307], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [218, 295, 307, 306], "score": 0.91, "content": "[\\pi^{\\prime}/\\rho]=[\\pi/\\rho^{\\prime}]=-1", "type": "inline_equation", "height": 11, "width": 89}, {"bbox": [307, 294, 310, 307], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 9, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [125, 150, 487, 307]}, {"type": "text", "bbox": [124, 306, 487, 384], "lines": [{"bbox": [136, 307, 486, 320], "spans": [{"bbox": [136, 307, 158, 320], "score": 1.0, "content": "Put ", "type": "text"}, {"bbox": [159, 309, 228, 320], "score": 0.93, "content": "K_{1}\\;=\\;k_{1}(\\sqrt{\\pi\\rho}\\,)", "type": "inline_equation", "height": 11, "width": 69}, {"bbox": [229, 307, 253, 320], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [253, 307, 326, 319], "score": 0.91, "content": "\\tilde{K}_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\rho^{\\prime}})", "type": "inline_equation", "height": 12, "width": 73}, {"bbox": [326, 307, 400, 320], "score": 1.0, "content": "; we claim that ", "type": "text"}, {"bbox": [400, 307, 454, 319], "score": 0.94, "content": "h_{2}(\\tilde{K}_{1})\\;=\\;2", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [455, 307, 486, 320], "score": 1.0, "content": ". This", "type": "text"}], "index": 16}, {"bbox": [125, 321, 486, 334], "spans": [{"bbox": [125, 321, 198, 334], "score": 1.0, "content": "is equivalent to ", "type": "text"}, {"bbox": [198, 321, 249, 333], "score": 0.92, "content": "h_{2}(\\widetilde{L}_{1})\\,=\\,1", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [250, 321, 286, 334], "score": 1.0, "content": ", where", "type": "text"}, {"bbox": [286, 321, 368, 333], "score": 0.93, "content": "\\tilde{L}_{1}\\,=\\,k_{1}(\\sqrt{\\pi},\\sqrt{\\rho^{\\prime}})", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [369, 321, 486, 334], "score": 1.0, "content": " is a quad r atic unramified", "type": "text"}], "index": 17}, {"bbox": [126, 333, 486, 349], "spans": [{"bbox": [126, 333, 180, 349], "score": 1.0, "content": "extension of", "type": "text"}, {"bbox": [180, 334, 194, 345], "score": 0.9, "content": "\\widetilde{K}_{1}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [194, 333, 219, 349], "score": 1.0, "content": ". Put ", "type": "text"}, {"bbox": [219, 334, 277, 347], "score": 0.93, "content": "\\widetilde{F}_{1}=k_{1}(\\sqrt{\\pi}\\,)", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [278, 333, 486, 349], "score": 1.0, "content": " a n d apply the ambiguous class number formula", "type": "text"}], "index": 18}, {"bbox": [125, 347, 486, 361], "spans": [{"bbox": [125, 347, 138, 361], "score": 1.0, "content": "to", "type": "text"}, {"bbox": [138, 347, 164, 360], "score": 0.95, "content": "\\widetilde{F}_{1}/k_{1}", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [165, 347, 188, 361], "score": 1.0, "content": " an d ", "type": "text"}, {"bbox": [188, 347, 216, 360], "score": 0.94, "content": "\\widetilde{L}_{1}/\\widetilde{F}_{1}", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [216, 347, 486, 361], "score": 1.0, "content": ": since there is only one ramified prime in each of these two", "type": "text"}], "index": 19}, {"bbox": [125, 360, 486, 373], "spans": [{"bbox": [125, 361, 214, 373], "score": 1.0, "content": "ext e nsions, w e fin d ", "type": "text"}, {"bbox": [215, 360, 353, 373], "score": 0.94, "content": "\\mathrm{Am}(\\widetilde{F}_{1}/k_{1})\\,=\\,\\mathrm{Am}(\\widetilde{L}_{1}/\\widetilde{F}_{1})\\,=\\,1", "type": "inline_equation", "height": 13, "width": 138}, {"bbox": [354, 361, 486, 373], "score": 1.0, "content": "; note that we have used the", "type": "text"}], "index": 20}, {"bbox": [126, 374, 415, 386], "spans": [{"bbox": [126, 374, 200, 386], "score": 1.0, "content": "assumption that ", "type": "text"}, {"bbox": [200, 375, 251, 386], "score": 0.94, "content": "[\\pi/\\rho^{\\prime}]=-1", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [251, 374, 329, 386], "score": 1.0, "content": " in deducing th a t ", "type": "text"}, {"bbox": [329, 375, 336, 384], "score": 0.88, "content": "{\\mathfrak{r}}^{\\prime}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [336, 374, 384, 386], "score": 1.0, "content": " is inert in", "type": "text"}, {"bbox": [385, 374, 410, 386], "score": 0.94, "content": "\\widetilde{F}_{1}/k_{1}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [411, 374, 415, 386], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 18.5, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [125, 307, 486, 386]}, {"type": "table", "bbox": [189, 412, 422, 468], "blocks": [{"type": "table_caption", "bbox": [125, 385, 486, 408], "group_id": 0, "lines": [{"bbox": [137, 386, 485, 399], "spans": [{"bbox": [137, 386, 485, 399], "score": 1.0, "content": "In our proof of Theorem 1 we have seen that there are th e following possibilities", "type": "text"}], "index": 22}, {"bbox": [126, 398, 200, 411], "spans": [{"bbox": [126, 398, 151, 411], "score": 1.0, "content": "when", "type": "text"}, {"bbox": [152, 399, 183, 410], "score": 0.76, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [183, 398, 200, 411], "score": 1.0, "content": " \| 4:", "type": "text"}], "index": 23}], "index": 22.5}, {"type": "table_body", "bbox": [189, 412, 422, 468], "group_id": 0, "lines": [{"bbox": [189, 412, 422, 468], "spans": [{"bbox": [189, 412, 422, 468], "score": 0.975, "html": "<html><body><table><tr><td>q2</td><td>Cl2(K2)</td><td>q1</td><td>h2(K1)</td><td>q2</td><td>Cl2(K2)</td><td>h2(L)</td></tr><tr><td>2</td><td>(2)</td><td>1</td><td>2</td><td>2</td><td>(2)</td><td>2m+1</td></tr><tr><td>2</td><td>(4)</td><td>1</td><td>4</td><td>\uff1f</td><td>(2,2)</td><td>2m+3</td></tr></table></body></html>", "type": "table", "image_path": "dfdcf3e7e1dbc3d44e3061767f642a3e9778b62e79babc37d43af68dae85a584.jpg"}]}], "index": 26, "virtual_lines": [{"bbox": [189, 412, 422, 425], "spans": [], "index": 24}, {"bbox": [189, 425, 422, 438], "spans": [], "index": 25}, {"bbox": [189, 438, 422, 451], "spans": [], "index": 26}, {"bbox": [189, 451, 422, 464], "spans": [], "index": 27}, {"bbox": [189, 464, 422, 477], "spans": [], "index": 28}]}], "index": 24.25, "page_num": "page_12", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [125, 470, 486, 520], "lines": [{"bbox": [124, 471, 486, 486], "spans": [{"bbox": [124, 471, 245, 486], "score": 1.0, "content": "In order to decide whether", "type": "text"}, {"bbox": [246, 474, 274, 484], "score": 0.93, "content": "\\widetilde{q}_{2}=1", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [274, 471, 289, 486], "score": 1.0, "content": " or", "type": "text"}, {"bbox": [289, 474, 317, 484], "score": 0.93, "content": "\\widetilde{q}_{2}=2", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [318, 471, 410, 486], "score": 1.0, "content": ", recall that we have ", "type": "text"}, {"bbox": [410, 474, 460, 484], "score": 0.93, "content": "h_{2}(K_{1})=4", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [460, 471, 486, 486], "score": 1.0, "content": "; thus", "type": "text"}], "index": 29}, {"bbox": [126, 483, 487, 498], "spans": [{"bbox": [126, 485, 139, 496], "score": 0.92, "content": "\\widetilde{K}_{1}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [139, 483, 401, 498], "score": 1.0, "content": " must be the field with 2 -class nu mber 2, and this implies ", "type": "text"}, {"bbox": [402, 484, 465, 497], "score": 0.92, "content": "h_{2}(\\widetilde{L})\\,=\\,2^{m+2}", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [465, 483, 487, 498], "score": 1.0, "content": " and", "type": "text"}], "index": 30}, {"bbox": [126, 497, 487, 511], "spans": [{"bbox": [126, 499, 155, 509], "score": 0.92, "content": "\\widetilde{q}_{2}=1", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [156, 497, 278, 511], "score": 1.0, "content": ". In particular we see that ", "type": "text"}, {"bbox": [278, 499, 325, 509], "score": 0.91, "content": "4\\mid h_{2}(K_{2})", "type": "inline_equation", "height": 10, "width": 47}, {"bbox": [325, 497, 392, 511], "score": 1.0, "content": " if and only if ", "type": "text"}, {"bbox": [392, 499, 437, 509], "score": 0.86, "content": "4\\mid h_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [437, 497, 487, 511], "score": 1.0, "content": " as long as", "type": "text"}], "index": 31}, {"bbox": [126, 510, 268, 522], "spans": [{"bbox": [126, 511, 191, 522], "score": 0.94, "content": "K_{1}=k_{1}(\\sqrt{\\pi\\rho}\\,)", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [191, 510, 216, 522], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [217, 511, 265, 521], "score": 0.94, "content": "[\\pi/\\rho]=+1", "type": "inline_equation", "height": 10, "width": 48}, {"bbox": [265, 510, 268, 522], "score": 1.0, "content": ".", "type": "text"}], "index": 32}], "index": 30.5, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [124, 471, 487, 522]}, {"type": "text", "bbox": [124, 520, 486, 651], "lines": [{"bbox": [137, 521, 487, 534], "spans": [{"bbox": [137, 521, 365, 534], "score": 1.0, "content": "The ambiguous class number formula shows that ", "type": "text"}, {"bbox": [365, 523, 400, 533], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [400, 521, 487, 534], "score": 1.0, "content": " is cyclic, thus 4 \|", "type": "text"}], "index": 33}, {"bbox": [126, 534, 486, 545], "spans": [{"bbox": [126, 535, 157, 545], "score": 0.94, "content": "h_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [157, 534, 221, 545], "score": 1.0, "content": " if and only if ", "type": "text"}, {"bbox": [221, 535, 265, 545], "score": 0.93, "content": "2\\mid h_{2}(L_{1})", "type": "inline_equation", "height": 10, "width": 44}, {"bbox": [265, 534, 299, 545], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [300, 534, 362, 545], "score": 0.94, "content": "L_{1}=K_{1}(\\sqrt{\\pi}\\,)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [362, 534, 486, 545], "score": 1.0, "content": " is the quadratic unramified", "type": "text"}], "index": 34}, {"bbox": [125, 545, 487, 558], "spans": [{"bbox": [125, 545, 182, 558], "score": 1.0, "content": "extension of ", "type": "text"}, {"bbox": [182, 547, 195, 556], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [196, 545, 425, 558], "score": 1.0, "content": ". Applying the ambiguous class number formula to ", "type": "text"}, {"bbox": [425, 547, 452, 557], "score": 0.95, "content": "L_{1}/F_{1}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [453, 545, 487, 558], "score": 1.0, "content": ", where", "type": "text"}], "index": 35}, {"bbox": [126, 558, 487, 570], "spans": [{"bbox": [126, 559, 188, 569], "score": 0.95, "content": "F_{1}\\;=\\;k_{1}(\\sqrt{\\pi}\\,)", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [189, 558, 254, 570], "score": 1.0, "content": ", we see that ", "type": "text"}, {"bbox": [254, 559, 300, 569], "score": 0.89, "content": "2\\:\\:\|\\:\\:h_{2}(L_{1})", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [300, 558, 369, 570], "score": 1.0, "content": " if and only if ", "type": "text"}, {"bbox": [369, 559, 430, 569], "score": 0.92, "content": "(E\\,:\\,H)\\;=\\;1", "type": "inline_equation", "height": 10, "width": 61}, {"bbox": [430, 558, 465, 570], "score": 1.0, "content": ". Now ", "type": "text"}, {"bbox": [465, 559, 474, 567], "score": 0.9, "content": "E", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [474, 558, 487, 570], "score": 1.0, "content": " is", "type": "text"}], "index": 36}, {"bbox": [125, 569, 487, 582], "spans": [{"bbox": [125, 569, 487, 582], "score": 1.0, "content": "generated by a root of unity (which always is a norm residue at primes dividing", "type": "text"}], "index": 37}, {"bbox": [126, 582, 485, 593], "spans": [{"bbox": [126, 583, 184, 591], "score": 0.8, "content": "r\\,\\equiv\\,1\\;\\mathrm{mod}\\;4", "type": "inline_equation", "height": 8, "width": 58}, {"bbox": [185, 582, 304, 593], "score": 1.0, "content": ") and a fundamental unit ", "type": "text"}, {"bbox": [304, 586, 310, 591], "score": 0.89, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [310, 582, 368, 593], "score": 1.0, "content": ". Therefore ", "type": "text"}, {"bbox": [368, 583, 430, 593], "score": 0.91, "content": "(E\\,:\\,H)\\;=\\;1", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [430, 582, 485, 593], "score": 1.0, "content": " if and only", "type": "text"}], "index": 38}, {"bbox": [125, 594, 487, 606], "spans": [{"bbox": [125, 594, 136, 606], "score": 1.0, "content": "if ", "type": "text"}, {"bbox": [136, 595, 247, 605], "score": 0.92, "content": "\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\,=\\,\\{\\varepsilon/\\mathfrak{R}_{2}\\}\\,=\\,+1", "type": "inline_equation", "height": 10, "width": 111}, {"bbox": [248, 594, 284, 606], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [284, 595, 348, 605], "score": 0.91, "content": "\\mathfrak{r}{\\mathcal{O}}_{F_{1}}\\,=\\,\\mathfrak{R}_{1}\\mathfrak{R}_{2}", "type": "inline_equation", "height": 10, "width": 64}, {"bbox": [348, 594, 402, 606], "score": 1.0, "content": " and where ", "type": "text"}, {"bbox": [402, 595, 430, 605], "score": 0.93, "content": "\\{\\,\\cdot\\,/\\,\\cdot\\,\\}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [430, 594, 487, 606], "score": 1.0, "content": " denotes the", "type": "text"}], "index": 39}, {"bbox": [125, 606, 487, 617], "spans": [{"bbox": [125, 606, 247, 617], "score": 1.0, "content": "quadratic residue symbol in ", "type": "text"}, {"bbox": [248, 607, 258, 616], "score": 0.91, "content": "F_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [259, 606, 291, 617], "score": 1.0, "content": ". Since", "type": "text"}, {"bbox": [291, 606, 414, 617], "score": 0.92, "content": "\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\{\\varepsilon/\\mathfrak{R}_{2}\\}=[\\varepsilon/\\mathfrak{r}]=+1", "type": "inline_equation", "height": 11, "width": 123}, {"bbox": [415, 606, 487, 617], "score": 1.0, "content": ", we have proved", "type": "text"}], "index": 40}, {"bbox": [126, 617, 487, 630], "spans": [{"bbox": [126, 617, 147, 630], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 618, 195, 629], "score": 0.91, "content": "4\\mid h_{2}(K_{1})", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [195, 617, 332, 630], "score": 1.0, "content": " if and only if the prime ideal ", "type": "text"}, {"bbox": [333, 619, 345, 628], "score": 0.91, "content": "\\Re_{1}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [346, 617, 378, 630], "score": 1.0, "content": " above ", "type": "text"}, {"bbox": [378, 621, 383, 626], "score": 0.65, "content": "\\mathfrak{r}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [383, 617, 487, 630], "score": 1.0, "content": " splits in the quadratic", "type": "text"}], "index": 41}, {"bbox": [126, 630, 486, 641], "spans": [{"bbox": [126, 630, 169, 641], "score": 1.0, "content": "extension ", "type": "text"}, {"bbox": [170, 630, 203, 641], "score": 0.93, "content": "F_{1}(\\sqrt{\\varepsilon})", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [203, 630, 266, 641], "score": 1.0, "content": ". But if we fix ", "type": "text"}, {"bbox": [267, 633, 272, 640], "score": 0.88, "content": "p", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [272, 630, 293, 641], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [294, 633, 299, 640], "score": 0.88, "content": "q", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [299, 630, 486, 641], "score": 1.0, "content": ", this happens for exactly half of the values", "type": "text"}], "index": 42}, {"bbox": [125, 641, 297, 654], "spans": [{"bbox": [125, 641, 137, 654], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [137, 646, 142, 650], "score": 0.89, "content": "r", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [142, 641, 190, 654], "score": 1.0, "content": " satisfying ", "type": "text"}, {"bbox": [190, 642, 239, 653], "score": 0.92, "content": "(p/r)=-1", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [239, 641, 244, 654], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [244, 642, 293, 653], "score": 0.91, "content": "(q/r)=+1", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [293, 641, 297, 654], "score": 1.0, "content": ".", "type": "text"}], "index": 43}], "index": 38, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [125, 521, 487, 654]}, {"type": "text", "bbox": [125, 651, 486, 699], "lines": [{"bbox": [136, 652, 487, 666], "spans": [{"bbox": [136, 652, 147, 666], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [148, 655, 176, 664], "score": 0.93, "content": "d_{2}=8", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [176, 652, 198, 666], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [198, 655, 222, 664], "score": 0.9, "content": "p=2", "type": "inline_equation", "height": 9, "width": 24}, {"bbox": [222, 652, 250, 666], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [250, 654, 298, 665], "score": 0.92, "content": "2\\mathcal{O}_{k_{1}}=22^{\\prime}", "type": "inline_equation", "height": 11, "width": 48}, {"bbox": [299, 652, 405, 666], "score": 1.0, "content": ", and we have to choose ", "type": "text"}, {"bbox": [405, 653, 442, 665], "score": 0.94, "content": "2^{h}=(\\pi)", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [443, 652, 487, 666], "score": 1.0, "content": " in such a", "type": "text"}], "index": 44}, {"bbox": [127, 665, 486, 677], "spans": [{"bbox": [127, 665, 166, 677], "score": 1.0, "content": "way that ", "type": "text"}, {"bbox": [167, 666, 214, 677], "score": 0.94, "content": "k_{1}(\\sqrt{\\pi}\\,)/k_{1}", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [215, 665, 309, 677], "score": 1.0, "content": " is unramified outside", "type": "text"}, {"bbox": [310, 669, 315, 676], "score": 0.74, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [316, 665, 412, 677], "score": 1.0, "content": ". The residue symbols", "type": "text"}, {"bbox": [413, 666, 435, 677], "score": 0.92, "content": "\\left[\\alpha/2\\right]", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [435, 665, 486, 677], "score": 1.0, "content": " are defined", "type": "text"}], "index": 45}, {"bbox": [125, 677, 486, 689], "spans": [{"bbox": [125, 677, 307, 689], "score": 1.0, "content": "as Kronecker symbols via the splitting of ", "type": "text"}, {"bbox": [308, 681, 313, 686], "score": 0.41, "content": "^{2}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [313, 677, 433, 689], "score": 1.0, "content": " in the quadratic extension ", "type": "text"}, {"bbox": [433, 678, 482, 689], "score": 0.93, "content": "k_{1}(\\sqrt{\\alpha}\\,)/k_{1}", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [482, 677, 486, 689], "score": 1.0, "content": ".", "type": "text"}], "index": 46}, {"bbox": [126, 689, 390, 702], "spans": [{"bbox": [126, 689, 390, 702], "score": 1.0, "content": "With these modifactions, the above arguments remain valid.", "type": "text"}], "index": 47}], "index": 45.5, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [125, 652, 487, 702]}]}	{"layout_dets": [{"category_id": 1, "poly": [347, 415, 1352, 415, 1352, 847, 347, 847], "score": 0.982}, {"category_id": 1, "poly": [347, 1446, 1352, 1446, 1352, 1810, 347, 1810], "score": 0.981}, {"category_id": 1, "poly": [347, 850, 1353, 850, 1353, 1069, 347, 1069], "score": 0.976}, {"category_id": 5, "poly": [525, 1145, 1174, 1145, 1174, 1300, 525, 1300], "score": 0.975, "html": "<html><body><table><tr><td>q2</td><td>Cl2(K2)</td><td>q1</td><td>h2(K1)</td><td>q2</td><td>Cl2(K2)</td><td>h2(L)</td></tr><tr><td>2</td><td>(2)</td><td>1</td><td>2</td><td>2</td><td>(2)</td><td>2m+1</td></tr><tr><td>2</td><td>(4)</td><td>1</td><td>4</td><td>\uff1f</td><td>(2,2)</td><td>2m+3</td></tr></table></body></html>"}, {"category_id": 1, "poly": [349, 1307, 1352, 1307, 1352, 1446, 349, 1446], 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1109.0, 382.0, 1109.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [351.0, 1106.0, 422.0, 1106.0, 422.0, 1142.0, 351.0, 1142.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [511.0, 1106.0, 558.0, 1106.0, 558.0, 1142.0, 511.0, 1142.0], "score": 1.0, "text": ""}], "page_info": {"page_no": 12, "height": 2200, "width": 1700}}	IMAGINARY QUADRATIC FIELDS 13 The referee (whom we’d like to thank for a couple of helpful remarks) asked whether and infinitely often. Let us show how to prove that both possibilities occur with equal density. Before we can do this, we have to study the quadratic extensions and of more closely. We assume that and are odd primes in t he following, and then say how to modify the arguments in the case or . The primes and split in as and . Let denote the odd class number of and write and for primary elements and (this is can easily be proved directly, but it is also a very special case of Hilbert’s first supplementary law for quadratic reciprocity in fields with odd class number (see [7]): if for an ideal with odd norm, then can be chosen primary (i.e. congruent to a square mod ) if and only if is primary (i.e. for all units , where denotes the quadratic residue symbol in )). Let denote the quadratic residue symbol in . Then , so we may choose the conjugates in such a way that and . Put and ; we claim that . This is equivalent to , where is a quad r atic unramified extension of . Put a n d apply the ambiguous class number formula to an d : since there is only one ramified prime in each of these two ext e nsions, w e fin d ; note that we have used the assumption that in deducing th a t is inert in . In our proof of Theorem 1 we have seen that there are th e following possibilities when \| 4: ``` In our proof of Theorem 1 we have seen that there are th e following possibilities when \| 4: ``` In order to decide whether or , recall that we have ; thus must be the field with 2 -class nu mber 2, and this implies and . In particular we see that if and only if as long as with . The ambiguous class number formula shows that is cyclic, thus 4 \| if and only if , where is the quadratic unramified extension of . Applying the ambiguous class number formula to , where , we see that if and only if . Now is generated by a root of unity (which always is a norm residue at primes dividing ) and a fundamental unit . Therefore if and only if , where and where denotes the quadratic residue symbol in . Since , we have proved that if and only if the prime ideal above splits in the quadratic extension . But if we fix and , this happens for exactly half of the values of satisfying , . If and , then , and we have to choose in such a way that is unramified outside . The residue symbols are defined as Kronecker symbols via the splitting of in the quadratic extension . With these modifactions, the above arguments remain valid.	<div class="pdf-page"> <p>The referee (whom we’d like to thank for a couple of helpful remarks) asked whether and infinitely often. Let us show how to prove that both possibilities occur with equal density.</p> <p>Before we can do this, we have to study the quadratic extensions and of more closely. We assume that and are odd primes in t he following, and then say how to modify the arguments in the case or . The primes and split in as and . Let denote the odd class number of and write and for primary elements and (this is can easily be proved directly, but it is also a very special case of Hilbert’s first supplementary law for quadratic reciprocity in fields with odd class number (see [7]): if for an ideal with odd norm, then can be chosen primary (i.e. congruent to a square mod ) if and only if is primary (i.e. for all units , where denotes the quadratic residue symbol in )). Let denote the quadratic residue symbol in . Then , so we may choose the conjugates in such a way that and .</p> <p>Put and ; we claim that . This is equivalent to , where is a quad r atic unramified extension of . Put a n d apply the ambiguous class number formula to an d : since there is only one ramified prime in each of these two ext e nsions, w e fin d ; note that we have used the assumption that in deducing th a t is inert in .</p> <h3>In our proof of Theorem 1 we have seen that there are th e following possibilities when \| 4:</h3> <p>In order to decide whether or , recall that we have ; thus must be the field with 2 -class nu mber 2, and this implies and . In particular we see that if and only if as long as with .</p> <p>The ambiguous class number formula shows that is cyclic, thus 4 \| if and only if , where is the quadratic unramified extension of . Applying the ambiguous class number formula to , where , we see that if and only if . Now is generated by a root of unity (which always is a norm residue at primes dividing ) and a fundamental unit . Therefore if and only if , where and where denotes the quadratic residue symbol in . Since , we have proved that if and only if the prime ideal above splits in the quadratic extension . But if we fix and , this happens for exactly half of the values of satisfying , .</p> <p>If and , then , and we have to choose in such a way that is unramified outside . The residue symbols are defined as Kronecker symbols via the splitting of in the quadratic extension . With these modifactions, the above arguments remain valid.</p> </div>	<div class="pdf-page"> <div class="pdf-discarded" data-x="398" data-y="116" data-width="226" data-height="12" style="opacity: 0.5;">IMAGINARY QUADRATIC FIELDS</div> <div class="pdf-discarded" data-x="796" data-y="117" data-width="17" data-height="11" style="opacity: 0.5;">13</div> <p class="pdf-text" data-x="209" data-y="144" data-width="604" data-height="47">The referee (whom we’d like to thank for a couple of helpful remarks) asked whether and infinitely often. Let us show how to prove that both possibilities occur with equal density.</p> <p class="pdf-text" data-x="207" data-y="192" data-width="606" data-height="201">Before we can do this, we have to study the quadratic extensions and of more closely. We assume that and are odd primes in t he following, and then say how to modify the arguments in the case or . The primes and split in as and . Let denote the odd class number of and write and for primary elements and (this is can easily be proved directly, but it is also a very special case of Hilbert’s first supplementary law for quadratic reciprocity in fields with odd class number (see [7]): if for an ideal with odd norm, then can be chosen primary (i.e. congruent to a square mod ) if and only if is primary (i.e. for all units , where denotes the quadratic residue symbol in )). Let denote the quadratic residue symbol in . Then , so we may choose the conjugates in such a way that and .</p> <p class="pdf-text" data-x="207" data-y="395" data-width="607" data-height="101">Put and ; we claim that . This is equivalent to , where is a quad r atic unramified extension of . Put a n d apply the ambiguous class number formula to an d : since there is only one ramified prime in each of these two ext e nsions, w e fin d ; note that we have used the assumption that in deducing th a t is inert in .</p> <caption class="pdf-table-caption" data-x="209" data-y="497" data-width="604" data-height="30">In our proof of Theorem 1 we have seen that there are th e following possibilities when \| 4:</caption> <p class="pdf-text" data-x="209" data-y="607" data-width="604" data-height="65">In order to decide whether or , recall that we have ; thus must be the field with 2 -class nu mber 2, and this implies and . In particular we see that if and only if as long as with .</p> <p class="pdf-text" data-x="207" data-y="672" data-width="606" data-height="169">The ambiguous class number formula shows that is cyclic, thus 4 \| if and only if , where is the quadratic unramified extension of . Applying the ambiguous class number formula to , where , we see that if and only if . Now is generated by a root of unity (which always is a norm residue at primes dividing ) and a fundamental unit . Therefore if and only if , where and where denotes the quadratic residue symbol in . Since , we have proved that if and only if the prime ideal above splits in the quadratic extension . But if we fix and , this happens for exactly half of the values of satisfying , .</p> <p class="pdf-text" data-x="209" data-y="841" data-width="604" data-height="62">If and , then , and we have to choose in such a way that is unramified outside . The residue symbols are defined as Kronecker symbols via the splitting of in the quadratic extension . With these modifactions, the above arguments remain valid.</p> </div>	{ "type": [ "text", "inline_equation", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "inline_equation", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text" ], "coordinates": [ [ 229, 147, 813, 164 ], [ 210, 162, 811, 178 ], [ 210, 178, 523, 193 ], [ 229, 193, 809, 210 ], [ 209, 208, 814, 227 ], [ 209, 224, 814, 241 ], [ 210, 241, 814, 257 ], [ 209, 256, 814, 272 ], [ 210, 271, 814, 288 ], [ 209, 288, 813, 303 ], [ 209, 302, 811, 319 ], [ 210, 319, 814, 334 ], [ 209, 333, 814, 351 ], [ 210, 350, 813, 365 ], [ 210, 364, 811, 381 ], [ 209, 380, 518, 396 ], [ 227, 396, 813, 413 ], [ 209, 415, 813, 431 ], [ 210, 430, 813, 451 ], [ 209, 448, 813, 466 ], [ 209, 465, 813, 482 ], [ 210, 483, 694, 499 ], [ 229, 499, 811, 515 ], [ 210, 514, 334, 531 ], [ 316, 532, 706, 605 ], [ 207, 608, 813, 628 ], [ 210, 624, 814, 643 ], [ 210, 642, 814, 660 ], [ 210, 659, 448, 674 ], [ 229, 673, 814, 690 ], [ 210, 690, 813, 704 ], [ 209, 704, 814, 721 ], [ 210, 721, 814, 736 ], [ 209, 735, 814, 752 ], [ 210, 752, 811, 766 ], [ 209, 768, 814, 783 ], [ 209, 783, 814, 797 ], [ 210, 797, 814, 814 ], [ 210, 814, 813, 828 ], [ 209, 828, 496, 845 ], [ 227, 842, 814, 861 ], [ 212, 859, 813, 875 ], [ 209, 875, 813, 890 ], [ 210, 890, 652, 907 ] ], "content": [ "The referee (whom we’d like to thank for a couple of helpful remarks) asked", "whether h_{2}(K)=2 and h_{2}(K)>2 infinitely often. Let us show how to prove that", "both possibilities occur with equal density.", "Before we can do this, we have to study the quadratic extensions K_{1} and \\widetilde{K}_{1}", "of k_{1} more closely. We assume that d_{2}\\,=\\,p and d_{3}~=~r are odd primes in t he", "following, and then say how to modify the arguments in the case d_{2}=8 or d_{3}=-8 .", "The primes p and r split in k_{1} as p\\mathcal{O}_{1}\\,=\\,\\mathfrak{p p}^{\\prime} and r\\mathcal{O}_{1}\\,=\\,\\mathfrak{r r}^{\\prime} . Let h denote the", "odd class number of k_{1} and write {\\mathfrak{p}}^{h}\\,=\\,(\\pi) and \\mathfrak{r}^{h}\\,=\\,(\\rho) for primary elements", "\\pi and \\rho (this is can easily be proved directly, but it is also a very special case", "of Hilbert’s first supplementary law for quadratic reciprocity in fields K with odd", "class number h (see [7]): if {\\mathfrak{a}}^{h}\\;=\\;\\alpha{\\mathcal{O}}_{K} for an ideal \\mathfrak{p} with odd norm, then \\alpha", "can be chosen primary (i.e. congruent to a square mod 4{\\cal O}_{K} ) if and only if \\mathfrak{a} is", "primary (i.e. [\\varepsilon/{\\mathfrak a}]=+1 for all units \\varepsilon\\in{\\mathcal{O}}_{K}^{\\times} , where [\\,\\cdot\\,/\\,\\cdot\\,] denotes the quadratic", "residue symbol in K )). Let \\big[\\cdot\\big/\\cdot\\big] denote the quadratic residue symbol in k_{1} . Then", "[\\pi/\\rho][\\pi^{\\prime}/\\rho]=[p/\\rho]=(p/r)=-1 , so we may choose the conjugates in such a way", "that [\\pi/\\rho]=+1 and [\\pi^{\\prime}/\\rho]=[\\pi/\\rho^{\\prime}]=-1 .", "Put K_{1}\\;=\\;k_{1}(\\sqrt{\\pi\\rho}\\,) and \\tilde{K}_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\rho^{\\prime}}) ; we claim that h_{2}(\\tilde{K}_{1})\\;=\\;2 . This", "is equivalent to h_{2}(\\widetilde{L}_{1})\\,=\\,1 , where \\tilde{L}_{1}\\,=\\,k_{1}(\\sqrt{\\pi},\\sqrt{\\rho^{\\prime}}) is a quad r atic unramified", "extension of \\widetilde{K}_{1} . Put \\widetilde{F}_{1}=k_{1}(\\sqrt{\\pi}\\,) a n d apply the ambiguous class number formula", "to \\widetilde{F}_{1}/k_{1} an d \\widetilde{L}_{1}/\\widetilde{F}_{1} : since there is only one ramified prime in each of these two", "ext e nsions, w e fin d \\mathrm{Am}(\\widetilde{F}_{1}/k_{1})\\,=\\,\\mathrm{Am}(\\widetilde{L}_{1}/\\widetilde{F}_{1})\\,=\\,1 ; note that we have used the", "assumption that [\\pi/\\rho^{\\prime}]=-1 in deducing th a t {\\mathfrak{r}}^{\\prime} is inert in \\widetilde{F}_{1}/k_{1} .", "In our proof of Theorem 1 we have seen that there are th e following possibilities", "when h_{2}(K_{2}) \| 4:", "", "In order to decide whether \\widetilde{q}_{2}=1 or \\widetilde{q}_{2}=2 , recall that we have h_{2}(K_{1})=4 ; thus", "\\widetilde{K}_{1} must be the field with 2 -class nu mber 2, and this implies h_{2}(\\widetilde{L})\\,=\\,2^{m+2} and", "\\widetilde{q}_{2}=1 . In particular we see that 4\\mid h_{2}(K_{2}) if and only if 4\\mid h_{2}(K_{1}) as long as", "K_{1}=k_{1}(\\sqrt{\\pi\\rho}\\,) with [\\pi/\\rho]=+1 .", "The ambiguous class number formula shows that \\mathrm{Cl}_{2}(K_{1}) is cyclic, thus 4 \|", "h_{2}(K_{1}) if and only if 2\\mid h_{2}(L_{1}) , where L_{1}=K_{1}(\\sqrt{\\pi}\\,) is the quadratic unramified", "extension of K_{1} . Applying the ambiguous class number formula to L_{1}/F_{1} , where", "F_{1}\\;=\\;k_{1}(\\sqrt{\\pi}\\,) , we see that 2\\:\\:\|\\:\\:h_{2}(L_{1}) if and only if (E\\,:\\,H)\\;=\\;1 . Now E is", "generated by a root of unity (which always is a norm residue at primes dividing", "r\\,\\equiv\\,1\\;\\mathrm{mod}\\;4 ) and a fundamental unit \\varepsilon . Therefore (E\\,:\\,H)\\;=\\;1 if and only", "if \\{\\varepsilon/\\mathfrak{R}_{1}\\}\\,=\\,\\{\\varepsilon/\\mathfrak{R}_{2}\\}\\,=\\,+1 , where \\mathfrak{r}{\\mathcal{O}}_{F_{1}}\\,=\\,\\mathfrak{R}_{1}\\mathfrak{R}_{2} and where \\{\\,\\cdot\\,/\\,\\cdot\\,\\} denotes the", "quadratic residue symbol in F_{1} . Since \\{\\varepsilon/\\mathfrak{R}_{1}\\}\\{\\varepsilon/\\mathfrak{R}_{2}\\}=[\\varepsilon/\\mathfrak{r}]=+1 , we have proved", "that 4\\mid h_{2}(K_{1}) if and only if the prime ideal \\Re_{1} above \\mathfrak{r} splits in the quadratic", "extension F_{1}(\\sqrt{\\varepsilon}) . But if we fix p and q , this happens for exactly half of the values", "of r satisfying (p/r)=-1 , (q/r)=+1 .", "If d_{2}=8 and p=2 , then 2\\mathcal{O}_{k_{1}}=22^{\\prime} , and we have to choose 2^{h}=(\\pi) in such a", "way that k_{1}(\\sqrt{\\pi}\\,)/k_{1} is unramified outside \\mathfrak{p} . The residue symbols \\left[\\alpha/2\\right] are defined", "as Kronecker symbols via the splitting of ^{2} in the quadratic extension k_{1}(\\sqrt{\\alpha}\\,)/k_{1} .", "With these modifactions, the above arguments remain valid." ], "index": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 19, 20, 21, 22, 23, 24, 41, 42, 65, 90, 91, 92, 93, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 159, 160, 161, 162 ] }	{ "coordinates": [], "index": [], "caption": [], "caption_coordinates": [] }	{ "type": [], "coordinates": [], "content": [], "index": [] }	{ "type": [ "table", "table_caption", "table_body" ], "coordinates": [ [ 316, 532, 706, 605 ], [ 209, 497, 813, 527 ], [ 316, 532, 706, 605 ] ], "content": [ "", "In our proof of Theorem 1 we have seen that there are th e following possibilities when \| 4:", "" ], "index": [ 0, 1, 2 ] }	{"document_info": {"total_pages": 14, "parse_type": "txt", "version": "1.0.1", "total_content_list_items": 155}, "content_statistics": {"total_blocks": 182, "total_lines": 526, "total_images": 4, "total_equations": 28, "total_tables": 11, "total_characters": 29658, "total_words": 5331, "block_type_distribution": {"title": 7, "text": 126, "interline_equation": 14, "discarded": 24, "table_caption": 3, "table_body": 4, "image_body": 2, "list": 2}}, "model_statistics": {"total_layout_detections": 2826, "avg_confidence": 0.9470955414012739, "min_confidence": 0.26, "max_confidence": 1.0}, "page_statistics": [{"page_number": 0, "page_size": [612.0, 792.0], "blocks": 11, "lines": 38, "images": 0, "equations": 2, "tables": 0, "characters": 2362, "words": 398, "discarded_blocks": 1}, {"page_number": 1, "page_size": [612.0, 792.0], "blocks": 10, "lines": 30, "images": 0, "equations": 0, "tables": 3, "characters": 1688, "words": 305, "discarded_blocks": 1}, {"page_number": 2, "page_size": [612.0, 792.0], "blocks": 15, "lines": 44, "images": 0, "equations": 4, "tables": 0, "characters": 2417, "words": 416, "discarded_blocks": 2}, {"page_number": 3, "page_size": [612.0, 792.0], "blocks": 7, "lines": 27, "images": 0, "equations": 0, "tables": 3, "characters": 1591, "words": 291, "discarded_blocks": 1}, {"page_number": 4, "page_size": [612.0, 792.0], "blocks": 23, "lines": 38, "images": 0, "equations": 6, "tables": 0, "characters": 1604, "words": 303, "discarded_blocks": 4}, {"page_number": 5, "page_size": [612.0, 792.0], "blocks": 17, "lines": 40, "images": 0, "equations": 6, "tables": 0, "characters": 2131, "words": 393, "discarded_blocks": 1}, {"page_number": 6, "page_size": [612.0, 792.0], "blocks": 15, "lines": 36, "images": 2, "equations": 4, "tables": 0, "characters": 1791, "words": 324, "discarded_blocks": 2}, {"page_number": 7, "page_size": [612.0, 792.0], "blocks": 12, "lines": 49, "images": 0, "equations": 0, "tables": 0, "characters": 3049, "words": 571, "discarded_blocks": 1}, {"page_number": 8, "page_size": [612.0, 792.0], "blocks": 11, "lines": 42, "images": 0, "equations": 0, "tables": 0, "characters": 2636, "words": 479, "discarded_blocks": 2}, {"page_number": 9, "page_size": [612.0, 792.0], "blocks": 14, "lines": 35, "images": 2, "equations": 0, "tables": 0, "characters": 2036, "words": 375, "discarded_blocks": 1}, {"page_number": 10, "page_size": [612.0, 792.0], "blocks": 17, "lines": 35, "images": 0, "equations": 6, "tables": 0, "characters": 1789, "words": 327, "discarded_blocks": 2}, {"page_number": 11, "page_size": [612.0, 792.0], "blocks": 14, "lines": 40, "images": 0, "equations": 0, "tables": 2, "characters": 2094, "words": 399, "discarded_blocks": 2}, {"page_number": 12, "page_size": [612.0, 792.0], "blocks": 10, "lines": 44, "images": 0, "equations": 0, "tables": 3, "characters": 2572, "words": 487, "discarded_blocks": 2}, {"page_number": 13, "page_size": [612.0, 792.0], "blocks": 6, "lines": 28, "images": 0, "equations": 0, "tables": 0, "characters": 1898, "words": 263, "discarded_blocks": 2}]}
0003244v1	13	[ 612, 792 ]	{ "type": [ "title", "list", "text", "text", "discarded", "discarded" ], "coordinates": [ [ 461, 146, 562, 159 ], [ 227, 166, 814, 465 ], [ 229, 475, 607, 501 ], [ 230, 510, 652, 536 ], [ 210, 117, 225, 128 ], [ 212, 545, 809, 584 ] ], "content": [ "References", "[1] E. Benjamin, F. Lemmermeyer, C. Snyder, Imaginary Quadratic Fields k with Cyclic , J. Number Theory 67 (1997), 229–245. [2] N. Blackburn, On Prime-Power Groups in which the Derived Group has Two Generators, Proc. Cambridge Phil. Soc. 53 (1957), 19–27. [3] N. Blackburn, On Prime Power Groups with Two Generators, Proc. Cambridge Phil. Soc. 54 (1958), 327–337. [4] G. Gras, Sur les ℓ-classes d’ide´aux dans les extensions cycliques relatives de degre´ premier ℓ, Ann. Inst. Fourier 23 (1973), 1–48. [5] P. Hall, A Contribution to the Theory of Groups of Prime Power Order, Proc. London Math. Soc. 36 (1933), 29–95. [6]H.Hasse,Bericht iber neuere Untersuchungen und Probleme aus der Theorie der alge- braischen Zahlkorper, Teil I: Reziprozitatsgesetz, Physica Verlag, Wurzburg 1965. [7] D. Hilbert, Uber die Theorie des relativ-quadratischen Zahlkorpers, Math. Ann. 51 (1899), 1–127. [8] B. Huppert, Endliche Gruppen I, Springer Verlag, Heidelberg, 1967. [9] S. Lang, Cyclotomic Fields. I, II, Springer Verlag 1990. [10] F. Lemmermeyer, Kuroda’s Class Number Formula, Acta Arith. 66.3 (1994), 245–260. [11] M. Moriya, Uber die Klassenzahl eines relativzyklischen Zahlkorpers von Primzahlgrad, Jap. J. Math. 10 (1933), 1–18. [12] L. R´edei, H. Reichardt, Die Anzahl der durch 4 teilbaren Invarianten der Klassengruppe eines beliebigen quadratischen Zahlkorpers, J. Reine Angew. Math. 170 (1933), 69-74. [13] B. Schmithals, Konstruktion imaginarquadratischer Korper mit unendlichem Klassenkor- perturm, Arch. Math. 34 (1980), 307–312.", "(E. Benjamin) Mathematics Department, Unity College -mail address: [email protected]", "(F. Lemmermeyer) Erwin-Rohde-Str. 19, Heidelberg, Germany -mail address: [email protected]", "14", "(C. Snyder) Department of Mathematics and Statistics, University of Maine, and, Research Institute of Mathematics, Orono E-mail address: [email protected]" ], "index": [ 0, 1, 2, 3, 4, 5 ] }	[{"type": "text", "text": "References ", "text_level": 1, "page_idx": 13}, {"type": "text", "text": "[1] E. Benjamin, F. Lemmermeyer, C. Snyder, Imaginary Quadratic Fields k with Cyclic $\\mathrm{Cl}_{2}(k^{1})$ , J. Number Theory 67 (1997), 229\u2013245. \n[2] N. Blackburn, On Prime-Power Groups in which the Derived Group has Two Generators, Proc. Cambridge Phil. Soc. 53 (1957), 19\u201327. \n[3] N. Blackburn, On Prime Power Groups with Two Generators, Proc. Cambridge Phil. Soc. 54 (1958), 327\u2013337. \n[4] G. Gras, Sur les \u2113-classes d\u2019ide\u00b4aux dans les extensions cycliques relatives de degre\u00b4 premier \u2113, Ann. Inst. Fourier 23 (1973), 1\u201348. \n[5] P. Hall, A Contribution to the Theory of Groups of Prime Power Order, Proc. London Math. Soc. 36 (1933), 29\u201395. \n[6]H.Hasse,Bericht iber neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkorper, Teil I: Reziprozitatsgesetz, Physica Verlag, Wurzburg 1965. \n[7] D. Hilbert, Uber die Theorie des relativ-quadratischen Zahlkorpers, Math. Ann. 51 (1899), 1\u2013127. \n[8] B. Huppert, Endliche Gruppen I, Springer Verlag, Heidelberg, 1967. \n[9] S. Lang, Cyclotomic Fields. I, II, Springer Verlag 1990. \n[10] F. Lemmermeyer, Kuroda\u2019s Class Number Formula, Acta Arith. 66.3 (1994), 245\u2013260. \n[11] M. Moriya, Uber die Klassenzahl eines relativzyklischen Zahlkorpers von Primzahlgrad, Jap. J. Math. 10 (1933), 1\u201318. \n[12] L. R\u00b4edei, H. Reichardt, Die Anzahl der durch 4 teilbaren Invarianten der Klassengruppe eines beliebigen quadratischen Zahlkorpers, J. Reine Angew. Math. 170 (1933), 69-74. \n[13] B. Schmithals, Konstruktion imaginarquadratischer Korper mit unendlichem Klassenkorperturm, Arch. Math. 34 (1980), 307\u2013312. ", "page_idx": 13}, {"type": "text", "text": "(E. Benjamin) Mathematics Department, Unity College $E$ -mail address: [email protected] ", "page_idx": 13}, {"type": "text", "text": "(F. Lemmermeyer) Erwin-Rohde-Str. 19, Heidelberg, Germany $E$ -mail address: [email protected] ", "page_idx": 13}]	{"preproc_blocks": [{"type": "title", "bbox": [276, 113, 336, 123], "lines": [{"bbox": [276, 114, 336, 126], "spans": [{"bbox": [276, 114, 336, 126], "score": 1.0, "content": "References", "type": "text"}], "index": 0}], "index": 0}, {"type": "text", "bbox": [136, 129, 487, 360], "lines": [{"bbox": [138, 131, 486, 143], "spans": [{"bbox": [138, 131, 486, 143], "score": 1.0, "content": "[1] E. Benjamin, F. 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403.0, 1084.0], "score": 1.0, "text": ""}], "page_info": {"page_no": 13, "height": 2200, "width": 1700}}	14 # References - [1] E. Benjamin, F. Lemmermeyer, C. Snyder, Imaginary Quadratic Fields k with Cyclic , J. Number Theory 67 (1997), 229–245. [2] N. Blackburn, On Prime-Power Groups in which the Derived Group has Two Generators, Proc. Cambridge Phil. Soc. 53 (1957), 19–27. [3] N. Blackburn, On Prime Power Groups with Two Generators, Proc. Cambridge Phil. Soc. 54 (1958), 327–337. [4] G. Gras, Sur les ℓ-classes d’ide´aux dans les extensions cycliques relatives de degre´ premier ℓ, Ann. Inst. Fourier 23 (1973), 1–48. [5] P. Hall, A Contribution to the Theory of Groups of Prime Power Order, Proc. London Math. Soc. 36 (1933), 29–95. [6]H.Hasse,Bericht iber neuere Untersuchungen und Probleme aus der Theorie der alge- braischen Zahlkorper, Teil I: Reziprozitatsgesetz, Physica Verlag, Wurzburg 1965. [7] D. Hilbert, Uber die Theorie des relativ-quadratischen Zahlkorpers, Math. Ann. 51 (1899), 1–127. [8] B. Huppert, Endliche Gruppen I, Springer Verlag, Heidelberg, 1967. [9] S. Lang, Cyclotomic Fields. I, II, Springer Verlag 1990. [10] F. Lemmermeyer, Kuroda’s Class Number Formula, Acta Arith. 66.3 (1994), 245–260. [11] M. Moriya, Uber die Klassenzahl eines relativzyklischen Zahlkorpers von Primzahlgrad, Jap. J. Math. 10 (1933), 1–18. [12] L. R´edei, H. Reichardt, Die Anzahl der durch 4 teilbaren Invarianten der Klassengruppe eines beliebigen quadratischen Zahlkorpers, J. Reine Angew. Math. 170 (1933), 69-74. [13] B. Schmithals, Konstruktion imaginarquadratischer Korper mit unendlichem Klassenkor- perturm, Arch. Math. 34 (1980), 307–312. (E. Benjamin) Mathematics Department, Unity College -mail address: [email protected] (F. Lemmermeyer) Erwin-Rohde-Str. 19, Heidelberg, Germany -mail address: [email protected] (C. Snyder) Department of Mathematics and Statistics, University of Maine, and, Research Institute of Mathematics, Orono E-mail address: [email protected]	<div class="pdf-page"> <h1>References</h1> <ul> <li>[1] E. Benjamin, F. Lemmermeyer, C. Snyder, Imaginary Quadratic Fields k with Cyclic , J. Number Theory 67 (1997), 229–245. [2] N. Blackburn, On Prime-Power Groups in which the Derived Group has Two Generators, Proc. Cambridge Phil. Soc. 53 (1957), 19–27. [3] N. Blackburn, On Prime Power Groups with Two Generators, Proc. Cambridge Phil. Soc. 54 (1958), 327–337. [4] G. Gras, Sur les ℓ-classes d’ide´aux dans les extensions cycliques relatives de degre´ premier ℓ, Ann. Inst. Fourier 23 (1973), 1–48. [5] P. Hall, A Contribution to the Theory of Groups of Prime Power Order, Proc. London Math. Soc. 36 (1933), 29–95. [6]H.Hasse,Bericht iber neuere Untersuchungen und Probleme aus der Theorie der alge- braischen Zahlkorper, Teil I: Reziprozitatsgesetz, Physica Verlag, Wurzburg 1965. [7] D. Hilbert, Uber die Theorie des relativ-quadratischen Zahlkorpers, Math. Ann. 51 (1899), 1–127. [8] B. Huppert, Endliche Gruppen I, Springer Verlag, Heidelberg, 1967. [9] S. Lang, Cyclotomic Fields. I, II, Springer Verlag 1990. [10] F. Lemmermeyer, Kuroda’s Class Number Formula, Acta Arith. 66.3 (1994), 245–260. [11] M. Moriya, Uber die Klassenzahl eines relativzyklischen Zahlkorpers von Primzahlgrad, Jap. J. Math. 10 (1933), 1–18. [12] L. R´edei, H. Reichardt, Die Anzahl der durch 4 teilbaren Invarianten der Klassengruppe eines beliebigen quadratischen Zahlkorpers, J. Reine Angew. Math. 170 (1933), 69-74. [13] B. Schmithals, Konstruktion imaginarquadratischer Korper mit unendlichem Klassenkor- perturm, Arch. Math. 34 (1980), 307–312.</li> </ul> <p>(E. Benjamin) Mathematics Department, Unity College -mail address: [email protected]</p> <p>(F. Lemmermeyer) Erwin-Rohde-Str. 19, Heidelberg, Germany -mail address: [email protected]</p> </div>	<div class="pdf-page"> <div class="pdf-discarded" data-x="210" data-y="117" data-width="15" data-height="11" style="opacity: 0.5;">14</div> <h1 class="pdf-title" data-x="461" data-y="146" data-width="101" data-height="13">References</h1> <ul class="pdf-list" data-x="227" data-y="166" data-width="587" data-height="299"> <li>[1] E. Benjamin, F. Lemmermeyer, C. Snyder, Imaginary Quadratic Fields k with Cyclic , J. Number Theory 67 (1997), 229–245. [2] N. Blackburn, On Prime-Power Groups in which the Derived Group has Two Generators, Proc. Cambridge Phil. Soc. 53 (1957), 19–27. [3] N. Blackburn, On Prime Power Groups with Two Generators, Proc. Cambridge Phil. Soc. 54 (1958), 327–337. [4] G. Gras, Sur les ℓ-classes d’ide´aux dans les extensions cycliques relatives de degre´ premier ℓ, Ann. Inst. Fourier 23 (1973), 1–48. [5] P. Hall, A Contribution to the Theory of Groups of Prime Power Order, Proc. London Math. Soc. 36 (1933), 29–95. [6]H.Hasse,Bericht iber neuere Untersuchungen und Probleme aus der Theorie der alge- braischen Zahlkorper, Teil I: Reziprozitatsgesetz, Physica Verlag, Wurzburg 1965. [7] D. Hilbert, Uber die Theorie des relativ-quadratischen Zahlkorpers, Math. Ann. 51 (1899), 1–127. [8] B. Huppert, Endliche Gruppen I, Springer Verlag, Heidelberg, 1967. [9] S. Lang, Cyclotomic Fields. I, II, Springer Verlag 1990. [10] F. Lemmermeyer, Kuroda’s Class Number Formula, Acta Arith. 66.3 (1994), 245–260. [11] M. Moriya, Uber die Klassenzahl eines relativzyklischen Zahlkorpers von Primzahlgrad, Jap. J. Math. 10 (1933), 1–18. [12] L. R´edei, H. Reichardt, Die Anzahl der durch 4 teilbaren Invarianten der Klassengruppe eines beliebigen quadratischen Zahlkorpers, J. Reine Angew. Math. 170 (1933), 69-74. [13] B. Schmithals, Konstruktion imaginarquadratischer Korper mit unendlichem Klassenkor- perturm, Arch. Math. 34 (1980), 307–312.</li> </ul> <p class="pdf-text" data-x="229" data-y="475" data-width="378" data-height="26">(E. Benjamin) Mathematics Department, Unity College -mail address: [email protected]</p> <p class="pdf-text" data-x="230" data-y="510" data-width="422" data-height="26">(F. Lemmermeyer) Erwin-Rohde-Str. 19, Heidelberg, Germany -mail address: [email protected]</p> <div class="pdf-discarded" data-x="212" data-y="545" data-width="597" data-height="39" style="opacity: 0.5;">(C. Snyder) Department of Mathematics and Statistics, University of Maine, and, Research Institute of Mathematics, Orono E-mail address: [email protected]</div> </div>	{ "type": [ "text", "text", "inline_equation", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "inline_equation", "text", "inline_equation" ], "coordinates": [ [ 461, 147, 562, 162 ], [ 230, 169, 813, 184 ], [ 254, 183, 552, 196 ], [ 229, 195, 813, 210 ], [ 252, 209, 538, 222 ], [ 230, 221, 811, 235 ], [ 252, 235, 378, 248 ], [ 230, 246, 814, 261 ], [ 252, 259, 488, 274 ], [ 229, 271, 813, 287 ], [ 252, 285, 436, 299 ], [ 230, 299, 811, 311 ], [ 252, 310, 769, 325 ], [ 229, 323, 813, 340 ], [ 252, 338, 294, 350 ], [ 229, 349, 680, 364 ], [ 229, 362, 600, 377 ], [ 224, 376, 794, 390 ], [ 222, 389, 813, 403 ], [ 252, 402, 445, 415 ], [ 224, 415, 814, 430 ], [ 254, 429, 793, 442 ], [ 224, 440, 813, 456 ], [ 254, 453, 515, 466 ], [ 230, 478, 610, 491 ], [ 230, 491, 456, 504 ], [ 230, 513, 652, 526 ], [ 230, 526, 533, 539 ] ], "content": [ "References", "[1] E. Benjamin, F. Lemmermeyer, C. Snyder, Imaginary Quadratic Fields k with Cyclic", "\\mathrm{Cl}_{2}(k^{1}) , J. Number Theory 67 (1997), 229–245.", "[2] N. Blackburn, On Prime-Power Groups in which the Derived Group has Two Generators,", "Proc. Cambridge Phil. Soc. 53 (1957), 19–27.", "[3] N. Blackburn, On Prime Power Groups with Two Generators, Proc. Cambridge Phil. Soc.", "54 (1958), 327–337.", "[4] G. Gras, Sur les ℓ-classes d’ide´aux dans les extensions cycliques relatives de degre´ premier", "ℓ, Ann. Inst. Fourier 23 (1973), 1–48.", "[5] P. Hall, A Contribution to the Theory of Groups of Prime Power Order, Proc. London", "Math. Soc. 36 (1933), 29–95.", "[6]H.Hasse,Bericht iber neuere Untersuchungen und Probleme aus der Theorie der alge-", "braischen Zahlkorper, Teil I: Reziprozitatsgesetz, Physica Verlag, Wurzburg 1965.", "[7] D. Hilbert, Uber die Theorie des relativ-quadratischen Zahlkorpers, Math. Ann. 51 (1899),", "1–127.", "[8] B. Huppert, Endliche Gruppen I, Springer Verlag, Heidelberg, 1967.", "[9] S. Lang, Cyclotomic Fields. I, II, Springer Verlag 1990.", "[10] F. Lemmermeyer, Kuroda’s Class Number Formula, Acta Arith. 66.3 (1994), 245–260.", "[11] M. Moriya, Uber die Klassenzahl eines relativzyklischen Zahlkorpers von Primzahlgrad,", "Jap. J. Math. 10 (1933), 1–18.", "[12] L. R´edei, H. Reichardt, Die Anzahl der durch 4 teilbaren Invarianten der Klassengruppe", "eines beliebigen quadratischen Zahlkorpers, J. Reine Angew. Math. 170 (1933), 69-74.", "[13] B. Schmithals, Konstruktion imaginarquadratischer Korper mit unendlichem Klassenkor-", "perturm, Arch. Math. 34 (1980), 307–312.", "(E. Benjamin) Mathematics Department, Unity College", "E -mail address: [email protected]", "(F. Lemmermeyer) Erwin-Rohde-Str. 19, Heidelberg, Germany", "E -mail address: [email protected]" ], "index": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 51, 52 ] }	{ "coordinates": [], "index": [], "caption": [], "caption_coordinates": [] }	{ "type": [], "coordinates": [], "content": [], "index": [] }	{ "type": [], "coordinates": [], "content": [], "index": [] }	{"document_info": {"total_pages": 14, "parse_type": "txt", "version": "1.0.1", "total_content_list_items": 155}, "content_statistics": {"total_blocks": 182, "total_lines": 526, "total_images": 4, "total_equations": 28, "total_tables": 11, "total_characters": 29658, "total_words": 5331, "block_type_distribution": {"title": 7, "text": 126, "interline_equation": 14, "discarded": 24, "table_caption": 3, "table_body": 4, "image_body": 2, "list": 2}}, "model_statistics": {"total_layout_detections": 2826, "avg_confidence": 0.9470955414012739, "min_confidence": 0.26, "max_confidence": 1.0}, "page_statistics": [{"page_number": 0, "page_size": [612.0, 792.0], "blocks": 11, "lines": 38, "images": 0, "equations": 2, "tables": 0, "characters": 2362, "words": 398, "discarded_blocks": 1}, {"page_number": 1, "page_size": [612.0, 792.0], "blocks": 10, "lines": 30, "images": 0, "equations": 0, "tables": 3, "characters": 1688, "words": 305, "discarded_blocks": 1}, {"page_number": 2, "page_size": [612.0, 792.0], "blocks": 15, "lines": 44, "images": 0, "equations": 4, "tables": 0, "characters": 2417, "words": 416, "discarded_blocks": 2}, {"page_number": 3, "page_size": [612.0, 792.0], "blocks": 7, "lines": 27, "images": 0, "equations": 0, "tables": 3, "characters": 1591, "words": 291, "discarded_blocks": 1}, {"page_number": 4, "page_size": [612.0, 792.0], "blocks": 23, "lines": 38, "images": 0, "equations": 6, "tables": 0, "characters": 1604, "words": 303, "discarded_blocks": 4}, {"page_number": 5, "page_size": [612.0, 792.0], "blocks": 17, "lines": 40, "images": 0, "equations": 6, "tables": 0, "characters": 2131, "words": 393, "discarded_blocks": 1}, {"page_number": 6, "page_size": [612.0, 792.0], "blocks": 15, "lines": 36, "images": 2, "equations": 4, "tables": 0, "characters": 1791, "words": 324, "discarded_blocks": 2}, {"page_number": 7, "page_size": [612.0, 792.0], "blocks": 12, "lines": 49, "images": 0, "equations": 0, "tables": 0, "characters": 3049, "words": 571, "discarded_blocks": 1}, {"page_number": 8, "page_size": [612.0, 792.0], "blocks": 11, "lines": 42, "images": 0, "equations": 0, "tables": 0, "characters": 2636, "words": 479, "discarded_blocks": 2}, {"page_number": 9, "page_size": [612.0, 792.0], "blocks": 14, "lines": 35, "images": 2, "equations": 0, "tables": 0, "characters": 2036, "words": 375, "discarded_blocks": 1}, {"page_number": 10, "page_size": [612.0, 792.0], "blocks": 17, "lines": 35, "images": 0, "equations": 6, "tables": 0, "characters": 1789, "words": 327, "discarded_blocks": 2}, {"page_number": 11, "page_size": [612.0, 792.0], "blocks": 14, "lines": 40, "images": 0, "equations": 0, "tables": 2, "characters": 2094, "words": 399, "discarded_blocks": 2}, {"page_number": 12, "page_size": [612.0, 792.0], "blocks": 10, "lines": 44, "images": 0, "equations": 0, "tables": 3, "characters": 2572, "words": 487, "discarded_blocks": 2}, {"page_number": 13, "page_size": [612.0, 792.0], "blocks": 6, "lines": 28, "images": 0, "equations": 0, "tables": 0, "characters": 1898, "words": 263, "discarded_blocks": 2}]}
0002004v1	0	[ 612, 792 ]	{ "type": [ "title", "text", "text", "title", "text", "text", "interline_equation", "text", "interline_equation", "text", "discarded" ], "coordinates": [ [ 249, 218, 747, 280 ], [ 354, 303, 639, 324 ], [ 351, 338, 642, 374 ], [ 456, 413, 537, 430 ], [ 219, 440, 774, 509 ], [ 168, 526, 824, 694 ], [ 418, 708, 575, 747 ], [ 168, 758, 808, 778 ], [ 168, 792, 836, 854 ], [ 168, 883, 823, 921 ], [ 23, 267, 60, 665 ] ], "content": [ "Three generation neutrino mixing is compatible with all experiments", "B. Hoeneisen and C. Marı´n", "Universidad San Francisco de Quito 2 February 2000", "Abstract", "We consider the minimal extension of the Standard Model with three generations of massive neutrinos that mix. We then determine the parameters of the model that satisfy all experimental constraints. PACS 14.60.Pq, 12.15.Ff", "Three observables in disagreement with the Standard Model of Quarks and Leptons are: i) A deficit of electron-type solar neutrinos; ii) A deficit of muon-type atmospheric neutrinos; and, possibly, iii) The observation of the apearance of in a beam of by the LSND Collaboration. The invisible width of the implies that the number of massless, or light Dirac, or light Majorana neutrino species is .[1] To account for these observations we consider the minimal extension of the Standard Model with three massive neutrinos that mix. The neutrino interaction eigenstates are superpositions of the neutrino mass eigenstates :", "", "We consider the “standard” parametrization of the unitary matrix [1]:", "", "where , , and . The probability that an ultrarelativistic neutrino produced as decays as", "arXiv:hep-ex/0002004v1 2 Feb 2000" ], "index": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ] }	[{"type": "text", "text": "Three generation neutrino mixing is compatible with all experiments ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "B. Hoeneisen and C. Mar\u0131\u00b4n ", "page_idx": 0}, {"type": "text", "text": "Universidad San Francisco de Quito 2 February 2000 ", "page_idx": 0}, {"type": "text", "text": "Abstract ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "We consider the minimal extension of the Standard Model with three generations of massive neutrinos that mix. We then determine the parameters of the model that satisfy all experimental constraints. PACS 14.60.Pq, 12.15.Ff ", "page_idx": 0}, {"type": "text", "text": "Three observables in disagreement with the Standard Model of Quarks and Leptons are: i) A deficit of electron-type solar neutrinos; ii) A deficit of muon-type atmospheric neutrinos; and, possibly, iii) The observation of the apearance of $\\nu_{e}$ in a beam of $\\nu_{\\mu}$ by the LSND Collaboration. The invisible width of the $Z$ implies that the number of massless, or light Dirac, or light Majorana neutrino species is $N_{\\nu}\\,=\\,2.993\\pm0.011.$ .[1] To account for these observations we consider the minimal extension of the Standard Model with three massive neutrinos that mix. The neutrino interaction eigenstates $\\nu_{l}$ are superpositions of the neutrino mass eigenstates $\\nu_{m}$ : ", "page_idx": 0}, {"type": "equation", "text": "$$\n\|\\nu_{l}\\rangle=\\sum_{m}U_{l m}\|\\nu_{m}\\rangle\n$$", "text_format": "latex", "page_idx": 0}, {"type": "text", "text": "We consider the \u201cstandard\u201d parametrization of the unitary matrix $U_{l m}$ [1]: ", "page_idx": 0}, {"type": "equation", "text": "$$\n\\left(\\begin{array}{c}{{\\nu_{e}}}\\\\ {{\\nu_{\\mu}}}\\\\ {{\\nu_{\\tau}}}\\end{array}\\right)=\\left(\\begin{array}{c c c}{{c_{12}c_{13}}}&{{s_{12}c_{13}}}&{{s_{13}e^{-i\\delta}}}\\\\ {{-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\\delta}}}&{{c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\\delta}}}&{{s_{23}c_{13}}}\\\\ {{s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\\delta}}}&{{-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\\delta}}}&{{c_{23}c_{13}}}\\end{array}\\right)\\left(\\begin{array}{c}{{\\nu_{1}}}\\\\ {{\\nu_{2}}}\\\\ {{\\nu_{3}}}\\end{array}\\right)\n$$", "text_format": "latex", "page_idx": 0}, {"type": "text", "text": "where $c_{i j}~\\equiv~c o s{\\theta}_{i j}$ , $s_{i j}~\\equiv~s i n\\theta_{i j}$ , $\\begin{array}{r}{0\\ \\leq\\ \\theta_{i j}\\ \\leq\\ \\frac{\\pi}{2}}\\end{array}$ and $-\\pi\\ \\leq\\ \\delta\\ <\\ \\pi$ . The probability that an ultrarelativistic neutrino produced as $\\nu_{l}$ decays as $\\nu_{l^{\\prime}}$ ", "page_idx": 0}]	{"preproc_blocks": [{"type": "title", "bbox": [149, 169, 447, 217], "lines": [{"bbox": [147, 172, 447, 195], "spans": [{"bbox": [147, 172, 447, 195], "score": 1.0, "content": "Three generation neutrino mixing is", "type": "text"}], "index": 0}, {"bbox": [164, 198, 430, 217], "spans": [{"bbox": [164, 198, 430, 217], "score": 1.0, "content": "compatible with all experiments", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [212, 235, 382, 251], "lines": [{"bbox": [212, 238, 380, 251], "spans": [{"bbox": [212, 238, 380, 251], "score": 1.0, "content": "B. Hoeneisen and C. 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We then determine the", "type": "text"}], "index": 7}, {"bbox": [130, 371, 446, 383], "spans": [{"bbox": [130, 371, 446, 383], "score": 1.0, "content": "parameters of the model that satisfy all experimental constraints.", "type": "text"}], "index": 8}, {"bbox": [131, 384, 251, 397], "spans": [{"bbox": [131, 384, 251, 397], "score": 1.0, "content": "PACS 14.60.Pq, 12.15.Ff", "type": "text"}], "index": 9}], "index": 7.5}, {"type": "text", "bbox": [101, 407, 493, 537], "lines": [{"bbox": [119, 409, 492, 424], "spans": [{"bbox": [119, 409, 492, 424], "score": 1.0, "content": "Three observables in disagreement with the Standard Model of Quarks", "type": "text"}], "index": 10}, {"bbox": [101, 425, 493, 438], "spans": [{"bbox": [101, 425, 493, 438], "score": 1.0, "content": "and Leptons are: i) A deficit of electron-type solar neutrinos; ii) A deficit of", "type": "text"}], "index": 11}, {"bbox": [101, 439, 492, 452], "spans": [{"bbox": [101, 439, 492, 452], "score": 1.0, "content": "muon-type atmospheric neutrinos; and, possibly, iii) The observation of the", "type": "text"}], "index": 12}, {"bbox": [101, 453, 492, 466], "spans": [{"bbox": [101, 453, 171, 466], "score": 1.0, "content": "apearance of ", "type": "text"}, {"bbox": [172, 456, 182, 465], "score": 0.91, "content": "\\nu_{e}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [182, 453, 256, 466], "score": 1.0, "content": " in a beam of ", "type": "text"}, {"bbox": [256, 456, 267, 466], "score": 0.92, "content": "\\nu_{\\mu}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [268, 453, 492, 466], "score": 1.0, "content": " by the LSND Collaboration. 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""}, {"category_id": 15, "poly": [585.0, 735.0, 1064.0, 735.0, 1064.0, 773.0, 585.0, 773.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [715.0, 778.0, 934.0, 778.0, 934.0, 807.0, 715.0, 807.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [758.0, 896.0, 896.0, 896.0, 896.0, 930.0, 758.0, 930.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [40.0, 584.0, 105.0, 584.0, 105.0, 1432.0, 40.0, 1432.0], "score": 1.0, "text": ""}], "page_info": {"page_no": 0, "height": 2200, "width": 1700}}	# Three generation neutrino mixing is compatible with all experiments arXiv:hep-ex/0002004v1 2 Feb 2000 B. Hoeneisen and C. Marı´n Universidad San Francisco de Quito 2 February 2000 # Abstract We consider the minimal extension of the Standard Model with three generations of massive neutrinos that mix. We then determine the parameters of the model that satisfy all experimental constraints. PACS 14.60.Pq, 12.15.Ff Three observables in disagreement with the Standard Model of Quarks and Leptons are: i) A deficit of electron-type solar neutrinos; ii) A deficit of muon-type atmospheric neutrinos; and, possibly, iii) The observation of the apearance of in a beam of by the LSND Collaboration. The invisible width of the implies that the number of massless, or light Dirac, or light Majorana neutrino species is .[1] To account for these observations we consider the minimal extension of the Standard Model with three massive neutrinos that mix. The neutrino interaction eigenstates are superpositions of the neutrino mass eigenstates : $$ \|\nu_{l}\rangle=\sum_{m}U_{l m}\|\nu_{m}\rangle $$ We consider the “standard” parametrization of the unitary matrix [1]: $$ \left(\begin{array}{c}{{\nu_{e}}}\\ {{\nu_{\mu}}}\\ {{\nu_{\tau}}}\end{array}\right)=\left(\begin{array}{c c c}{{c_{12}c_{13}}}&{{s_{12}c_{13}}}&{{s_{13}e^{-i\delta}}}\\ {{-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}}}&{{c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta}}}&{{s_{23}c_{13}}}\\ {{s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}}}&{{-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta}}}&{{c_{23}c_{13}}}\end{array}\right)\left(\begin{array}{c}{{\nu_{1}}}\\ {{\nu_{2}}}\\ {{\nu_{3}}}\end{array}\right) $$ where , , and . The probability that an ultrarelativistic neutrino produced as decays as	<div class="pdf-page"> <h1>Three generation neutrino mixing is compatible with all experiments</h1> <p>B. Hoeneisen and C. Marı´n</p> <p>Universidad San Francisco de Quito 2 February 2000</p> <h1>Abstract</h1> <p>We consider the minimal extension of the Standard Model with three generations of massive neutrinos that mix. We then determine the parameters of the model that satisfy all experimental constraints. PACS 14.60.Pq, 12.15.Ff</p> <p>Three observables in disagreement with the Standard Model of Quarks and Leptons are: i) A deficit of electron-type solar neutrinos; ii) A deficit of muon-type atmospheric neutrinos; and, possibly, iii) The observation of the apearance of in a beam of by the LSND Collaboration. The invisible width of the implies that the number of massless, or light Dirac, or light Majorana neutrino species is .[1] To account for these observations we consider the minimal extension of the Standard Model with three massive neutrinos that mix. The neutrino interaction eigenstates are superpositions of the neutrino mass eigenstates :</p> <p>We consider the “standard” parametrization of the unitary matrix [1]:</p> <p>where , , and . The probability that an ultrarelativistic neutrino produced as decays as</p> </div>	<div class="pdf-page"> <h1 class="pdf-title" data-x="249" data-y="218" data-width="498" data-height="62">Three generation neutrino mixing is compatible with all experiments</h1> <div class="pdf-discarded" data-x="23" data-y="267" data-width="37" data-height="398" style="opacity: 0.5;">arXiv:hep-ex/0002004v1 2 Feb 2000</div> <p class="pdf-text" data-x="354" data-y="303" data-width="285" data-height="21">B. Hoeneisen and C. Marı´n</p> <p class="pdf-text" data-x="351" data-y="338" data-width="291" data-height="36">Universidad San Francisco de Quito 2 February 2000</p> <h1 class="pdf-title" data-x="456" data-y="413" data-width="81" data-height="17">Abstract</h1> <p class="pdf-text" data-x="219" data-y="440" data-width="555" data-height="69">We consider the minimal extension of the Standard Model with three generations of massive neutrinos that mix. We then determine the parameters of the model that satisfy all experimental constraints. PACS 14.60.Pq, 12.15.Ff</p> <p class="pdf-text" data-x="168" data-y="526" data-width="656" data-height="168">Three observables in disagreement with the Standard Model of Quarks and Leptons are: i) A deficit of electron-type solar neutrinos; ii) A deficit of muon-type atmospheric neutrinos; and, possibly, iii) The observation of the apearance of in a beam of by the LSND Collaboration. The invisible width of the implies that the number of massless, or light Dirac, or light Majorana neutrino species is .[1] To account for these observations we consider the minimal extension of the Standard Model with three massive neutrinos that mix. The neutrino interaction eigenstates are superpositions of the neutrino mass eigenstates :</p> <p class="pdf-text" data-x="168" data-y="758" data-width="640" data-height="20">We consider the “standard” parametrization of the unitary matrix [1]:</p> <p class="pdf-text" data-x="168" data-y="883" data-width="655" data-height="38">where , , and . The probability that an ultrarelativistic neutrino produced as decays as</p> </div>	{ "type": [ "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "inline_equation", "inline_equation", "inline_equation", "text", "inline_equation", "inline_equation", "interline_equation", "inline_equation", "interline_equation", "inline_equation", "inline_equation" ], "coordinates": [ [ 245, 222, 747, 252 ], [ 274, 256, 719, 280 ], [ 354, 307, 635, 324 ], [ 351, 341, 640, 359 ], [ 430, 362, 562, 374 ], [ 455, 416, 538, 431 ], [ 220, 444, 774, 458 ], [ 219, 461, 774, 477 ], [ 217, 479, 746, 495 ], [ 219, 496, 419, 513 ], [ 199, 528, 823, 548 ], [ 168, 549, 824, 566 ], [ 168, 567, 823, 584 ], [ 168, 585, 823, 602 ], [ 170, 605, 821, 621 ], [ 170, 623, 823, 641 ], [ 170, 641, 823, 658 ], [ 168, 660, 823, 677 ], [ 168, 678, 612, 698 ], [ 418, 708, 575, 747 ], [ 168, 760, 804, 780 ], [ 168, 792, 836, 854 ], [ 168, 885, 824, 906 ], [ 170, 905, 819, 923 ] ], "content": [ "Three generation neutrino mixing is", "compatible with all experiments", "B. 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The", "probability that an ultrarelativistic neutrino produced as \\nu_{l} decays as \\nu_{l^{\\prime}}" ], "index": [ 0, 1, 2, 5, 6, 10, 16, 17, 18, 19, 26, 27, 28, 29, 30, 31, 32, 33, 34, 45, 65, 86, 108, 109 ] }	{ "coordinates": [], "index": [], "caption": [], "caption_coordinates": [] }	{ "type": [ "interline_equation", "interline_equation", "interline_equation", "interline_equation" ], "coordinates": [ [ 418, 708, 575, 747 ], [ 168, 792, 836, 854 ], [ 418, 708, 575, 747 ], [ 168, 792, 836, 854 ] ], "content": [ "", "", "\|\\nu_{l}\\rangle=\\sum_{m}U_{l m}\|\\nu_{m}\\rangle", "\\left(\\begin{array}{c}{{\\nu_{e}}}\\\\ {{\\nu_{\\mu}}}\\\\ {{\\nu_{\\tau}}}\\end{array}\\right)=\\left(\\begin{array}{c c c}{{c_{12}c_{13}}}&{{s_{12}c_{13}}}&{{s_{13}e^{-i\\delta}}}\\\\ {{-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\\delta}}}&{{c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\\delta}}}&{{s_{23}c_{13}}}\\\\ {{s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\\delta}}}&{{-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\\delta}}}&{{c_{23}c_{13}}}\\end{array}\\right)\\left(\\begin{array}{c}{{\\nu_{1}}}\\\\ {{\\nu_{2}}}\\\\ {{\\nu_{3}}}\\end{array}\\right)" ], "index": [ 0, 1, 2, 3 ] }	{ "type": [], "coordinates": [], "content": [], "index": [] }	{"document_info": {"total_pages": 7, "parse_type": "txt", "version": "1.0.1", "total_content_list_items": 37}, "content_statistics": {"total_blocks": 48, "total_lines": 162, "total_images": 9, "total_equations": 6, "total_tables": 9, "total_characters": 9294, "total_words": 1493, "block_type_distribution": {"title": 3, "text": 22, "interline_equation": 3, "discarded": 7, "table_body": 4, "table_caption": 1, "image_body": 3, "image_caption": 3, "list": 2}}, "model_statistics": {"total_layout_detections": 488, "avg_confidence": 0.9117827868852458, "min_confidence": 0.25, "max_confidence": 1.0}, "page_statistics": [{"page_number": 0, "page_size": [612.0, 792.0], "blocks": 11, "lines": 24, "images": 0, "equations": 4, "tables": 0, "characters": 1202, "words": 182, "discarded_blocks": 1}, {"page_number": 1, "page_size": [612.0, 792.0], "blocks": 8, "lines": 30, "images": 0, "equations": 2, "tables": 2, "characters": 1638, "words": 275, "discarded_blocks": 1}, {"page_number": 2, "page_size": [612.0, 792.0], "blocks": 6, "lines": 34, "images": 0, "equations": 0, "tables": 2, "characters": 2275, "words": 376, "discarded_blocks": 1}, {"page_number": 3, "page_size": [612.0, 792.0], "blocks": 7, "lines": 8, "images": 3, "equations": 0, "tables": 5, "characters": 249, "words": 37, "discarded_blocks": 1}, {"page_number": 4, "page_size": [612.0, 792.0], "blocks": 7, "lines": 29, "images": 3, "equations": 0, "tables": 0, "characters": 1840, "words": 288, "discarded_blocks": 1}, {"page_number": 5, "page_size": [612.0, 792.0], "blocks": 7, "lines": 37, "images": 3, "equations": 0, "tables": 0, "characters": 2089, "words": 334, "discarded_blocks": 1}, {"page_number": 6, "page_size": [612.0, 792.0], "blocks": 2, "lines": 0, "images": 0, "equations": 0, "tables": 0, "characters": 1, "words": 1, "discarded_blocks": 1}]}
0002004v1	1	[ 612, 792 ]	{ "type": [ "table_body", "text", "text", "interline_equation", "text", "text", "text", "discarded" ], "coordinates": [ [ 187, 161, 804, 298 ], [ 167, 323, 823, 417 ], [ 168, 440, 212, 461 ], [ 237, 473, 754, 513 ], [ 168, 519, 823, 800 ], [ 168, 801, 823, 911 ], [ 195, 914, 821, 932 ], [ 490, 954, 503, 967 ] ], "content": [ "", "Table 1: Observed solar electron-type neutrino flux, compared to the Stan- dard Solar Model (SSM) predictions asuming no neutrino mixing[7], and their ratio. The Solar Neutrino Unit (SNU) is captures per atom per second. For Kamiokande (Super Kamiokande) the flux is in units of at Earth above 7MeV (6.5MeV).", "is[1]:", "", "where and are the energy and traveling distance of , and is the mass of . We choose . This extension of the Standard Model introduces six parameters: , , , , and two mass-squared differences, e.g. and . We vary these parameters to minimize a . This has 14 terms obtained from the solar neutrino data summarized in Table 1, the atmospheric neutrino data shown in Table 2, and the LSND data[9]: for (here cor- responds to “large” , and corresponds to , see dis- cussion in [1]). Because one author[10] of the LSND Collaboration is in disagreement with the conclusion, and because the result has not been con- firmed by an independent experiment, we multiply the error by 1.5 and take . We require that the astrophysical, reactor and accelerator limits be satisfied. The most stringent of these limits are listed in Table 3.", "The has 8 degrees of freedom (14 terms minus 6 parameters). Varying the parameters we obtain minimums of , a few of which are listed in Table 4. With confidence the neutrino mass-squared differences lie within the dots shown in Figure 1. Note that one of the mass-squared differences is determined by the solar neutrino experiments and the other one by the atmospheric neutrino observations.", "If neutrinos have a hierarchy of masses (as the charged leptons, up quarks,", "2" ], "index": [ 0, 1, 2, 3, 4, 5, 6, 7 ] }	[{"type": "table", "img_path": "images/58f4e515357520a89a5c257eeb2486ac2b1e29825cfbe284d732c4c616013e28.jpg", "table_caption": [], "table_footnote": [], "table_body": "\n\n<html><body><table><tr><td>Experiment</td><td>Energy [MeV]</td><td>Observed flux (SNU)</td><td>SSM1 predic- tion (SNU)</td><td>Ratio</td></tr><tr><td>Homestake[2]</td><td>0.87</td><td>2.56 \u00b1 0.23</td><td>21-2 2</td><td>0.33\u00b10.05</td></tr><tr><td>Sage[3]</td><td>0.233 - 0.4</td><td>67\u00b18</td><td></td><td>0.52 \u00b1 0.07</td></tr><tr><td>Gallex[4]</td><td>0.233 - 0.4</td><td>78\u00b18</td><td></td><td>0.60 \u00b1 0.07</td></tr><tr><td>Kamiokande[5]</td><td>7-13</td><td>2.80 \u00b1 0.38</td><td></td><td>0.53 \u00b1 0.11</td></tr><tr><td>Super-Kam.[6]</td><td>6 - 13</td><td>2.42 +0.12 -0.09</td><td></td><td>0.46 \u00b1 0.08</td></tr></table></body></html>\n\n", "page_idx": 1}, {"type": "text", "text": "Table 1: Observed solar electron-type neutrino flux, compared to the Standard Solar Model (SSM) predictions asuming no neutrino mixing[7], and their ratio. The Solar Neutrino Unit (SNU) is $10^{-36}$ captures per atom per second. For Kamiokande (Super Kamiokande) the flux is in units of $10^{6}\\mathrm{cm}^{-2}\\mathrm{s}^{-1}$ at Earth above 7MeV (6.5MeV). ", "page_idx": 1}, {"type": "text", "text": "is[1]: ", "page_idx": 1}, {"type": "equation", "text": "$$\nP(\\nu_{l}\\to\\nu_{l^{\\prime}})=\|\\sum_{m}U_{l m}e x p(-i L M_{m}^{2}/2E)U_{l^{\\prime}m}^{*}\|^{2}=P(\\bar{\\nu}_{l^{\\prime}}\\to\\bar{\\nu}_{l})\n$$", "text_format": "latex", "page_idx": 1}, {"type": "text", "text": "where $E$ and $L$ are the energy and traveling distance of $\\nu_{l}$ , and $M_{m}$ is the mass of $\\nu_{m}$ . We choose $M_{1}\\,\\leq\\,M_{2}\\,\\leq\\,M_{3}$ . This extension of the Standard Model introduces six parameters: $S_{12}$ , $s_{23}$ , $s_{13}$ , $\\delta$ , and two mass-squared differences, e.g. $\\Delta M_{21}^{2}\\equiv M_{2}^{2}-M_{1}^{2}$ and $\\Delta M_{32}^{2}\\equiv M_{3}^{2}-M_{2}^{2}$ . We vary these parameters to minimize a $\\chi^{2}$ . This $\\chi^{2}$ has 14 terms obtained from the solar neutrino data summarized in Table 1, the atmospheric neutrino data shown in Table 2, and the LSND data[9]: $P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})\\,=\\,\\textstyle{\\frac{1}{2}}\\sin^{2}(2\\theta)\\,=\\,0.0031\\pm0.0013$ for $L[\\mathrm{km}]/E[\\mathrm{GeV}]\\!=[P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})]^{1/2}/1.27\\cdot\\Delta M^{2}[\\mathrm{eV^{2}}]\\!\\approx\\,0.73$ (here $\\sin^{2}(2\\theta)$ corresponds to \u201clarge\u201d $\\Delta M^{2}$ , and $\\Delta M^{2}$ corresponds to $\\sin^{2}(2\\theta)\\,=\\,1$ , see discussion in [1]). Because one author[10] of the LSND Collaboration is in disagreement with the conclusion, and because the result has not been confirmed by an independent experiment, we multiply the error by 1.5 and take $P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})=0.0031\\pm0.0020$ . We require that the astrophysical, reactor and accelerator limits be satisfied. The most stringent of these limits are listed in Table 3. ", "page_idx": 1}, {"type": "text", "text": "The $\\chi^{2}$ has 8 degrees of freedom (14 terms minus 6 parameters). Varying the parameters we obtain minimums of $\\chi^{2}$ , a few of which are listed in Table 4. With $90\\%$ confidence the neutrino mass-squared differences lie within the dots shown in Figure 1. Note that one of the mass-squared differences is determined by the solar neutrino experiments and the other one by the atmospheric neutrino observations. ", "page_idx": 1}, {"type": "text", "text": "If neutrinos have a hierarchy of masses (as the charged leptons, up quarks, ", "page_idx": 1}]	{"preproc_blocks": [{"type": "table", "bbox": [112, 125, 481, 231], "blocks": [{"type": "table_body", "bbox": [112, 125, 481, 231], "group_id": 0, "lines": [{"bbox": [112, 125, 481, 231], "spans": [{"bbox": [112, 125, 481, 231], "score": 0.976, "html": "<html><body><table><tr><td>Experiment</td><td>Energy [MeV]</td><td>Observed flux (SNU)</td><td>SSM1 predic- tion (SNU)</td><td>Ratio</td></tr><tr><td>Homestake[2]</td><td>0.87</td><td>2.56 \u00b1 0.23</td><td>21-2 2</td><td>0.33\u00b10.05</td></tr><tr><td>Sage[3]</td><td>0.233 - 0.4</td><td>67\u00b18</td><td></td><td>0.52 \u00b1 0.07</td></tr><tr><td>Gallex[4]</td><td>0.233 - 0.4</td><td>78\u00b18</td><td></td><td>0.60 \u00b1 0.07</td></tr><tr><td>Kamiokande[5]</td><td>7-13</td><td>2.80 \u00b1 0.38</td><td></td><td>0.53 \u00b1 0.11</td></tr><tr><td>Super-Kam.[6]</td><td>6 - 13</td><td>2.42 +0.12 -0.09</td><td></td><td>0.46 \u00b1 0.08</td></tr></table></body></html>", "type": "table", "image_path": "58f4e515357520a89a5c257eeb2486ac2b1e29825cfbe284d732c4c616013e28.jpg"}]}], "index": 1, "virtual_lines": [{"bbox": [112, 125, 481, 160.33333333333334], "spans": [], "index": 0}, {"bbox": [112, 160.33333333333334, 481, 195.66666666666669], "spans": [], "index": 1}, {"bbox": [112, 195.66666666666669, 481, 231.00000000000003], "spans": [], "index": 2}]}], "index": 1}, {"type": "text", "bbox": [100, 250, 492, 323], "lines": [{"bbox": [101, 252, 492, 268], "spans": [{"bbox": [101, 252, 492, 268], "score": 1.0, "content": "Table 1: Observed solar electron-type neutrino flux, compared to the Stan-", "type": "text"}], "index": 3}, {"bbox": [102, 268, 492, 281], "spans": [{"bbox": [102, 268, 492, 281], "score": 1.0, "content": "dard Solar Model (SSM) predictions asuming no neutrino mixing[7], and their", "type": "text"}], "index": 4}, {"bbox": [101, 281, 492, 297], "spans": [{"bbox": [101, 281, 308, 297], "score": 1.0, "content": "ratio. The Solar Neutrino Unit (SNU) is ", "type": "text"}, {"bbox": [309, 283, 336, 293], "score": 0.92, "content": "10^{-36}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [336, 281, 492, 297], "score": 1.0, "content": " captures per atom per second.", "type": "text"}], "index": 5}, {"bbox": [101, 296, 493, 310], "spans": [{"bbox": [101, 296, 417, 310], "score": 1.0, "content": "For Kamiokande (Super Kamiokande) the flux is in units of ", "type": "text"}, {"bbox": [417, 297, 476, 307], "score": 0.92, "content": "10^{6}\\mathrm{cm}^{-2}\\mathrm{s}^{-1}", "type": "inline_equation", "height": 10, "width": 59}, {"bbox": [477, 296, 493, 310], "score": 1.0, "content": " at", "type": "text"}], "index": 6}, {"bbox": [101, 311, 256, 325], "spans": [{"bbox": [101, 311, 256, 325], "score": 1.0, "content": "Earth above 7MeV (6.5MeV).", "type": "text"}], "index": 7}], "index": 5}, {"type": "text", "bbox": [101, 341, 127, 357], "lines": [{"bbox": [100, 342, 128, 361], "spans": [{"bbox": [100, 342, 128, 361], "score": 1.0, "content": "is[1]:", "type": "text"}], "index": 8}], "index": 8}, {"type": "interline_equation", "bbox": [142, 366, 451, 397], "lines": [{"bbox": [142, 366, 451, 397], "spans": [{"bbox": [142, 366, 451, 397], "score": 0.92, "content": "P(\\nu_{l}\\to\\nu_{l^{\\prime}})=\|\\sum_{m}U_{l m}e x p(-i L M_{m}^{2}/2E)U_{l^{\\prime}m}^{}\|^{2}=P(\\bar{\\nu}_{l^{\\prime}}\\to\\bar{\\nu}_{l})", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [101, 402, 492, 619], "lines": [{"bbox": [102, 406, 492, 418], "spans": [{"bbox": [102, 406, 134, 418], "score": 1.0, "content": "where", "type": "text"}, {"bbox": [135, 407, 144, 415], "score": 0.9, "content": "E", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [145, 406, 168, 418], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [168, 407, 177, 415], "score": 0.9, "content": "L", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [177, 406, 378, 418], "score": 1.0, "content": " are the energy and traveling distance of ", "type": "text"}, {"bbox": [378, 410, 387, 417], "score": 0.89, "content": "\\nu_{l}", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [387, 406, 414, 418], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [415, 407, 434, 417], "score": 0.93, "content": "M_{m}", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [434, 406, 492, 418], "score": 1.0, "content": " is the mass", "type": "text"}], "index": 10}, {"bbox": [101, 418, 492, 433], "spans": [{"bbox": [101, 418, 116, 433], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [116, 424, 130, 432], "score": 0.9, "content": "\\nu_{m}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [131, 418, 198, 433], "score": 1.0, "content": ". We choose ", "type": "text"}, {"bbox": [198, 421, 283, 432], "score": 0.95, "content": "M_{1}\\,\\leq\\,M_{2}\\,\\leq\\,M_{3}", "type": "inline_equation", "height": 11, "width": 85}, {"bbox": [283, 418, 492, 433], "score": 1.0, "content": ". This extension of the Standard Model", "type": "text"}], "index": 11}, {"bbox": [101, 434, 493, 449], "spans": [{"bbox": [101, 434, 241, 449], "score": 1.0, "content": "introduces six parameters: ", "type": "text"}, {"bbox": [241, 439, 256, 446], "score": 0.85, "content": "S_{12}", "type": "inline_equation", "height": 7, "width": 15}, {"bbox": [256, 434, 262, 449], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [263, 439, 277, 446], "score": 0.83, "content": "s_{23}", "type": "inline_equation", "height": 7, "width": 14}, {"bbox": [278, 434, 284, 449], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [284, 439, 299, 446], "score": 0.78, "content": "s_{13}", "type": "inline_equation", "height": 7, "width": 15}, {"bbox": [299, 434, 305, 449], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [306, 436, 312, 445], "score": 0.8, "content": "\\delta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [312, 434, 493, 449], "score": 1.0, "content": ", and two mass-squared differences,", "type": "text"}], "index": 12}, {"bbox": [99, 443, 495, 469], "spans": [{"bbox": [99, 443, 126, 469], "score": 1.0, "content": "e.g. ", "type": "text"}, {"bbox": [126, 449, 223, 462], "score": 0.93, "content": "\\Delta M_{21}^{2}\\equiv M_{2}^{2}-M_{1}^{2}", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [224, 443, 250, 469], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [250, 449, 347, 462], "score": 0.93, "content": "\\Delta M_{32}^{2}\\equiv M_{3}^{2}-M_{2}^{2}", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [348, 443, 495, 469], "score": 1.0, "content": ". We vary these parameters", "type": "text"}], "index": 13}, {"bbox": [100, 462, 493, 477], "spans": [{"bbox": [100, 462, 179, 477], "score": 1.0, "content": "to minimize a ", "type": "text"}, {"bbox": [179, 464, 191, 476], "score": 0.91, "content": "\\chi^{2}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [192, 462, 230, 477], "score": 1.0, "content": ". This ", "type": "text"}, {"bbox": [230, 463, 243, 476], "score": 0.91, "content": "\\chi^{2}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [243, 462, 493, 477], "score": 1.0, "content": " has 14 terms obtained from the solar neutrino", "type": "text"}], "index": 14}, {"bbox": [101, 477, 492, 491], "spans": [{"bbox": [101, 477, 492, 491], "score": 1.0, "content": "data summarized in Table 1, the atmospheric neutrino data shown in Table", "type": "text"}], "index": 15}, {"bbox": [100, 490, 493, 506], "spans": [{"bbox": [100, 490, 242, 506], "score": 1.0, "content": "2, and the LSND data[9]: ", "type": "text"}, {"bbox": [243, 492, 472, 505], "score": 0.86, "content": "P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})\\,=\\,\\textstyle{\\frac{1}{2}}\\sin^{2}(2\\theta)\\,=\\,0.0031\\pm0.0013", "type": "inline_equation", "height": 13, "width": 229}, {"bbox": [473, 490, 493, 506], "score": 1.0, "content": " for", "type": "text"}], "index": 16}, {"bbox": [102, 506, 492, 520], "spans": [{"bbox": [102, 506, 392, 520], "score": 0.86, "content": "L[\\mathrm{km}]/E[\\mathrm{GeV}]\\!=[P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})]^{1/2}/1.27\\cdot\\Delta M^{2}[\\mathrm{eV^{2}}]\\!\\approx\\,0.73", "type": "inline_equation", "height": 14, "width": 290}, {"bbox": [392, 506, 426, 520], "score": 1.0, "content": " (here ", "type": "text"}, {"bbox": [426, 506, 466, 520], "score": 0.85, "content": "\\sin^{2}(2\\theta)", "type": "inline_equation", "height": 14, "width": 40}, {"bbox": [467, 506, 492, 520], "score": 1.0, "content": " cor-", "type": "text"}], "index": 17}, {"bbox": [102, 521, 491, 534], "spans": [{"bbox": [102, 521, 208, 534], "score": 1.0, "content": "responds to \u201clarge\u201d ", "type": "text"}, {"bbox": [208, 521, 235, 531], "score": 0.89, "content": "\\Delta M^{2}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [236, 521, 267, 534], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [267, 521, 294, 531], "score": 0.9, "content": "\\Delta M^{2}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [295, 521, 379, 534], "score": 1.0, "content": " corresponds to ", "type": "text"}, {"bbox": [380, 521, 444, 534], "score": 0.93, "content": "\\sin^{2}(2\\theta)\\,=\\,1", "type": "inline_equation", "height": 13, "width": 64}, {"bbox": [444, 521, 491, 534], "score": 1.0, "content": ", see dis-", "type": "text"}], "index": 18}, {"bbox": [101, 536, 492, 548], "spans": [{"bbox": [101, 536, 492, 548], "score": 1.0, "content": "cussion in [1]). Because one author[10] of the LSND Collaboration is in", "type": "text"}], "index": 19}, {"bbox": [102, 551, 492, 563], "spans": [{"bbox": [102, 551, 492, 563], "score": 1.0, "content": "disagreement with the conclusion, and because the result has not been con-", "type": "text"}], "index": 20}, {"bbox": [101, 564, 492, 578], "spans": [{"bbox": [101, 564, 492, 578], "score": 1.0, "content": "firmed by an independent experiment, we multiply the error by 1.5 and take", "type": "text"}], "index": 21}, {"bbox": [102, 578, 492, 593], "spans": [{"bbox": [102, 579, 262, 592], "score": 0.93, "content": "P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})=0.0031\\pm0.0020", "type": "inline_equation", "height": 13, "width": 160}, {"bbox": [263, 578, 492, 593], "score": 1.0, "content": ". We require that the astrophysical, reactor", "type": "text"}], "index": 22}, {"bbox": [101, 593, 492, 606], "spans": [{"bbox": [101, 593, 492, 606], "score": 1.0, "content": "and accelerator limits be satisfied. The most stringent of these limits are", "type": "text"}], "index": 23}, {"bbox": [101, 608, 189, 620], "spans": [{"bbox": [101, 608, 189, 620], "score": 1.0, "content": "listed in Table 3.", "type": "text"}], "index": 24}], "index": 17}, {"type": "text", "bbox": [101, 620, 492, 705], "lines": [{"bbox": [119, 620, 492, 637], "spans": [{"bbox": [119, 620, 143, 637], "score": 1.0, "content": "The ", "type": "text"}, {"bbox": [144, 622, 155, 635], "score": 0.92, "content": "\\chi^{2}", "type": "inline_equation", "height": 13, "width": 11}, {"bbox": [156, 620, 492, 637], "score": 1.0, "content": " has 8 degrees of freedom (14 terms minus 6 parameters). Varying", "type": "text"}], "index": 25}, {"bbox": [102, 636, 492, 650], "spans": [{"bbox": [102, 636, 304, 650], "score": 1.0, "content": "the parameters we obtain minimums of ", "type": "text"}, {"bbox": [304, 636, 316, 649], "score": 0.92, "content": "\\chi^{2}", "type": "inline_equation", "height": 13, "width": 12}, {"bbox": [317, 636, 492, 650], "score": 1.0, "content": ", a few of which are listed in Table", "type": "text"}], "index": 26}, {"bbox": [101, 650, 492, 665], "spans": [{"bbox": [101, 650, 153, 665], "score": 1.0, "content": "4. With ", "type": "text"}, {"bbox": [153, 651, 176, 662], "score": 0.28, "content": "90\\%", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [176, 650, 492, 665], "score": 1.0, "content": " confidence the neutrino mass-squared differences lie within", "type": "text"}], "index": 27}, {"bbox": [101, 666, 492, 679], "spans": [{"bbox": [101, 666, 492, 679], "score": 1.0, "content": "the dots shown in Figure 1. Note that one of the mass-squared differences", "type": "text"}], "index": 28}, {"bbox": [101, 680, 492, 694], "spans": [{"bbox": [101, 680, 492, 694], "score": 1.0, "content": "is determined by the solar neutrino experiments and the other one by the", "type": "text"}], "index": 29}, {"bbox": [102, 695, 281, 708], "spans": [{"bbox": [102, 695, 281, 708], "score": 1.0, "content": "atmospheric neutrino observations.", "type": "text"}], "index": 30}], "index": 27.5}, {"type": "text", "bbox": [117, 707, 491, 721], "lines": [{"bbox": [118, 708, 490, 722], "spans": [{"bbox": [118, 708, 490, 722], "score": 1.0, "content": "If neutrinos have a hierarchy of masses (as the charged leptons, up quarks,", "type": "text"}], "index": 31}], "index": 31}], "layout_bboxes": [], "page_idx": 1, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [{"type": "table", "bbox": [112, 125, 481, 231], "blocks": [{"type": "table_body", "bbox": [112, 125, 481, 231], "group_id": 0, "lines": [{"bbox": [112, 125, 481, 231], "spans": [{"bbox": [112, 125, 481, 231], "score": 0.976, "html": "<html><body><table><tr><td>Experiment</td><td>Energy [MeV]</td><td>Observed flux (SNU)</td><td>SSM1 predic- tion (SNU)</td><td>Ratio</td></tr><tr><td>Homestake[2]</td><td>0.87</td><td>2.56 \u00b1 0.23</td><td>21-2 2</td><td>0.33\u00b10.05</td></tr><tr><td>Sage[3]</td><td>0.233 - 0.4</td><td>67\u00b18</td><td></td><td>0.52 \u00b1 0.07</td></tr><tr><td>Gallex[4]</td><td>0.233 - 0.4</td><td>78\u00b18</td><td></td><td>0.60 \u00b1 0.07</td></tr><tr><td>Kamiokande[5]</td><td>7-13</td><td>2.80 \u00b1 0.38</td><td></td><td>0.53 \u00b1 0.11</td></tr><tr><td>Super-Kam.[6]</td><td>6 - 13</td><td>2.42 +0.12 -0.09</td><td></td><td>0.46 \u00b1 0.08</td></tr></table></body></html>", "type": "table", "image_path": "58f4e515357520a89a5c257eeb2486ac2b1e29825cfbe284d732c4c616013e28.jpg"}]}], "index": 1, "virtual_lines": [{"bbox": [112, 125, 481, 160.33333333333334], "spans": [], "index": 0}, {"bbox": [112, 160.33333333333334, 481, 195.66666666666669], "spans": [], "index": 1}, {"bbox": [112, 195.66666666666669, 481, 231.00000000000003], "spans": [], "index": 2}]}], "index": 1}], "interline_equations": [{"type": "interline_equation", "bbox": [142, 366, 451, 397], "lines": [{"bbox": [142, 366, 451, 397], "spans": [{"bbox": [142, 366, 451, 397], "score": 0.92, "content": "P(\\nu_{l}\\to\\nu_{l^{\\prime}})=\|\\sum_{m}U_{l m}e x p(-i L M_{m}^{2}/2E)U_{l^{\\prime}m}^{}\|^{2}=P(\\bar{\\nu}_{l^{\\prime}}\\to\\bar{\\nu}_{l})", "type": "interline_equation"}], "index": 9}], "index": 9}], "discarded_blocks": [{"type": "discarded", "bbox": [293, 738, 301, 748], "lines": [{"bbox": [293, 739, 301, 751], "spans": [{"bbox": [293, 739, 301, 751], "score": 1.0, "content": "2", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "table", "bbox": [112, 125, 481, 231], "blocks": [{"type": "table_body", "bbox": [112, 125, 481, 231], "group_id": 0, "lines": [{"bbox": [112, 125, 481, 231], "spans": [{"bbox": [112, 125, 481, 231], "score": 0.976, "html": "<html><body><table><tr><td>Experiment</td><td>Energy [MeV]</td><td>Observed flux (SNU)</td><td>SSM1 predic- tion (SNU)</td><td>Ratio</td></tr><tr><td>Homestake[2]</td><td>0.87</td><td>2.56 \u00b1 0.23</td><td>21-2 2</td><td>0.33\u00b10.05</td></tr><tr><td>Sage[3]</td><td>0.233 - 0.4</td><td>67\u00b18</td><td></td><td>0.52 \u00b1 0.07</td></tr><tr><td>Gallex[4]</td><td>0.233 - 0.4</td><td>78\u00b18</td><td></td><td>0.60 \u00b1 0.07</td></tr><tr><td>Kamiokande[5]</td><td>7-13</td><td>2.80 \u00b1 0.38</td><td></td><td>0.53 \u00b1 0.11</td></tr><tr><td>Super-Kam.[6]</td><td>6 - 13</td><td>2.42 +0.12 -0.09</td><td></td><td>0.46 \u00b1 0.08</td></tr></table></body></html>", "type": "table", "image_path": "58f4e515357520a89a5c257eeb2486ac2b1e29825cfbe284d732c4c616013e28.jpg"}]}], "index": 1, "virtual_lines": [{"bbox": [112, 125, 481, 160.33333333333334], "spans": [], "index": 0}, {"bbox": [112, 160.33333333333334, 481, 195.66666666666669], "spans": [], "index": 1}, {"bbox": [112, 195.66666666666669, 481, 231.00000000000003], "spans": [], "index": 2}]}], "index": 1, "page_num": "page_1", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [100, 250, 492, 323], "lines": [{"bbox": [101, 252, 492, 268], "spans": [{"bbox": [101, 252, 492, 268], "score": 1.0, "content": "Table 1: Observed solar electron-type neutrino flux, compared to the Stan-", "type": "text"}], "index": 3}, {"bbox": [102, 268, 492, 281], "spans": [{"bbox": [102, 268, 492, 281], "score": 1.0, "content": "dard Solar Model (SSM) predictions asuming no neutrino mixing[7], and their", "type": "text"}], "index": 4}, {"bbox": [101, 281, 492, 297], "spans": [{"bbox": [101, 281, 308, 297], "score": 1.0, "content": "ratio. 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We choose ", "type": "text"}, {"bbox": [198, 421, 283, 432], "score": 0.95, "content": "M_{1}\\,\\leq\\,M_{2}\\,\\leq\\,M_{3}", "type": "inline_equation", "height": 11, "width": 85}, {"bbox": [283, 418, 492, 433], "score": 1.0, "content": ". 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We vary these parameters", "type": "text"}], "index": 13}, {"bbox": [100, 462, 493, 477], "spans": [{"bbox": [100, 462, 179, 477], "score": 1.0, "content": "to minimize a ", "type": "text"}, {"bbox": [179, 464, 191, 476], "score": 0.91, "content": "\\chi^{2}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [192, 462, 230, 477], "score": 1.0, "content": ". This ", "type": "text"}, {"bbox": [230, 463, 243, 476], "score": 0.91, "content": "\\chi^{2}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [243, 462, 493, 477], "score": 1.0, "content": " has 14 terms obtained from the solar neutrino", "type": "text"}], "index": 14}, {"bbox": [101, 477, 492, 491], "spans": [{"bbox": [101, 477, 492, 491], "score": 1.0, "content": "data summarized in Table 1, the atmospheric neutrino data shown in Table", "type": "text"}], "index": 15}, {"bbox": [100, 490, 493, 506], "spans": [{"bbox": [100, 490, 242, 506], "score": 1.0, "content": "2, and the LSND data[9]: ", "type": "text"}, {"bbox": [243, 492, 472, 505], "score": 0.86, "content": "P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})\\,=\\,\\textstyle{\\frac{1}{2}}\\sin^{2}(2\\theta)\\,=\\,0.0031\\pm0.0013", "type": "inline_equation", "height": 13, "width": 229}, {"bbox": [473, 490, 493, 506], "score": 1.0, "content": " for", "type": "text"}], "index": 16}, {"bbox": [102, 506, 492, 520], "spans": [{"bbox": [102, 506, 392, 520], "score": 0.86, "content": "L[\\mathrm{km}]/E[\\mathrm{GeV}]\\!=[P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})]^{1/2}/1.27\\cdot\\Delta M^{2}[\\mathrm{eV^{2}}]\\!\\approx\\,0.73", "type": "inline_equation", "height": 14, "width": 290}, {"bbox": [392, 506, 426, 520], "score": 1.0, "content": " (here ", "type": "text"}, {"bbox": [426, 506, 466, 520], "score": 0.85, "content": "\\sin^{2}(2\\theta)", "type": "inline_equation", "height": 14, "width": 40}, {"bbox": [467, 506, 492, 520], "score": 1.0, "content": " cor-", "type": "text"}], "index": 17}, {"bbox": [102, 521, 491, 534], "spans": [{"bbox": [102, 521, 208, 534], "score": 1.0, "content": "responds to \u201clarge\u201d ", "type": "text"}, {"bbox": [208, 521, 235, 531], "score": 0.89, "content": "\\Delta M^{2}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [236, 521, 267, 534], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [267, 521, 294, 531], "score": 0.9, "content": "\\Delta M^{2}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [295, 521, 379, 534], "score": 1.0, "content": " corresponds to ", "type": "text"}, {"bbox": [380, 521, 444, 534], "score": 0.93, "content": "\\sin^{2}(2\\theta)\\,=\\,1", "type": "inline_equation", "height": 13, "width": 64}, {"bbox": [444, 521, 491, 534], "score": 1.0, "content": ", see dis-", "type": "text"}], "index": 18}, {"bbox": [101, 536, 492, 548], "spans": [{"bbox": [101, 536, 492, 548], "score": 1.0, "content": "cussion in [1]). 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1.0, "text": ""}, {"category_id": 15, "poly": [1325.0, 824.0, 1371.0, 824.0, 1371.0, 863.0, 1325.0, 863.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [283.0, 866.0, 713.0, 866.0, 713.0, 903.0, 283.0, 903.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [278.0, 952.0, 357.0, 952.0, 357.0, 1004.0, 278.0, 1004.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [330.0, 1968.0, 1363.0, 1968.0, 1363.0, 2008.0, 330.0, 2008.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [815.0, 2054.0, 837.0, 2054.0, 837.0, 2087.0, 815.0, 2087.0], "score": 1.0, "text": ""}], "page_info": {"page_no": 1, "height": 2200, "width": 1700}}	Table 1: Observed solar electron-type neutrino flux, compared to the Stan- dard Solar Model (SSM) predictions asuming no neutrino mixing[7], and their ratio. The Solar Neutrino Unit (SNU) is captures per atom per second. For Kamiokande (Super Kamiokande) the flux is in units of at Earth above 7MeV (6.5MeV). is[1]: $$ P(\nu_{l}\to\nu_{l^{\prime}})=\|\sum_{m}U_{l m}e x p(-i L M_{m}^{2}/2E)U_{l^{\prime}m}^{*}\|^{2}=P(\bar{\nu}_{l^{\prime}}\to\bar{\nu}_{l}) $$ where and are the energy and traveling distance of , and is the mass of . We choose . This extension of the Standard Model introduces six parameters: , , , , and two mass-squared differences, e.g. and . We vary these parameters to minimize a . This has 14 terms obtained from the solar neutrino data summarized in Table 1, the atmospheric neutrino data shown in Table 2, and the LSND data[9]: for (here cor- responds to “large” , and corresponds to , see dis- cussion in [1]). Because one author[10] of the LSND Collaboration is in disagreement with the conclusion, and because the result has not been con- firmed by an independent experiment, we multiply the error by 1.5 and take . We require that the astrophysical, reactor and accelerator limits be satisfied. The most stringent of these limits are listed in Table 3. The has 8 degrees of freedom (14 terms minus 6 parameters). Varying the parameters we obtain minimums of , a few of which are listed in Table 4. With confidence the neutrino mass-squared differences lie within the dots shown in Figure 1. Note that one of the mass-squared differences is determined by the solar neutrino experiments and the other one by the atmospheric neutrino observations. If neutrinos have a hierarchy of masses (as the charged leptons, up quarks, 2	<div class="pdf-page"> <p>Table 1: Observed solar electron-type neutrino flux, compared to the Stan- dard Solar Model (SSM) predictions asuming no neutrino mixing[7], and their ratio. The Solar Neutrino Unit (SNU) is captures per atom per second. For Kamiokande (Super Kamiokande) the flux is in units of at Earth above 7MeV (6.5MeV).</p> <p>is[1]:</p> <p>where and are the energy and traveling distance of , and is the mass of . We choose . This extension of the Standard Model introduces six parameters: , , , , and two mass-squared differences, e.g. and . We vary these parameters to minimize a . This has 14 terms obtained from the solar neutrino data summarized in Table 1, the atmospheric neutrino data shown in Table 2, and the LSND data[9]: for (here cor- responds to “large” , and corresponds to , see dis- cussion in [1]). Because one author[10] of the LSND Collaboration is in disagreement with the conclusion, and because the result has not been con- firmed by an independent experiment, we multiply the error by 1.5 and take . We require that the astrophysical, reactor and accelerator limits be satisfied. The most stringent of these limits are listed in Table 3.</p> <p>The has 8 degrees of freedom (14 terms minus 6 parameters). Varying the parameters we obtain minimums of , a few of which are listed in Table 4. With confidence the neutrino mass-squared differences lie within the dots shown in Figure 1. Note that one of the mass-squared differences is determined by the solar neutrino experiments and the other one by the atmospheric neutrino observations.</p> <p>If neutrinos have a hierarchy of masses (as the charged leptons, up quarks,</p> </div>	<div class="pdf-page"> <p class="pdf-text" data-x="167" data-y="323" data-width="656" data-height="94">Table 1: Observed solar electron-type neutrino flux, compared to the Stan- dard Solar Model (SSM) predictions asuming no neutrino mixing[7], and their ratio. The Solar Neutrino Unit (SNU) is captures per atom per second. For Kamiokande (Super Kamiokande) the flux is in units of at Earth above 7MeV (6.5MeV).</p> <p class="pdf-text" data-x="168" data-y="440" data-width="44" data-height="21">is[1]:</p> <p class="pdf-text" data-x="168" data-y="519" data-width="655" data-height="281">where and are the energy and traveling distance of , and is the mass of . We choose . This extension of the Standard Model introduces six parameters: , , , , and two mass-squared differences, e.g. and . We vary these parameters to minimize a . This has 14 terms obtained from the solar neutrino data summarized in Table 1, the atmospheric neutrino data shown in Table 2, and the LSND data[9]: for (here cor- responds to “large” , and corresponds to , see dis- cussion in [1]). Because one author[10] of the LSND Collaboration is in disagreement with the conclusion, and because the result has not been con- firmed by an independent experiment, we multiply the error by 1.5 and take . We require that the astrophysical, reactor and accelerator limits be satisfied. The most stringent of these limits are listed in Table 3.</p> <p class="pdf-text" data-x="168" data-y="801" data-width="655" data-height="110">The has 8 degrees of freedom (14 terms minus 6 parameters). Varying the parameters we obtain minimums of , a few of which are listed in Table 4. With confidence the neutrino mass-squared differences lie within the dots shown in Figure 1. Note that one of the mass-squared differences is determined by the solar neutrino experiments and the other one by the atmospheric neutrino observations.</p> <p class="pdf-text" data-x="195" data-y="914" data-width="626" data-height="18">If neutrinos have a hierarchy of masses (as the charged leptons, up quarks,</p> <div class="pdf-discarded" data-x="490" data-y="954" data-width="13" data-height="13" style="opacity: 0.5;">2</div> </div>	{ "type": [ "text", "text", "text", "inline_equation", "inline_equation", "text", "text", "interline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "inline_equation", "inline_equation", "inline_equation", "text", "text", "text", "inline_equation", "text", "text", "inline_equation", "inline_equation", "inline_equation", "text", "text", "text", "text" ], "coordinates": [ [ 187, 161, 804, 298 ], [ 168, 325, 823, 346 ], [ 170, 346, 823, 363 ], [ 168, 363, 823, 384 ], [ 168, 382, 824, 400 ], [ 168, 402, 428, 420 ], [ 167, 442, 214, 466 ], [ 237, 473, 754, 513 ], [ 170, 524, 823, 540 ], [ 168, 540, 823, 559 ], [ 168, 561, 824, 580 ], [ 165, 572, 828, 606 ], [ 167, 597, 824, 616 ], [ 168, 616, 823, 634 ], [ 167, 633, 824, 654 ], [ 170, 654, 823, 672 ], [ 170, 673, 821, 690 ], [ 168, 693, 823, 708 ], [ 170, 712, 823, 727 ], [ 168, 729, 823, 747 ], [ 170, 747, 823, 766 ], [ 168, 766, 823, 783 ], [ 168, 786, 316, 801 ], [ 199, 801, 823, 823 ], [ 170, 822, 823, 840 ], [ 168, 840, 823, 859 ], [ 168, 861, 823, 877 ], [ 168, 879, 823, 897 ], [ 170, 898, 470, 915 ], [ 197, 915, 819, 933 ] ], "content": [ "", "Table 1: Observed solar electron-type neutrino flux, compared to the Stan-", "dard Solar Model (SSM) predictions asuming no neutrino mixing[7], and their", "ratio. The Solar Neutrino Unit (SNU) is 10^{-36} captures per atom per second.", "For Kamiokande (Super Kamiokande) the flux is in units of 10^{6}\\mathrm{cm}^{-2}\\mathrm{s}^{-1} at", "Earth above 7MeV (6.5MeV).", "is[1]:", "P(\\nu_{l}\\to\\nu_{l^{\\prime}})=\|\\sum_{m}U_{l m}e x p(-i L M_{m}^{2}/2E)U_{l^{\\prime}m}^{*}\|^{2}=P(\\bar{\\nu}_{l^{\\prime}}\\to\\bar{\\nu}_{l})", "where E and L are the energy and traveling distance of \\nu_{l} , and M_{m} is the mass", "of \\nu_{m} . We choose M_{1}\\,\\leq\\,M_{2}\\,\\leq\\,M_{3} . This extension of the Standard Model", "introduces six parameters: S_{12} , s_{23} , s_{13} , \\delta , and two mass-squared differences,", "e.g. \\Delta M_{21}^{2}\\equiv M_{2}^{2}-M_{1}^{2} and \\Delta M_{32}^{2}\\equiv M_{3}^{2}-M_{2}^{2} . We vary these parameters", "to minimize a \\chi^{2} . This \\chi^{2} has 14 terms obtained from the solar neutrino", "data summarized in Table 1, the atmospheric neutrino data shown in Table", "2, and the LSND data[9]: P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})\\,=\\,\\textstyle{\\frac{1}{2}}\\sin^{2}(2\\theta)\\,=\\,0.0031\\pm0.0013 for", "L[\\mathrm{km}]/E[\\mathrm{GeV}]\\!=[P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})]^{1/2}/1.27\\cdot\\Delta M^{2}[\\mathrm{eV^{2}}]\\!\\approx\\,0.73 (here \\sin^{2}(2\\theta) cor-", "responds to “large” \\Delta M^{2} , and \\Delta M^{2} corresponds to \\sin^{2}(2\\theta)\\,=\\,1 , see dis-", "cussion in [1]). Because one author[10] of the LSND Collaboration is in", "disagreement with the conclusion, and because the result has not been con-", "firmed by an independent experiment, we multiply the error by 1.5 and take", "P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})=0.0031\\pm0.0020 . We require that the astrophysical, reactor", "and accelerator limits be satisfied. The most stringent of these limits are", "listed in Table 3.", "The \\chi^{2} has 8 degrees of freedom (14 terms minus 6 parameters). Varying", "the parameters we obtain minimums of \\chi^{2} , a few of which are listed in Table", "4. With 90\\% confidence the neutrino mass-squared differences lie within", "the dots shown in Figure 1. Note that one of the mass-squared differences", "is determined by the solar neutrino experiments and the other one by the", "atmospheric neutrino observations.", "If neutrinos have a hierarchy of masses (as the charged leptons, up quarks," ], "index": [ 0, 1, 2, 3, 4, 5, 7, 14, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 45, 46, 47, 48, 49, 50, 74 ] }	{ "coordinates": [], "index": [], "caption": [], "caption_coordinates": [] }	{ "type": [ "interline_equation", "interline_equation" ], "coordinates": [ [ 237, 473, 754, 513 ], [ 237, 473, 754, 513 ] ], "content": [ "", "P(\\nu_{l}\\to\\nu_{l^{\\prime}})=\|\\sum_{m}U_{l m}e x p(-i L M_{m}^{2}/2E)U_{l^{\\prime}m}^{*}\|^{2}=P(\\bar{\\nu}_{l^{\\prime}}\\to\\bar{\\nu}_{l})" ], "index": [ 0, 1 ] }	{ "type": [ "table", "table_body" ], "coordinates": [ [ 187, 161, 804, 298 ], [ 187, 161, 804, 298 ] ], "content": [ "", "" ], "index": [ 0, 1 ] }	{"document_info": {"total_pages": 7, "parse_type": "txt", "version": "1.0.1", "total_content_list_items": 37}, "content_statistics": {"total_blocks": 48, "total_lines": 162, "total_images": 9, "total_equations": 6, "total_tables": 9, "total_characters": 9294, "total_words": 1493, "block_type_distribution": {"title": 3, "text": 22, "interline_equation": 3, "discarded": 7, "table_body": 4, "table_caption": 1, "image_body": 3, "image_caption": 3, "list": 2}}, "model_statistics": {"total_layout_detections": 488, "avg_confidence": 0.9117827868852458, "min_confidence": 0.25, "max_confidence": 1.0}, "page_statistics": [{"page_number": 0, "page_size": [612.0, 792.0], "blocks": 11, "lines": 24, "images": 0, "equations": 4, "tables": 0, "characters": 1202, "words": 182, "discarded_blocks": 1}, {"page_number": 1, "page_size": [612.0, 792.0], "blocks": 8, "lines": 30, "images": 0, "equations": 2, "tables": 2, "characters": 1638, "words": 275, "discarded_blocks": 1}, {"page_number": 2, "page_size": [612.0, 792.0], "blocks": 6, "lines": 34, "images": 0, "equations": 0, "tables": 2, "characters": 2275, "words": 376, "discarded_blocks": 1}, {"page_number": 3, "page_size": [612.0, 792.0], "blocks": 7, "lines": 8, "images": 3, "equations": 0, "tables": 5, "characters": 249, "words": 37, "discarded_blocks": 1}, {"page_number": 4, "page_size": [612.0, 792.0], "blocks": 7, "lines": 29, "images": 3, "equations": 0, "tables": 0, "characters": 1840, "words": 288, "discarded_blocks": 1}, {"page_number": 5, "page_size": [612.0, 792.0], "blocks": 7, "lines": 37, "images": 3, "equations": 0, "tables": 0, "characters": 2089, "words": 334, "discarded_blocks": 1}, {"page_number": 6, "page_size": [612.0, 792.0], "blocks": 2, "lines": 0, "images": 0, "equations": 0, "tables": 0, "characters": 1, "words": 1, "discarded_blocks": 1}]}
0002004v1	2	[ 612, 792 ]	{ "type": [ "table_body", "text", "text", "text", "text", "discarded" ], "coordinates": [ [ 302, 161, 689, 261 ], [ 168, 285, 823, 380 ], [ 168, 407, 823, 443 ], [ 167, 444, 823, 704 ], [ 167, 705, 824, 929 ], [ 490, 954, 503, 967 ] ], "content": [ "", "Table 2: Ratio of the numbers of observed and predicted electron-type and muon-type neutrinos as a function of the flight length-to-energy ratio as mea- sured by the Super-Kamiokande Collaboration.[8] Because of the uncertainty on the absolute neutrino flux and because is observed to be independent of , we divide the numbers in this table by 1.15 so that .", "or down quarks), then the “upper island” in Figure 1 applies, and 0.07eV, and , with large uncertainties.", "Note in Table 1 that the ratio of the observed-to-predicted solar neu- trino flux is significantly lower for the Homestake experiment than for Sage, Gallex, Kamiokande and Super-Kamiokande which are all compatible with 0.5. These latter experiments observe neutrinos within wide energy bands, while the chlorine detector in the Homestake mine observes monochromatic electron-type neutrinos from a line. For the Homestake experiment the spread in is due to the spread in , which in turn is due to the excen- tricity of the orbit of the Earth. Therefore the interference is coherent for up to oscillations from the Sun to the Earth (here is either or ). Due to this coherence at “small” it is possible to find acceptable solutions with as shown in Figure 1. For larger values of coherence is lost and we find solutions with which are unacceptable if the Homestake experimental and theoretical errors are correct.", "An important test of the model would be to observe seasonal variations of the neutrino flux of the line. If the lower ratio measured by the Homestake experiment is real, we expect that the electron-type neutrino flux of the line is near a minimum of the oscillation at the average Sun-Earth distance. In other words, there are an odd number of half-wavelengths from Sun to Earth. Then we expect a modulation of the neutrino flux with a period of half a year, with maximums occurring at the perihelion and aphelion of the Earth orbit. We see no statistically significant Fourier component of the time dependent Homestake data from 1970.281 to 1994.388.[12] In particular the amplitude relative to the mean of a Fourier component of period 0.5 years is . This observation implies that there are periods of oscillation from Sun to Earth at 90% confidence level. With a", "3" ], "index": [ 0, 1, 2, 3, 4, 5 ] }	[{"type": "table", "img_path": "images/77d9acb6b86f9c9a84ea1fbf27649993b710e71540165032c1de6955bb3136fb.jpg", "table_caption": [], "table_footnote": [], "table_body": "\n\n<html><body><table><tr><td>L/E [km/GeV]</td><td>Re</td><td>R\u03bc</td></tr><tr><td>10</td><td>1.20 \u00b1 0.15</td><td>1.00 \u00b1 0.15</td></tr><tr><td>100</td><td>1.20 \u00b1 0.15</td><td>0.85 \u00b1 0.12</td></tr><tr><td>1000</td><td>1.20 \u00b1 0.15</td><td>0.70\u00b1 0.10</td></tr><tr><td>10000</td><td>1.20 \u00b1 0.15</td><td>0.60\u00b1 0.08</td></tr></table></body></html>\n\n", "page_idx": 2}, {"type": "text", "text": "Table 2: Ratio of the numbers of observed and predicted electron-type and muon-type neutrinos as a function of the flight length-to-energy ratio as measured by the Super-Kamiokande Collaboration.[8] Because of the uncertainty on the absolute neutrino flux and because $R_{e}$ is observed to be independent of $L/E$ , we divide the numbers in this table by 1.15 so that $R_{e}\\approx1$ . ", "page_idx": 2}, {"type": "text", "text": "or down quarks), then the \u201cupper island\u201d in Figure 1 applies, and $M_{3}\\,\\approx$ 0.07eV, $M_{2}\\approx10^{-5}\\mathrm{eV}$ and $M_{1}<M_{2}$ , with large uncertainties. ", "page_idx": 2}, {"type": "text", "text": "Note in Table 1 that the ratio of the observed-to-predicted solar neutrino flux is significantly lower for the Homestake experiment than for Sage, Gallex, Kamiokande and Super-Kamiokande which are all compatible with 0.5. These latter experiments observe neutrinos within wide energy bands, while the chlorine detector in the Homestake mine observes monochromatic electron-type neutrinos from a $^7B e$ line. For the Homestake experiment the spread in $L/E$ is due to the spread in $L$ , which in turn is due to the excentricity of the orbit of the Earth. Therefore the interference is coherent for up to $L\\Delta M^{2}/(2E\\!\\cdot\\!2\\pi)\\approx30$ oscillations from the Sun to the Earth (here $\\Delta M^{2}$ is either $\\Delta M_{21}^{2}$ or $\\Delta M_{32}^{2}$ ). Due to this coherence at \u201csmall\u201d $\\Delta M^{2}$ it is possible to find acceptable solutions with $\\chi^{2}<13.4$ as shown in Figure 1. For larger values of $\\Delta M^{2}$ coherence is lost and we find solutions with $\\chi^{2}>18$ which are unacceptable if the Homestake experimental and theoretical errors are correct. ", "page_idx": 2}, {"type": "text", "text": "An important test of the model would be to observe seasonal variations of the neutrino flux of the $^7B e$ line. If the lower ratio measured by the Homestake experiment is real, we expect that the electron-type neutrino flux of the $^7B e$ line is near a minimum of the oscillation at the average Sun-Earth distance. In other words, there are an odd number of half-wavelengths from Sun to Earth. Then we expect a modulation of the $^7B e$ neutrino flux with a period of half a year, with maximums occurring at the perihelion and aphelion of the Earth orbit. We see no statistically significant Fourier component of the time dependent Homestake data from 1970.281 to 1994.388.[12] In particular the amplitude relative to the mean of a Fourier component of period 0.5 years is $0.09\\pm0.10$ . This observation implies that there are $\\leq8.5$ periods of oscillation from Sun to Earth at 90% confidence level. With a $\\chi^{2}$ ", "page_idx": 2}]	{"preproc_blocks": [{"type": "table", "bbox": [181, 125, 412, 202], "blocks": [{"type": "table_body", "bbox": [181, 125, 412, 202], "group_id": 0, "lines": [{"bbox": [181, 125, 412, 202], "spans": [{"bbox": [181, 125, 412, 202], "score": 0.976, "html": "<html><body><table><tr><td>L/E [km/GeV]</td><td>Re</td><td>R\u03bc</td></tr><tr><td>10</td><td>1.20 \u00b1 0.15</td><td>1.00 \u00b1 0.15</td></tr><tr><td>100</td><td>1.20 \u00b1 0.15</td><td>0.85 \u00b1 0.12</td></tr><tr><td>1000</td><td>1.20 \u00b1 0.15</td><td>0.70\u00b1 0.10</td></tr><tr><td>10000</td><td>1.20 \u00b1 0.15</td><td>0.60\u00b1 0.08</td></tr></table></body></html>", "type": "table", "image_path": "77d9acb6b86f9c9a84ea1fbf27649993b710e71540165032c1de6955bb3136fb.jpg"}]}], "index": 2.5, "virtual_lines": [{"bbox": [181, 125, 412, 139], "spans": [], "index": 0}, {"bbox": [181, 139, 412, 153], "spans": [], "index": 1}, {"bbox": [181, 153, 412, 167], "spans": [], "index": 2}, {"bbox": [181, 167, 412, 181], "spans": [], "index": 3}, {"bbox": [181, 181, 412, 195], "spans": [], "index": 4}, {"bbox": [181, 195, 412, 209], "spans": [], "index": 5}]}], "index": 2.5}, {"type": "text", "bbox": [101, 221, 492, 294], "lines": [{"bbox": [101, 224, 492, 238], "spans": [{"bbox": [101, 224, 492, 238], "score": 1.0, "content": "Table 2: Ratio of the numbers of observed and predicted electron-type and", "type": "text"}], "index": 6}, {"bbox": [101, 238, 492, 254], "spans": [{"bbox": [101, 238, 492, 254], "score": 1.0, "content": "muon-type neutrinos as a function of the flight length-to-energy ratio as mea-", "type": "text"}], "index": 7}, {"bbox": [102, 254, 491, 267], "spans": [{"bbox": [102, 254, 491, 267], "score": 1.0, "content": "sured by the Super-Kamiokande Collaboration.[8] Because of the uncertainty", "type": "text"}], "index": 8}, {"bbox": [102, 268, 491, 281], "spans": [{"bbox": [102, 268, 321, 281], "score": 1.0, "content": "on the absolute neutrino flux and because ", "type": "text"}, {"bbox": [321, 270, 334, 280], "score": 0.93, "content": "R_{e}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [335, 268, 491, 281], "score": 1.0, "content": " is observed to be independent", "type": "text"}], "index": 9}, {"bbox": [102, 282, 452, 296], "spans": [{"bbox": [102, 282, 115, 296], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [115, 283, 139, 296], "score": 0.94, "content": "L/E", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [139, 282, 411, 296], "score": 1.0, "content": ", we divide the numbers in this table by 1.15 so that ", "type": "text"}, {"bbox": [412, 284, 447, 294], "score": 0.92, "content": "R_{e}\\approx1", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [447, 282, 452, 296], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 8}, {"type": "text", "bbox": [101, 315, 492, 343], "lines": [{"bbox": [102, 317, 492, 331], "spans": [{"bbox": [102, 317, 460, 331], "score": 1.0, "content": "or down quarks), then the \u201cupper island\u201d in Figure 1 applies, and ", "type": "text"}, {"bbox": [460, 318, 492, 329], "score": 0.88, "content": "M_{3}\\,\\approx", "type": "inline_equation", "height": 11, "width": 32}], "index": 11}, {"bbox": [101, 331, 418, 345], "spans": [{"bbox": [101, 331, 144, 345], "score": 1.0, "content": "0.07eV, ", "type": "text"}, {"bbox": [144, 332, 213, 344], "score": 0.92, "content": "M_{2}\\approx10^{-5}\\mathrm{eV}", "type": "inline_equation", "height": 12, "width": 69}, {"bbox": [214, 331, 239, 345], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [240, 333, 288, 344], "score": 0.94, "content": "M_{1}<M_{2}", "type": "inline_equation", "height": 11, "width": 48}, {"bbox": [288, 331, 418, 345], "score": 1.0, "content": ", with large uncertainties.", "type": "text"}], "index": 12}], "index": 11.5}, {"type": "text", "bbox": [100, 344, 492, 545], "lines": [{"bbox": [119, 345, 491, 359], "spans": [{"bbox": [119, 345, 491, 359], "score": 1.0, "content": "Note in Table 1 that the ratio of the observed-to-predicted solar neu-", "type": "text"}], "index": 13}, {"bbox": [101, 359, 492, 375], "spans": [{"bbox": [101, 359, 492, 375], "score": 1.0, "content": "trino flux is significantly lower for the Homestake experiment than for Sage,", "type": "text"}], "index": 14}, {"bbox": [102, 376, 491, 388], "spans": [{"bbox": [102, 376, 491, 388], "score": 1.0, "content": "Gallex, Kamiokande and Super-Kamiokande which are all compatible with", "type": "text"}], "index": 15}, {"bbox": [101, 390, 491, 403], "spans": [{"bbox": [101, 390, 491, 403], "score": 1.0, "content": "0.5. These latter experiments observe neutrinos within wide energy bands,", "type": "text"}], "index": 16}, {"bbox": [101, 403, 492, 417], "spans": [{"bbox": [101, 403, 492, 417], "score": 1.0, "content": "while the chlorine detector in the Homestake mine observes monochromatic", "type": "text"}], "index": 17}, {"bbox": [101, 418, 492, 432], "spans": [{"bbox": [101, 418, 262, 432], "score": 1.0, "content": "electron-type neutrinos from a ", "type": "text"}, {"bbox": [262, 419, 282, 428], "score": 0.92, "content": "^7B e", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [282, 418, 492, 432], "score": 1.0, "content": " line. For the Homestake experiment the", "type": "text"}], "index": 18}, {"bbox": [101, 433, 492, 447], "spans": [{"bbox": [101, 433, 153, 447], "score": 1.0, "content": "spread in ", "type": "text"}, {"bbox": [153, 434, 177, 446], "score": 0.94, "content": "L/E", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [177, 433, 300, 447], "score": 1.0, "content": " is due to the spread in ", "type": "text"}, {"bbox": [300, 434, 309, 443], "score": 0.9, "content": "L", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [309, 433, 492, 447], "score": 1.0, "content": ", which in turn is due to the excen-", "type": "text"}], "index": 19}, {"bbox": [101, 446, 492, 462], "spans": [{"bbox": [101, 446, 492, 462], "score": 1.0, "content": "tricity of the orbit of the Earth. Therefore the interference is coherent for up", "type": "text"}], "index": 20}, {"bbox": [101, 461, 492, 475], "spans": [{"bbox": [101, 461, 115, 475], "score": 1.0, "content": "to ", "type": "text"}, {"bbox": [115, 462, 227, 475], "score": 0.93, "content": "L\\Delta M^{2}/(2E\\!\\cdot\\!2\\pi)\\approx30", "type": "inline_equation", "height": 13, "width": 112}, {"bbox": [227, 461, 452, 475], "score": 1.0, "content": " oscillations from the Sun to the Earth (here ", "type": "text"}, {"bbox": [453, 462, 479, 472], "score": 0.93, "content": "\\Delta M^{2}", "type": "inline_equation", "height": 10, "width": 26}, {"bbox": [480, 461, 492, 475], "score": 1.0, "content": " is", "type": "text"}], "index": 21}, {"bbox": [101, 475, 492, 490], "spans": [{"bbox": [101, 475, 135, 490], "score": 1.0, "content": "either ", "type": "text"}, {"bbox": [135, 477, 165, 489], "score": 0.94, "content": "\\Delta M_{21}^{2}", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [165, 475, 183, 490], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [183, 477, 213, 489], "score": 0.92, "content": "\\Delta M_{32}^{2}", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [214, 475, 396, 490], "score": 1.0, "content": "). Due to this coherence at \u201csmall\u201d ", "type": "text"}, {"bbox": [397, 477, 424, 486], "score": 0.92, "content": "\\Delta M^{2}", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [424, 475, 492, 490], "score": 1.0, "content": " it is possible", "type": "text"}], "index": 22}, {"bbox": [101, 490, 492, 504], "spans": [{"bbox": [101, 490, 272, 504], "score": 1.0, "content": "to find acceptable solutions with ", "type": "text"}, {"bbox": [273, 491, 322, 503], "score": 0.93, "content": "\\chi^{2}<13.4", "type": "inline_equation", "height": 12, "width": 49}, {"bbox": [322, 490, 492, 504], "score": 1.0, "content": " as shown in Figure 1. For larger", "type": "text"}], "index": 23}, {"bbox": [102, 505, 491, 518], "spans": [{"bbox": [102, 505, 151, 518], "score": 1.0, "content": "values of ", "type": "text"}, {"bbox": [152, 505, 179, 515], "score": 0.92, "content": "\\Delta M^{2}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [179, 505, 414, 518], "score": 1.0, "content": " coherence is lost and we find solutions with ", "type": "text"}, {"bbox": [415, 505, 457, 518], "score": 0.95, "content": "\\chi^{2}>18", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [457, 505, 491, 518], "score": 1.0, "content": " which", "type": "text"}], "index": 24}, {"bbox": [102, 519, 492, 532], "spans": [{"bbox": [102, 519, 492, 532], "score": 1.0, "content": "are unacceptable if the Homestake experimental and theoretical errors are", "type": "text"}], "index": 25}, {"bbox": [101, 535, 141, 546], "spans": [{"bbox": [101, 535, 141, 546], "score": 1.0, "content": "correct.", "type": "text"}], "index": 26}], "index": 19.5}, {"type": "text", "bbox": [100, 546, 493, 719], "lines": [{"bbox": [120, 549, 492, 561], "spans": [{"bbox": [120, 549, 492, 561], "score": 1.0, "content": "An important test of the model would be to observe seasonal variations", "type": "text"}], "index": 27}, {"bbox": [101, 562, 492, 576], "spans": [{"bbox": [101, 562, 248, 576], "score": 1.0, "content": "of the neutrino flux of the ", "type": "text"}, {"bbox": [249, 563, 269, 573], "score": 0.92, "content": "^7B e", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [269, 562, 492, 576], "score": 1.0, "content": " line. 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With a ", "type": "text"}, {"bbox": [479, 708, 491, 720], "score": 0.91, "content": "\\chi^{2}", "type": "inline_equation", "height": 12, "width": 12}], "index": 38}], "index": 32.5}], "layout_bboxes": [], "page_idx": 2, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [{"type": "table", "bbox": [181, 125, 412, 202], "blocks": [{"type": "table_body", "bbox": [181, 125, 412, 202], "group_id": 0, "lines": [{"bbox": [181, 125, 412, 202], "spans": [{"bbox": [181, 125, 412, 202], "score": 0.976, "html": "<html><body><table><tr><td>L/E [km/GeV]</td><td>Re</td><td>R\u03bc</td></tr><tr><td>10</td><td>1.20 \u00b1 0.15</td><td>1.00 \u00b1 0.15</td></tr><tr><td>100</td><td>1.20 \u00b1 0.15</td><td>0.85 \u00b1 0.12</td></tr><tr><td>1000</td><td>1.20 \u00b1 0.15</td><td>0.70\u00b1 0.10</td></tr><tr><td>10000</td><td>1.20 \u00b1 0.15</td><td>0.60\u00b1 0.08</td></tr></table></body></html>", "type": "table", "image_path": 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"<html><body><table><tr><td>L/E [km/GeV]</td><td>Re</td><td>R\u03bc</td></tr><tr><td>10</td><td>1.20 \u00b1 0.15</td><td>1.00 \u00b1 0.15</td></tr><tr><td>100</td><td>1.20 \u00b1 0.15</td><td>0.85 \u00b1 0.12</td></tr><tr><td>1000</td><td>1.20 \u00b1 0.15</td><td>0.70\u00b1 0.10</td></tr><tr><td>10000</td><td>1.20 \u00b1 0.15</td><td>0.60\u00b1 0.08</td></tr></table></body></html>", "type": "table", "image_path": "77d9acb6b86f9c9a84ea1fbf27649993b710e71540165032c1de6955bb3136fb.jpg"}]}], "index": 2.5, "virtual_lines": [{"bbox": [181, 125, 412, 139], "spans": [], "index": 0}, {"bbox": [181, 139, 412, 153], "spans": [], "index": 1}, {"bbox": [181, 153, 412, 167], "spans": [], "index": 2}, {"bbox": [181, 167, 412, 181], "spans": [], "index": 3}, {"bbox": [181, 181, 412, 195], "spans": [], "index": 4}, {"bbox": [181, 195, 412, 209], "spans": [], "index": 5}]}], "index": 2.5, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [101, 221, 492, 294], "lines": [{"bbox": [101, 224, 492, 238], "spans": [{"bbox": [101, 224, 492, 238], "score": 1.0, "content": "Table 2: Ratio of the numbers of observed and predicted electron-type and", "type": "text"}], "index": 6}, {"bbox": [101, 238, 492, 254], "spans": [{"bbox": [101, 238, 492, 254], "score": 1.0, "content": "muon-type neutrinos as a function of the flight length-to-energy ratio as mea-", "type": "text"}], "index": 7}, {"bbox": [102, 254, 491, 267], "spans": [{"bbox": [102, 254, 491, 267], "score": 1.0, "content": "sured by the Super-Kamiokande Collaboration.[8] Because of the uncertainty", "type": "text"}], "index": 8}, {"bbox": [102, 268, 491, 281], "spans": [{"bbox": [102, 268, 321, 281], "score": 1.0, "content": "on the absolute neutrino flux and because ", "type": "text"}, {"bbox": [321, 270, 334, 280], "score": 0.93, "content": "R_{e}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [335, 268, 491, 281], "score": 1.0, "content": " is observed to be independent", "type": "text"}], "index": 9}, {"bbox": [102, 282, 452, 296], "spans": [{"bbox": [102, 282, 115, 296], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [115, 283, 139, 296], "score": 0.94, "content": "L/E", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [139, 282, 411, 296], "score": 1.0, "content": ", we divide the numbers in this table by 1.15 so that ", "type": "text"}, {"bbox": [412, 284, 447, 294], "score": 0.92, "content": "R_{e}\\approx1", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [447, 282, 452, 296], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 8, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [101, 224, 492, 296]}, {"type": "text", "bbox": [101, 315, 492, 343], "lines": [{"bbox": [102, 317, 492, 331], "spans": [{"bbox": [102, 317, 460, 331], "score": 1.0, "content": "or down quarks), then the \u201cupper island\u201d in Figure 1 applies, and ", "type": "text"}, {"bbox": [460, 318, 492, 329], 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ratio of the observed-to-predicted solar neu-", "type": "text"}], "index": 13}, {"bbox": [101, 359, 492, 375], "spans": [{"bbox": [101, 359, 492, 375], "score": 1.0, "content": "trino flux is significantly lower for the Homestake experiment than for Sage,", "type": "text"}], "index": 14}, {"bbox": [102, 376, 491, 388], "spans": [{"bbox": [102, 376, 491, 388], "score": 1.0, "content": "Gallex, Kamiokande and Super-Kamiokande which are all compatible with", "type": "text"}], "index": 15}, {"bbox": [101, 390, 491, 403], "spans": [{"bbox": [101, 390, 491, 403], "score": 1.0, "content": "0.5. These latter experiments observe neutrinos within wide energy bands,", "type": "text"}], "index": 16}, {"bbox": [101, 403, 492, 417], "spans": [{"bbox": [101, 403, 492, 417], "score": 1.0, "content": "while the chlorine detector in the Homestake mine observes monochromatic", "type": "text"}], "index": 17}, {"bbox": [101, 418, 492, 432], "spans": [{"bbox": [101, 418, 262, 432], "score": 1.0, "content": "electron-type neutrinos from a ", "type": "text"}, {"bbox": [262, 419, 282, 428], "score": 0.92, "content": "^7B e", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [282, 418, 492, 432], "score": 1.0, "content": " line. For the Homestake experiment the", "type": "text"}], "index": 18}, {"bbox": [101, 433, 492, 447], "spans": [{"bbox": [101, 433, 153, 447], "score": 1.0, "content": "spread in ", "type": "text"}, {"bbox": [153, 434, 177, 446], "score": 0.94, "content": "L/E", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [177, 433, 300, 447], "score": 1.0, "content": " is due to the spread in ", "type": "text"}, {"bbox": [300, 434, 309, 443], "score": 0.9, "content": "L", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [309, 433, 492, 447], "score": 1.0, "content": ", which in turn is due to the excen-", "type": "text"}], "index": 19}, {"bbox": [101, 446, 492, 462], "spans": [{"bbox": [101, 446, 492, 462], "score": 1.0, "content": "tricity of the orbit of the Earth. 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For larger", "type": "text"}], "index": 23}, {"bbox": [102, 505, 491, 518], "spans": [{"bbox": [102, 505, 151, 518], "score": 1.0, "content": "values of ", "type": "text"}, {"bbox": [152, 505, 179, 515], "score": 0.92, "content": "\\Delta M^{2}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [179, 505, 414, 518], "score": 1.0, "content": " coherence is lost and we find solutions with ", "type": "text"}, {"bbox": [415, 505, 457, 518], "score": 0.95, "content": "\\chi^{2}>18", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [457, 505, 491, 518], "score": 1.0, "content": " which", "type": "text"}], "index": 24}, {"bbox": [102, 519, 492, 532], "spans": [{"bbox": [102, 519, 492, 532], "score": 1.0, "content": "are unacceptable if the Homestake experimental and theoretical errors are", "type": "text"}], "index": 25}, {"bbox": [101, 535, 141, 546], "spans": [{"bbox": [101, 535, 141, 546], "score": 1.0, "content": "correct.", "type": "text"}], "index": 26}], "index": 19.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [101, 345, 492, 546]}, {"type": "text", "bbox": [100, 546, 493, 719], "lines": [{"bbox": [120, 549, 492, 561], "spans": [{"bbox": [120, 549, 492, 561], "score": 1.0, "content": "An important test of the model would be to observe seasonal variations", "type": "text"}], "index": 27}, {"bbox": [101, 562, 492, 576], "spans": [{"bbox": [101, 562, 248, 576], "score": 1.0, "content": "of the neutrino flux of the ", "type": "text"}, {"bbox": [249, 563, 269, 573], "score": 0.92, "content": "^7B e", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [269, 562, 492, 576], "score": 1.0, "content": " line. 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Super-Kamiokande Collaboration.[8] Because of the uncertainty on the absolute neutrino flux and because is observed to be independent of , we divide the numbers in this table by 1.15 so that . or down quarks), then the “upper island” in Figure 1 applies, and 0.07eV, and , with large uncertainties. Note in Table 1 that the ratio of the observed-to-predicted solar neu- trino flux is significantly lower for the Homestake experiment than for Sage, Gallex, Kamiokande and Super-Kamiokande which are all compatible with 0.5. These latter experiments observe neutrinos within wide energy bands, while the chlorine detector in the Homestake mine observes monochromatic electron-type neutrinos from a line. For the Homestake experiment the spread in is due to the spread in , which in turn is due to the excen- tricity of the orbit of the Earth. Therefore the interference is coherent for up to oscillations from the Sun to the Earth (here is either or ). Due to this coherence at “small” it is possible to find acceptable solutions with as shown in Figure 1. For larger values of coherence is lost and we find solutions with which are unacceptable if the Homestake experimental and theoretical errors are correct. An important test of the model would be to observe seasonal variations of the neutrino flux of the line. If the lower ratio measured by the Homestake experiment is real, we expect that the electron-type neutrino flux of the line is near a minimum of the oscillation at the average Sun-Earth distance. In other words, there are an odd number of half-wavelengths from Sun to Earth. Then we expect a modulation of the neutrino flux with a period of half a year, with maximums occurring at the perihelion and aphelion of the Earth orbit. We see no statistically significant Fourier component of the time dependent Homestake data from 1970.281 to 1994.388.[12] In particular the amplitude relative to the mean of a Fourier component of period 0.5 years is . This observation implies that there are periods of oscillation from Sun to Earth at 90% confidence level. With a 3	<div class="pdf-page"> <p>Table 2: Ratio of the numbers of observed and predicted electron-type and muon-type neutrinos as a function of the flight length-to-energy ratio as mea- sured by the Super-Kamiokande Collaboration.[8] Because of the uncertainty on the absolute neutrino flux and because is observed to be independent of , we divide the numbers in this table by 1.15 so that .</p> <p>or down quarks), then the “upper island” in Figure 1 applies, and 0.07eV, and , with large uncertainties.</p> <p>Note in Table 1 that the ratio of the observed-to-predicted solar neu- trino flux is significantly lower for the Homestake experiment than for Sage, Gallex, Kamiokande and Super-Kamiokande which are all compatible with 0.5. These latter experiments observe neutrinos within wide energy bands, while the chlorine detector in the Homestake mine observes monochromatic electron-type neutrinos from a line. For the Homestake experiment the spread in is due to the spread in , which in turn is due to the excen- tricity of the orbit of the Earth. Therefore the interference is coherent for up to oscillations from the Sun to the Earth (here is either or ). Due to this coherence at “small” it is possible to find acceptable solutions with as shown in Figure 1. For larger values of coherence is lost and we find solutions with which are unacceptable if the Homestake experimental and theoretical errors are correct.</p> <p>An important test of the model would be to observe seasonal variations of the neutrino flux of the line. If the lower ratio measured by the Homestake experiment is real, we expect that the electron-type neutrino flux of the line is near a minimum of the oscillation at the average Sun-Earth distance. In other words, there are an odd number of half-wavelengths from Sun to Earth. Then we expect a modulation of the neutrino flux with a period of half a year, with maximums occurring at the perihelion and aphelion of the Earth orbit. We see no statistically significant Fourier component of the time dependent Homestake data from 1970.281 to 1994.388.[12] In particular the amplitude relative to the mean of a Fourier component of period 0.5 years is . This observation implies that there are periods of oscillation from Sun to Earth at 90% confidence level. With a</p> </div>	<div class="pdf-page"> <p class="pdf-text" data-x="168" data-y="285" data-width="655" data-height="95">Table 2: Ratio of the numbers of observed and predicted electron-type and muon-type neutrinos as a function of the flight length-to-energy ratio as mea- sured by the Super-Kamiokande Collaboration.[8] Because of the uncertainty on the absolute neutrino flux and because is observed to be independent of , we divide the numbers in this table by 1.15 so that .</p> <p class="pdf-text" data-x="168" data-y="407" data-width="655" data-height="36">or down quarks), then the “upper island” in Figure 1 applies, and 0.07eV, and , with large uncertainties.</p> <p class="pdf-text" data-x="167" data-y="444" data-width="656" data-height="260">Note in Table 1 that the ratio of the observed-to-predicted solar neu- trino flux is significantly lower for the Homestake experiment than for Sage, Gallex, Kamiokande and Super-Kamiokande which are all compatible with 0.5. These latter experiments observe neutrinos within wide energy bands, while the chlorine detector in the Homestake mine observes monochromatic electron-type neutrinos from a line. For the Homestake experiment the spread in is due to the spread in , which in turn is due to the excen- tricity of the orbit of the Earth. Therefore the interference is coherent for up to oscillations from the Sun to the Earth (here is either or ). Due to this coherence at “small” it is possible to find acceptable solutions with as shown in Figure 1. For larger values of coherence is lost and we find solutions with which are unacceptable if the Homestake experimental and theoretical errors are correct.</p> <p class="pdf-text" data-x="167" data-y="705" data-width="657" data-height="224">An important test of the model would be to observe seasonal variations of the neutrino flux of the line. If the lower ratio measured by the Homestake experiment is real, we expect that the electron-type neutrino flux of the line is near a minimum of the oscillation at the average Sun-Earth distance. In other words, there are an odd number of half-wavelengths from Sun to Earth. Then we expect a modulation of the neutrino flux with a period of half a year, with maximums occurring at the perihelion and aphelion of the Earth orbit. We see no statistically significant Fourier component of the time dependent Homestake data from 1970.281 to 1994.388.[12] In particular the amplitude relative to the mean of a Fourier component of period 0.5 years is . This observation implies that there are periods of oscillation from Sun to Earth at 90% confidence level. With a</p> <div class="pdf-discarded" data-x="490" data-y="954" data-width="13" data-height="13" style="opacity: 0.5;">3</div> </div>	{ "type": [ "text", "text", "text", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "text", "text", "text", "text", "inline_equation", "inline_equation", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "text", "text", "inline_equation", "text", "inline_equation", "text", "inline_equation", "text", "text", "text", "text", "inline_equation", "inline_equation" ], "coordinates": [ [ 302, 161, 689, 261 ], [ 168, 289, 823, 307 ], [ 168, 307, 823, 328 ], [ 170, 328, 821, 345 ], [ 170, 346, 821, 363 ], [ 170, 364, 756, 382 ], [ 170, 409, 823, 427 ], [ 168, 427, 699, 446 ], [ 199, 446, 821, 464 ], [ 168, 464, 823, 484 ], [ 170, 486, 821, 501 ], [ 168, 504, 821, 521 ], [ 168, 521, 823, 539 ], [ 168, 540, 823, 558 ], [ 168, 559, 823, 577 ], [ 168, 576, 823, 597 ], [ 168, 596, 823, 614 ], [ 168, 614, 823, 633 ], [ 168, 633, 823, 651 ], [ 170, 652, 821, 669 ], [ 170, 671, 823, 687 ], [ 168, 691, 235, 705 ], [ 200, 709, 823, 725 ], [ 168, 726, 823, 744 ], [ 168, 746, 823, 764 ], [ 168, 764, 823, 782 ], [ 168, 782, 823, 800 ], [ 168, 801, 824, 818 ], [ 167, 821, 824, 839 ], [ 168, 839, 823, 857 ], [ 168, 858, 823, 875 ], [ 168, 877, 826, 893 ], [ 168, 896, 821, 914 ], [ 168, 912, 821, 932 ] ], "content": [ "", "Table 2: Ratio of the numbers of observed and predicted electron-type and", "muon-type neutrinos as a function of the flight length-to-energy ratio as mea-", "sured by the Super-Kamiokande Collaboration.[8] Because of the uncertainty", "on the absolute neutrino flux and because R_{e} is observed to be independent", "of L/E , we divide the numbers in this table by 1.15 so that R_{e}\\approx1 .", "or down quarks), then the “upper island” in Figure 1 applies, and M_{3}\\,\\approx", "0.07eV, M_{2}\\approx10^{-5}\\mathrm{eV} and M_{1}<M_{2} , with large uncertainties.", "Note in Table 1 that the ratio of the observed-to-predicted solar neu-", "trino flux is significantly lower for the Homestake experiment than for Sage,", "Gallex, Kamiokande and Super-Kamiokande which are all compatible with", "0.5. These latter experiments observe neutrinos within wide energy bands,", "while the chlorine detector in the Homestake mine observes monochromatic", "electron-type neutrinos from a ^7B e line. For the Homestake experiment the", "spread in L/E is due to the spread in L , which in turn is due to the excen-", "tricity of the orbit of the Earth. Therefore the interference is coherent for up", "to L\\Delta M^{2}/(2E\\!\\cdot\\!2\\pi)\\approx30 oscillations from the Sun to the Earth (here \\Delta M^{2} is", "either \\Delta M_{21}^{2} or \\Delta M_{32}^{2} ). Due to this coherence at “small” \\Delta M^{2} it is possible", "to find acceptable solutions with \\chi^{2}<13.4 as shown in Figure 1. For larger", "values of \\Delta M^{2} coherence is lost and we find solutions with \\chi^{2}>18 which", "are unacceptable if the Homestake experimental and theoretical errors are", "correct.", "An important test of the model would be to observe seasonal variations", "of the neutrino flux of the ^7B e line. If the lower ratio measured by the", "Homestake experiment is real, we expect that the electron-type neutrino flux", "of the ^7B e line is near a minimum of the oscillation at the average Sun-Earth", "distance. In other words, there are an odd number of half-wavelengths from", "Sun to Earth. Then we expect a modulation of the ^7B e neutrino flux with a", "period of half a year, with maximums occurring at the perihelion and aphelion", "of the Earth orbit. We see no statistically significant Fourier component", "of the time dependent Homestake data from 1970.281 to 1994.388.[12] In", "particular the amplitude relative to the mean of a Fourier component of", "period 0.5 years is 0.09\\pm0.10 . This observation implies that there are \\leq8.5", "periods of oscillation from Sun to Earth at 90% confidence level. 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0002004v1	3	[ 612, 792 ]	{ "type": [ "table_body", "text", "table_body", "table_caption", "image_body", "image_caption", "discarded" ], "coordinates": [ [ 271, 182, 722, 374 ], [ 168, 399, 824, 438 ], [ 205, 486, 788, 566 ], [ 204, 593, 788, 612 ], [ 297, 650, 679, 852 ], [ 167, 855, 826, 894 ], [ 490, 954, 505, 968 ] ], "content": [ "", "Table 3: Limits on the mixing probabilities from astrophysical, accelerator and reactor experiments.[1]", "", "Table 4: Parameters at local minima of for 8 degrees of freedom.", "", "Figure 1: The mass-squared differences lie within the dots with confidence.", "4" ], "index": [ 0, 1, 2, 3, 4, 5, 6 ] }	[{"type": "table", "img_path": "images/e34693f6a4630b6d50b7fea14902eda53eacb639273a3b7d28541f4a689db91c.jpg", "table_caption": [], "table_footnote": [], "table_body": "\n\n<html><body><table><tr><td>Probability</td><td>L/E [km/GeV]</td></tr><tr><td>P(ve \u2192 ve) > 0.99 P(v\u03bc\u2192v\u03bc) >0.99 P(v\u03bc(v\u03bc)\u2192 ve(ve)) < 0.90 \u00b7 10-3 P(v\u03bc\u2192v)< 0.002 P(v\u03bc \u2190\u2192v)<0.35 P(v\u03bc \u2192v)< 0.022 P(ve \u2192v)< 0.125 P(ve \u2190> v\u03bc)< 0.25 P(ve \u2192ve) > 0.75[11]</td><td>88 0.34 0.31 0.039 31 0.053 0.031 40 232</td></tr></table></body></html>\n\n", "page_idx": 3}, {"type": "text", "text": "Table 3: Limits on the mixing probabilities from astrophysical, accelerator and reactor experiments.[1] ", "page_idx": 3}, {"type": "table", "img_path": "images/44d04858a3b47b2947d8c65076bdf3a711a7804e0d9fd409e81311b27e95fc98.jpg", "table_caption": ["Table 4: Parameters at local minima of $\\chi^{2}$ for 8 degrees of freedom. 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{"category_id": 15, "poly": [284.0, 909.0, 673.0, 909.0, 673.0, 946.0, 284.0, 946.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [283.0, 866.0, 1367.0, 866.0, 1367.0, 904.0, 283.0, 904.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [284.0, 909.0, 673.0, 909.0, 673.0, 946.0, 284.0, 946.0], "score": 1.0, "text": ""}], "page_info": {"page_no": 3, "height": 2200, "width": 1700}}	Table 3: Limits on the mixing probabilities from astrophysical, accelerator and reactor experiments.[1] Table 4: Parameters at local minima of for 8 degrees of freedom. ``` Table 4: Parameters at local minima of for 8 degrees of freedom. ``` Figure 1: The mass-squared differences lie within the dots with confidence. ![Image]() Figure 1: The mass-squared differences lie within the dots with confidence. 4	<div class="pdf-page"> <p>Table 3: Limits on the mixing probabilities from astrophysical, accelerator and reactor experiments.[1]</p> <h3>Table 4: Parameters at local minima of for 8 degrees of freedom.</h3> <em>Figure 1: The mass-squared differences lie within the dots with confidence.</em> </div>	<div class="pdf-page"> <p class="pdf-text" data-x="168" data-y="399" data-width="656" data-height="39">Table 3: Limits on the mixing probabilities from astrophysical, accelerator and reactor experiments.[1]</p> <caption class="pdf-table-caption" data-x="204" data-y="593" data-width="584" data-height="19">Table 4: Parameters at local minima of for 8 degrees of freedom.</caption> <figcaption class="pdf-image-caption" data-x="167" data-y="855" data-width="659" data-height="39">Figure 1: The mass-squared differences lie within the dots with confidence.</figcaption> <div class="pdf-discarded" data-x="490" data-y="954" data-width="15" data-height="14" style="opacity: 0.5;">4</div> </div>	{ "type": [ "text", "text", "text", "text", "inline_equation", "text", "inline_equation", "inline_equation" ], "coordinates": [ [ 271, 182, 722, 374 ], [ 168, 402, 823, 420 ], [ 170, 422, 404, 439 ], 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0002004v1	4	[ 612, 792 ]	{ "type": [ "image_body", "image_caption", "text", "text", "text", "text", "discarded" ], "coordinates": [ [ 299, 153, 680, 352 ], [ 167, 358, 824, 433 ], [ 170, 473, 821, 528 ], [ 168, 530, 823, 603 ], [ 168, 605, 823, 696 ], [ 168, 698, 824, 924 ], [ 490, 954, 503, 967 ] ], "content": [ "", "Figure 2: Detail of the “upper island” of Figure 1 for the fit with 116 degrees of freedom (see text) at confidence level. The “lower island” is symmet- rical. The vertical bands correspond, from left to right, to 2.5, 3.5, 4.5, 6.5 and 7.5 oscillations from Sun to Earth of the line.", "with 116 degrees of freedom, including the 8 discussed earlier plus the 108 measurements by the Homestake Collaboration from 1970.281 to 1994.388[12] we obtain the allowed region shown in Figure 2.", "The reliability of depends on the correctness of the error as- signed to the Homestake observed-to-predicted flux ratio. For example, if the Homestake error listed in Table 1 is doubled we obtain the solutions shown in Figure 3.", "In view of the preceeding results let us assume that neutrinos indeed have mass. The question then arizes wether neutrinos are distinct from an- tineutrinos (Dirac neutrinos) or wether neutrinos are their own antiparticles (Majorana neutrinos). This latter possibility arizes because neutrinos have no electric charge.", "Let us consider Big-Bang nucleosynthesis that determines the abundances of the light elements , , and . These abundances are determined by the temperatures of freezout when the reaction rates become comparable to the expansion rate . Here is the equivalent number of massless neutrino flavors that are ultrarelativistic at and are still in thermal equilibrium with photons and electrons at that temperature. The calculated abundances of the light elements are in agreement with observations if at confidence level.[1] For three generations of Majorana neutrinos, . For three generations of Dirac neutrinos, while in thermal equilibrium. However, in the Standard Model only the left-handed component of neutrinos couple to , and . Right-handed neutrinos are not in thermal equilibrium at", "5" ], "index": [ 0, 1, 2, 3, 4, 5, 6 ] }	[{"type": "image", "img_path": "images/49ddd88f34f410230ca5004ca24c8d7cd3aae9d8c2efdbfc0b80f607e482766c.jpg", "img_caption": ["Figure 2: Detail of the \u201cupper island\u201d of Figure 1 for the fit with 116 degrees of freedom (see text) at $90\\%$ confidence level. The \u201clower island\u201d is symmetrical. The vertical bands correspond, from left to right, to 2.5, 3.5, 4.5, 6.5 and 7.5 oscillations from Sun to Earth of the $^7B e$ line. "], "img_footnote": [], "page_idx": 4}, {"type": "text", "text": "with 116 degrees of freedom, including the 8 discussed earlier plus the 108 measurements by the Homestake Collaboration from 1970.281 to 1994.388[12] we obtain the allowed region shown in Figure 2. ", "page_idx": 4}, {"type": "text", "text": "The reliability of $M_{2}^{2}\\mathrm{~-~}M_{1}^{2}$ depends on the correctness of the error assigned to the Homestake observed-to-predicted flux ratio. For example, if the Homestake error listed in Table 1 is doubled we obtain the solutions shown in Figure 3. ", "page_idx": 4}, {"type": "text", "text": "In view of the preceeding results let us assume that neutrinos indeed have mass. The question then arizes wether neutrinos are distinct from antineutrinos (Dirac neutrinos) or wether neutrinos are their own antiparticles (Majorana neutrinos). This latter possibility arizes because neutrinos have no electric charge. ", "page_idx": 4}, {"type": "text", "text": "Let us consider Big-Bang nucleosynthesis that determines the abundances of the light elements $D$ , $^{3}H e$ , $^4H e$ and $^7L i$ . These abundances are determined by the temperatures of freezout $T_{f}\\approx1M e V$ when the reaction rates $\\propto T_{f}^{5}$ become comparable to the expansion rate $\\propto T_{f}^{2}\\times(5.5+\\textstyle{\\frac{7}{4}}{N_{\\nu}})^{1/2}$ . Here $N_{\\nu}$ is the equivalent number of massless neutrino flavors that are ultrarelativistic at $T_{f}$ and are still in thermal equilibrium with photons and electrons at that temperature. The calculated abundances of the light elements are in agreement with observations if $1.6\\,\\leq\\,N_{\\nu}\\,\\leq\\,4.0$ at $95\\%$ confidence level.[1] For three generations of Majorana neutrinos, $N_{\\nu}=3$ . For three generations of Dirac neutrinos, $N_{\\nu}\\,=\\,6$ while in thermal equilibrium. However, in the Standard Model only the left-handed component of neutrinos couple to $Z$ , $W^{+}$ and $W^{-}$ . 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[731.0, 1143.0, 1368.0, 1143.0, 1368.0, 1184.0, 731.0, 1184.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [285.0, 1187.0, 1366.0, 1187.0, 1366.0, 1222.0, 285.0, 1222.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [284.0, 1226.0, 1366.0, 1226.0, 1366.0, 1260.0, 284.0, 1260.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [283.0, 1266.0, 454.0, 1266.0, 454.0, 1303.0, 283.0, 1303.0], "score": 1.0, "text": ""}], "page_info": {"page_no": 4, "height": 2200, "width": 1700}}	Figure 2: Detail of the “upper island” of Figure 1 for the fit with 116 degrees of freedom (see text) at confidence level. The “lower island” is symmet- rical. The vertical bands correspond, from left to right, to 2.5, 3.5, 4.5, 6.5 and 7.5 oscillations from Sun to Earth of the line. ![Image]() Figure 2: Detail of the “upper island” of Figure 1 for the fit with 116 degrees of freedom (see text) at confidence level. The “lower island” is symmet- rical. The vertical bands correspond, from left to right, to 2.5, 3.5, 4.5, 6.5 and 7.5 oscillations from Sun to Earth of the line. with 116 degrees of freedom, including the 8 discussed earlier plus the 108 measurements by the Homestake Collaboration from 1970.281 to 1994.388[12] we obtain the allowed region shown in Figure 2. The reliability of depends on the correctness of the error as- signed to the Homestake observed-to-predicted flux ratio. For example, if the Homestake error listed in Table 1 is doubled we obtain the solutions shown in Figure 3. In view of the preceeding results let us assume that neutrinos indeed have mass. The question then arizes wether neutrinos are distinct from an- tineutrinos (Dirac neutrinos) or wether neutrinos are their own antiparticles (Majorana neutrinos). This latter possibility arizes because neutrinos have no electric charge. Let us consider Big-Bang nucleosynthesis that determines the abundances of the light elements , , and . These abundances are determined by the temperatures of freezout when the reaction rates become comparable to the expansion rate . Here is the equivalent number of massless neutrino flavors that are ultrarelativistic at and are still in thermal equilibrium with photons and electrons at that temperature. The calculated abundances of the light elements are in agreement with observations if at confidence level.[1] For three generations of Majorana neutrinos, . For three generations of Dirac neutrinos, while in thermal equilibrium. However, in the Standard Model only the left-handed component of neutrinos couple to , and . Right-handed neutrinos are not in thermal equilibrium at 5	<div class="pdf-page"> <em>Figure 2: Detail of the “upper island” of Figure 1 for the fit with 116 degrees of freedom (see text) at confidence level. The “lower island” is symmet- rical. The vertical bands correspond, from left to right, to 2.5, 3.5, 4.5, 6.5 and 7.5 oscillations from Sun to Earth of the line.</em> <p>with 116 degrees of freedom, including the 8 discussed earlier plus the 108 measurements by the Homestake Collaboration from 1970.281 to 1994.388[12] we obtain the allowed region shown in Figure 2.</p> <p>The reliability of depends on the correctness of the error as- signed to the Homestake observed-to-predicted flux ratio. For example, if the Homestake error listed in Table 1 is doubled we obtain the solutions shown in Figure 3.</p> <p>In view of the preceeding results let us assume that neutrinos indeed have mass. The question then arizes wether neutrinos are distinct from an- tineutrinos (Dirac neutrinos) or wether neutrinos are their own antiparticles (Majorana neutrinos). This latter possibility arizes because neutrinos have no electric charge.</p> <p>Let us consider Big-Bang nucleosynthesis that determines the abundances of the light elements , , and . These abundances are determined by the temperatures of freezout when the reaction rates become comparable to the expansion rate . Here is the equivalent number of massless neutrino flavors that are ultrarelativistic at and are still in thermal equilibrium with photons and electrons at that temperature. The calculated abundances of the light elements are in agreement with observations if at confidence level.[1] For three generations of Majorana neutrinos, . For three generations of Dirac neutrinos, while in thermal equilibrium. However, in the Standard Model only the left-handed component of neutrinos couple to , and . Right-handed neutrinos are not in thermal equilibrium at</p> </div>	<div class="pdf-page"> <figcaption class="pdf-image-caption" data-x="167" data-y="358" data-width="657" data-height="75">Figure 2: Detail of the “upper island” of Figure 1 for the fit with 116 degrees of freedom (see text) at confidence level. The “lower island” is symmet- rical. The vertical bands correspond, from left to right, to 2.5, 3.5, 4.5, 6.5 and 7.5 oscillations from Sun to Earth of the line.</figcaption> <p class="pdf-text" data-x="170" data-y="473" data-width="651" data-height="55">with 116 degrees of freedom, including the 8 discussed earlier plus the 108 measurements by the Homestake Collaboration from 1970.281 to 1994.388[12] we obtain the allowed region shown in Figure 2.</p> <p class="pdf-text" data-x="168" data-y="530" data-width="655" data-height="73">The reliability of depends on the correctness of the error as- signed to the Homestake observed-to-predicted flux ratio. For example, if the Homestake error listed in Table 1 is doubled we obtain the solutions shown in Figure 3.</p> <p class="pdf-text" data-x="168" data-y="605" data-width="655" data-height="91">In view of the preceeding results let us assume that neutrinos indeed have mass. The question then arizes wether neutrinos are distinct from an- tineutrinos (Dirac neutrinos) or wether neutrinos are their own antiparticles (Majorana neutrinos). This latter possibility arizes because neutrinos have no electric charge.</p> <p class="pdf-text" data-x="168" data-y="698" data-width="656" data-height="226">Let us consider Big-Bang nucleosynthesis that determines the abundances of the light elements , , and . These abundances are determined by the temperatures of freezout when the reaction rates become comparable to the expansion rate . Here is the equivalent number of massless neutrino flavors that are ultrarelativistic at and are still in thermal equilibrium with photons and electrons at that temperature. The calculated abundances of the light elements are in agreement with observations if at confidence level.[1] For three generations of Majorana neutrinos, . For three generations of Dirac neutrinos, while in thermal equilibrium. However, in the Standard Model only the left-handed component of neutrinos couple to , and . 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0002004v1	5	[ 612, 792 ]	{ "type": [ "image_body", "image_caption", "text", "text", "title", "list", "discarded" ], "coordinates": [ [ 299, 153, 680, 352 ], [ 168, 359, 823, 395 ], [ 168, 438, 823, 531 ], [ 168, 532, 823, 643 ], [ 168, 672, 324, 695 ], [ 167, 711, 824, 919 ], [ 490, 955, 503, 967 ] ], "content": [ "", "Figure 3: Same as Figure 1 but we have doubled the error of the Homestake experiment, i.e. .", ": their temperature has lagged below the temperature of photons due to the anihilation of particle-antiparticle pairs after the decoupling of the right- handed neutrinos. Therefore for Dirac neutrinos at we have . So we can not distinguish Dirac from Majorana neutrinos using available data on nucleosynthesis.", "In conclusion, the minimal extension of the Standard Model with three massive Majorana or Dirac neutrinos that mix is in good agreement with all experimental constraints. However, confirmation of the model is needed, e.g. by the observation of seasonal variations of the spectral lines with a period of 0.5 years, or spectral distortions and seasonal variations of the low energy neutrinos from the solar reaction.", "References", "] Review of Particle Physics, The European Physical Journal C3 (1998) 1 [2] B.T. Cleveland et al., Astrophys. J. 496 (1998) 505; B.T. Cleveland et al., Nucl. Phys. B (Proc. Suppl.) 38 (1995) 47; R. Davis, Prog. Part. Nucl. Phys. 32 (1994) 13 [3] SAGE Collab., V. Gavrin et al., in Neutrino 98, Proceedings of the XVIII International Conference on Neutrino Physics and Astrophysics, Takayama, Japan, 1998, edited by Y. Suzuki and Y. Totsuca. [4] GALLEX Collab., P. Anselmann et al., Phys. Lett. B342, (1995) 440; GALLEX Collaboration, W. Hampel et al., Phys. Lett. B388 (1996) 364 [5] KAMIOKANDE Collab., Y. Fukuda et al., Phys. Rev. Lett. 77 (1996) 1683 [6] Super-Kamiokande Collab., Phys. Rev. Lett. 81 (1998) 1158 [7] J.N. Bahcall, S. Basu, and M.H. Pinsonneault, Phys. Lett. B433 (1998) 1 [8] Super-Kamiokande Collab., Phys. Rev. Lett. 81 (1998) 1562 [9] LSND Collab., Phys. Rev. Lett. 75 (1995) 2650 [10] Hill, Phys. Rev. Lett. 75 (1995) 2654 [11] “Neutrino Oscillation Experiments at Nuclear Reactors”, G. Gratta. 17th International Workshop on Weak Interactions and Neutrinos : WIN ’99 Cape Town, South Africa ; 24-30 Jan 1999 . Palo Verde reactor disa- pearence experiment, unpublished. [12] Cleveland et.al., Astrophysical Journal 496 (1998), 505", "6" ], "index": [ 0, 1, 2, 3, 4, 5, 6 ] }	[{"type": "image", "img_path": "images/2ce2465af6c296f9c4eea471476bfc3a9c2bf97bb24d3b8eb832c5d95051d7f5.jpg", "img_caption": ["Figure 3: Same as Figure 1 but we have doubled the error of the Homestake experiment, i.e. $R=0.33\\pm0.10$ . "], "img_footnote": [], "page_idx": 5}, {"type": "text", "text": "$T_{f}$ : their temperature has lagged below the temperature of photons due to the anihilation of particle-antiparticle pairs after the decoupling of the righthanded neutrinos. Therefore for Dirac neutrinos at $T_{f}$ we have $N_{\\nu}\\approx3$ . So we can not distinguish Dirac from Majorana neutrinos using available data on nucleosynthesis. ", "page_idx": 5}, {"type": "text", "text": "In conclusion, the minimal extension of the Standard Model with three massive Majorana or Dirac neutrinos that mix is in good agreement with all experimental constraints. However, confirmation of the model is needed, e.g. by the observation of seasonal variations of the $^7B e$ spectral lines with a period of 0.5 years, or spectral distortions and seasonal variations of the low energy neutrinos from the solar $p p$ reaction. ", "page_idx": 5}, {"type": "text", "text": "References ", "text_level": 1, "page_idx": 5}, {"type": "text", "text": "] Review of Particle Physics, The European Physical Journal C3 (1998) 1 \n[2] B.T. Cleveland et al., Astrophys. J. 496 (1998) 505; B.T. Cleveland et al., Nucl. Phys. B (Proc. Suppl.) 38 (1995) 47; R. Davis, Prog. Part. Nucl. Phys. 32 (1994) 13 \n[3] SAGE Collab., V. Gavrin et al., in Neutrino 98, Proceedings of the XVIII International Conference on Neutrino Physics and Astrophysics, Takayama, Japan, 1998, edited by Y. Suzuki and Y. Totsuca. \n[4] GALLEX Collab., P. Anselmann et al., Phys. Lett. B342, (1995) 440; GALLEX Collaboration, W. Hampel et al., Phys. Lett. B388 (1996) 364 \n[5] KAMIOKANDE Collab., Y. Fukuda et al., Phys. Rev. Lett. 77 (1996) 1683 \n[6] Super-Kamiokande Collab., Phys. Rev. Lett. 81 (1998) 1158 \n[7] J.N. Bahcall, S. Basu, and M.H. Pinsonneault, Phys. Lett. B433 (1998) 1 \n[8] Super-Kamiokande Collab., Phys. Rev. Lett. 81 (1998) 1562 \n[9] LSND Collab., Phys. Rev. Lett. 75 (1995) 2650 \n[10] Hill, Phys. Rev. Lett. 75 (1995) 2654 \n[11] \u201cNeutrino Oscillation Experiments at Nuclear Reactors\u201d, G. Gratta. 17th International Workshop on Weak Interactions and Neutrinos : WIN \u201999 Cape Town, South Africa ; 24-30 Jan 1999 . Palo Verde reactor disapearence experiment, unpublished. \n[12] Cleveland et.al., Astrophysical Journal 496 (1998), 505 ", "page_idx": 5}]	{"preproc_blocks": [{"type": "image", "bbox": [179, 119, 407, 273], "blocks": [{"type": "image_body", "bbox": [179, 119, 407, 273], "group_id": 0, "lines": [{"bbox": [179, 119, 407, 273], "spans": [{"bbox": [179, 119, 407, 273], "score": 0.953, "type": "image", "image_path": "2ce2465af6c296f9c4eea471476bfc3a9c2bf97bb24d3b8eb832c5d95051d7f5.jpg"}]}], "index": 5.5, "virtual_lines": [{"bbox": [179, 119, 407, 133], "spans": [], "index": 0}, {"bbox": [179, 133, 407, 147], "spans": [], "index": 1}, {"bbox": [179, 147, 407, 161], "spans": [], "index": 2}, {"bbox": [179, 161, 407, 175], "spans": [], "index": 3}, {"bbox": [179, 175, 407, 189], "spans": [], "index": 4}, {"bbox": [179, 189, 407, 203], "spans": [], "index": 5}, {"bbox": [179, 203, 407, 217], "spans": [], "index": 6}, {"bbox": [179, 217, 407, 231], "spans": [], "index": 7}, {"bbox": [179, 231, 407, 245], "spans": [], "index": 8}, {"bbox": [179, 245, 407, 259], "spans": [], "index": 9}, {"bbox": [179, 259, 407, 273], "spans": [], "index": 10}, {"bbox": [179, 273, 407, 287], "spans": [], "index": 11}]}, {"type": "image_caption", "bbox": [101, 278, 492, 306], "group_id": 0, "lines": [{"bbox": [102, 280, 492, 294], "spans": [{"bbox": [102, 280, 492, 294], "score": 1.0, "content": "Figure 3: Same as Figure 1 but we have doubled the error of the Homestake", "type": "text"}], "index": 12}, {"bbox": [102, 295, 273, 308], "spans": [{"bbox": [102, 295, 187, 308], "score": 1.0, "content": "experiment, i.e. ", "type": "text"}, {"bbox": [188, 296, 269, 306], "score": 0.89, "content": "R=0.33\\pm0.10", "type": "inline_equation", "height": 10, "width": 81}, {"bbox": [270, 295, 273, 308], "score": 1.0, "content": ".", "type": "text"}], "index": 13}], "index": 12.5}], "index": 9.0}, {"type": "text", "bbox": [101, 339, 492, 411], "lines": [{"bbox": [102, 342, 492, 356], "spans": [{"bbox": [102, 343, 115, 356], "score": 0.92, "content": "T_{f}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [115, 342, 492, 355], "score": 1.0, "content": ": their temperature has lagged below the temperature of photons due to", "type": "text"}], "index": 14}, {"bbox": [102, 357, 491, 370], "spans": [{"bbox": [102, 357, 491, 370], "score": 1.0, "content": "the anihilation of particle-antiparticle pairs after the decoupling of the right-", "type": "text"}], "index": 15}, {"bbox": [101, 371, 492, 385], "spans": [{"bbox": [101, 371, 370, 385], "score": 1.0, "content": "handed neutrinos. 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. ![Image]() Figure 3: Same as Figure 1 but we have doubled the error of the Homestake experiment, i.e. . : their temperature has lagged below the temperature of photons due to the anihilation of particle-antiparticle pairs after the decoupling of the right- handed neutrinos. Therefore for Dirac neutrinos at we have . So we can not distinguish Dirac from Majorana neutrinos using available data on nucleosynthesis. In conclusion, the minimal extension of the Standard Model with three massive Majorana or Dirac neutrinos that mix is in good agreement with all experimental constraints. However, confirmation of the model is needed, e.g. by the observation of seasonal variations of the spectral lines with a period of 0.5 years, or spectral distortions and seasonal variations of the low energy neutrinos from the solar reaction. # References - ] Review of Particle Physics, The European Physical Journal C3 (1998) 1 [2] B.T. Cleveland et al., Astrophys. J. 496 (1998) 505; B.T. Cleveland et al., Nucl. Phys. B (Proc. Suppl.) 38 (1995) 47; R. Davis, Prog. Part. Nucl. Phys. 32 (1994) 13 [3] SAGE Collab., V. Gavrin et al., in Neutrino 98, Proceedings of the XVIII International Conference on Neutrino Physics and Astrophysics, Takayama, Japan, 1998, edited by Y. Suzuki and Y. Totsuca. [4] GALLEX Collab., P. Anselmann et al., Phys. Lett. B342, (1995) 440; GALLEX Collaboration, W. Hampel et al., Phys. Lett. B388 (1996) 364 [5] KAMIOKANDE Collab., Y. Fukuda et al., Phys. Rev. Lett. 77 (1996) 1683 [6] Super-Kamiokande Collab., Phys. Rev. Lett. 81 (1998) 1158 [7] J.N. Bahcall, S. Basu, and M.H. Pinsonneault, Phys. Lett. B433 (1998) 1 [8] Super-Kamiokande Collab., Phys. Rev. Lett. 81 (1998) 1562 [9] LSND Collab., Phys. Rev. Lett. 75 (1995) 2650 [10] Hill, Phys. Rev. Lett. 75 (1995) 2654 [11] “Neutrino Oscillation Experiments at Nuclear Reactors”, G. Gratta. 17th International Workshop on Weak Interactions and Neutrinos : WIN ’99 Cape Town, South Africa ; 24-30 Jan 1999 . Palo Verde reactor disa- pearence experiment, unpublished. [12] Cleveland et.al., Astrophysical Journal 496 (1998), 505 6	<div class="pdf-page"> <em>Figure 3: Same as Figure 1 but we have doubled the error of the Homestake experiment, i.e. .</em> <p>: their temperature has lagged below the temperature of photons due to the anihilation of particle-antiparticle pairs after the decoupling of the right- handed neutrinos. Therefore for Dirac neutrinos at we have . So we can not distinguish Dirac from Majorana neutrinos using available data on nucleosynthesis.</p> <p>In conclusion, the minimal extension of the Standard Model with three massive Majorana or Dirac neutrinos that mix is in good agreement with all experimental constraints. However, confirmation of the model is needed, e.g. by the observation of seasonal variations of the spectral lines with a period of 0.5 years, or spectral distortions and seasonal variations of the low energy neutrinos from the solar reaction.</p> <h1>References</h1> <ul> <li>] Review of Particle Physics, The European Physical Journal C3 (1998) 1 [2] B.T. Cleveland et al., Astrophys. J. 496 (1998) 505; B.T. Cleveland et al., Nucl. Phys. B (Proc. Suppl.) 38 (1995) 47; R. Davis, Prog. Part. Nucl. Phys. 32 (1994) 13 [3] SAGE Collab., V. Gavrin et al., in Neutrino 98, Proceedings of the XVIII International Conference on Neutrino Physics and Astrophysics, Takayama, Japan, 1998, edited by Y. Suzuki and Y. Totsuca. [4] GALLEX Collab., P. Anselmann et al., Phys. Lett. B342, (1995) 440; GALLEX Collaboration, W. Hampel et al., Phys. Lett. B388 (1996) 364 [5] KAMIOKANDE Collab., Y. Fukuda et al., Phys. Rev. Lett. 77 (1996) 1683 [6] Super-Kamiokande Collab., Phys. Rev. Lett. 81 (1998) 1158 [7] J.N. Bahcall, S. Basu, and M.H. Pinsonneault, Phys. Lett. B433 (1998) 1 [8] Super-Kamiokande Collab., Phys. Rev. Lett. 81 (1998) 1562 [9] LSND Collab., Phys. Rev. Lett. 75 (1995) 2650 [10] Hill, Phys. Rev. Lett. 75 (1995) 2654 [11] “Neutrino Oscillation Experiments at Nuclear Reactors”, G. Gratta. 17th International Workshop on Weak Interactions and Neutrinos : WIN ’99 Cape Town, South Africa ; 24-30 Jan 1999 . Palo Verde reactor disa- pearence experiment, unpublished. [12] Cleveland et.al., Astrophysical Journal 496 (1998), 505</li> </ul> </div>	<div class="pdf-page"> <figcaption class="pdf-image-caption" data-x="168" data-y="359" data-width="655" data-height="36">Figure 3: Same as Figure 1 but we have doubled the error of the Homestake experiment, i.e. .</figcaption> <p class="pdf-text" data-x="168" data-y="438" data-width="655" data-height="93">: their temperature has lagged below the temperature of photons due to the anihilation of particle-antiparticle pairs after the decoupling of the right- handed neutrinos. Therefore for Dirac neutrinos at we have . So we can not distinguish Dirac from Majorana neutrinos using available data on nucleosynthesis.</p> <p class="pdf-text" data-x="168" data-y="532" data-width="655" data-height="111">In conclusion, the minimal extension of the Standard Model with three massive Majorana or Dirac neutrinos that mix is in good agreement with all experimental constraints. However, confirmation of the model is needed, e.g. by the observation of seasonal variations of the spectral lines with a period of 0.5 years, or spectral distortions and seasonal variations of the low energy neutrinos from the solar reaction.</p> <h1 class="pdf-title" data-x="168" data-y="672" data-width="156" data-height="23">References</h1> <ul class="pdf-list" data-x="167" data-y="711" data-width="657" data-height="208"> <li>] Review of Particle Physics, The European Physical Journal C3 (1998) 1 [2] B.T. Cleveland et al., Astrophys. J. 496 (1998) 505; B.T. Cleveland et al., Nucl. Phys. B (Proc. Suppl.) 38 (1995) 47; R. Davis, Prog. Part. Nucl. Phys. 32 (1994) 13 [3] SAGE Collab., V. Gavrin et al., in Neutrino 98, Proceedings of the XVIII International Conference on Neutrino Physics and Astrophysics, Takayama, Japan, 1998, edited by Y. Suzuki and Y. Totsuca. [4] GALLEX Collab., P. Anselmann et al., Phys. Lett. B342, (1995) 440; GALLEX Collaboration, W. Hampel et al., Phys. Lett. B388 (1996) 364 [5] KAMIOKANDE Collab., Y. Fukuda et al., Phys. Rev. Lett. 77 (1996) 1683 [6] Super-Kamiokande Collab., Phys. Rev. Lett. 81 (1998) 1158 [7] J.N. Bahcall, S. Basu, and M.H. Pinsonneault, Phys. Lett. B433 (1998) 1 [8] Super-Kamiokande Collab., Phys. Rev. Lett. 81 (1998) 1562 [9] LSND Collab., Phys. Rev. Lett. 75 (1995) 2650 [10] Hill, Phys. Rev. Lett. 75 (1995) 2654 [11] “Neutrino Oscillation Experiments at Nuclear Reactors”, G. Gratta. 17th International Workshop on Weak Interactions and Neutrinos : WIN ’99 Cape Town, South Africa ; 24-30 Jan 1999 . Palo Verde reactor disa- pearence experiment, unpublished. 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Therefore for Dirac neutrinos at T_{f} we have N_{\\nu}\\approx3 . So", "we can not distinguish Dirac from Majorana neutrinos using available data", "on nucleosynthesis.", "In conclusion, the minimal extension of the Standard Model with three", "massive Majorana or Dirac neutrinos that mix is in good agreement with", "all experimental constraints. However, confirmation of the model is needed,", "e.g. by the observation of seasonal variations of the ^7B e spectral lines with", "a period of 0.5 years, or spectral distortions and seasonal variations of the", "low energy neutrinos from the solar p p reaction.", "References", "] Review of Particle Physics, The European Physical Journal C3 (1998) 1", "[2] B.T. Cleveland et al., Astrophys. J. 496 (1998) 505; B.T. Cleveland et", "al., Nucl. Phys. B (Proc. Suppl.) 38 (1995) 47; R. Davis, Prog. Part.", "Nucl. Phys. 32 (1994) 13", "[3] SAGE Collab., V. Gavrin et al., in Neutrino 98, Proceedings of the", "XVIII International Conference on Neutrino Physics and Astrophysics,", "Takayama, Japan, 1998, edited by Y. Suzuki and Y. Totsuca.", "[4] GALLEX Collab., P. Anselmann et al., Phys. Lett. B342, (1995) 440;", "GALLEX Collaboration, W. Hampel et al., Phys. Lett. B388 (1996) 364", "[5] KAMIOKANDE Collab., Y. Fukuda et al., Phys. Rev. Lett. 77 (1996)", "1683", "[6] Super-Kamiokande Collab., Phys. Rev. Lett. 81 (1998) 1158", "[7] J.N. Bahcall, S. Basu, and M.H. Pinsonneault, Phys. Lett. B433 (1998)", "1", "[8] Super-Kamiokande Collab., Phys. Rev. Lett. 81 (1998) 1562", "[9] LSND Collab., Phys. Rev. Lett. 75 (1995) 2650", "[10] Hill, Phys. Rev. Lett. 75 (1995) 2654", "[11] “Neutrino Oscillation Experiments at Nuclear Reactors”, G. Gratta.", "17th International Workshop on Weak Interactions and Neutrinos : WIN", "’99 Cape Town, South Africa ; 24-30 Jan 1999 . 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0001060v1	0	[ 612, 792 ]	{ "type": [ "title", "text", "title", "text", "text", "text", "title", "text", "discarded", "discarded" ], "coordinates": [ [ 205, 214, 813, 277 ], [ 332, 298, 687, 368 ], [ 470, 424, 550, 440 ], [ 232, 449, 788, 572 ], [ 212, 588, 769, 624 ], [ 414, 640, 605, 656 ], [ 184, 683, 411, 709 ], [ 184, 724, 836, 836 ], [ 207, 845, 366, 859 ], [ 23, 212, 63, 724 ] ], "content": [ "Special Lagrangian Geometry in irreducible symplectic 4-folds", "Alessandro Arsie S.I.S.S.A. - I.S.A.S. Via Beirut 4 - 34013 Trieste, Italy", "Abstract", "Having fixed a Kaehler class and the unique corresponding hyper- kaehler metric, we prove that all special Lagrangian submanifolds of an irreducible symplectic 4-fold are obtained by complex submani- folds via a generalization of the so called hyperkaehler rotation trick; thus they retain part of the rigidity of the complex submanifolds: in- deed all special Lagrangian submanifolds of turn out to be real analytic.", "MSC (1991): Primary: 53C15, Secondary: 53A40, 51P05, 53C20 Keywords: special Lagrangian submanifolds, hyperkaehler structures.", "REF.: 75/99/FM/GEO", "1 Introduction", "Under the flourishing research activity on D-branes in string theory, the role of special Lagrangian submanifolds in physics has become more and more relevant (see for example [1]) untill it was eventually conjectured in [11] that they can be considered as the cornerstones of the mirror phenomenon. In- deed, D-branes are special Lagrangian submanifolds equipped with a flat line bundle. In physical literature, special Lagrangian submanifolds of the compactification space are related to physical states which retain part of the vacuum supersymmetry: for this reason they are often called supersym- metric cycles or BPS states.", "∗e-mail: [email protected]", "arXiv:math/0001060v1 [math.DG] 11 Jan 2000" ], "index": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ] }	[{"type": "text", "text": "Special Lagrangian Geometry in irreducible symplectic 4-folds ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "Alessandro Arsie S.I.S.S.A. - I.S.A.S. Via Beirut 4 - 34013 Trieste, Italy ", "page_idx": 0}, {"type": "text", "text": "Abstract ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "Having fixed a Kaehler class and the unique corresponding hyperkaehler metric, we prove that all special Lagrangian submanifolds of an irreducible symplectic 4-fold $X$ are obtained by complex submanifolds via a generalization of the so called hyperkaehler rotation trick; thus they retain part of the rigidity of the complex submanifolds: indeed all special Lagrangian submanifolds of $X$ turn out to be real analytic. ", "page_idx": 0}, {"type": "text", "text": "MSC (1991): Primary: 53C15, Secondary: 53A40, 51P05, 53C20 Keywords: special Lagrangian submanifolds, hyperkaehler structures. ", "page_idx": 0}, {"type": "text", "text": "REF.: 75/99/FM/GEO ", "page_idx": 0}, {"type": "text", "text": "1 Introduction ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "Under the flourishing research activity on D-branes in string theory, the role of special Lagrangian submanifolds in physics has become more and more relevant (see for example [1]) untill it was eventually conjectured in [11] that they can be considered as the cornerstones of the mirror phenomenon. Indeed, D-branes are special Lagrangian submanifolds equipped with a flat $U(1)$ line bundle. In physical literature, special Lagrangian submanifolds of the compactification space are related to physical states which retain part of the vacuum supersymmetry: for this reason they are often called supersymmetric cycles or BPS states. 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[380.0, 1480.0, 685.0, 1480.0, 685.0, 1525.0, 380.0, 1525.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [690.0, 1381.0, 1002.0, 1381.0, 1002.0, 1415.0, 690.0, 1415.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [780.0, 917.0, 918.0, 917.0, 918.0, 951.0, 780.0, 951.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [342.0, 1819.0, 615.0, 1819.0, 615.0, 1855.0, 342.0, 1855.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [43.0, 462.0, 105.0, 462.0, 105.0, 1559.0, 43.0, 1559.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [42.0, 461.0, 106.0, 461.0, 106.0, 1561.0, 42.0, 1561.0], "score": 1.0, "text": ""}], "page_info": {"page_no": 0, "height": 2200, "width": 1700}}	arXiv:math/0001060v1 [math.DG] 11 Jan 2000 # Special Lagrangian Geometry in irreducible symplectic 4-folds Alessandro Arsie S.I.S.S.A. - I.S.A.S. Via Beirut 4 - 34013 Trieste, Italy # Abstract Having fixed a Kaehler class and the unique corresponding hyper- kaehler metric, we prove that all special Lagrangian submanifolds of an irreducible symplectic 4-fold are obtained by complex submani- folds via a generalization of the so called hyperkaehler rotation trick; thus they retain part of the rigidity of the complex submanifolds: in- deed all special Lagrangian submanifolds of turn out to be real analytic. MSC (1991): Primary: 53C15, Secondary: 53A40, 51P05, 53C20 Keywords: special Lagrangian submanifolds, hyperkaehler structures. REF.: 75/99/FM/GEO # 1 Introduction Under the flourishing research activity on D-branes in string theory, the role of special Lagrangian submanifolds in physics has become more and more relevant (see for example [1]) untill it was eventually conjectured in [11] that they can be considered as the cornerstones of the mirror phenomenon. In- deed, D-branes are special Lagrangian submanifolds equipped with a flat line bundle. In physical literature, special Lagrangian submanifolds of the compactification space are related to physical states which retain part of the vacuum supersymmetry: for this reason they are often called supersym- metric cycles or BPS states. ∗e-mail: [email protected]	<div class="pdf-page"> <h1>Special Lagrangian Geometry in irreducible symplectic 4-folds</h1> <p>Alessandro Arsie S.I.S.S.A. - I.S.A.S. Via Beirut 4 - 34013 Trieste, Italy</p> <h1>Abstract</h1> <p>Having fixed a Kaehler class and the unique corresponding hyper- kaehler metric, we prove that all special Lagrangian submanifolds of an irreducible symplectic 4-fold are obtained by complex submani- folds via a generalization of the so called hyperkaehler rotation trick; thus they retain part of the rigidity of the complex submanifolds: in- deed all special Lagrangian submanifolds of turn out to be real analytic.</p> <p>MSC (1991): Primary: 53C15, Secondary: 53A40, 51P05, 53C20 Keywords: special Lagrangian submanifolds, hyperkaehler structures.</p> <p>REF.: 75/99/FM/GEO</p> <h1>1 Introduction</h1> <p>Under the flourishing research activity on D-branes in string theory, the role of special Lagrangian submanifolds in physics has become more and more relevant (see for example [1]) untill it was eventually conjectured in [11] that they can be considered as the cornerstones of the mirror phenomenon. In- deed, D-branes are special Lagrangian submanifolds equipped with a flat line bundle. In physical literature, special Lagrangian submanifolds of the compactification space are related to physical states which retain part of the vacuum supersymmetry: for this reason they are often called supersym- metric cycles or BPS states.</p> </div>	<div class="pdf-page"> <div class="pdf-discarded" data-x="23" data-y="212" data-width="40" data-height="512" style="opacity: 0.5;">arXiv:math/0001060v1 [math.DG] 11 Jan 2000</div> <h1 class="pdf-title" data-x="205" data-y="214" data-width="608" data-height="63">Special Lagrangian Geometry in irreducible symplectic 4-folds</h1> <p class="pdf-text" data-x="332" data-y="298" data-width="355" data-height="70">Alessandro Arsie S.I.S.S.A. - I.S.A.S. Via Beirut 4 - 34013 Trieste, Italy</p> <h1 class="pdf-title" data-x="470" data-y="424" data-width="80" data-height="16">Abstract</h1> <p class="pdf-text" data-x="232" data-y="449" data-width="556" data-height="123">Having fixed a Kaehler class and the unique corresponding hyper- kaehler metric, we prove that all special Lagrangian submanifolds of an irreducible symplectic 4-fold are obtained by complex submani- folds via a generalization of the so called hyperkaehler rotation trick; thus they retain part of the rigidity of the complex submanifolds: in- deed all special Lagrangian submanifolds of turn out to be real analytic.</p> <p class="pdf-text" data-x="212" data-y="588" data-width="557" data-height="36">MSC (1991): Primary: 53C15, Secondary: 53A40, 51P05, 53C20 Keywords: special Lagrangian submanifolds, hyperkaehler structures.</p> <p class="pdf-text" data-x="414" data-y="640" data-width="191" data-height="16">REF.: 75/99/FM/GEO</p> <h1 class="pdf-title" data-x="184" data-y="683" data-width="227" data-height="26">1 Introduction</h1> <p class="pdf-text" data-x="184" data-y="724" data-width="652" data-height="112">Under the flourishing research activity on D-branes in string theory, the role of special Lagrangian submanifolds in physics has become more and more relevant (see for example [1]) untill it was eventually conjectured in [11] that they can be considered as the cornerstones of the mirror phenomenon. In- deed, D-branes are special Lagrangian submanifolds equipped with a flat line bundle. In physical literature, special Lagrangian submanifolds of the compactification space are related to physical states which retain part of the vacuum supersymmetry: for this reason they are often called supersym- metric cycles or BPS states.</p> <div class="pdf-discarded" data-x="207" data-y="845" data-width="159" data-height="14" style="opacity: 0.5;">∗e-mail: [email protected]</div> </div>	{ "type": [ "text", "text", "text", "text", "text", "text", "text", "text", "inline_equation", "text", "text", "inline_equation", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "inline_equation", "text", "text", "text" ], "coordinates": [ [ 207, 219, 813, 246 ], [ 384, 253, 634, 276 ], [ 419, 302, 600, 320 ], [ 408, 325, 607, 343 ], [ 332, 346, 684, 369 ], [ 468, 426, 552, 442 ], [ 259, 453, 786, 469 ], [ 230, 470, 789, 488 ], [ 234, 490, 786, 504 ], [ 230, 505, 786, 522 ], [ 232, 523, 786, 539 ], [ 232, 541, 788, 557 ], [ 232, 559, 301, 574 ], [ 212, 592, 731, 608 ], [ 212, 611, 769, 628 ], [ 414, 642, 602, 658 ], [ 184, 687, 411, 709 ], [ 184, 727, 836, 746 ], [ 184, 746, 838, 765 ], [ 184, 765, 836, 782 ], [ 184, 783, 834, 801 ], [ 184, 802, 836, 821 ], [ 184, 819, 839, 840 ], [ 185, 165, 838, 182 ], [ 182, 183, 833, 202 ], [ 184, 201, 424, 219 ] ], "content": [ "Special Lagrangian Geometry in irreducible", "symplectic 4-folds", "Alessandro Arsie", "S.I.S.S.A. - I.S.A.S.", "Via Beirut 4 - 34013 Trieste, Italy", "Abstract", "Having fixed a Kaehler class and the unique corresponding hyper-", "kaehler metric, we prove that all special Lagrangian submanifolds of", "an irreducible symplectic 4-fold X are obtained by complex submani-", "folds via a generalization of the so called hyperkaehler rotation trick;", "thus they retain part of the rigidity of the complex submanifolds: in-", "deed all special Lagrangian submanifolds of X turn out to be real", "analytic.", "MSC (1991): Primary: 53C15, Secondary: 53A40, 51P05, 53C20", "Keywords: special Lagrangian submanifolds, hyperkaehler structures.", "REF.: 75/99/FM/GEO", "1 Introduction", "Under the flourishing research activity on D-branes in string theory, the role", "of special Lagrangian submanifolds in physics has become more and more", "relevant (see for example [1]) untill it was eventually conjectured in [11] that", "they can be considered as the cornerstones of the mirror phenomenon. 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0001060v1	1	[ 612, 792 ]	{ "type": [ "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "discarded" ], "coordinates": [ [ 184, 161, 836, 217 ], [ 182, 218, 836, 329 ], [ 212, 329, 456, 347 ], [ 184, 349, 838, 385 ], [ 215, 386, 473, 403 ], [ 214, 405, 438, 422 ], [ 212, 422, 806, 442 ], [ 182, 443, 836, 665 ], [ 210, 667, 580, 683 ], [ 184, 685, 836, 758 ], [ 215, 760, 649, 778 ], [ 182, 779, 839, 815 ], [ 212, 815, 416, 833 ], [ 184, 835, 836, 871 ], [ 501, 893, 515, 907 ] ], "content": [ "", "Despite their importance, there are very few explicit examples of special Lagrangian submanifolds, especially in Calabi-Yau 3-folds. However, in an irreducible symplectic 4-fold (realized as a hyperkaehler manifold) we have a complete control of the special Lagrangian geometry of its submanifolds, via a sort of hyperkaehler trick; moreover this enables us to prove that special Lagrangian submanifolds retain part of the rigidity of complex submanifolds.", "We first recall the following:", "Definition 1.1: A complex manifold is called irreducible symplectic if it satisfies the following three conditions:", "1) is compact and Kaehler;", ") is simply connected;", "3) is spanned by an everywhere non-degenerate 2-form .", "In particular, irreducible symplectic manifolds are special cases of Calabi- Yau manifolds (the top holomorphic form which trivializes the canonical line bundle is given by a suitable power of the holomorphic 2-form ). In dimen- sion 2, K3 surfaces are the only irreducible symplectic manifolds, and indeed irreducible symplectic manifolds appear as higher-dimensional analogues of K3 surfaces, as strongly suggested in [5]. Unfortunately, up to now there are very few explicit examples of irreducible symplectic manifolds. Indeed almost all known examples turn out to be birational to two standard series of examples: Hilbert schemes of points on K3 surfaces and generalized Kummer varieties (both series were first studied in [2]), but quite recently O’Grady has constructed irreducible symplectic manifolds which are not birational to any of the elements of the two groups (see [10]).", "Finally, let us recall from [4] the following:", "Definition 1.2: Let be a Calabi-Yau n-fold, with Kaehler form and holomorphic nowhere vanishing n-form . (real) -dimensional sub- manifold of is called special Lagrangian if the following two conditions are satisfied:", "1) is Lagrangian with respect to , i.e. ;", "2) there exists a multiple of such that ; one can prove (see [4]) that both conditions are equivalent to:", "’) .", "The condition ) in the previous definition means that the real part of restricts to the volume form of , induced by the Calabi-Yau Riemannian metric . In this way special Lagrangian submanifolds are considered as a type of calibrated submanifolds (see [4] for further details on this point).", "2" ], "index": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 ] }	[{"type": "text", "text": "", "page_idx": 1}, {"type": "text", "text": "Despite their importance, there are very few explicit examples of special Lagrangian submanifolds, especially in Calabi-Yau 3-folds. However, in an irreducible symplectic 4-fold (realized as a hyperkaehler manifold) we have a complete control of the special Lagrangian geometry of its submanifolds, via a sort of hyperkaehler trick; moreover this enables us to prove that special Lagrangian submanifolds retain part of the rigidity of complex submanifolds. ", "page_idx": 1}, {"type": "text", "text": "We first recall the following: ", "page_idx": 1}, {"type": "text", "text": "Definition 1.1: A complex manifold $X$ is called irreducible symplectic if it satisfies the following three conditions: ", "page_idx": 1}, {"type": "text", "text": "1) $X$ is compact and Kaehler; ", "page_idx": 1}, {"type": "text", "text": "$\\boldsymbol{\\mathcal{Q}}$ ) $X$ is simply connected; ", "page_idx": 1}, {"type": "text", "text": "3) $H^{0}(X,\\Omega_{X}^{2})$ is spanned by an everywhere non-degenerate 2-form $\\omega$ . ", "page_idx": 1}, {"type": "text", "text": "In particular, irreducible symplectic manifolds are special cases of CalabiYau manifolds (the top holomorphic form which trivializes the canonical line bundle is given by a suitable power of the holomorphic 2-form $\\omega$ ). In dimension 2, K3 surfaces are the only irreducible symplectic manifolds, and indeed irreducible symplectic manifolds appear as higher-dimensional analogues of K3 surfaces, as strongly suggested in [5]. Unfortunately, up to now there are very few explicit examples of irreducible symplectic manifolds. Indeed almost all known examples turn out to be birational to two standard series of examples: Hilbert schemes of points on K3 surfaces and generalized Kummer varieties (both series were first studied in [2]), but quite recently O\u2019Grady has constructed irreducible symplectic manifolds which are not birational to any of the elements of the two groups (see [10]). ", "page_idx": 1}, {"type": "text", "text": "Finally, let us recall from [4] the following: ", "page_idx": 1}, {"type": "text", "text": "Definition 1.2: Let $X$ be a Calabi-Yau n-fold, with Kaehler form $\\omega$ and holomorphic nowhere vanishing n-form $\\Omega$ . $A$ (real) $\\boldsymbol{n}$ -dimensional submanifold $j:\\Lambda\\hookrightarrow X$ of $X$ is called special Lagrangian if the following two conditions are satisfied: ", "page_idx": 1}, {"type": "text", "text": "1) $\\Lambda$ is Lagrangian with respect to $\\omega$ , i.e. $j^{}\\omega=0$ ; ", "page_idx": 1}, {"type": "text", "text": "2) there exists a multiple $\\Omega^{\\prime}$ of $\\Omega$ such that $j^{}\\mathrm{Im}(\\Omega^{\\prime})\\!=\\!0_{.}$ ; one can prove (see [4]) that both conditions are equivalent to: ", "page_idx": 1}, {"type": "text", "text": "$\\mathit{1}$ \u2019) $j^{*}\\mathrm{Re}(\\Omega^{\\prime})=V o l_{g}(\\Lambda)$ . ", "page_idx": 1}, {"type": "text", "text": "The condition $1^{\\prime}$ ) in the previous definition means that the real part of $\\Omega^{\\prime}$ restricts to the volume form of $\\Lambda$ , induced by the Calabi-Yau Riemannian metric $g$ . In this way special Lagrangian submanifolds are considered as a type of calibrated submanifolds (see [4] for further details on this point). ", "page_idx": 1}]	{"preproc_blocks": [{"type": "text", "bbox": [110, 125, 500, 168], "lines": [{"bbox": [111, 128, 501, 141], "spans": [{"bbox": [111, 128, 501, 141], "score": 1.0, "content": "the compactification space are related to physical states which retain part of", "type": "text"}], "index": 0}, {"bbox": [109, 142, 498, 157], "spans": [{"bbox": [109, 142, 498, 157], "score": 1.0, "content": "the vacuum supersymmetry: for this reason they are often called supersym-", "type": "text"}], "index": 1}, {"bbox": [110, 156, 254, 170], "spans": [{"bbox": [110, 156, 254, 170], "score": 1.0, "content": "metric cycles or BPS states.", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [109, 169, 500, 255], "lines": [{"bbox": [127, 171, 500, 186], "spans": [{"bbox": [127, 171, 500, 186], "score": 1.0, "content": "Despite their importance, there are very few explicit examples of special", "type": "text"}], "index": 3}, {"bbox": [110, 186, 501, 200], "spans": [{"bbox": [110, 186, 501, 200], "score": 1.0, "content": "Lagrangian submanifolds, especially in Calabi-Yau 3-folds. 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In dimen-", "type": "text"}], "index": 17}, {"bbox": [110, 388, 500, 402], "spans": [{"bbox": [110, 388, 500, 402], "score": 1.0, "content": "sion 2, K3 surfaces are the only irreducible symplectic manifolds, and indeed", "type": "text"}], "index": 18}, {"bbox": [109, 402, 502, 416], "spans": [{"bbox": [109, 402, 502, 416], "score": 1.0, "content": "irreducible symplectic manifolds appear as higher-dimensional analogues of", "type": "text"}], "index": 19}, {"bbox": [110, 417, 500, 431], "spans": [{"bbox": [110, 417, 500, 431], "score": 1.0, "content": "K3 surfaces, as strongly suggested in [5]. Unfortunately, up to now there", "type": "text"}], "index": 20}, {"bbox": [109, 432, 500, 444], "spans": [{"bbox": [109, 432, 500, 444], "score": 1.0, "content": "are very few explicit examples of irreducible symplectic manifolds. Indeed", "type": "text"}], "index": 21}, {"bbox": [110, 446, 502, 459], "spans": [{"bbox": [110, 446, 502, 459], "score": 1.0, "content": "almost all known examples turn out to be birational to two standard series of", "type": "text"}], "index": 22}, {"bbox": [110, 461, 500, 474], "spans": [{"bbox": [110, 461, 500, 474], "score": 1.0, "content": "examples: Hilbert schemes of points on K3 surfaces and generalized Kummer", "type": "text"}], "index": 23}, {"bbox": [110, 475, 499, 488], "spans": [{"bbox": [110, 475, 499, 488], "score": 1.0, "content": "varieties (both series were first studied in [2]), but quite recently O\u2019Grady", "type": "text"}], "index": 24}, {"bbox": [109, 489, 500, 503], "spans": [{"bbox": [109, 489, 500, 503], "score": 1.0, "content": "has constructed irreducible symplectic manifolds which are not birational to", "type": "text"}], "index": 25}, {"bbox": [110, 503, 357, 517], "spans": [{"bbox": [110, 503, 357, 517], "score": 1.0, "content": "any of the elements of the two groups (see [10]).", "type": "text"}], "index": 26}], "index": 20.5}, {"type": "text", "bbox": [126, 516, 347, 529], "lines": [{"bbox": [127, 517, 346, 531], "spans": [{"bbox": [127, 517, 346, 531], "score": 1.0, "content": "Finally, let us recall from [4] the following:", "type": "text"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [110, 530, 500, 587], "lines": [{"bbox": [127, 531, 498, 546], "spans": [{"bbox": [127, 531, 243, 546], "score": 1.0, "content": "Definition 1.2: Let ", "type": "text"}, {"bbox": [243, 534, 254, 542], "score": 0.87, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [255, 531, 491, 546], "score": 1.0, "content": " be a Calabi-Yau n-fold, with Kaehler form ", "type": "text"}, {"bbox": [491, 537, 498, 542], "score": 0.33, "content": "\\omega", "type": "inline_equation", "height": 5, "width": 7}], "index": 28}, {"bbox": [111, 547, 499, 560], "spans": [{"bbox": [111, 547, 338, 560], "score": 1.0, "content": "and holomorphic nowhere vanishing n-form ", "type": "text"}, {"bbox": [338, 548, 347, 557], "score": 0.45, "content": "\\Omega", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [347, 547, 356, 560], "score": 1.0, "content": ". ", "type": "text"}, {"bbox": [356, 547, 365, 557], "score": 0.4, "content": "A", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [366, 547, 402, 560], "score": 1.0, "content": " (real) ", "type": "text"}, {"bbox": [402, 551, 409, 557], "score": 0.57, "content": "\\boldsymbol{n}", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [410, 547, 499, 560], "score": 1.0, "content": "-dimensional sub-", "type": "text"}], "index": 29}, {"bbox": [111, 561, 500, 575], "spans": [{"bbox": [111, 561, 159, 575], "score": 1.0, "content": "manifold ", "type": "text"}, {"bbox": [159, 563, 219, 573], "score": 0.9, "content": "j:\\Lambda\\hookrightarrow X", "type": "inline_equation", "height": 10, "width": 60}, {"bbox": [219, 561, 237, 575], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [238, 563, 249, 571], "score": 0.87, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [249, 561, 500, 575], "score": 1.0, "content": " is called special Lagrangian if the following two", "type": "text"}], "index": 30}, {"bbox": [111, 576, 232, 589], "spans": [{"bbox": [111, 576, 232, 589], "score": 1.0, "content": "conditions are satisfied:", "type": "text"}], "index": 31}], "index": 29.5}, {"type": "text", "bbox": [129, 588, 388, 602], "lines": [{"bbox": [129, 590, 387, 604], "spans": [{"bbox": [129, 590, 142, 604], "score": 1.0, "content": "1) ", "type": "text"}, {"bbox": [142, 591, 151, 601], "score": 0.61, "content": "\\Lambda", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [152, 590, 305, 604], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [306, 594, 314, 600], "score": 0.67, "content": "\\omega", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [314, 590, 342, 604], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [343, 592, 383, 603], "score": 0.91, "content": "j^{}\\omega=0", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [383, 590, 387, 604], "score": 1.0, "content": ";", "type": "text"}], "index": 32}], "index": 32}, {"type": "text", "bbox": [109, 603, 502, 631], "lines": [{"bbox": [127, 604, 501, 620], "spans": [{"bbox": [127, 604, 262, 620], "score": 1.0, "content": "2) there exists a multiple ", "type": "text"}, {"bbox": [262, 606, 274, 615], "score": 0.88, "content": "\\Omega^{\\prime}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [274, 604, 292, 620], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [293, 606, 302, 615], "score": 0.79, "content": "\\Omega", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [302, 604, 357, 620], "score": 1.0, "content": "such that ", "type": "text"}, {"bbox": [358, 605, 419, 618], "score": 0.67, "content": "j^{}\\mathrm{Im}(\\Omega^{\\prime})\\!=\\!0_{.}", "type": "inline_equation", "height": 13, "width": 61}, {"bbox": [420, 604, 501, 620], "score": 1.0, "content": "; one can prove", "type": "text"}], "index": 33}, {"bbox": [111, 619, 351, 634], "spans": [{"bbox": [111, 619, 351, 634], "score": 1.0, "content": "(see [4]) that both conditions are equivalent to:", "type": "text"}], "index": 34}], "index": 33.5}, {"type": "text", "bbox": [127, 631, 249, 645], "lines": [{"bbox": [128, 631, 249, 648], "spans": [{"bbox": [128, 634, 134, 644], "score": 0.26, "content": "\\mathit{1}", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [135, 631, 146, 648], "score": 1.0, "content": "\u2019) ", "type": "text"}, {"bbox": [146, 634, 248, 647], "score": 0.92, "content": "j^{}\\mathrm{Re}(\\Omega^{\\prime})=V o l_{g}(\\Lambda)", "type": "inline_equation", "height": 13, "width": 102}, {"bbox": [248, 631, 249, 648], "score": 1.0, "content": ".", "type": "text"}], "index": 35}], "index": 35}, {"type": "text", "bbox": [110, 646, 500, 674], "lines": [{"bbox": [127, 646, 501, 663], "spans": [{"bbox": [127, 646, 204, 663], "score": 1.0, "content": "The condition ", "type": "text"}, {"bbox": [204, 649, 213, 658], "score": 0.7, "content": "1^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [214, 646, 501, 663], "score": 1.0, "content": ") in the previous definition means that the real part of", "type": "text"}], "index": 36}, {"bbox": [110, 661, 500, 676], "spans": [{"bbox": [110, 663, 121, 672], "score": 0.89, "content": "\\Omega^{\\prime}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [122, 661, 282, 676], "score": 1.0, "content": " restricts to the volume form of", "type": "text"}, {"bbox": [283, 664, 291, 672], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [292, 661, 500, 676], "score": 1.0, "content": ", induced by the Calabi-Yau Riemannian", "type": "text"}], "index": 37}], "index": 36.5}], "layout_bboxes": [], "page_idx": 1, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [300, 691, 308, 702], "lines": [{"bbox": [300, 692, 309, 705], "spans": [{"bbox": [300, 692, 309, 705], "score": 1.0, "content": "2", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [110, 125, 500, 168], "lines": [], "index": 1, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [109, 128, 501, 170], "lines_deleted": true}, {"type": "text", "bbox": [109, 169, 500, 255], "lines": [{"bbox": [127, 171, 500, 186], "spans": [{"bbox": [127, 171, 500, 186], "score": 1.0, "content": "Despite their importance, there are very few explicit examples of special", "type": "text"}], "index": 3}, {"bbox": [110, 186, 501, 200], "spans": [{"bbox": [110, 186, 501, 200], "score": 1.0, "content": "Lagrangian submanifolds, especially in Calabi-Yau 3-folds. However, in an", "type": "text"}], "index": 4}, {"bbox": [109, 200, 501, 214], "spans": [{"bbox": [109, 200, 501, 214], "score": 1.0, "content": "irreducible symplectic 4-fold (realized as a hyperkaehler manifold) we have a", "type": "text"}], "index": 5}, {"bbox": [109, 214, 501, 230], "spans": [{"bbox": [109, 214, 501, 230], "score": 1.0, "content": "complete control of the special Lagrangian geometry of its submanifolds, via", "type": "text"}], "index": 6}, {"bbox": [110, 229, 499, 243], "spans": [{"bbox": [110, 229, 499, 243], "score": 1.0, "content": "a sort of hyperkaehler trick; moreover this enables us to prove that special", "type": "text"}], "index": 7}, {"bbox": [109, 243, 498, 258], "spans": [{"bbox": [109, 243, 498, 258], "score": 1.0, "content": "Lagrangian submanifolds retain part of the rigidity of complex submanifolds.", "type": "text"}], "index": 8}], "index": 5.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [109, 171, 501, 258]}, {"type": "text", "bbox": [127, 255, 273, 269], "lines": [{"bbox": [127, 256, 272, 271], "spans": [{"bbox": [127, 256, 272, 271], "score": 1.0, "content": "We first recall the following:", "type": "text"}], "index": 9}], "index": 9, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [127, 256, 272, 271]}, {"type": "text", "bbox": [110, 270, 501, 298], "lines": [{"bbox": [127, 272, 502, 287], "spans": [{"bbox": [127, 272, 320, 287], "score": 1.0, "content": "Definition 1.1: A complex manifold ", "type": "text"}, {"bbox": [320, 274, 331, 282], "score": 0.67, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [331, 272, 502, 287], "score": 1.0, "content": " is called irreducible symplectic if", "type": "text"}], "index": 10}, {"bbox": [111, 288, 321, 300], "spans": [{"bbox": [111, 288, 321, 300], "score": 1.0, "content": "it satisfies the following three conditions:", "type": "text"}], "index": 11}], "index": 10.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [111, 272, 502, 300]}, {"type": "text", "bbox": [129, 299, 283, 312], "lines": [{"bbox": [129, 300, 282, 314], "spans": [{"bbox": [129, 300, 142, 314], "score": 1.0, "content": "1) ", "type": "text"}, {"bbox": [142, 303, 154, 311], "score": 0.89, "content": "X", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [154, 300, 282, 314], "score": 1.0, "content": " is compact and Kaehler;", "type": "text"}], "index": 12}], "index": 12, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [129, 300, 282, 314]}, {"type": "text", "bbox": [128, 314, 262, 327], "lines": [{"bbox": [128, 316, 260, 328], "spans": [{"bbox": [128, 317, 135, 326], "score": 0.64, "content": "\\boldsymbol{\\mathcal{Q}}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [135, 316, 142, 328], "score": 1.0, "content": ") ", "type": "text"}, {"bbox": [142, 317, 154, 326], "score": 0.89, "content": "X", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [154, 316, 260, 328], "score": 1.0, "content": " is simply connected;", "type": "text"}], "index": 13}], "index": 13, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [128, 316, 260, 328]}, {"type": "text", "bbox": [127, 327, 482, 342], "lines": [{"bbox": [128, 329, 483, 344], "spans": [{"bbox": [128, 329, 142, 344], "score": 1.0, "content": "3) ", "type": "text"}, {"bbox": [142, 330, 199, 343], "score": 0.94, "content": "H^{0}(X,\\Omega_{X}^{2})", "type": "inline_equation", "height": 13, "width": 57}, {"bbox": [199, 329, 470, 344], "score": 1.0, "content": " is spanned by an everywhere non-degenerate 2-form ", "type": "text"}, {"bbox": [470, 335, 479, 340], "score": 0.41, "content": "\\omega", "type": "inline_equation", "height": 5, "width": 9}, {"bbox": [479, 329, 483, 344], "score": 1.0, "content": ".", "type": "text"}], "index": 14}], "index": 14, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [128, 329, 483, 344]}, {"type": "text", "bbox": [109, 343, 500, 515], "lines": [{"bbox": [127, 345, 499, 357], "spans": [{"bbox": [127, 345, 499, 357], "score": 1.0, "content": "In particular, irreducible symplectic manifolds are special cases of Calabi-", "type": "text"}], "index": 15}, {"bbox": [110, 359, 500, 372], "spans": [{"bbox": [110, 359, 500, 372], "score": 1.0, "content": "Yau manifolds (the top holomorphic form which trivializes the canonical line", "type": "text"}], "index": 16}, {"bbox": [110, 374, 500, 387], "spans": [{"bbox": [110, 374, 428, 387], "score": 1.0, "content": "bundle is given by a suitable power of the holomorphic 2-form ", "type": "text"}, {"bbox": [429, 378, 437, 384], "score": 0.87, "content": "\\omega", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [437, 374, 500, 387], "score": 1.0, "content": "). In dimen-", "type": "text"}], "index": 17}, {"bbox": [110, 388, 500, 402], "spans": [{"bbox": [110, 388, 500, 402], "score": 1.0, "content": "sion 2, K3 surfaces are the only irreducible symplectic manifolds, and indeed", "type": "text"}], "index": 18}, {"bbox": [109, 402, 502, 416], "spans": [{"bbox": [109, 402, 502, 416], "score": 1.0, "content": "irreducible symplectic manifolds appear as higher-dimensional analogues of", "type": "text"}], "index": 19}, {"bbox": [110, 417, 500, 431], "spans": [{"bbox": [110, 417, 500, 431], "score": 1.0, "content": "K3 surfaces, as strongly suggested in [5]. Unfortunately, up to now there", "type": "text"}], "index": 20}, {"bbox": [109, 432, 500, 444], "spans": [{"bbox": [109, 432, 500, 444], "score": 1.0, "content": "are very few explicit examples of irreducible symplectic manifolds. Indeed", "type": "text"}], "index": 21}, {"bbox": [110, 446, 502, 459], "spans": [{"bbox": [110, 446, 502, 459], "score": 1.0, "content": "almost all known examples turn out to be birational to two standard series of", "type": "text"}], "index": 22}, {"bbox": [110, 461, 500, 474], "spans": [{"bbox": [110, 461, 500, 474], "score": 1.0, "content": "examples: Hilbert schemes of points on K3 surfaces and generalized Kummer", "type": "text"}], "index": 23}, {"bbox": [110, 475, 499, 488], "spans": [{"bbox": [110, 475, 499, 488], "score": 1.0, "content": "varieties (both series were first studied in [2]), but quite recently O\u2019Grady", "type": "text"}], "index": 24}, {"bbox": [109, 489, 500, 503], "spans": [{"bbox": [109, 489, 500, 503], "score": 1.0, "content": "has constructed irreducible symplectic manifolds which are not birational to", "type": "text"}], "index": 25}, {"bbox": [110, 503, 357, 517], "spans": [{"bbox": [110, 503, 357, 517], "score": 1.0, "content": "any of the elements of the two groups (see [10]).", "type": "text"}], "index": 26}], "index": 20.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [109, 345, 502, 517]}, {"type": "text", "bbox": [126, 516, 347, 529], "lines": [{"bbox": [127, 517, 346, 531], "spans": [{"bbox": [127, 517, 346, 531], "score": 1.0, "content": "Finally, let us recall from [4] the following:", "type": "text"}], "index": 27}], "index": 27, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [127, 517, 346, 531]}, {"type": "text", "bbox": [110, 530, 500, 587], "lines": [{"bbox": [127, 531, 498, 546], "spans": [{"bbox": [127, 531, 243, 546], "score": 1.0, "content": "Definition 1.2: Let ", "type": "text"}, {"bbox": [243, 534, 254, 542], "score": 0.87, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [255, 531, 491, 546], "score": 1.0, "content": " be a Calabi-Yau n-fold, with Kaehler form ", "type": "text"}, {"bbox": [491, 537, 498, 542], "score": 0.33, "content": "\\omega", "type": "inline_equation", "height": 5, "width": 7}], "index": 28}, {"bbox": [111, 547, 499, 560], "spans": [{"bbox": [111, 547, 338, 560], "score": 1.0, "content": "and holomorphic nowhere vanishing n-form ", "type": "text"}, {"bbox": [338, 548, 347, 557], "score": 0.45, "content": "\\Omega", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [347, 547, 356, 560], "score": 1.0, "content": ". ", "type": "text"}, {"bbox": [356, 547, 365, 557], "score": 0.4, "content": "A", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [366, 547, 402, 560], "score": 1.0, "content": " (real) ", "type": "text"}, {"bbox": [402, 551, 409, 557], "score": 0.57, "content": "\\boldsymbol{n}", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [410, 547, 499, 560], "score": 1.0, "content": "-dimensional sub-", "type": "text"}], "index": 29}, {"bbox": [111, 561, 500, 575], "spans": [{"bbox": [111, 561, 159, 575], "score": 1.0, "content": "manifold ", "type": "text"}, {"bbox": [159, 563, 219, 573], "score": 0.9, "content": "j:\\Lambda\\hookrightarrow X", "type": "inline_equation", "height": 10, "width": 60}, {"bbox": [219, 561, 237, 575], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [238, 563, 249, 571], "score": 0.87, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [249, 561, 500, 575], "score": 1.0, "content": " is called special Lagrangian if the following two", "type": "text"}], "index": 30}, {"bbox": [111, 576, 232, 589], "spans": [{"bbox": [111, 576, 232, 589], "score": 1.0, "content": "conditions are satisfied:", "type": "text"}], "index": 31}], "index": 29.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [111, 531, 500, 589]}, {"type": "text", "bbox": [129, 588, 388, 602], "lines": [{"bbox": [129, 590, 387, 604], "spans": [{"bbox": [129, 590, 142, 604], "score": 1.0, "content": "1) ", "type": "text"}, {"bbox": [142, 591, 151, 601], "score": 0.61, "content": "\\Lambda", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [152, 590, 305, 604], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [306, 594, 314, 600], "score": 0.67, "content": "\\omega", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [314, 590, 342, 604], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [343, 592, 383, 603], "score": 0.91, "content": "j^{}\\omega=0", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [383, 590, 387, 604], "score": 1.0, "content": ";", "type": "text"}], "index": 32}], "index": 32, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [129, 590, 387, 604]}, {"type": "text", "bbox": [109, 603, 502, 631], "lines": [{"bbox": [127, 604, 501, 620], "spans": [{"bbox": [127, 604, 262, 620], "score": 1.0, "content": "2) there exists a multiple ", "type": "text"}, {"bbox": [262, 606, 274, 615], "score": 0.88, "content": "\\Omega^{\\prime}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [274, 604, 292, 620], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [293, 606, 302, 615], "score": 0.79, "content": "\\Omega", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [302, 604, 357, 620], "score": 1.0, "content": "such that ", "type": "text"}, {"bbox": [358, 605, 419, 618], "score": 0.67, "content": "j^{}\\mathrm{Im}(\\Omega^{\\prime})\\!=\\!0_{.}", "type": "inline_equation", "height": 13, "width": 61}, {"bbox": [420, 604, 501, 620], "score": 1.0, "content": "; 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956.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [555.0, 916.0, 1307.0, 916.0, 1307.0, 956.0, 555.0, 956.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1332.0, 916.0, 1344.0, 916.0, 1344.0, 956.0, 1332.0, 956.0], "score": 1.0, "text": ""}], "page_info": {"page_no": 1, "height": 2200, "width": 1700}}	Despite their importance, there are very few explicit examples of special Lagrangian submanifolds, especially in Calabi-Yau 3-folds. However, in an irreducible symplectic 4-fold (realized as a hyperkaehler manifold) we have a complete control of the special Lagrangian geometry of its submanifolds, via a sort of hyperkaehler trick; moreover this enables us to prove that special Lagrangian submanifolds retain part of the rigidity of complex submanifolds. We first recall the following: Definition 1.1: A complex manifold is called irreducible symplectic if it satisfies the following three conditions: 1) is compact and Kaehler; ) is simply connected; 3) is spanned by an everywhere non-degenerate 2-form . In particular, irreducible symplectic manifolds are special cases of Calabi- Yau manifolds (the top holomorphic form which trivializes the canonical line bundle is given by a suitable power of the holomorphic 2-form ). In dimen- sion 2, K3 surfaces are the only irreducible symplectic manifolds, and indeed irreducible symplectic manifolds appear as higher-dimensional analogues of K3 surfaces, as strongly suggested in [5]. Unfortunately, up to now there are very few explicit examples of irreducible symplectic manifolds. Indeed almost all known examples turn out to be birational to two standard series of examples: Hilbert schemes of points on K3 surfaces and generalized Kummer varieties (both series were first studied in [2]), but quite recently O’Grady has constructed irreducible symplectic manifolds which are not birational to any of the elements of the two groups (see [10]). Finally, let us recall from [4] the following: Definition 1.2: Let be a Calabi-Yau n-fold, with Kaehler form and holomorphic nowhere vanishing n-form . (real) -dimensional sub- manifold of is called special Lagrangian if the following two conditions are satisfied: 1) is Lagrangian with respect to , i.e. ; 2) there exists a multiple of such that ; one can prove (see [4]) that both conditions are equivalent to: ’) . The condition ) in the previous definition means that the real part of restricts to the volume form of , induced by the Calabi-Yau Riemannian metric . In this way special Lagrangian submanifolds are considered as a type of calibrated submanifolds (see [4] for further details on this point). 2	<div class="pdf-page"> <p>Despite their importance, there are very few explicit examples of special Lagrangian submanifolds, especially in Calabi-Yau 3-folds. However, in an irreducible symplectic 4-fold (realized as a hyperkaehler manifold) we have a complete control of the special Lagrangian geometry of its submanifolds, via a sort of hyperkaehler trick; moreover this enables us to prove that special Lagrangian submanifolds retain part of the rigidity of complex submanifolds.</p> <p>We first recall the following:</p> <p>Definition 1.1: A complex manifold is called irreducible symplectic if it satisfies the following three conditions:</p> <p>1) is compact and Kaehler;</p> <p>) is simply connected;</p> <p>3) is spanned by an everywhere non-degenerate 2-form .</p> <p>In particular, irreducible symplectic manifolds are special cases of Calabi- Yau manifolds (the top holomorphic form which trivializes the canonical line bundle is given by a suitable power of the holomorphic 2-form ). In dimen- sion 2, K3 surfaces are the only irreducible symplectic manifolds, and indeed irreducible symplectic manifolds appear as higher-dimensional analogues of K3 surfaces, as strongly suggested in [5]. Unfortunately, up to now there are very few explicit examples of irreducible symplectic manifolds. Indeed almost all known examples turn out to be birational to two standard series of examples: Hilbert schemes of points on K3 surfaces and generalized Kummer varieties (both series were first studied in [2]), but quite recently O’Grady has constructed irreducible symplectic manifolds which are not birational to any of the elements of the two groups (see [10]).</p> <p>Finally, let us recall from [4] the following:</p> <p>Definition 1.2: Let be a Calabi-Yau n-fold, with Kaehler form and holomorphic nowhere vanishing n-form . (real) -dimensional sub- manifold of is called special Lagrangian if the following two conditions are satisfied:</p> <p>1) is Lagrangian with respect to , i.e. ;</p> <p>2) there exists a multiple of such that ; one can prove (see [4]) that both conditions are equivalent to:</p> <p>’) .</p> <p>The condition ) in the previous definition means that the real part of restricts to the volume form of , induced by the Calabi-Yau Riemannian metric . In this way special Lagrangian submanifolds are considered as a type of calibrated submanifolds (see [4] for further details on this point).</p> </div>	<div class="pdf-page"> <p class="pdf-text" data-x="182" data-y="218" data-width="654" data-height="111">Despite their importance, there are very few explicit examples of special Lagrangian submanifolds, especially in Calabi-Yau 3-folds. However, in an irreducible symplectic 4-fold (realized as a hyperkaehler manifold) we have a complete control of the special Lagrangian geometry of its submanifolds, via a sort of hyperkaehler trick; moreover this enables us to prove that special Lagrangian submanifolds retain part of the rigidity of complex submanifolds.</p> <p class="pdf-text" data-x="212" data-y="329" data-width="244" data-height="18">We first recall the following:</p> <p class="pdf-text" data-x="184" data-y="349" data-width="654" data-height="36">Definition 1.1: A complex manifold is called irreducible symplectic if it satisfies the following three conditions:</p> <p class="pdf-text" data-x="215" data-y="386" data-width="258" data-height="17">1) is compact and Kaehler;</p> <p class="pdf-text" data-x="214" data-y="405" data-width="224" data-height="17">) is simply connected;</p> <p class="pdf-text" data-x="212" data-y="422" data-width="594" data-height="20">3) is spanned by an everywhere non-degenerate 2-form .</p> <p class="pdf-text" data-x="182" data-y="443" data-width="654" data-height="222">In particular, irreducible symplectic manifolds are special cases of Calabi- Yau manifolds (the top holomorphic form which trivializes the canonical line bundle is given by a suitable power of the holomorphic 2-form ). In dimen- sion 2, K3 surfaces are the only irreducible symplectic manifolds, and indeed irreducible symplectic manifolds appear as higher-dimensional analogues of K3 surfaces, as strongly suggested in [5]. Unfortunately, up to now there are very few explicit examples of irreducible symplectic manifolds. Indeed almost all known examples turn out to be birational to two standard series of examples: Hilbert schemes of points on K3 surfaces and generalized Kummer varieties (both series were first studied in [2]), but quite recently O’Grady has constructed irreducible symplectic manifolds which are not birational to any of the elements of the two groups (see [10]).</p> <p class="pdf-text" data-x="210" data-y="667" data-width="370" data-height="16">Finally, let us recall from [4] the following:</p> <p class="pdf-text" data-x="184" data-y="685" data-width="652" data-height="73">Definition 1.2: Let be a Calabi-Yau n-fold, with Kaehler form and holomorphic nowhere vanishing n-form . (real) -dimensional sub- manifold of is called special Lagrangian if the following two conditions are satisfied:</p> <p class="pdf-text" data-x="215" data-y="760" data-width="434" data-height="18">1) is Lagrangian with respect to , i.e. ;</p> <p class="pdf-text" data-x="182" data-y="779" data-width="657" data-height="36">2) there exists a multiple of such that ; one can prove (see [4]) that both conditions are equivalent to:</p> <p class="pdf-text" data-x="212" data-y="815" data-width="204" data-height="18">’) .</p> <p class="pdf-text" data-x="184" data-y="835" data-width="652" data-height="36">The condition ) in the previous definition means that the real part of restricts to the volume form of , induced by the Calabi-Yau Riemannian metric . In this way special Lagrangian submanifolds are considered as a type of calibrated submanifolds (see [4] for further details on this point).</p> <div class="pdf-discarded" data-x="501" data-y="893" data-width="14" data-height="14" style="opacity: 0.5;">2</div> </div>	{ "type": [ "text", "text", "text", "text", "text", "text", "text", "inline_equation", "text", "inline_equation", "inline_equation", "inline_equation", "text", "text", "inline_equation", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "inline_equation", "inline_equation", "inline_equation", "text", "inline_equation", "inline_equation", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text" ], "coordinates": [ [ 212, 221, 836, 240 ], [ 184, 240, 838, 258 ], [ 182, 258, 838, 276 ], [ 182, 276, 838, 297 ], [ 184, 296, 834, 314 ], [ 182, 314, 833, 333 ], [ 212, 330, 455, 350 ], [ 212, 351, 839, 371 ], [ 185, 372, 537, 387 ], [ 215, 387, 471, 405 ], [ 214, 408, 435, 424 ], [ 214, 425, 808, 444 ], [ 212, 446, 834, 461 ], [ 184, 464, 836, 480 ], [ 184, 483, 836, 500 ], [ 184, 501, 836, 519 ], [ 182, 519, 839, 537 ], [ 184, 539, 836, 557 ], [ 182, 558, 836, 574 ], [ 184, 576, 839, 593 ], [ 184, 596, 836, 612 ], [ 184, 614, 834, 630 ], [ 182, 632, 836, 650 ], [ 184, 650, 597, 668 ], [ 212, 668, 578, 686 ], [ 212, 686, 833, 705 ], [ 185, 707, 834, 724 ], [ 185, 725, 836, 743 ], [ 185, 744, 388, 761 ], [ 215, 762, 647, 780 ], [ 212, 780, 838, 801 ], [ 185, 800, 587, 819 ], [ 214, 815, 416, 837 ], [ 212, 835, 838, 857 ], [ 184, 854, 836, 874 ], [ 182, 165, 839, 183 ], [ 184, 183, 808, 201 ] ], "content": [ "Despite their importance, there are very few explicit examples of special", "Lagrangian submanifolds, especially in Calabi-Yau 3-folds. However, in an", "irreducible symplectic 4-fold (realized as a hyperkaehler manifold) we have a", "complete control of the special Lagrangian geometry of its submanifolds, via", "a sort of hyperkaehler trick; moreover this enables us to prove that special", "Lagrangian submanifolds retain part of the rigidity of complex submanifolds.", "We first recall the following:", "Definition 1.1: A complex manifold X is called irreducible symplectic if", "it satisfies the following three conditions:", "1) X is compact and Kaehler;", "\\boldsymbol{\\mathcal{Q}} ) X is simply connected;", "3) H^{0}(X,\\Omega_{X}^{2}) is spanned by an everywhere non-degenerate 2-form \\omega .", "In particular, irreducible symplectic manifolds are special cases of Calabi-", "Yau manifolds (the top holomorphic form which trivializes the canonical line", "bundle is given by a suitable power of the holomorphic 2-form \\omega ). In dimen-", "sion 2, K3 surfaces are the only irreducible symplectic manifolds, and indeed", "irreducible symplectic manifolds appear as higher-dimensional analogues of", "K3 surfaces, as strongly suggested in [5]. Unfortunately, up to now there", "are very few explicit examples of irreducible symplectic manifolds. Indeed", "almost all known examples turn out to be birational to two standard series of", "examples: Hilbert schemes of points on K3 surfaces and generalized Kummer", "varieties (both series were first studied in [2]), but quite recently O’Grady", "has constructed irreducible symplectic manifolds which are not birational to", "any of the elements of the two groups (see [10]).", "Finally, let us recall from [4] the following:", "Definition 1.2: Let X be a Calabi-Yau n-fold, with Kaehler form \\omega", "and holomorphic nowhere vanishing n-form \\Omega . A (real) \\boldsymbol{n} -dimensional sub-", "manifold j:\\Lambda\\hookrightarrow X of X is called special Lagrangian if the following two", "conditions are satisfied:", "1) \\Lambda is Lagrangian with respect to \\omega , i.e. j^{}\\omega=0 ;", "2) there exists a multiple \\Omega^{\\prime} of \\Omega such that j^{}\\mathrm{Im}(\\Omega^{\\prime})\\!=\\!0_{.} ; one can prove", "(see [4]) that both conditions are equivalent to:", "\\mathit{1} ’) j^{*}\\mathrm{Re}(\\Omega^{\\prime})=V o l_{g}(\\Lambda) .", "The condition 1^{\\prime} ) in the previous definition means that the real part of", "\\Omega^{\\prime} restricts to the volume form of \\Lambda , induced by the Calabi-Yau Riemannian", "metric g . 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0001060v1	2	[ 612, 792 ]	{ "type": [ "text", "title", "text", "text", "text", "text", "interline_equation", "text", "interline_equation", "text", "text", "discarded" ], "coordinates": [ [ 184, 161, 838, 199 ], [ 185, 226, 836, 279 ], [ 184, 293, 836, 350 ], [ 184, 351, 836, 405 ], [ 182, 407, 836, 519 ], [ 182, 519, 836, 593 ], [ 418, 593, 600, 628 ], [ 182, 630, 762, 650 ], [ 322, 664, 696, 699 ], [ 182, 702, 836, 814 ], [ 184, 815, 836, 852 ], [ 501, 893, 515, 907 ] ], "content": [ "", "2 Characterization of special Lagrangian sub- manifolds", "In this section we will describe all special Lagrangian submanifolds of an ir- reducible symplectic 4-fold (having fixed a Kaehler class in the Kaehler cone). The key result is the following:", "Theorem 2.1: Every connected special Lagrangian submanifold of an irreducible symplectic 4-fold is also bi-Lagrangian, in the sense that it is Lagrangian with respect to two different symplectic structures.", "Proof: Let us a fix a Kaehler class on the irreducible symplectic 4- fold . By Yau’s Theorem this determines a unique hyperkaehler metric . Choose a hyperkaehler structure compatible with the metric (notice that the triple is not uniquely determined) and consider the associated symplectic structures , and .", "Consider a special Lagrangian submanifold in the complex structure (this is not restrictive, since is not uniquely determined); that is assume that is calibrated by the real part of the holomorphic (in the structure ) 4-form:", "", "Notice that the real and immaginary part of are then given by:", "", "Obviously, by the property of being special Lagrangian we have that is Lagrangian with respect to . We will prove that having fixed the calibration, if is not Lagrangian also with respect to , then it is nec- essarily Lagrangian with respect to . First we work locally and consider , spanned by . Since is assumed not to be Lagrangian with respect to , we have to deal with two cases.", "First case: is a symplectic vector space for the structure . In this case we can choose a symplectic basis for and this can always be chosen to be of the form . Then is Lagrangian in the symplectic struc- ture ; indeed ; analogously for ; since belong to a Lagrangian subspace of , and analogously for . Thus is also Lagrangian for the symplectic structure .", "3" ], "index": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ] }	[{"type": "text", "text": "", "page_idx": 2}, {"type": "text", "text": "2 Characterization of special Lagrangian submanifolds ", "text_level": 1, "page_idx": 2}, {"type": "text", "text": "In this section we will describe all special Lagrangian submanifolds of an irreducible symplectic 4-fold $X$ (having fixed a Kaehler class $[\\omega]$ in the Kaehler cone). The key result is the following: ", "page_idx": 2}, {"type": "text", "text": "Theorem 2.1: Every connected special Lagrangian submanifold of an irreducible symplectic 4-fold is also bi-Lagrangian, in the sense that it is Lagrangian with respect to two different symplectic structures. ", "page_idx": 2}, {"type": "text", "text": "Proof: Let us a fix a Kaehler class on the irreducible symplectic 4- fold $X$ . By Yau\u2019s Theorem this determines a unique hyperkaehler metric $g$ . Choose a hyperkaehler structure $(I,J,K)$ compatible with the metric $g$ (notice that the triple $(I,J,K)$ is not uniquely determined) and consider the associated symplectic structures $\\omega_{I}(.,.):=g(I.,.)$ , $\\omega_{J}(.,.):=g(J.,.)$ and $\\omega_{K}(.,.):=g(K.,.)$ . ", "page_idx": 2}, {"type": "text", "text": "Consider a special Lagrangian submanifold $\\Lambda$ in the complex structure $K$ (this is not restrictive, since $(I,J,K)$ is not uniquely determined); that is assume that $\\Lambda$ is calibrated by the real part of the holomorphic (in the structure $K$ ) 4-form: ", "page_idx": 2}, {"type": "equation", "text": "$$\n\\Omega_{K}:=\\frac{1}{2!}(\\omega_{I}+i\\omega_{J})^{2}.\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "Notice that the real and immaginary part of $\\Omega_{K}$ are then given by: ", "page_idx": 2}, {"type": "equation", "text": "$$\n\\mathrm{Re}(\\Omega_{K})=\\frac{1}{2}(\\omega_{I}^{2}-\\omega_{J}^{2})\\quad\\mathrm{Im}(\\Omega_{K})=\\omega_{I}\\wedge\\omega_{J}.\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "Obviously, by the property of being special Lagrangian we have that $\\Lambda$ is Lagrangian with respect to $\\omega_{K}$ . We will prove that having fixed the calibration, if $\\Lambda$ is not Lagrangian also with respect to $\\omega_{I}$ , then it is necessarily Lagrangian with respect to $\\omega_{J}$ . First we work locally and consider $V:=T_{p}\\Lambda$ $y\\in\\Lambda)$ , spanned by $(w_{1},w_{2},w_{3},w_{4})$ . Since $\\Lambda$ is assumed not to be Lagrangian with respect to $\\omega_{I}$ , we have to deal with two cases. ", "page_idx": 2}, {"type": "text", "text": "First case: $V$ is a symplectic vector space for the structure $\\omega_{I}$ . In this case we can choose a symplectic basis for $V$ and this can always be chosen to be of the form $v_{1},I v_{1},v_{2},I v_{2}$ . Then $V$ is Lagrangian in the symplectic structure $\\omega_{J}$ ; indeed $\\omega_{J}(v_{1},I v_{1})=g(J v_{1},I v_{1})=g(I J v_{1},-v_{1})=-\\omega_{K}(v_{1},v_{1})=0$ ; analogously for $\\omega_{J}(v_{2},I v_{2})$ ; $\\omega_{J}(v_{1},I v_{2})\\;=\\;g(J v_{1},I v_{2})\\;=\\;-\\omega_{K}(v_{1},v_{2})\\;=\\;0$ since $v_{1},v_{2}$ belong to a Lagrangian subspace of $\\omega_{K}$ , and analogously for $\\omega_{J}(v_{2},I v_{1})=-\\omega_{K}(v_{2},v_{1})=0$ . Thus $V$ is also Lagrangian for the symplectic structure $\\omega_{J}$ . ", "page_idx": 2}]	{"preproc_blocks": [{"type": "text", "bbox": [110, 125, 501, 154], "lines": [{"bbox": [109, 128, 502, 142], "spans": [{"bbox": [109, 128, 147, 142], "score": 1.0, "content": "metric ", "type": "text"}, {"bbox": [147, 133, 154, 141], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [154, 128, 502, 142], "score": 1.0, "content": ". In this way special Lagrangian submanifolds are considered as a", "type": "text"}], "index": 0}, {"bbox": [110, 142, 483, 156], "spans": [{"bbox": [110, 142, 483, 156], "score": 1.0, "content": "type of calibrated submanifolds (see [4] for further details on this point).", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "title", "bbox": [111, 175, 500, 216], "lines": [{"bbox": [111, 178, 498, 197], "spans": [{"bbox": [111, 181, 122, 194], "score": 1.0, "content": "2", "type": "text"}, {"bbox": [138, 178, 498, 197], "score": 1.0, "content": "Characterization of special Lagrangian sub-", "type": "text"}], "index": 2}, {"bbox": [140, 201, 221, 218], "spans": [{"bbox": [140, 201, 221, 218], "score": 1.0, "content": "manifolds", "type": "text"}], "index": 3}], "index": 2.5}, {"type": "text", "bbox": [110, 227, 500, 271], "lines": [{"bbox": [109, 231, 498, 244], "spans": [{"bbox": [109, 231, 498, 244], "score": 1.0, "content": "In this section we will describe all special Lagrangian submanifolds of an ir-", "type": "text"}], "index": 4}, {"bbox": [110, 245, 499, 259], "spans": [{"bbox": [110, 245, 248, 259], "score": 1.0, "content": "reducible symplectic 4-fold ", "type": "text"}, {"bbox": [248, 247, 259, 255], "score": 0.9, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [259, 245, 409, 259], "score": 1.0, "content": " (having fixed a Kaehler class ", "type": "text"}, {"bbox": [410, 246, 424, 258], "score": 0.88, "content": "[\\omega]", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [424, 245, 499, 259], "score": 1.0, "content": " in the Kaehler", "type": "text"}], "index": 5}, {"bbox": [110, 259, 305, 273], "spans": [{"bbox": [110, 259, 305, 273], "score": 1.0, "content": "cone). The key result is the following:", "type": "text"}], "index": 6}], "index": 5}, {"type": "text", "bbox": [110, 272, 500, 314], "lines": [{"bbox": [127, 273, 501, 287], "spans": [{"bbox": [127, 273, 501, 287], "score": 1.0, "content": "Theorem 2.1: Every connected special Lagrangian submanifold of an", "type": "text"}], "index": 7}, {"bbox": [110, 288, 502, 302], "spans": [{"bbox": [110, 288, 502, 302], "score": 1.0, "content": "irreducible symplectic 4-fold is also bi-Lagrangian, in the sense that it is", "type": "text"}], "index": 8}, {"bbox": [111, 302, 427, 316], "spans": [{"bbox": [111, 302, 427, 316], "score": 1.0, "content": "Lagrangian with respect to two different symplectic structures.", "type": "text"}], "index": 9}], "index": 8}, {"type": "text", "bbox": [109, 315, 500, 402], "lines": [{"bbox": [127, 317, 499, 330], "spans": [{"bbox": [127, 317, 499, 330], "score": 1.0, "content": "Proof: Let us a fix a Kaehler class on the irreducible symplectic 4-", "type": "text"}], "index": 10}, {"bbox": [110, 332, 500, 345], "spans": [{"bbox": [110, 332, 134, 345], "score": 1.0, "content": "fold ", "type": "text"}, {"bbox": [134, 333, 145, 342], "score": 0.9, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [146, 332, 500, 345], "score": 1.0, "content": ". By Yau\u2019s Theorem this determines a unique hyperkaehler metric", "type": "text"}], "index": 11}, {"bbox": [110, 346, 499, 361], "spans": [{"bbox": [110, 351, 117, 359], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [117, 346, 299, 361], "score": 1.0, "content": ". Choose a hyperkaehler structure ", "type": "text"}, {"bbox": [299, 347, 342, 360], "score": 0.94, "content": "(I,J,K)", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [343, 346, 492, 361], "score": 1.0, "content": " compatible with the metric ", "type": "text"}, {"bbox": [493, 351, 499, 359], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 6}], "index": 12}, {"bbox": [110, 361, 500, 375], "spans": [{"bbox": [110, 361, 232, 375], "score": 1.0, "content": "(notice that the triple ", "type": "text"}, {"bbox": [232, 361, 275, 374], "score": 0.94, "content": "(I,J,K)", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [276, 361, 500, 375], "score": 1.0, "content": " is not uniquely determined) and consider", "type": "text"}], "index": 13}, {"bbox": [109, 375, 500, 389], "spans": [{"bbox": [109, 375, 298, 389], "score": 1.0, "content": "the associated symplectic structures ", "type": "text"}, {"bbox": [298, 376, 383, 388], "score": 0.84, "content": "\\omega_{I}(.,.):=g(I.,.)", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [384, 375, 389, 389], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [390, 376, 476, 388], "score": 0.91, "content": "\\omega_{J}(.,.):=g(J.,.)", "type": "inline_equation", "height": 12, "width": 86}, {"bbox": [477, 375, 500, 389], "score": 1.0, "content": " and", "type": "text"}], "index": 14}, {"bbox": [110, 388, 208, 404], "spans": [{"bbox": [110, 390, 203, 403], "score": 0.94, "content": "\\omega_{K}(.,.):=g(K.,.)", "type": "inline_equation", "height": 13, "width": 93}, {"bbox": [203, 388, 208, 404], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 12.5}, {"type": "text", "bbox": [109, 402, 500, 459], "lines": [{"bbox": [127, 403, 500, 418], "spans": [{"bbox": [127, 403, 356, 418], "score": 1.0, "content": "Consider a special Lagrangian submanifold ", "type": "text"}, {"bbox": [357, 405, 365, 414], "score": 0.89, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [366, 403, 500, 418], "score": 1.0, "content": " in the complex structure", "type": "text"}], "index": 16}, {"bbox": [110, 417, 500, 433], "spans": [{"bbox": [110, 420, 121, 428], "score": 0.89, "content": "K", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [122, 417, 276, 433], "score": 1.0, "content": " (this is not restrictive, since ", "type": "text"}, {"bbox": [277, 419, 320, 432], "score": 0.95, "content": "(I,J,K)", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [320, 417, 500, 433], "score": 1.0, "content": " is not uniquely determined); that", "type": "text"}], "index": 17}, {"bbox": [108, 431, 500, 447], "spans": [{"bbox": [108, 431, 190, 447], "score": 1.0, "content": "is assume that ", "type": "text"}, {"bbox": [190, 434, 199, 443], "score": 0.91, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [199, 431, 500, 447], "score": 1.0, "content": " is calibrated by the real part of the holomorphic (in the", "type": "text"}], "index": 18}, {"bbox": [109, 446, 218, 461], "spans": [{"bbox": [109, 446, 160, 461], "score": 1.0, "content": "structure ", "type": "text"}, {"bbox": [160, 449, 172, 458], "score": 0.88, "content": "K", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [172, 446, 218, 461], "score": 1.0, "content": ") 4-form:", "type": "text"}], "index": 19}], "index": 17.5}, {"type": "interline_equation", "bbox": [250, 459, 359, 486], "lines": [{"bbox": [250, 459, 359, 486], "spans": [{"bbox": [250, 459, 359, 486], "score": 0.95, "content": "\\Omega_{K}:=\\frac{1}{2!}(\\omega_{I}+i\\omega_{J})^{2}.", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [109, 488, 456, 503], "lines": [{"bbox": [109, 491, 455, 505], "spans": [{"bbox": [109, 491, 341, 505], "score": 1.0, "content": "Notice that the real and immaginary part of ", "type": "text"}, {"bbox": [341, 492, 358, 503], "score": 0.93, "content": "\\Omega_{K}", "type": "inline_equation", "height": 11, "width": 17}, {"bbox": [358, 491, 455, 505], "score": 1.0, "content": " are then given by:", "type": "text"}], "index": 21}], "index": 21}, {"type": "interline_equation", "bbox": [193, 514, 416, 541], "lines": [{"bbox": [193, 514, 416, 541], "spans": [{"bbox": [193, 514, 416, 541], "score": 0.94, "content": "\\mathrm{Re}(\\Omega_{K})=\\frac{1}{2}(\\omega_{I}^{2}-\\omega_{J}^{2})\\quad\\mathrm{Im}(\\Omega_{K})=\\omega_{I}\\wedge\\omega_{J}.", "type": "interline_equation"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [109, 543, 500, 630], "lines": [{"bbox": [127, 545, 501, 560], "spans": [{"bbox": [127, 545, 501, 560], "score": 1.0, "content": "Obviously, by the property of being special Lagrangian we have that", "type": "text"}], "index": 23}, {"bbox": [110, 560, 500, 575], "spans": [{"bbox": [110, 562, 119, 571], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [119, 560, 282, 575], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [283, 565, 299, 573], "score": 0.89, "content": "\\omega_{K}", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [299, 560, 500, 575], "score": 1.0, "content": ". We will prove that having fixed the", "type": "text"}], "index": 24}, {"bbox": [109, 575, 500, 589], "spans": [{"bbox": [109, 575, 185, 589], "score": 1.0, "content": "calibration, if ", "type": "text"}, {"bbox": [185, 577, 194, 586], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [194, 575, 403, 589], "score": 1.0, "content": " is not Lagrangian also with respect to ", "type": "text"}, {"bbox": [404, 580, 416, 587], "score": 0.88, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [417, 575, 500, 589], "score": 1.0, "content": ", then it is nec-", "type": "text"}], "index": 25}, {"bbox": [109, 590, 500, 603], "spans": [{"bbox": [109, 590, 296, 603], "score": 1.0, "content": "essarily Lagrangian with respect to ", "type": "text"}, {"bbox": [297, 594, 310, 602], "score": 0.88, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [310, 590, 500, 603], "score": 1.0, "content": ". First we work locally and consider", "type": "text"}], "index": 26}, {"bbox": [110, 604, 501, 618], "spans": [{"bbox": [110, 605, 161, 618], "score": 0.88, "content": "V:=T_{p}\\Lambda", "type": "inline_equation", "height": 13, "width": 51}, {"bbox": [161, 604, 167, 618], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [168, 605, 203, 617], "score": 0.65, "content": "y\\in\\Lambda)", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [204, 604, 275, 618], "score": 1.0, "content": ", spanned by ", "type": "text"}, {"bbox": [276, 604, 353, 617], "score": 0.91, "content": "(w_{1},w_{2},w_{3},w_{4})", "type": "inline_equation", "height": 13, "width": 77}, {"bbox": [353, 604, 394, 618], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [394, 605, 403, 614], "score": 0.91, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [403, 604, 501, 618], "score": 1.0, "content": " is assumed not to", "type": "text"}], "index": 27}, {"bbox": [110, 618, 448, 632], "spans": [{"bbox": [110, 618, 268, 632], "score": 1.0, "content": "be Lagrangian with respect to ", "type": "text"}, {"bbox": [268, 623, 281, 630], "score": 0.89, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [281, 618, 448, 632], "score": 1.0, "content": ", we have to deal with two cases.", "type": "text"}], "index": 28}], "index": 25.5}, {"type": "text", "bbox": [110, 631, 500, 659], "lines": [{"bbox": [126, 631, 500, 647], "spans": [{"bbox": [126, 631, 187, 647], "score": 1.0, "content": "First case: ", "type": "text"}, {"bbox": [188, 634, 197, 643], "score": 0.88, "content": "V", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [198, 631, 441, 647], "score": 1.0, "content": " is a symplectic vector space for the structure ", "type": "text"}, {"bbox": [442, 637, 454, 645], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [454, 631, 500, 647], "score": 1.0, "content": ". In this", "type": "text"}], "index": 29}, {"bbox": [109, 647, 499, 661], "spans": [{"bbox": [109, 647, 320, 661], "score": 1.0, "content": "case we can choose a symplectic basis for ", "type": "text"}, {"bbox": [321, 649, 330, 657], "score": 0.9, "content": "V", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [331, 647, 499, 661], "score": 1.0, "content": " and this can always be chosen to", "type": "text"}], "index": 30}], "index": 29.5}], "layout_bboxes": [], "page_idx": 2, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [250, 459, 359, 486], "lines": [{"bbox": [250, 459, 359, 486], "spans": [{"bbox": [250, 459, 359, 486], "score": 0.95, "content": "\\Omega_{K}:=\\frac{1}{2!}(\\omega_{I}+i\\omega_{J})^{2}.", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "interline_equation", "bbox": [193, 514, 416, 541], "lines": [{"bbox": [193, 514, 416, 541], "spans": [{"bbox": [193, 514, 416, 541], "score": 0.94, "content": "\\mathrm{Re}(\\Omega_{K})=\\frac{1}{2}(\\omega_{I}^{2}-\\omega_{J}^{2})\\quad\\mathrm{Im}(\\Omega_{K})=\\omega_{I}\\wedge\\omega_{J}.", "type": "interline_equation"}], "index": 22}], "index": 22}], "discarded_blocks": [{"type": "discarded", "bbox": [300, 691, 308, 702], "lines": [{"bbox": [300, 692, 309, 705], "spans": [{"bbox": [300, 692, 309, 705], "score": 1.0, "content": "3", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [110, 125, 501, 154], "lines": [], "index": 0.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [109, 128, 502, 156], "lines_deleted": true}, {"type": "title", "bbox": [111, 175, 500, 216], "lines": [{"bbox": [111, 178, 498, 197], "spans": [{"bbox": [111, 181, 122, 194], "score": 1.0, "content": "2", "type": "text"}, {"bbox": [138, 178, 498, 197], "score": 1.0, "content": "Characterization of special Lagrangian sub-", "type": "text"}], "index": 2}, {"bbox": [140, 201, 221, 218], "spans": [{"bbox": [140, 201, 221, 218], "score": 1.0, "content": "manifolds", "type": "text"}], "index": 3}], "index": 2.5, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [110, 227, 500, 271], "lines": [{"bbox": [109, 231, 498, 244], "spans": [{"bbox": [109, 231, 498, 244], "score": 1.0, "content": "In this section we will describe all special Lagrangian submanifolds of an ir-", "type": "text"}], "index": 4}, {"bbox": [110, 245, 499, 259], "spans": [{"bbox": [110, 245, 248, 259], "score": 1.0, "content": "reducible symplectic 4-fold ", "type": "text"}, {"bbox": [248, 247, 259, 255], "score": 0.9, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [259, 245, 409, 259], "score": 1.0, "content": " (having fixed a Kaehler class ", "type": "text"}, {"bbox": [410, 246, 424, 258], "score": 0.88, "content": "[\\omega]", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [424, 245, 499, 259], "score": 1.0, "content": " in the Kaehler", "type": "text"}], "index": 5}, {"bbox": [110, 259, 305, 273], "spans": [{"bbox": [110, 259, 305, 273], "score": 1.0, "content": "cone). The key result is the following:", "type": "text"}], "index": 6}], "index": 5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [109, 231, 499, 273]}, {"type": "text", "bbox": [110, 272, 500, 314], "lines": [{"bbox": [127, 273, 501, 287], "spans": [{"bbox": [127, 273, 501, 287], "score": 1.0, "content": "Theorem 2.1: Every connected special Lagrangian submanifold of an", "type": "text"}], "index": 7}, {"bbox": [110, 288, 502, 302], "spans": [{"bbox": [110, 288, 502, 302], "score": 1.0, "content": "irreducible symplectic 4-fold is also bi-Lagrangian, in the sense that it is", "type": "text"}], "index": 8}, {"bbox": [111, 302, 427, 316], "spans": [{"bbox": [111, 302, 427, 316], "score": 1.0, "content": "Lagrangian with respect to two different symplectic structures.", "type": "text"}], "index": 9}], "index": 8, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [110, 273, 502, 316]}, {"type": "text", "bbox": [109, 315, 500, 402], "lines": [{"bbox": [127, 317, 499, 330], "spans": [{"bbox": [127, 317, 499, 330], "score": 1.0, "content": "Proof: Let us a fix a Kaehler class on the irreducible symplectic 4-", "type": "text"}], "index": 10}, {"bbox": [110, 332, 500, 345], "spans": [{"bbox": [110, 332, 134, 345], "score": 1.0, "content": "fold ", "type": "text"}, {"bbox": [134, 333, 145, 342], "score": 0.9, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [146, 332, 500, 345], "score": 1.0, "content": ". By Yau\u2019s Theorem this determines a unique hyperkaehler metric", "type": "text"}], "index": 11}, {"bbox": [110, 346, 499, 361], "spans": [{"bbox": [110, 351, 117, 359], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [117, 346, 299, 361], "score": 1.0, "content": ". Choose a hyperkaehler structure ", "type": "text"}, {"bbox": [299, 347, 342, 360], "score": 0.94, "content": "(I,J,K)", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [343, 346, 492, 361], "score": 1.0, "content": " compatible with the metric ", "type": "text"}, {"bbox": [493, 351, 499, 359], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 6}], "index": 12}, {"bbox": [110, 361, 500, 375], "spans": [{"bbox": [110, 361, 232, 375], "score": 1.0, "content": "(notice that the triple ", "type": "text"}, {"bbox": [232, 361, 275, 374], "score": 0.94, "content": "(I,J,K)", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [276, 361, 500, 375], "score": 1.0, "content": " is not uniquely determined) and consider", "type": "text"}], "index": 13}, {"bbox": [109, 375, 500, 389], "spans": [{"bbox": [109, 375, 298, 389], "score": 1.0, "content": "the associated symplectic structures ", "type": "text"}, {"bbox": [298, 376, 383, 388], "score": 0.84, "content": "\\omega_{I}(.,.):=g(I.,.)", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [384, 375, 389, 389], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [390, 376, 476, 388], "score": 0.91, "content": "\\omega_{J}(.,.):=g(J.,.)", "type": "inline_equation", "height": 12, "width": 86}, {"bbox": [477, 375, 500, 389], "score": 1.0, "content": " and", "type": "text"}], "index": 14}, {"bbox": [110, 388, 208, 404], "spans": [{"bbox": [110, 390, 203, 403], "score": 0.94, "content": "\\omega_{K}(.,.):=g(K.,.)", "type": "inline_equation", "height": 13, "width": 93}, {"bbox": [203, 388, 208, 404], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 12.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [109, 317, 500, 404]}, {"type": "text", "bbox": [109, 402, 500, 459], "lines": [{"bbox": [127, 403, 500, 418], "spans": [{"bbox": [127, 403, 356, 418], "score": 1.0, "content": "Consider a special Lagrangian submanifold ", "type": "text"}, {"bbox": [357, 405, 365, 414], "score": 0.89, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [366, 403, 500, 418], "score": 1.0, "content": " in the complex structure", "type": "text"}], "index": 16}, {"bbox": [110, 417, 500, 433], "spans": [{"bbox": [110, 420, 121, 428], "score": 0.89, "content": "K", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [122, 417, 276, 433], "score": 1.0, "content": " (this is not restrictive, since ", "type": "text"}, {"bbox": [277, 419, 320, 432], "score": 0.95, "content": "(I,J,K)", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [320, 417, 500, 433], "score": 1.0, "content": " is not uniquely determined); that", "type": "text"}], "index": 17}, {"bbox": [108, 431, 500, 447], "spans": [{"bbox": [108, 431, 190, 447], "score": 1.0, "content": "is assume that ", "type": "text"}, {"bbox": [190, 434, 199, 443], "score": 0.91, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [199, 431, 500, 447], "score": 1.0, "content": " is calibrated by the real part of the holomorphic (in the", "type": "text"}], "index": 18}, {"bbox": [109, 446, 218, 461], "spans": [{"bbox": [109, 446, 160, 461], "score": 1.0, "content": "structure ", "type": "text"}, {"bbox": [160, 449, 172, 458], "score": 0.88, "content": "K", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [172, 446, 218, 461], "score": 1.0, "content": ") 4-form:", "type": "text"}], "index": 19}], "index": 17.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [108, 403, 500, 461]}, {"type": "interline_equation", "bbox": [250, 459, 359, 486], "lines": [{"bbox": [250, 459, 359, 486], "spans": [{"bbox": [250, 459, 359, 486], "score": 0.95, "content": "\\Omega_{K}:=\\frac{1}{2!}(\\omega_{I}+i\\omega_{J})^{2}.", "type": "interline_equation"}], "index": 20}], "index": 20, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [109, 488, 456, 503], "lines": [{"bbox": [109, 491, 455, 505], "spans": [{"bbox": [109, 491, 341, 505], "score": 1.0, "content": "Notice that the real and immaginary part of ", "type": "text"}, {"bbox": [341, 492, 358, 503], "score": 0.93, "content": "\\Omega_{K}", "type": "inline_equation", "height": 11, "width": 17}, {"bbox": [358, 491, 455, 505], "score": 1.0, "content": " are then given by:", "type": "text"}], "index": 21}], "index": 21, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [109, 491, 455, 505]}, {"type": "interline_equation", "bbox": [193, 514, 416, 541], "lines": [{"bbox": [193, 514, 416, 541], "spans": [{"bbox": [193, 514, 416, 541], "score": 0.94, "content": "\\mathrm{Re}(\\Omega_{K})=\\frac{1}{2}(\\omega_{I}^{2}-\\omega_{J}^{2})\\quad\\mathrm{Im}(\\Omega_{K})=\\omega_{I}\\wedge\\omega_{J}.", "type": "interline_equation"}], "index": 22}], "index": 22, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [109, 543, 500, 630], "lines": [{"bbox": [127, 545, 501, 560], "spans": [{"bbox": [127, 545, 501, 560], "score": 1.0, "content": "Obviously, by the property of being special Lagrangian we have that", "type": "text"}], "index": 23}, {"bbox": [110, 560, 500, 575], "spans": [{"bbox": [110, 562, 119, 571], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [119, 560, 282, 575], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [283, 565, 299, 573], "score": 0.89, "content": "\\omega_{K}", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [299, 560, 500, 575], "score": 1.0, "content": ". We will prove that having fixed the", "type": "text"}], "index": 24}, {"bbox": [109, 575, 500, 589], "spans": [{"bbox": [109, 575, 185, 589], "score": 1.0, "content": "calibration, if ", "type": "text"}, {"bbox": [185, 577, 194, 586], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [194, 575, 403, 589], "score": 1.0, "content": " is not Lagrangian also with respect to ", "type": "text"}, {"bbox": [404, 580, 416, 587], "score": 0.88, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [417, 575, 500, 589], "score": 1.0, "content": ", then it is nec-", "type": "text"}], "index": 25}, {"bbox": [109, 590, 500, 603], "spans": [{"bbox": [109, 590, 296, 603], "score": 1.0, "content": "essarily Lagrangian with respect to ", "type": "text"}, {"bbox": [297, 594, 310, 602], "score": 0.88, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [310, 590, 500, 603], "score": 1.0, "content": ". First we work locally and consider", "type": "text"}], "index": 26}, {"bbox": [110, 604, 501, 618], "spans": [{"bbox": [110, 605, 161, 618], "score": 0.88, "content": "V:=T_{p}\\Lambda", "type": "inline_equation", "height": 13, "width": 51}, {"bbox": [161, 604, 167, 618], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [168, 605, 203, 617], "score": 0.65, "content": "y\\in\\Lambda)", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [204, 604, 275, 618], "score": 1.0, "content": ", spanned by ", "type": "text"}, {"bbox": [276, 604, 353, 617], "score": 0.91, "content": "(w_{1},w_{2},w_{3},w_{4})", "type": "inline_equation", "height": 13, "width": 77}, {"bbox": [353, 604, 394, 618], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [394, 605, 403, 614], "score": 0.91, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [403, 604, 501, 618], "score": 1.0, "content": " is assumed not to", "type": "text"}], "index": 27}, {"bbox": [110, 618, 448, 632], "spans": [{"bbox": [110, 618, 268, 632], "score": 1.0, "content": "be Lagrangian with respect to ", "type": "text"}, {"bbox": [268, 623, 281, 630], "score": 0.89, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [281, 618, 448, 632], "score": 1.0, "content": ", we have to deal with two cases.", "type": "text"}], "index": 28}], "index": 25.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [109, 545, 501, 632]}, {"type": "text", "bbox": [110, 631, 500, 659], "lines": [{"bbox": [126, 631, 500, 647], "spans": [{"bbox": [126, 631, 187, 647], "score": 1.0, "content": "First case: ", "type": "text"}, {"bbox": [188, 634, 197, 643], "score": 0.88, "content": "V", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [198, 631, 441, 647], "score": 1.0, "content": " is a symplectic vector space for the structure ", "type": "text"}, {"bbox": [442, 637, 454, 645], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [454, 631, 500, 647], "score": 1.0, "content": ". In this", "type": "text"}], "index": 29}, {"bbox": [109, 647, 499, 661], "spans": [{"bbox": [109, 647, 320, 661], "score": 1.0, "content": "case we can choose a symplectic basis for ", "type": "text"}, {"bbox": [321, 649, 330, 657], "score": 0.9, "content": "V", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [331, 647, 499, 661], "score": 1.0, "content": " and this can always be chosen to", "type": "text"}], "index": 30}, {"bbox": [109, 127, 500, 142], "spans": [{"bbox": [109, 127, 185, 142], "score": 1.0, "content": "be of the form ", "type": "text", "cross_page": true}, {"bbox": [186, 129, 255, 141], "score": 0.93, "content": "v_{1},I v_{1},v_{2},I v_{2}", "type": "inline_equation", "height": 12, "width": 69, "cross_page": true}, {"bbox": [256, 127, 294, 142], "score": 1.0, "content": ". Then ", "type": "text", "cross_page": true}, {"bbox": [294, 129, 303, 138], "score": 0.91, "content": "V", "type": "inline_equation", "height": 9, "width": 9, "cross_page": true}, {"bbox": [304, 127, 500, 142], "score": 1.0, "content": " is Lagrangian in the symplectic struc-", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [109, 142, 501, 157], "spans": [{"bbox": [109, 142, 134, 157], "score": 1.0, "content": "ture ", "type": "text", "cross_page": true}, {"bbox": [135, 147, 148, 155], "score": 0.88, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13, "cross_page": true}, {"bbox": [149, 142, 192, 157], "score": 1.0, "content": "; indeed ", "type": "text", "cross_page": true}, {"bbox": [193, 144, 496, 156], "score": 0.94, "content": "\\omega_{J}(v_{1},I v_{1})=g(J v_{1},I v_{1})=g(I J v_{1},-v_{1})=-\\omega_{K}(v_{1},v_{1})=0", "type": "inline_equation", "height": 12, "width": 303, "cross_page": true}, {"bbox": [497, 142, 501, 157], "score": 1.0, "content": ";", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [109, 156, 498, 171], "spans": [{"bbox": [109, 156, 195, 171], "score": 1.0, "content": "analogously for ", "type": "text", "cross_page": true}, {"bbox": [195, 158, 250, 170], "score": 0.92, "content": "\\omega_{J}(v_{2},I v_{2})", "type": "inline_equation", "height": 12, "width": 55, "cross_page": true}, {"bbox": [250, 156, 257, 171], "score": 1.0, "content": "; ", "type": "text", "cross_page": true}, {"bbox": [257, 158, 498, 170], "score": 0.9, "content": "\\omega_{J}(v_{1},I v_{2})\\;=\\;g(J v_{1},I v_{2})\\;=\\;-\\omega_{K}(v_{1},v_{2})\\;=\\;0", "type": "inline_equation", "height": 12, "width": 241, "cross_page": true}], "index": 2}, {"bbox": [108, 171, 501, 186], "spans": [{"bbox": [108, 171, 140, 186], "score": 1.0, "content": "since ", "type": "text", "cross_page": true}, {"bbox": [141, 176, 167, 183], "score": 0.9, "content": "v_{1},v_{2}", "type": "inline_equation", "height": 7, "width": 26, "cross_page": true}, {"bbox": [167, 171, 369, 186], "score": 1.0, "content": " belong to a Lagrangian subspace of ", "type": "text", "cross_page": true}, {"bbox": [369, 176, 385, 183], "score": 0.89, "content": "\\omega_{K}", "type": "inline_equation", "height": 7, "width": 16, "cross_page": true}, {"bbox": [386, 171, 501, 186], "score": 1.0, "content": ", and analogously for", "type": "text", "cross_page": true}], "index": 3}, {"bbox": [110, 185, 500, 200], "spans": [{"bbox": [110, 186, 262, 199], "score": 0.93, "content": "\\omega_{J}(v_{2},I v_{1})=-\\omega_{K}(v_{2},v_{1})=0", "type": "inline_equation", "height": 13, "width": 152, "cross_page": true}, {"bbox": [262, 185, 299, 200], "score": 1.0, "content": ". 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835.0, 1959.0], "score": 1.0, "text": ""}], "page_info": {"page_no": 2, "height": 2200, "width": 1700}}	# 2 Characterization of special Lagrangian sub- manifolds In this section we will describe all special Lagrangian submanifolds of an ir- reducible symplectic 4-fold (having fixed a Kaehler class in the Kaehler cone). The key result is the following: Theorem 2.1: Every connected special Lagrangian submanifold of an irreducible symplectic 4-fold is also bi-Lagrangian, in the sense that it is Lagrangian with respect to two different symplectic structures. Proof: Let us a fix a Kaehler class on the irreducible symplectic 4- fold . By Yau’s Theorem this determines a unique hyperkaehler metric . Choose a hyperkaehler structure compatible with the metric (notice that the triple is not uniquely determined) and consider the associated symplectic structures , and . Consider a special Lagrangian submanifold in the complex structure (this is not restrictive, since is not uniquely determined); that is assume that is calibrated by the real part of the holomorphic (in the structure ) 4-form: $$ \Omega_{K}:=\frac{1}{2!}(\omega_{I}+i\omega_{J})^{2}. $$ Notice that the real and immaginary part of are then given by: $$ \mathrm{Re}(\Omega_{K})=\frac{1}{2}(\omega_{I}^{2}-\omega_{J}^{2})\quad\mathrm{Im}(\Omega_{K})=\omega_{I}\wedge\omega_{J}. $$ Obviously, by the property of being special Lagrangian we have that is Lagrangian with respect to . We will prove that having fixed the calibration, if is not Lagrangian also with respect to , then it is nec- essarily Lagrangian with respect to . First we work locally and consider , spanned by . Since is assumed not to be Lagrangian with respect to , we have to deal with two cases. First case: is a symplectic vector space for the structure . In this case we can choose a symplectic basis for and this can always be chosen to be of the form . Then is Lagrangian in the symplectic struc- ture ; indeed ; analogously for ; since belong to a Lagrangian subspace of , and analogously for . Thus is also Lagrangian for the symplectic structure . 3	<div class="pdf-page"> <h1>2 Characterization of special Lagrangian sub- manifolds</h1> <p>In this section we will describe all special Lagrangian submanifolds of an ir- reducible symplectic 4-fold (having fixed a Kaehler class in the Kaehler cone). The key result is the following:</p> <p>Theorem 2.1: Every connected special Lagrangian submanifold of an irreducible symplectic 4-fold is also bi-Lagrangian, in the sense that it is Lagrangian with respect to two different symplectic structures.</p> <p>Proof: Let us a fix a Kaehler class on the irreducible symplectic 4- fold . By Yau’s Theorem this determines a unique hyperkaehler metric . Choose a hyperkaehler structure compatible with the metric (notice that the triple is not uniquely determined) and consider the associated symplectic structures , and .</p> <p>Consider a special Lagrangian submanifold in the complex structure (this is not restrictive, since is not uniquely determined); that is assume that is calibrated by the real part of the holomorphic (in the structure ) 4-form:</p> <p>Notice that the real and immaginary part of are then given by:</p> <p>Obviously, by the property of being special Lagrangian we have that is Lagrangian with respect to . We will prove that having fixed the calibration, if is not Lagrangian also with respect to , then it is nec- essarily Lagrangian with respect to . First we work locally and consider , spanned by . Since is assumed not to be Lagrangian with respect to , we have to deal with two cases.</p> <p>First case: is a symplectic vector space for the structure . In this case we can choose a symplectic basis for and this can always be chosen to be of the form . Then is Lagrangian in the symplectic struc- ture ; indeed ; analogously for ; since belong to a Lagrangian subspace of , and analogously for . Thus is also Lagrangian for the symplectic structure .</p> </div>	<div class="pdf-page"> <h1 class="pdf-title" data-x="185" data-y="226" data-width="651" data-height="53">2 Characterization of special Lagrangian sub- manifolds</h1> <p class="pdf-text" data-x="184" data-y="293" data-width="652" data-height="57">In this section we will describe all special Lagrangian submanifolds of an ir- reducible symplectic 4-fold (having fixed a Kaehler class in the Kaehler cone). The key result is the following:</p> <p class="pdf-text" data-x="184" data-y="351" data-width="652" data-height="54">Theorem 2.1: Every connected special Lagrangian submanifold of an irreducible symplectic 4-fold is also bi-Lagrangian, in the sense that it is Lagrangian with respect to two different symplectic structures.</p> <p class="pdf-text" data-x="182" data-y="407" data-width="654" data-height="112">Proof: Let us a fix a Kaehler class on the irreducible symplectic 4- fold . By Yau’s Theorem this determines a unique hyperkaehler metric . Choose a hyperkaehler structure compatible with the metric (notice that the triple is not uniquely determined) and consider the associated symplectic structures , and .</p> <p class="pdf-text" data-x="182" data-y="519" data-width="654" data-height="74">Consider a special Lagrangian submanifold in the complex structure (this is not restrictive, since is not uniquely determined); that is assume that is calibrated by the real part of the holomorphic (in the structure ) 4-form:</p> <p class="pdf-text" data-x="182" data-y="630" data-width="580" data-height="20">Notice that the real and immaginary part of are then given by:</p> <p class="pdf-text" data-x="182" data-y="702" data-width="654" data-height="112">Obviously, by the property of being special Lagrangian we have that is Lagrangian with respect to . We will prove that having fixed the calibration, if is not Lagrangian also with respect to , then it is nec- essarily Lagrangian with respect to . First we work locally and consider , spanned by . Since is assumed not to be Lagrangian with respect to , we have to deal with two cases.</p> <p class="pdf-text" data-x="184" data-y="815" data-width="652" data-height="37">First case: is a symplectic vector space for the structure . In this case we can choose a symplectic basis for and this can always be chosen to be of the form . Then is Lagrangian in the symplectic struc- ture ; indeed ; analogously for ; since belong to a Lagrangian subspace of , and analogously for . Thus is also Lagrangian for the symplectic structure .</p> <div class="pdf-discarded" data-x="501" data-y="893" data-width="14" data-height="14" style="opacity: 0.5;">3</div> </div>	{ "type": [ "text", "text", "text", "inline_equation", "text", "text", "text", "text", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "interline_equation", "inline_equation", "interline_equation", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation" ], "coordinates": [ [ 185, 230, 833, 254 ], [ 234, 259, 369, 281 ], [ 182, 298, 833, 315 ], [ 184, 316, 834, 334 ], [ 184, 334, 510, 352 ], [ 212, 352, 838, 371 ], [ 184, 372, 839, 390 ], [ 185, 390, 714, 408 ], [ 212, 409, 834, 426 ], [ 184, 429, 836, 446 ], [ 184, 447, 834, 466 ], [ 184, 466, 836, 484 ], [ 182, 484, 836, 502 ], [ 184, 501, 348, 522 ], [ 212, 521, 836, 540 ], [ 184, 539, 836, 559 ], [ 180, 557, 836, 577 ], [ 182, 576, 364, 596 ], [ 418, 593, 600, 628 ], [ 182, 634, 761, 652 ], [ 322, 664, 696, 699 ], [ 212, 704, 838, 724 ], [ 184, 724, 836, 743 ], [ 182, 743, 836, 761 ], [ 182, 762, 836, 779 ], [ 184, 780, 838, 799 ], [ 184, 799, 749, 817 ], [ 210, 815, 836, 836 ], [ 182, 836, 834, 854 ], [ 182, 164, 836, 183 ], [ 182, 183, 838, 202 ], [ 182, 201, 833, 221 ], [ 180, 221, 838, 240 ], [ 184, 239, 836, 258 ], [ 182, 257, 297, 277 ] ], "content": [ "2 Characterization of special Lagrangian sub-", "manifolds", "In this section we will describe all special Lagrangian submanifolds of an ir-", "reducible symplectic 4-fold X (having fixed a Kaehler class [\\omega] in the Kaehler", "cone). The key result is the following:", "Theorem 2.1: Every connected special Lagrangian submanifold of an", "irreducible symplectic 4-fold is also bi-Lagrangian, in the sense that it is", "Lagrangian with respect to two different symplectic structures.", "Proof: Let us a fix a Kaehler class on the irreducible symplectic 4-", "fold X . By Yau’s Theorem this determines a unique hyperkaehler metric", "g . Choose a hyperkaehler structure (I,J,K) compatible with the metric g", "(notice that the triple (I,J,K) is not uniquely determined) and consider", "the associated symplectic structures \\omega_{I}(.,.):=g(I.,.) , \\omega_{J}(.,.):=g(J.,.) and", "\\omega_{K}(.,.):=g(K.,.) .", "Consider a special Lagrangian submanifold \\Lambda in the complex structure", "K (this is not restrictive, since (I,J,K) is not uniquely determined); that", "is assume that \\Lambda is calibrated by the real part of the holomorphic (in the", "structure K ) 4-form:", "\\Omega_{K}:=\\frac{1}{2!}(\\omega_{I}+i\\omega_{J})^{2}.", "Notice that the real and immaginary part of \\Omega_{K} are then given by:", "\\mathrm{Re}(\\Omega_{K})=\\frac{1}{2}(\\omega_{I}^{2}-\\omega_{J}^{2})\\quad\\mathrm{Im}(\\Omega_{K})=\\omega_{I}\\wedge\\omega_{J}.", "Obviously, by the property of being special Lagrangian we have that", "\\Lambda is Lagrangian with respect to \\omega_{K} . We will prove that having fixed the", "calibration, if \\Lambda is not Lagrangian also with respect to \\omega_{I} , then it is nec-", "essarily Lagrangian with respect to \\omega_{J} . First we work locally and consider", "V:=T_{p}\\Lambda y\\in\\Lambda) , spanned by (w_{1},w_{2},w_{3},w_{4}) . Since \\Lambda is assumed not to", "be Lagrangian with respect to \\omega_{I} , we have to deal with two cases.", "First case: V is a symplectic vector space for the structure \\omega_{I} . In this", "case we can choose a symplectic basis for V and this can always be chosen to", "be of the form v_{1},I v_{1},v_{2},I v_{2} . Then V is Lagrangian in the symplectic struc-", "ture \\omega_{J} ; indeed \\omega_{J}(v_{1},I v_{1})=g(J v_{1},I v_{1})=g(I J v_{1},-v_{1})=-\\omega_{K}(v_{1},v_{1})=0 ;", "analogously for \\omega_{J}(v_{2},I v_{2}) ; \\omega_{J}(v_{1},I v_{2})\\;=\\;g(J v_{1},I v_{2})\\;=\\;-\\omega_{K}(v_{1},v_{2})\\;=\\;0", "since v_{1},v_{2} belong to a Lagrangian subspace of \\omega_{K} , and analogously for", "\\omega_{J}(v_{2},I v_{1})=-\\omega_{K}(v_{2},v_{1})=0 . Thus V is also Lagrangian for the symplectic", "structure \\omega_{J} ." ], "index": [ 0, 1, 2, 3, 4, 7, 8, 9, 15, 16, 17, 18, 19, 20, 29, 30, 31, 32, 47, 66, 86, 107, 108, 109, 110, 111, 112, 134, 135, 136, 137, 138, 139, 140, 141 ] }	{ "coordinates": [], "index": [], "caption": [], "caption_coordinates": [] }	{ "type": [ "interline_equation", "interline_equation", "interline_equation", "interline_equation" ], "coordinates": [ [ 418, 593, 600, 628 ], [ 322, 664, 696, 699 ], [ 418, 593, 600, 628 ], [ 322, 664, 696, 699 ] ], "content": [ "", "", "\\Omega_{K}:=\\frac{1}{2!}(\\omega_{I}+i\\omega_{J})^{2}.", "\\mathrm{Re}(\\Omega_{K})=\\frac{1}{2}(\\omega_{I}^{2}-\\omega_{J}^{2})\\quad\\mathrm{Im}(\\Omega_{K})=\\omega_{I}\\wedge\\omega_{J}." ], "index": [ 0, 1, 2, 3 ] }	{ "type": [], "coordinates": [], "content": [], "index": [] }	{"document_info": {"total_pages": 9, "parse_type": "txt", "version": "1.0.1", "total_content_list_items": 72}, "content_statistics": {"total_blocks": 81, "total_lines": 269, "total_images": 0, "total_equations": 18, "total_tables": 0, "total_characters": 16112, "total_words": 2578, "block_type_distribution": {"title": 6, "text": 55, "discarded": 9, "interline_equation": 9, "list": 2}}, "model_statistics": {"total_layout_detections": 813, "avg_confidence": 0.9488487084870848, "min_confidence": 0.26, "max_confidence": 1.0}, "page_statistics": [{"page_number": 0, "page_size": [612.0, 792.0], "blocks": 10, "lines": 26, "images": 0, "equations": 0, "tables": 0, "characters": 1419, "words": 204, "discarded_blocks": 2}, {"page_number": 1, "page_size": [612.0, 792.0], "blocks": 15, "lines": 37, "images": 0, "equations": 0, "tables": 0, "characters": 2328, "words": 359, "discarded_blocks": 1}, {"page_number": 2, "page_size": [612.0, 792.0], "blocks": 12, "lines": 35, "images": 0, "equations": 4, "tables": 0, "characters": 1843, "words": 306, "discarded_blocks": 1}, {"page_number": 3, "page_size": [612.0, 792.0], "blocks": 8, "lines": 30, "images": 0, "equations": 6, "tables": 0, "characters": 1797, "words": 312, "discarded_blocks": 0}, {"page_number": 4, "page_size": [612.0, 792.0], "blocks": 12, "lines": 34, "images": 0, "equations": 4, "tables": 0, "characters": 1905, "words": 328, "discarded_blocks": 1}, {"page_number": 5, "page_size": [612.0, 792.0], "blocks": 11, "lines": 32, "images": 0, "equations": 4, "tables": 0, "characters": 1846, "words": 308, "discarded_blocks": 1}, {"page_number": 6, "page_size": [612.0, 792.0], "blocks": 6, "lines": 46, "images": 0, "equations": 0, "tables": 0, "characters": 3307, "words": 521, "discarded_blocks": 1}, {"page_number": 7, "page_size": [612.0, 792.0], "blocks": 5, "lines": 29, "images": 0, "equations": 0, "tables": 0, "characters": 1666, "words": 239, "discarded_blocks": 1}, {"page_number": 8, "page_size": [612.0, 792.0], "blocks": 2, "lines": 0, "images": 0, "equations": 0, "tables": 0, "characters": 1, "words": 1, "discarded_blocks": 1}]}
0001060v1	3	[ 612, 792 ]	{ "type": [ "text", "text", "text", "text", "interline_equation", "interline_equation", "interline_equation", "text" ], "coordinates": [ [ 182, 160, 836, 274 ], [ 182, 275, 836, 479 ], [ 182, 480, 836, 535 ], [ 182, 536, 836, 722 ], [ 220, 735, 801, 755 ], [ 205, 768, 814, 786 ], [ 291, 793, 726, 810 ], [ 184, 815, 838, 871 ] ], "content": [ "", "Second case: is neither symplectic nor Lagrangian for the structure . Notice can not be symplectic with respect to , otherwise by the first case it would be Lagrangian in the strucutre ; moreover we can assume that is not Lagrangian with respect to , otherwise there is nothing to prove. So in this case is neither Lagrangian nor symplectic in the structure and in the structure . This means that contains a symplectic 2-plane with respect to and a symplectic 2-plane with respect to . Indeed, consider ; since is not Lagrangian in the structure , there exists such that and this implies that the vector subspace spanned by is a symplectic vector space for , which can not be extended to all . The same reasoning applies in the structure .", "We prove that this can not happen, since it violates the calibration con- dition. We have to distinguish three different subcases according to the intersection of with .", "First subcase: and have zero intersection. If this happens we can always choose a basis of of the form . Write for the 2- plane spanned by and for that spanned by , so that . Indeed, since is not Lagrangian with respect to , it has to contain a symplectic 2-plane like , and similarly for and . Moreover, since is not symplectic with respect to , it turns out that the symplectic 2-plane can not be completed to a symplectic basis of , so that has to contain an isotropic 2-plane for , which is . The same reasoning (with the roles reversed) applies obviously to the symplectic structure . Hence, in this case we have:", "", "", "", "using the defining relations of , the quaternionic relation , the invariance of and the fact that is Lagrangian with respect to . So this subcase is not consistent with the calibration property." ], "index": [ 0, 1, 2, 3, 4, 5, 6, 7 ] }	[{"type": "text", "text": "", "page_idx": 3}, {"type": "text", "text": "Second case: $V$ is neither symplectic nor Lagrangian for the structure $\\omega_{I}$ . Notice $V$ can not be symplectic with respect to $\\omega_{J}$ , otherwise by the first case it would be Lagrangian in the strucutre $\\omega_{I}$ ; moreover we can assume that $V$ is not Lagrangian with respect to $\\omega_{J}$ , otherwise there is nothing to prove. So in this case $V$ is neither Lagrangian nor symplectic in the structure $\\omega_{I}$ and in the structure $\\omega_{J}$ . This means that $V$ contains a symplectic 2-plane $\\pi$ with respect to $\\omega_{I}$ and a symplectic 2-plane $\\rho$ with respect to $\\omega_{J}$ . Indeed, consider $v_{1}\\in V$ ; since $V$ is not Lagrangian in the structure $\\omega_{I}$ , there exists $v_{2}\\in V$ such that $\\omega_{I}(v_{1},v_{2})\\neq0$ and this implies that the vector subspace $\\pi$ spanned by $(v_{1},v_{2})$ is a symplectic vector space for $\\omega_{I}$ , which can not be extended to all $V$ . The same reasoning applies in the structure $\\omega_{J}$ . ", "page_idx": 3}, {"type": "text", "text": "We prove that this can not happen, since it violates the calibration condition. We have to distinguish three different subcases according to the intersection of $\\pi$ with $\\rho$ . ", "page_idx": 3}, {"type": "text", "text": "First subcase: $\\pi$ and $\\rho$ have zero intersection. If this happens we can always choose a basis of $V$ of the form $(v_{1},I v_{1},v_{2},J v_{2})$ . Write $\\pi$ for the 2- plane spanned by $v_{1},I v_{1}$ and $\\rho$ for that spanned by $v_{2},J v_{2}$ , so that $V=\\pi\\oplus\\rho$ . Indeed, since $V$ is not Lagrangian with respect to $\\omega_{I}$ , it has to contain a symplectic 2-plane like $\\pi$ , and similarly for $\\rho$ and $\\omega_{J}$ . Moreover, since $V$ is not symplectic with respect to $\\omega_{I}$ , it turns out that the symplectic 2-plane $\\pi$ can not be completed to a symplectic basis of $V$ , so that $V$ has to contain an isotropic 2-plane for $\\omega_{I}$ , which is $\\rho$ . The same reasoning (with the roles reversed) applies obviously to the symplectic structure $\\omega_{J}$ . Hence, in this case we have: ", "page_idx": 3}, {"type": "equation", "text": "$$\n2\\mathrm{Re}(\\Omega_{K})\|_{V}=(\\omega_{I}^{2}-\\omega_{J}^{2})(v_{1},I v_{1},v_{2},J v_{2})=\\omega_{I}(v_{1},I v_{1})\\omega_{I}(v_{2},J v_{2})-\n$$", "text_format": "latex", "page_idx": 3}, {"type": "equation", "text": "$$\n\\omega_{I}(v_{1},v_{2})\\omega_{I}(I v_{1},J v_{2})+\\omega_{I}(v_{1},J v_{2})\\omega_{I}(I v_{1},v_{2})-\\omega_{J}(v_{1},I v_{1})\\omega_{J}(v_{2},J v_{2})+\n$$", "text_format": "latex", "page_idx": 3}, {"type": "equation", "text": "$$\n\\omega_{J}(v_{1},v_{2})\\omega_{J}(I v_{1},J v_{2})-\\omega_{J}(v_{1},J v_{2})\\omega_{J}(I v_{2},v_{2})=0,\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "using the defining relations of $\\omega_{I},\\omega_{J},\\omega_{K}$ , the quaternionic relation $I J=K$ , the invariance of $g$ and the fact that $V$ is Lagrangian with respect to $\\omega_{K}$ . So this subcase is not consistent with the calibration property. ", "page_idx": 3}]	{"preproc_blocks": [{"type": "text", "bbox": [109, 124, 500, 212], "lines": [{"bbox": [109, 127, 500, 142], "spans": [{"bbox": [109, 127, 185, 142], "score": 1.0, "content": "be of the form ", "type": "text"}, {"bbox": [186, 129, 255, 141], "score": 0.93, "content": "v_{1},I v_{1},v_{2},I v_{2}", "type": "inline_equation", "height": 12, "width": 69}, {"bbox": [256, 127, 294, 142], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [294, 129, 303, 138], "score": 0.91, "content": "V", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [304, 127, 500, 142], "score": 1.0, "content": " is Lagrangian in the symplectic struc-", "type": "text"}], "index": 0}, {"bbox": [109, 142, 501, 157], "spans": [{"bbox": [109, 142, 134, 157], "score": 1.0, "content": "ture ", "type": "text"}, {"bbox": [135, 147, 148, 155], "score": 0.88, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [149, 142, 192, 157], "score": 1.0, "content": "; indeed ", "type": "text"}, {"bbox": [193, 144, 496, 156], "score": 0.94, "content": "\\omega_{J}(v_{1},I v_{1})=g(J v_{1},I v_{1})=g(I J v_{1},-v_{1})=-\\omega_{K}(v_{1},v_{1})=0", "type": "inline_equation", "height": 12, "width": 303}, {"bbox": [497, 142, 501, 157], "score": 1.0, "content": ";", "type": "text"}], "index": 1}, {"bbox": [109, 156, 498, 171], "spans": [{"bbox": [109, 156, 195, 171], "score": 1.0, "content": "analogously for ", "type": "text"}, {"bbox": [195, 158, 250, 170], "score": 0.92, "content": "\\omega_{J}(v_{2},I v_{2})", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [250, 156, 257, 171], "score": 1.0, "content": "; ", "type": "text"}, {"bbox": [257, 158, 498, 170], "score": 0.9, "content": "\\omega_{J}(v_{1},I v_{2})\\;=\\;g(J v_{1},I v_{2})\\;=\\;-\\omega_{K}(v_{1},v_{2})\\;=\\;0", "type": "inline_equation", "height": 12, "width": 241}], "index": 2}, {"bbox": [108, 171, 501, 186], "spans": [{"bbox": [108, 171, 140, 186], "score": 1.0, "content": "since ", "type": "text"}, {"bbox": [141, 176, 167, 183], "score": 0.9, "content": "v_{1},v_{2}", "type": "inline_equation", "height": 7, "width": 26}, {"bbox": [167, 171, 369, 186], "score": 1.0, "content": " belong to a Lagrangian subspace of ", "type": "text"}, {"bbox": [369, 176, 385, 183], "score": 0.89, "content": "\\omega_{K}", "type": "inline_equation", "height": 7, "width": 16}, {"bbox": [386, 171, 501, 186], "score": 1.0, "content": ", and analogously for", "type": "text"}], "index": 3}, {"bbox": [110, 185, 500, 200], "spans": [{"bbox": [110, 186, 262, 199], "score": 0.93, "content": "\\omega_{J}(v_{2},I v_{1})=-\\omega_{K}(v_{2},v_{1})=0", "type": "inline_equation", "height": 13, "width": 152}, {"bbox": [262, 185, 299, 200], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [299, 187, 309, 196], "score": 0.89, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [309, 185, 500, 200], "score": 1.0, "content": " is also Lagrangian for the symplectic", "type": "text"}], "index": 4}, {"bbox": [109, 199, 178, 215], "spans": [{"bbox": [109, 199, 160, 215], "score": 1.0, "content": "structure ", "type": "text"}, {"bbox": [160, 204, 174, 212], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [174, 199, 178, 215], "score": 1.0, "content": ".", "type": "text"}], "index": 5}], "index": 2.5}, {"type": "text", "bbox": [109, 213, 500, 371], "lines": [{"bbox": [127, 213, 500, 228], "spans": [{"bbox": [127, 213, 195, 228], "score": 1.0, "content": "Second case: ", "type": "text"}, {"bbox": [196, 216, 205, 225], "score": 0.88, "content": "V", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [205, 213, 483, 228], "score": 1.0, "content": " is neither symplectic nor Lagrangian for the structure ", "type": "text"}, {"bbox": [483, 219, 496, 226], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [496, 213, 500, 228], "score": 1.0, "content": ".", "type": "text"}], "index": 6}, {"bbox": [109, 228, 500, 243], "spans": [{"bbox": [109, 228, 147, 243], "score": 1.0, "content": "Notice ", "type": "text"}, {"bbox": [148, 231, 158, 240], "score": 0.89, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [158, 228, 365, 243], "score": 1.0, "content": " can not be symplectic with respect to ", "type": "text"}, {"bbox": [366, 234, 379, 241], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [379, 228, 500, 243], "score": 1.0, "content": ", otherwise by the first", "type": "text"}], "index": 7}, {"bbox": [110, 244, 500, 257], "spans": [{"bbox": [110, 244, 349, 257], "score": 1.0, "content": "case it would be Lagrangian in the strucutre ", "type": "text"}, {"bbox": [349, 248, 362, 255], "score": 0.87, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [362, 244, 500, 257], "score": 1.0, "content": "; moreover we can assume", "type": "text"}], "index": 8}, {"bbox": [110, 257, 500, 272], "spans": [{"bbox": [110, 257, 136, 272], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [136, 259, 146, 268], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [146, 257, 328, 272], "score": 1.0, "content": " is not Lagrangian with respect to ", "type": "text"}, {"bbox": [328, 262, 342, 270], "score": 0.91, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [342, 257, 500, 272], "score": 1.0, "content": ", otherwise there is nothing to", "type": "text"}], "index": 9}, {"bbox": [110, 273, 501, 286], "spans": [{"bbox": [110, 273, 220, 286], "score": 1.0, "content": "prove. So in this case ", "type": "text"}, {"bbox": [220, 274, 230, 283], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [230, 273, 501, 286], "score": 1.0, "content": " is neither Lagrangian nor symplectic in the structure", "type": "text"}], "index": 10}, {"bbox": [110, 286, 501, 301], "spans": [{"bbox": [110, 291, 122, 299], "score": 0.83, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [123, 286, 229, 301], "score": 1.0, "content": " and in the structure ", "type": "text"}, {"bbox": [230, 291, 243, 299], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [243, 286, 337, 301], "score": 1.0, "content": ". This means that ", "type": "text"}, {"bbox": [337, 288, 347, 297], "score": 0.89, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [347, 286, 501, 301], "score": 1.0, "content": " contains a symplectic 2-plane", "type": "text"}], "index": 11}, {"bbox": [110, 302, 500, 315], "spans": [{"bbox": [110, 306, 117, 311], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [118, 302, 201, 315], "score": 1.0, "content": " with respect to ", "type": "text"}, {"bbox": [202, 306, 214, 313], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [214, 302, 348, 315], "score": 1.0, "content": " and a symplectic 2-plane ", "type": "text"}, {"bbox": [349, 306, 355, 314], "score": 0.88, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [355, 302, 439, 315], "score": 1.0, "content": " with respect to ", "type": "text"}, {"bbox": [440, 306, 453, 313], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [453, 302, 500, 315], "score": 1.0, "content": ". Indeed,", "type": "text"}], "index": 12}, {"bbox": [110, 316, 500, 329], "spans": [{"bbox": [110, 316, 155, 329], "score": 1.0, "content": "consider ", "type": "text"}, {"bbox": [156, 317, 191, 327], "score": 0.93, "content": "v_{1}\\in V", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [192, 316, 227, 329], "score": 1.0, "content": "; since ", "type": "text"}, {"bbox": [228, 317, 237, 326], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [237, 316, 420, 329], "score": 1.0, "content": " is not Lagrangian in the structure ", "type": "text"}, {"bbox": [421, 320, 433, 327], "score": 0.89, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [433, 316, 500, 329], "score": 1.0, "content": ", there exists", "type": "text"}], "index": 13}, {"bbox": [110, 330, 500, 344], "spans": [{"bbox": [110, 331, 147, 342], "score": 0.93, "content": "v_{2}\\in V", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [148, 330, 205, 344], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [206, 331, 278, 343], "score": 0.94, "content": "\\omega_{I}(v_{1},v_{2})\\neq0", "type": "inline_equation", "height": 12, "width": 72}, {"bbox": [278, 330, 500, 344], "score": 1.0, "content": " and this implies that the vector subspace", "type": "text"}], "index": 14}, {"bbox": [110, 344, 500, 359], "spans": [{"bbox": [110, 349, 117, 355], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [118, 344, 184, 359], "score": 1.0, "content": " spanned by ", "type": "text"}, {"bbox": [185, 345, 219, 358], "score": 0.94, "content": "(v_{1},v_{2})", "type": "inline_equation", "height": 13, "width": 34}, {"bbox": [219, 344, 389, 359], "score": 1.0, "content": " is a symplectic vector space for ", "type": "text"}, {"bbox": [389, 349, 402, 357], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [402, 344, 500, 359], "score": 1.0, "content": ", which can not be", "type": "text"}], "index": 15}, {"bbox": [110, 358, 455, 374], "spans": [{"bbox": [110, 358, 190, 374], "score": 1.0, "content": "extended to all ", "type": "text"}, {"bbox": [190, 361, 200, 369], "score": 0.9, "content": "V", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [200, 358, 437, 374], "score": 1.0, "content": ". The same reasoning applies in the structure ", "type": "text"}, {"bbox": [437, 364, 450, 371], "score": 0.91, "content": "\\omega_{J}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [451, 358, 455, 374], "score": 1.0, "content": ".", "type": "text"}], "index": 16}], "index": 11}, {"type": "text", "bbox": [109, 372, 500, 414], "lines": [{"bbox": [127, 373, 499, 388], "spans": [{"bbox": [127, 373, 499, 388], "score": 1.0, "content": "We prove that this can not happen, since it violates the calibration con-", "type": "text"}], "index": 17}, {"bbox": [110, 387, 500, 402], "spans": [{"bbox": [110, 387, 500, 402], "score": 1.0, "content": "dition. We have to distinguish three different subcases according to the", "type": "text"}], "index": 18}, {"bbox": [110, 402, 235, 417], "spans": [{"bbox": [110, 402, 186, 417], "score": 1.0, "content": "intersection of ", "type": "text"}, {"bbox": [186, 407, 194, 412], "score": 0.89, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [194, 402, 223, 417], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [224, 407, 230, 415], "score": 0.9, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [230, 402, 235, 417], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18}, {"type": "text", "bbox": [109, 415, 500, 559], "lines": [{"bbox": [127, 416, 500, 430], "spans": [{"bbox": [127, 416, 207, 430], "score": 1.0, "content": "First subcase: ", "type": "text"}, {"bbox": [207, 421, 214, 427], "score": 0.89, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [214, 416, 243, 430], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [243, 421, 249, 429], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [250, 416, 500, 430], "score": 1.0, "content": " have zero intersection. If this happens we can", "type": "text"}], "index": 20}, {"bbox": [110, 431, 500, 445], "spans": [{"bbox": [110, 431, 239, 445], "score": 1.0, "content": "always choose a basis of ", "type": "text"}, {"bbox": [239, 433, 249, 441], "score": 0.9, "content": "V", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [249, 431, 315, 445], "score": 1.0, "content": " of the form ", "type": "text"}, {"bbox": [316, 432, 396, 444], "score": 0.92, "content": "(v_{1},I v_{1},v_{2},J v_{2})", "type": "inline_equation", "height": 12, "width": 80}, {"bbox": [396, 431, 438, 445], "score": 1.0, "content": ". Write ", "type": "text"}, {"bbox": [438, 436, 446, 441], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [446, 431, 500, 445], "score": 1.0, "content": " for the 2-", "type": "text"}], "index": 21}, {"bbox": [110, 445, 500, 460], "spans": [{"bbox": [110, 445, 200, 460], "score": 1.0, "content": "plane spanned by ", "type": "text"}, {"bbox": [200, 447, 232, 458], "score": 0.94, "content": "v_{1},I v_{1}", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [233, 445, 257, 460], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [258, 450, 264, 458], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [264, 445, 368, 460], "score": 1.0, "content": " for that spanned by ", "type": "text"}, {"bbox": [368, 447, 402, 458], "score": 0.92, "content": "v_{2},J v_{2}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [402, 445, 446, 460], "score": 1.0, "content": ", so that ", "type": "text"}, {"bbox": [446, 447, 496, 458], "score": 0.93, "content": "V=\\pi\\oplus\\rho", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [496, 445, 500, 460], "score": 1.0, "content": ".", "type": "text"}], "index": 22}, {"bbox": [109, 459, 501, 475], "spans": [{"bbox": [109, 459, 182, 475], "score": 1.0, "content": "Indeed, since ", "type": "text"}, {"bbox": [183, 462, 192, 470], "score": 0.91, "content": "V", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [192, 459, 379, 475], "score": 1.0, "content": " is not Lagrangian with respect to ", "type": "text"}, {"bbox": [379, 465, 392, 472], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [392, 459, 501, 475], "score": 1.0, "content": ", it has to contain a", "type": "text"}], "index": 23}, {"bbox": [109, 475, 501, 488], "spans": [{"bbox": [109, 475, 231, 488], "score": 1.0, "content": "symplectic 2-plane like ", "type": "text"}, {"bbox": [231, 479, 239, 485], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [239, 475, 336, 488], "score": 1.0, "content": ", and similarly for ", "type": "text"}, {"bbox": [336, 479, 342, 487], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [343, 475, 369, 488], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [370, 479, 383, 487], "score": 0.88, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [383, 475, 477, 488], "score": 1.0, "content": ". Moreover, since ", "type": "text"}, {"bbox": [477, 476, 487, 485], "score": 0.89, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [487, 475, 501, 488], "score": 1.0, "content": " is", "type": "text"}], "index": 24}, {"bbox": [110, 490, 500, 502], "spans": [{"bbox": [110, 490, 272, 502], "score": 1.0, "content": "not symplectic with respect to ", "type": "text"}, {"bbox": [272, 493, 285, 501], "score": 0.89, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [285, 490, 500, 502], "score": 1.0, "content": ", it turns out that the symplectic 2-plane", "type": "text"}], "index": 25}, {"bbox": [110, 504, 500, 516], "spans": [{"bbox": [110, 508, 117, 514], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [118, 504, 357, 516], "score": 1.0, "content": " can not be completed to a symplectic basis of ", "type": "text"}, {"bbox": [357, 505, 367, 514], "score": 0.91, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [367, 504, 413, 516], "score": 1.0, "content": ", so that ", "type": "text"}, {"bbox": [413, 505, 423, 514], "score": 0.89, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [423, 504, 500, 516], "score": 1.0, "content": " has to contain", "type": "text"}], "index": 26}, {"bbox": [110, 518, 500, 531], "spans": [{"bbox": [110, 518, 234, 531], "score": 1.0, "content": "an isotropic 2-plane for ", "type": "text"}, {"bbox": [234, 522, 247, 530], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [247, 518, 300, 531], "score": 1.0, "content": ", which is ", "type": "text"}, {"bbox": [300, 523, 307, 531], "score": 0.9, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [307, 518, 500, 531], "score": 1.0, "content": ". The same reasoning (with the roles", "type": "text"}], "index": 27}, {"bbox": [110, 533, 500, 546], "spans": [{"bbox": [110, 533, 401, 546], "score": 1.0, "content": "reversed) applies obviously to the symplectic structure ", "type": "text"}, {"bbox": [401, 537, 414, 545], "score": 0.89, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [415, 533, 500, 546], "score": 1.0, "content": ". Hence, in this", "type": "text"}], "index": 28}, {"bbox": [110, 547, 180, 560], "spans": [{"bbox": [110, 547, 180, 560], "score": 1.0, "content": "case we have:", "type": "text"}], "index": 29}], "index": 24.5}, {"type": "interline_equation", "bbox": [132, 569, 479, 584], "lines": [{"bbox": [132, 569, 479, 584], "spans": [{"bbox": [132, 569, 479, 584], "score": 0.82, "content": "2\\mathrm{Re}(\\Omega_{K})\|_{V}=(\\omega_{I}^{2}-\\omega_{J}^{2})(v_{1},I v_{1},v_{2},J v_{2})=\\omega_{I}(v_{1},I v_{1})\\omega_{I}(v_{2},J v_{2})-", "type": "interline_equation"}], "index": 30}], "index": 30}, {"type": "interline_equation", "bbox": [123, 594, 487, 608], "lines": [{"bbox": [123, 594, 487, 608], "spans": [{"bbox": [123, 594, 487, 608], "score": 0.87, "content": "\\omega_{I}(v_{1},v_{2})\\omega_{I}(I v_{1},J v_{2})+\\omega_{I}(v_{1},J v_{2})\\omega_{I}(I v_{1},v_{2})-\\omega_{J}(v_{1},I v_{1})\\omega_{J}(v_{2},J v_{2})+", "type": "interline_equation"}], "index": 31}], "index": 31}, {"type": "interline_equation", "bbox": [174, 614, 434, 627], "lines": [{"bbox": [174, 614, 434, 627], "spans": [{"bbox": [174, 614, 434, 627], "score": 0.87, "content": "\\omega_{J}(v_{1},v_{2})\\omega_{J}(I v_{1},J v_{2})-\\omega_{J}(v_{1},J v_{2})\\omega_{J}(I v_{2},v_{2})=0,", "type": "interline_equation"}], "index": 32}], "index": 32}, {"type": "text", "bbox": [110, 631, 501, 674], "lines": [{"bbox": [109, 633, 500, 648], "spans": [{"bbox": [109, 633, 266, 648], "score": 1.0, "content": "using the defining relations of ", "type": "text"}, {"bbox": [266, 638, 318, 646], "score": 0.91, "content": "\\omega_{I},\\omega_{J},\\omega_{K}", "type": "inline_equation", "height": 8, "width": 52}, {"bbox": [318, 633, 455, 648], "score": 1.0, "content": ", the quaternionic relation ", "type": "text"}, {"bbox": [455, 635, 496, 644], "score": 0.92, "content": "I J=K", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [496, 633, 500, 648], "score": 1.0, "content": ",", "type": "text"}], "index": 33}, {"bbox": [110, 648, 500, 662], "spans": [{"bbox": [110, 648, 198, 662], "score": 1.0, "content": "the invariance of ", "type": "text"}, {"bbox": [198, 653, 204, 660], "score": 0.89, "content": "g", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [204, 648, 297, 662], "score": 1.0, "content": " and the fact that ", "type": "text"}, {"bbox": [297, 649, 307, 658], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [307, 648, 462, 662], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [462, 653, 478, 660], "score": 0.91, "content": "\\omega_{K}", "type": "inline_equation", "height": 7, "width": 16}, {"bbox": [479, 648, 500, 662], "score": 1.0, "content": ". So", "type": "text"}], "index": 34}, {"bbox": [110, 662, 414, 675], "spans": [{"bbox": [110, 662, 414, 675], "score": 1.0, "content": "this subcase is not consistent with the calibration property.", "type": "text"}], "index": 35}], "index": 34}], "layout_bboxes": [], "page_idx": 3, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [132, 569, 479, 584], "lines": [{"bbox": [132, 569, 479, 584], "spans": [{"bbox": [132, 569, 479, 584], "score": 0.82, "content": "2\\mathrm{Re}(\\Omega_{K})\|_{V}=(\\omega_{I}^{2}-\\omega_{J}^{2})(v_{1},I v_{1},v_{2},J v_{2})=\\omega_{I}(v_{1},I v_{1})\\omega_{I}(v_{2},J v_{2})-", "type": "interline_equation"}], "index": 30}], "index": 30}, {"type": "interline_equation", "bbox": [123, 594, 487, 608], "lines": [{"bbox": [123, 594, 487, 608], "spans": [{"bbox": [123, 594, 487, 608], "score": 0.87, "content": "\\omega_{I}(v_{1},v_{2})\\omega_{I}(I v_{1},J v_{2})+\\omega_{I}(v_{1},J v_{2})\\omega_{I}(I v_{1},v_{2})-\\omega_{J}(v_{1},I v_{1})\\omega_{J}(v_{2},J v_{2})+", "type": "interline_equation"}], "index": 31}], "index": 31}, {"type": "interline_equation", "bbox": [174, 614, 434, 627], "lines": [{"bbox": [174, 614, 434, 627], "spans": [{"bbox": [174, 614, 434, 627], "score": 0.87, "content": "\\omega_{J}(v_{1},v_{2})\\omega_{J}(I v_{1},J v_{2})-\\omega_{J}(v_{1},J v_{2})\\omega_{J}(I v_{2},v_{2})=0,", "type": "interline_equation"}], "index": 32}], "index": 32}], "discarded_blocks": [], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [109, 124, 500, 212], "lines": [], "index": 2.5, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [108, 127, 501, 215], "lines_deleted": true}, {"type": "text", "bbox": [109, 213, 500, 371], "lines": [{"bbox": [127, 213, 500, 228], "spans": [{"bbox": [127, 213, 195, 228], "score": 1.0, "content": "Second case: ", "type": "text"}, {"bbox": [196, 216, 205, 225], "score": 0.88, "content": "V", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [205, 213, 483, 228], "score": 1.0, "content": " is neither symplectic nor Lagrangian for the structure ", "type": "text"}, {"bbox": [483, 219, 496, 226], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [496, 213, 500, 228], "score": 1.0, "content": ".", "type": "text"}], "index": 6}, {"bbox": [109, 228, 500, 243], "spans": [{"bbox": [109, 228, 147, 243], "score": 1.0, "content": "Notice ", "type": "text"}, {"bbox": [148, 231, 158, 240], "score": 0.89, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [158, 228, 365, 243], "score": 1.0, "content": " can not be symplectic with respect to ", "type": "text"}, {"bbox": [366, 234, 379, 241], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [379, 228, 500, 243], "score": 1.0, "content": ", otherwise by the first", "type": "text"}], "index": 7}, {"bbox": [110, 244, 500, 257], "spans": [{"bbox": [110, 244, 349, 257], "score": 1.0, "content": "case it would be Lagrangian in the strucutre ", "type": "text"}, {"bbox": [349, 248, 362, 255], "score": 0.87, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [362, 244, 500, 257], "score": 1.0, "content": "; moreover we can assume", "type": "text"}], "index": 8}, {"bbox": [110, 257, 500, 272], "spans": [{"bbox": [110, 257, 136, 272], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [136, 259, 146, 268], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [146, 257, 328, 272], "score": 1.0, "content": " is not Lagrangian with respect to ", "type": "text"}, {"bbox": [328, 262, 342, 270], "score": 0.91, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [342, 257, 500, 272], "score": 1.0, "content": ", otherwise there is nothing to", "type": "text"}], "index": 9}, {"bbox": [110, 273, 501, 286], "spans": [{"bbox": [110, 273, 220, 286], "score": 1.0, "content": "prove. So in this case ", "type": "text"}, {"bbox": [220, 274, 230, 283], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [230, 273, 501, 286], "score": 1.0, "content": " is neither Lagrangian nor symplectic in the structure", "type": "text"}], "index": 10}, {"bbox": [110, 286, 501, 301], "spans": [{"bbox": [110, 291, 122, 299], "score": 0.83, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [123, 286, 229, 301], "score": 1.0, "content": " and in the structure ", "type": "text"}, {"bbox": [230, 291, 243, 299], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [243, 286, 337, 301], "score": 1.0, "content": ". This means that ", "type": "text"}, {"bbox": [337, 288, 347, 297], "score": 0.89, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [347, 286, 501, 301], "score": 1.0, "content": " contains a symplectic 2-plane", "type": "text"}], "index": 11}, {"bbox": [110, 302, 500, 315], "spans": [{"bbox": [110, 306, 117, 311], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [118, 302, 201, 315], "score": 1.0, "content": " with respect to ", "type": "text"}, {"bbox": [202, 306, 214, 313], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [214, 302, 348, 315], "score": 1.0, "content": " and a symplectic 2-plane ", "type": "text"}, {"bbox": [349, 306, 355, 314], "score": 0.88, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [355, 302, 439, 315], "score": 1.0, "content": " with respect to ", "type": "text"}, {"bbox": [440, 306, 453, 313], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [453, 302, 500, 315], "score": 1.0, "content": ". Indeed,", "type": "text"}], "index": 12}, {"bbox": [110, 316, 500, 329], "spans": [{"bbox": [110, 316, 155, 329], "score": 1.0, "content": "consider ", "type": "text"}, {"bbox": [156, 317, 191, 327], "score": 0.93, "content": "v_{1}\\in V", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [192, 316, 227, 329], "score": 1.0, "content": "; since ", "type": "text"}, {"bbox": [228, 317, 237, 326], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [237, 316, 420, 329], "score": 1.0, "content": " is not Lagrangian in the structure ", "type": "text"}, {"bbox": [421, 320, 433, 327], "score": 0.89, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [433, 316, 500, 329], "score": 1.0, "content": ", there exists", "type": "text"}], "index": 13}, {"bbox": [110, 330, 500, 344], "spans": [{"bbox": [110, 331, 147, 342], "score": 0.93, "content": "v_{2}\\in V", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [148, 330, 205, 344], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [206, 331, 278, 343], "score": 0.94, "content": "\\omega_{I}(v_{1},v_{2})\\neq0", "type": "inline_equation", "height": 12, "width": 72}, {"bbox": [278, 330, 500, 344], "score": 1.0, "content": " and this implies that the vector subspace", "type": "text"}], "index": 14}, {"bbox": [110, 344, 500, 359], "spans": [{"bbox": [110, 349, 117, 355], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [118, 344, 184, 359], "score": 1.0, "content": " spanned by ", "type": "text"}, {"bbox": [185, 345, 219, 358], "score": 0.94, "content": "(v_{1},v_{2})", "type": "inline_equation", "height": 13, "width": 34}, {"bbox": [219, 344, 389, 359], "score": 1.0, "content": " is a symplectic vector space for ", "type": "text"}, {"bbox": [389, 349, 402, 357], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [402, 344, 500, 359], "score": 1.0, "content": ", which can not be", "type": "text"}], "index": 15}, {"bbox": [110, 358, 455, 374], "spans": [{"bbox": [110, 358, 190, 374], "score": 1.0, "content": "extended to all ", "type": "text"}, {"bbox": [190, 361, 200, 369], "score": 0.9, "content": "V", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [200, 358, 437, 374], "score": 1.0, "content": ". The same reasoning applies in the structure ", "type": "text"}, {"bbox": [437, 364, 450, 371], "score": 0.91, "content": "\\omega_{J}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [451, 358, 455, 374], "score": 1.0, "content": ".", "type": "text"}], "index": 16}], "index": 11, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [109, 213, 501, 374]}, {"type": "text", "bbox": [109, 372, 500, 414], "lines": [{"bbox": [127, 373, 499, 388], "spans": [{"bbox": [127, 373, 499, 388], "score": 1.0, "content": "We prove that this can not happen, since it violates the calibration con-", "type": "text"}], "index": 17}, {"bbox": [110, 387, 500, 402], "spans": [{"bbox": [110, 387, 500, 402], "score": 1.0, "content": "dition. We have to distinguish three different subcases according to the", "type": "text"}], "index": 18}, {"bbox": [110, 402, 235, 417], "spans": [{"bbox": [110, 402, 186, 417], "score": 1.0, "content": "intersection of ", "type": "text"}, {"bbox": [186, 407, 194, 412], "score": 0.89, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [194, 402, 223, 417], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [224, 407, 230, 415], "score": 0.9, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [230, 402, 235, 417], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [110, 373, 500, 417]}, {"type": "text", "bbox": [109, 415, 500, 559], "lines": [{"bbox": [127, 416, 500, 430], "spans": [{"bbox": [127, 416, 207, 430], "score": 1.0, "content": "First subcase: ", "type": "text"}, {"bbox": [207, 421, 214, 427], "score": 0.89, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [214, 416, 243, 430], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [243, 421, 249, 429], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [250, 416, 500, 430], "score": 1.0, "content": " have zero intersection. If this happens we can", "type": "text"}], "index": 20}, {"bbox": [110, 431, 500, 445], "spans": [{"bbox": [110, 431, 239, 445], "score": 1.0, "content": "always choose a basis of ", "type": "text"}, {"bbox": [239, 433, 249, 441], "score": 0.9, "content": "V", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [249, 431, 315, 445], "score": 1.0, "content": " of the form ", "type": "text"}, {"bbox": [316, 432, 396, 444], "score": 0.92, "content": "(v_{1},I v_{1},v_{2},J v_{2})", "type": "inline_equation", "height": 12, "width": 80}, {"bbox": [396, 431, 438, 445], "score": 1.0, "content": ". Write ", "type": "text"}, {"bbox": [438, 436, 446, 441], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [446, 431, 500, 445], "score": 1.0, "content": " for the 2-", "type": "text"}], "index": 21}, {"bbox": [110, 445, 500, 460], "spans": [{"bbox": [110, 445, 200, 460], "score": 1.0, "content": "plane spanned by ", "type": "text"}, {"bbox": [200, 447, 232, 458], "score": 0.94, "content": "v_{1},I v_{1}", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [233, 445, 257, 460], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [258, 450, 264, 458], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [264, 445, 368, 460], "score": 1.0, "content": " for that spanned by ", "type": "text"}, {"bbox": [368, 447, 402, 458], "score": 0.92, "content": "v_{2},J v_{2}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [402, 445, 446, 460], "score": 1.0, "content": ", so that ", "type": "text"}, {"bbox": [446, 447, 496, 458], "score": 0.93, "content": "V=\\pi\\oplus\\rho", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [496, 445, 500, 460], "score": 1.0, "content": ".", "type": "text"}], "index": 22}, {"bbox": [109, 459, 501, 475], "spans": [{"bbox": [109, 459, 182, 475], "score": 1.0, "content": "Indeed, since ", "type": "text"}, {"bbox": [183, 462, 192, 470], "score": 0.91, "content": "V", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [192, 459, 379, 475], "score": 1.0, "content": " is not Lagrangian with respect to ", "type": "text"}, {"bbox": [379, 465, 392, 472], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [392, 459, 501, 475], "score": 1.0, "content": ", it has to contain a", "type": "text"}], "index": 23}, {"bbox": [109, 475, 501, 488], "spans": [{"bbox": [109, 475, 231, 488], "score": 1.0, "content": "symplectic 2-plane like ", "type": "text"}, {"bbox": [231, 479, 239, 485], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [239, 475, 336, 488], "score": 1.0, "content": ", and similarly for ", "type": "text"}, {"bbox": [336, 479, 342, 487], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [343, 475, 369, 488], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [370, 479, 383, 487], "score": 0.88, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [383, 475, 477, 488], "score": 1.0, "content": ". 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The same reasoning (with the roles", "type": "text"}], "index": 27}, {"bbox": [110, 533, 500, 546], "spans": [{"bbox": [110, 533, 401, 546], "score": 1.0, "content": "reversed) applies obviously to the symplectic structure ", "type": "text"}, {"bbox": [401, 537, 414, 545], "score": 0.89, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [415, 533, 500, 546], "score": 1.0, "content": ". 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Notice can not be symplectic with respect to , otherwise by the first case it would be Lagrangian in the strucutre ; moreover we can assume that is not Lagrangian with respect to , otherwise there is nothing to prove. So in this case is neither Lagrangian nor symplectic in the structure and in the structure . This means that contains a symplectic 2-plane with respect to and a symplectic 2-plane with respect to . Indeed, consider ; since is not Lagrangian in the structure , there exists such that and this implies that the vector subspace spanned by is a symplectic vector space for , which can not be extended to all . The same reasoning applies in the structure . We prove that this can not happen, since it violates the calibration con- dition. We have to distinguish three different subcases according to the intersection of with . First subcase: and have zero intersection. If this happens we can always choose a basis of of the form . Write for the 2- plane spanned by and for that spanned by , so that . Indeed, since is not Lagrangian with respect to , it has to contain a symplectic 2-plane like , and similarly for and . Moreover, since is not symplectic with respect to , it turns out that the symplectic 2-plane can not be completed to a symplectic basis of , so that has to contain an isotropic 2-plane for , which is . The same reasoning (with the roles reversed) applies obviously to the symplectic structure . Hence, in this case we have: $$ 2\mathrm{Re}(\Omega_{K})\|_{V}=(\omega_{I}^{2}-\omega_{J}^{2})(v_{1},I v_{1},v_{2},J v_{2})=\omega_{I}(v_{1},I v_{1})\omega_{I}(v_{2},J v_{2})- $$ $$ \omega_{I}(v_{1},v_{2})\omega_{I}(I v_{1},J v_{2})+\omega_{I}(v_{1},J v_{2})\omega_{I}(I v_{1},v_{2})-\omega_{J}(v_{1},I v_{1})\omega_{J}(v_{2},J v_{2})+ $$ $$ \omega_{J}(v_{1},v_{2})\omega_{J}(I v_{1},J v_{2})-\omega_{J}(v_{1},J v_{2})\omega_{J}(I v_{2},v_{2})=0, $$ using the defining relations of , the quaternionic relation , the invariance of and the fact that is Lagrangian with respect to . So this subcase is not consistent with the calibration property.	<div class="pdf-page"> <p>Second case: is neither symplectic nor Lagrangian for the structure . Notice can not be symplectic with respect to , otherwise by the first case it would be Lagrangian in the strucutre ; moreover we can assume that is not Lagrangian with respect to , otherwise there is nothing to prove. So in this case is neither Lagrangian nor symplectic in the structure and in the structure . This means that contains a symplectic 2-plane with respect to and a symplectic 2-plane with respect to . Indeed, consider ; since is not Lagrangian in the structure , there exists such that and this implies that the vector subspace spanned by is a symplectic vector space for , which can not be extended to all . The same reasoning applies in the structure .</p> <p>We prove that this can not happen, since it violates the calibration con- dition. We have to distinguish three different subcases according to the intersection of with .</p> <p>First subcase: and have zero intersection. If this happens we can always choose a basis of of the form . Write for the 2- plane spanned by and for that spanned by , so that . Indeed, since is not Lagrangian with respect to , it has to contain a symplectic 2-plane like , and similarly for and . Moreover, since is not symplectic with respect to , it turns out that the symplectic 2-plane can not be completed to a symplectic basis of , so that has to contain an isotropic 2-plane for , which is . The same reasoning (with the roles reversed) applies obviously to the symplectic structure . Hence, in this case we have:</p> <p>using the defining relations of , the quaternionic relation , the invariance of and the fact that is Lagrangian with respect to . So this subcase is not consistent with the calibration property.</p> </div>	<div class="pdf-page"> <p class="pdf-text" data-x="182" data-y="275" data-width="654" data-height="204">Second case: is neither symplectic nor Lagrangian for the structure . Notice can not be symplectic with respect to , otherwise by the first case it would be Lagrangian in the strucutre ; moreover we can assume that is not Lagrangian with respect to , otherwise there is nothing to prove. So in this case is neither Lagrangian nor symplectic in the structure and in the structure . This means that contains a symplectic 2-plane with respect to and a symplectic 2-plane with respect to . Indeed, consider ; since is not Lagrangian in the structure , there exists such that and this implies that the vector subspace spanned by is a symplectic vector space for , which can not be extended to all . The same reasoning applies in the structure .</p> <p class="pdf-text" data-x="182" data-y="480" data-width="654" data-height="55">We prove that this can not happen, since it violates the calibration con- dition. We have to distinguish three different subcases according to the intersection of with .</p> <p class="pdf-text" data-x="182" data-y="536" data-width="654" data-height="186">First subcase: and have zero intersection. If this happens we can always choose a basis of of the form . Write for the 2- plane spanned by and for that spanned by , so that . Indeed, since is not Lagrangian with respect to , it has to contain a symplectic 2-plane like , and similarly for and . Moreover, since is not symplectic with respect to , it turns out that the symplectic 2-plane can not be completed to a symplectic basis of , so that has to contain an isotropic 2-plane for , which is . The same reasoning (with the roles reversed) applies obviously to the symplectic structure . Hence, in this case we have:</p> <p class="pdf-text" data-x="184" data-y="815" data-width="654" data-height="56">using the defining relations of , the quaternionic relation , the invariance of and the fact that is Lagrangian with respect to . So this subcase is not consistent with the calibration property.</p> </div>	{ "type": [ "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "interline_equation", "interline_equation", "interline_equation", "inline_equation", "inline_equation", "text" ], "coordinates": [ [ 212, 275, 836, 294 ], [ 182, 294, 836, 314 ], [ 184, 315, 836, 332 ], [ 184, 332, 836, 351 ], [ 184, 352, 838, 369 ], [ 184, 369, 838, 389 ], [ 184, 390, 836, 407 ], [ 184, 408, 836, 425 ], [ 184, 426, 836, 444 ], [ 184, 444, 836, 464 ], [ 184, 462, 761, 483 ], [ 212, 482, 834, 501 ], [ 184, 500, 836, 519 ], [ 184, 519, 393, 539 ], [ 212, 537, 836, 555 ], [ 184, 557, 836, 575 ], [ 184, 575, 836, 594 ], [ 182, 593, 838, 614 ], [ 182, 614, 838, 630 ], [ 184, 633, 836, 649 ], [ 184, 651, 836, 667 ], [ 184, 669, 836, 686 ], [ 184, 689, 836, 705 ], [ 184, 707, 301, 724 ], [ 220, 735, 801, 755 ], [ 205, 768, 814, 786 ], [ 291, 793, 726, 810 ], [ 182, 818, 836, 837 ], [ 184, 837, 836, 855 ], [ 184, 855, 692, 872 ] ], "content": [ "Second case: V is neither symplectic nor Lagrangian for the structure \\omega_{I} .", "Notice V can not be symplectic with respect to \\omega_{J} , otherwise by the first", "case it would be Lagrangian in the strucutre \\omega_{I} ; moreover we can assume", "that V is not Lagrangian with respect to \\omega_{J} , otherwise there is nothing to", "prove. So in this case V is neither Lagrangian nor symplectic in the structure", "\\omega_{I} and in the structure \\omega_{J} . This means that V contains a symplectic 2-plane", "\\pi with respect to \\omega_{I} and a symplectic 2-plane \\rho with respect to \\omega_{J} . Indeed,", "consider v_{1}\\in V ; since V is not Lagrangian in the structure \\omega_{I} , there exists", "v_{2}\\in V such that \\omega_{I}(v_{1},v_{2})\\neq0 and this implies that the vector subspace", "\\pi spanned by (v_{1},v_{2}) is a symplectic vector space for \\omega_{I} , which can not be", "extended to all V . The same reasoning applies in the structure \\omega_{J} .", "We prove that this can not happen, since it violates the calibration con-", "dition. We have to distinguish three different subcases according to the", "intersection of \\pi with \\rho .", "First subcase: \\pi and \\rho have zero intersection. If this happens we can", "always choose a basis of V of the form (v_{1},I v_{1},v_{2},J v_{2}) . Write \\pi for the 2-", "plane spanned by v_{1},I v_{1} and \\rho for that spanned by v_{2},J v_{2} , so that V=\\pi\\oplus\\rho .", "Indeed, since V is not Lagrangian with respect to \\omega_{I} , it has to contain a", "symplectic 2-plane like \\pi , and similarly for \\rho and \\omega_{J} . Moreover, since V is", "not symplectic with respect to \\omega_{I} , it turns out that the symplectic 2-plane", "\\pi can not be completed to a symplectic basis of V , so that V has to contain", "an isotropic 2-plane for \\omega_{I} , which is \\rho . The same reasoning (with the roles", "reversed) applies obviously to the symplectic structure \\omega_{J} . Hence, in this", "case we have:", "2\\mathrm{Re}(\\Omega_{K})\|_{V}=(\\omega_{I}^{2}-\\omega_{J}^{2})(v_{1},I v_{1},v_{2},J v_{2})=\\omega_{I}(v_{1},I v_{1})\\omega_{I}(v_{2},J v_{2})-", "\\omega_{I}(v_{1},v_{2})\\omega_{I}(I v_{1},J v_{2})+\\omega_{I}(v_{1},J v_{2})\\omega_{I}(I v_{1},v_{2})-\\omega_{J}(v_{1},I v_{1})\\omega_{J}(v_{2},J v_{2})+", "\\omega_{J}(v_{1},v_{2})\\omega_{J}(I v_{1},J v_{2})-\\omega_{J}(v_{1},J v_{2})\\omega_{J}(I v_{2},v_{2})=0,", "using the defining relations of \\omega_{I},\\omega_{J},\\omega_{K} , the quaternionic relation I J=K ,", "the invariance of g and the fact that V is Lagrangian with respect to \\omega_{K} . So", "this subcase is not consistent with the calibration property." ], "index": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 49, 74, 100, 127, 128, 129 ] }	{ "coordinates": [], "index": [], "caption": [], "caption_coordinates": [] }	{ "type": [ "interline_equation", "interline_equation", "interline_equation", "interline_equation", "interline_equation", "interline_equation" ], "coordinates": [ [ 220, 735, 801, 755 ], [ 205, 768, 814, 786 ], [ 291, 793, 726, 810 ], [ 220, 735, 801, 755 ], [ 205, 768, 814, 786 ], [ 291, 793, 726, 810 ] ], "content": [ "", "", "", "2\\mathrm{Re}(\\Omega_{K})\|_{V}=(\\omega_{I}^{2}-\\omega_{J}^{2})(v_{1},I v_{1},v_{2},J v_{2})=\\omega_{I}(v_{1},I v_{1})\\omega_{I}(v_{2},J v_{2})-", "\\omega_{I}(v_{1},v_{2})\\omega_{I}(I v_{1},J v_{2})+\\omega_{I}(v_{1},J v_{2})\\omega_{I}(I v_{1},v_{2})-\\omega_{J}(v_{1},I v_{1})\\omega_{J}(v_{2},J v_{2})+", "\\omega_{J}(v_{1},v_{2})\\omega_{J}(I v_{1},J v_{2})-\\omega_{J}(v_{1},J v_{2})\\omega_{J}(I v_{2},v_{2})=0," ], "index": [ 0, 1, 2, 3, 4, 5 ] }	{ "type": [], "coordinates": [], "content": [], "index": [] }	{"document_info": {"total_pages": 9, "parse_type": "txt", "version": "1.0.1", "total_content_list_items": 72}, "content_statistics": {"total_blocks": 81, "total_lines": 269, "total_images": 0, "total_equations": 18, "total_tables": 0, "total_characters": 16112, "total_words": 2578, "block_type_distribution": {"title": 6, "text": 55, "discarded": 9, "interline_equation": 9, "list": 2}}, "model_statistics": {"total_layout_detections": 813, "avg_confidence": 0.9488487084870848, "min_confidence": 0.26, "max_confidence": 1.0}, "page_statistics": [{"page_number": 0, "page_size": [612.0, 792.0], "blocks": 10, "lines": 26, "images": 0, "equations": 0, "tables": 0, "characters": 1419, "words": 204, "discarded_blocks": 2}, {"page_number": 1, "page_size": [612.0, 792.0], "blocks": 15, "lines": 37, "images": 0, "equations": 0, "tables": 0, "characters": 2328, "words": 359, "discarded_blocks": 1}, {"page_number": 2, "page_size": [612.0, 792.0], "blocks": 12, "lines": 35, "images": 0, "equations": 4, "tables": 0, "characters": 1843, "words": 306, "discarded_blocks": 1}, {"page_number": 3, "page_size": [612.0, 792.0], "blocks": 8, "lines": 30, "images": 0, "equations": 6, "tables": 0, "characters": 1797, "words": 312, "discarded_blocks": 0}, {"page_number": 4, "page_size": [612.0, 792.0], "blocks": 12, "lines": 34, "images": 0, "equations": 4, "tables": 0, "characters": 1905, "words": 328, "discarded_blocks": 1}, {"page_number": 5, "page_size": [612.0, 792.0], "blocks": 11, "lines": 32, "images": 0, "equations": 4, "tables": 0, "characters": 1846, "words": 308, "discarded_blocks": 1}, {"page_number": 6, "page_size": [612.0, 792.0], "blocks": 6, "lines": 46, "images": 0, "equations": 0, "tables": 0, "characters": 3307, "words": 521, "discarded_blocks": 1}, {"page_number": 7, "page_size": [612.0, 792.0], "blocks": 5, "lines": 29, "images": 0, "equations": 0, "tables": 0, "characters": 1666, "words": 239, "discarded_blocks": 1}, {"page_number": 8, "page_size": [612.0, 792.0], "blocks": 2, "lines": 0, "images": 0, "equations": 0, "tables": 0, "characters": 1, "words": 1, "discarded_blocks": 1}]}
0001060v1	4	[ 612, 792 ]	{ "type": [ "text", "text", "text", "text", "interline_equation", "text", "text", "text", "text", "interline_equation", "text", "discarded" ], "coordinates": [ [ 182, 161, 836, 254 ], [ 182, 256, 836, 329 ], [ 182, 329, 834, 367 ], [ 182, 367, 836, 479 ], [ 408, 487, 610, 531 ], [ 182, 533, 838, 629 ], [ 210, 636, 727, 654 ], [ 184, 655, 838, 727 ], [ 182, 729, 836, 784 ], [ 430, 793, 590, 830 ], [ 184, 833, 838, 872 ], [ 501, 893, 517, 907 ] ], "content": [ "Second subcase: and have a 1-dimensional intersection spanned by a vector . In this case we can choose a basis of of the form ( is spanned by , while is spanned by . Again by the same computation of the previous subcase one shows that this configuration is not compatible with the calibration.", "Third subcase: Finally can not clearly happen, since otherwise one can choose a basis of equal to , but then, in this basis is iden- tically vanishing, contrary to the assumption that is a symplectic 2-plane also for .", "Since the second case can never happen has to be Lagrangian also with respect to .", "Up to now, we have worked only locally; to conclude the proof it is necessary to show that if is Lagrangian with respect to , then it can not be possible that is Lagrangian with respect to , for a different . Notice that any tangent space to can not be Lagrangian with respect to both and , otherwise it would violates the calibration condition. Consider now the following smooth sections of :", "", "and the zero section . Obviously, is closed in , and by the previous reasoning can be decomposed as , that is as the disjoint union of two proper closed subsets. But this is clearly impossible, since is connected, and this implies that one of the two closed subset is empty, so is bi-Lagrangian. 口", "The previous theorem is important in view of the following:", "Corollary 2.1: Every (connected, compact and without border) special Lagrangian submanifold Λ of a hyperkaehler 4-fold can be realized as complex submanifold, via hyperkaehler rotation of the complex structure of .", "Proof: Let be a special Lagrangian submanifold of in the complex structure . Then by definition , but by the previous theorem, since this means:", "", "By Wirtinger’s theorem, since is assumed to be compact and without border, condition (3) is equivalent to say that is a complex submanifold of", "5" ], "index": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ] }	[{"type": "text", "text": "Second subcase: $\\pi$ and $\\rho$ have a 1-dimensional intersection spanned by a vector $v_{1}$ . In this case we can choose a basis of $V$ of the form $(v_{1},I v_{1},J v_{1},w)$ ( $\\pi$ is spanned by $(v_{1},I v_{1})$ , while $\\rho$ is spanned by $\\left(v_{1},J v_{1}\\right))$ . Again by the same computation of the previous subcase one shows that this configuration is not compatible with the calibration. ", "page_idx": 4}, {"type": "text", "text": "Third subcase: Finally $\\pi{=}\\rho$ can not clearly happen, since otherwise one can choose a basis of $\\pi$ equal to $(v_{1},I v_{1})$ , but then, in this basis $\\omega_{J}$ is identically vanishing, contrary to the assumption that $\\rho~=~\\pi$ is a symplectic 2-plane also for $\\omega_{J}$ . ", "page_idx": 4}, {"type": "text", "text": "Since the second case can never happen $V$ has to be Lagrangian also with respect to $\\omega_{J}$ . ", "page_idx": 4}, {"type": "text", "text": "Up to now, we have worked only locally; to conclude the proof it is necessary to show that if $T_{p}\\Lambda$ is Lagrangian with respect to $\\omega_{J}$ , then it can not be possible that $T_{q}\\Lambda$ is Lagrangian with respect to $\\omega_{I}$ , for a different $q\\in\\Lambda$ . Notice that any tangent space to $\\Lambda$ can not be Lagrangian with respect to both $\\omega_{I}$ and $\\omega_{J}$ , otherwise it would violates the calibration condition. Consider now the following smooth sections of $\\Lambda^{2}T^{}\\Lambda$ : ", "page_idx": 4}, {"type": "equation", "text": "$$\n\\begin{array}{c c c c}{{\\alpha_{I,J}:\\ \\Lambda}}&{{\\to}}&{{\\Lambda^{2}\\,T^{}\\Lambda}}\\\\ {{p}}&{{\\mapsto}}&{{\\omega_{I,J}\|T_{p}\\Lambda}}\\end{array}\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "and the zero section $s_{0}:\\Lambda\\to\\Lambda^{2}\\,T^{}\\Lambda$ . Obviously, $s_{0}(\\Lambda)$ is closed in $\\Lambda^{2}T^{}\\Lambda$ , and by the previous reasoning $\\Lambda$ can be decomposed as $\\Lambda\\,=\\,\\alpha_{I}^{-1}(s_{0}(\\Lambda))\\cup$ $\\alpha_{J}^{-1}(s_{0}(\\Lambda))$ , that is as the disjoint union of two proper closed subsets. But this is clearly impossible, since $\\Lambda$ is connected, and this implies that one of the two closed subset is empty, so $\\Lambda$ is bi-Lagrangian. \u53e3 ", "page_idx": 4}, {"type": "text", "text": "The previous theorem is important in view of the following: ", "page_idx": 4}, {"type": "text", "text": "Corollary 2.1: Every (connected, compact and without border) special Lagrangian submanifold \u039b of a hyperkaehler 4-fold $X$ can be realized as $a$ complex submanifold, via hyperkaehler rotation of the complex structure of $X$ . ", "page_idx": 4}, {"type": "text", "text": "Proof: Let $\\Lambda$ be a special Lagrangian submanifold of $X$ in the complex structure $K$ . Then by definition $\\mathrm{Re}(\\Omega_{K})_{\|\\Lambda}=\\mathrm{Vol}_{g}(\\Lambda)$ , but by the previous theorem, since $\\omega_{J}\|_{\\Lambda}=0$ this means: ", "page_idx": 4}, {"type": "equation", "text": "$$\n\\mathrm{Vol}_{g}(\\Lambda)=\\frac{1}{2}\\int_{\\Lambda}\\omega_{I}^{2}.\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "By Wirtinger\u2019s theorem, since $\\Lambda$ is assumed to be compact and without border, condition (3) is equivalent to say that $\\Lambda$ is a complex submanifold of ", "page_idx": 4}]	{"preproc_blocks": [{"type": "text", "bbox": [109, 125, 500, 197], "lines": [{"bbox": [127, 127, 501, 142], "spans": [{"bbox": [127, 127, 214, 142], "score": 1.0, "content": "Second subcase: ", "type": "text"}, {"bbox": [214, 133, 222, 138], "score": 0.89, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [222, 127, 248, 142], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [248, 133, 254, 141], "score": 0.9, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [255, 127, 501, 142], "score": 1.0, "content": " have a 1-dimensional intersection spanned by a", "type": "text"}], "index": 0}, {"bbox": [110, 142, 499, 156], "spans": [{"bbox": [110, 142, 144, 156], "score": 1.0, "content": "vector ", "type": "text"}, {"bbox": [145, 147, 155, 154], "score": 0.91, "content": "v_{1}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [155, 142, 348, 156], "score": 1.0, "content": ". 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Again by the", "type": "text"}], "index": 2}, {"bbox": [110, 172, 499, 185], "spans": [{"bbox": [110, 172, 499, 185], "score": 1.0, "content": "same computation of the previous subcase one shows that this configuration", "type": "text"}], "index": 3}, {"bbox": [109, 186, 307, 198], "spans": [{"bbox": [109, 186, 307, 198], "score": 1.0, "content": "is not compatible with the calibration.", "type": "text"}], "index": 4}], "index": 2}, {"type": "text", "bbox": [109, 198, 500, 255], "lines": [{"bbox": [127, 199, 500, 215], "spans": [{"bbox": [127, 199, 248, 215], "score": 1.0, "content": "Third subcase: Finally ", "type": "text"}, {"bbox": [248, 204, 271, 213], "score": 0.88, "content": "\\pi{=}\\rho", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [271, 199, 500, 215], "score": 1.0, "content": " can not clearly happen, since otherwise one", "type": "text"}], "index": 5}, {"bbox": [110, 214, 499, 228], "spans": [{"bbox": [110, 214, 221, 228], "score": 1.0, "content": "can choose a basis of ", "type": "text"}, {"bbox": [221, 219, 228, 225], "score": 0.89, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [229, 214, 278, 228], "score": 1.0, "content": " equal to ", "type": "text"}, {"bbox": [278, 215, 320, 228], "score": 0.94, "content": "(v_{1},I v_{1})", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [320, 214, 444, 228], "score": 1.0, "content": ", but then, in this basis ", "type": "text"}, {"bbox": [444, 219, 457, 227], "score": 0.91, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [458, 214, 499, 228], "score": 1.0, "content": " is iden-", "type": "text"}], "index": 6}, {"bbox": [109, 227, 500, 245], "spans": [{"bbox": [109, 227, 380, 245], "score": 1.0, "content": "tically vanishing, contrary to the assumption that ", "type": "text"}, {"bbox": [380, 234, 415, 242], "score": 0.91, "content": "\\rho~=~\\pi", "type": "inline_equation", "height": 8, "width": 35}, {"bbox": [415, 227, 500, 245], "score": 1.0, "content": " is a symplectic", "type": "text"}], "index": 7}, {"bbox": [110, 241, 211, 259], "spans": [{"bbox": [110, 241, 192, 259], "score": 1.0, "content": "2-plane also for ", "type": "text"}, {"bbox": [192, 248, 206, 256], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [206, 241, 211, 259], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 6.5}, {"type": "text", "bbox": [109, 255, 499, 284], "lines": [{"bbox": [126, 257, 499, 271], "spans": [{"bbox": [126, 257, 330, 271], "score": 1.0, "content": "Since the second case can never happen ", "type": "text"}, {"bbox": [331, 259, 340, 268], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [340, 257, 499, 271], "score": 1.0, "content": " has to be Lagrangian also with", "type": "text"}], "index": 9}, {"bbox": [109, 271, 182, 287], "spans": [{"bbox": [109, 271, 164, 287], "score": 1.0, "content": "respect to ", "type": "text"}, {"bbox": [164, 277, 178, 285], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [178, 271, 182, 287], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 9.5}, {"type": "text", "bbox": [109, 284, 500, 371], "lines": [{"bbox": [128, 286, 501, 301], "spans": [{"bbox": [128, 286, 501, 301], "score": 1.0, "content": "Up to now, we have worked only locally; to conclude the proof it is", "type": "text"}], "index": 11}, {"bbox": [110, 302, 501, 315], "spans": [{"bbox": [110, 302, 235, 315], "score": 1.0, "content": "necessary to show that if ", "type": "text"}, {"bbox": [235, 303, 255, 315], "score": 0.93, "content": "T_{p}\\Lambda", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [256, 302, 406, 315], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [406, 306, 420, 313], "score": 0.89, "content": "\\omega_{J}", "type": "inline_equation", "height": 7, "width": 14}, {"bbox": [420, 302, 501, 315], "score": 1.0, "content": ", then it can not", "type": "text"}], "index": 12}, {"bbox": [110, 315, 500, 330], "spans": [{"bbox": [110, 315, 195, 330], "score": 1.0, "content": "be possible that ", "type": "text"}, {"bbox": [195, 317, 216, 329], "score": 0.94, "content": "T_{q}\\Lambda", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [216, 315, 373, 330], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [374, 320, 386, 327], "score": 0.89, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [386, 315, 466, 330], "score": 1.0, "content": ", for a different ", "type": "text"}, {"bbox": [467, 317, 496, 328], "score": 0.93, "content": "q\\in\\Lambda", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [496, 315, 500, 330], "score": 1.0, "content": ".", "type": "text"}], "index": 13}, {"bbox": [109, 330, 500, 344], "spans": [{"bbox": [109, 330, 275, 344], "score": 1.0, "content": "Notice that any tangent space to ", "type": "text"}, {"bbox": [275, 331, 284, 340], "score": 0.89, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [284, 330, 500, 344], "score": 1.0, "content": " can not be Lagrangian with respect to both", "type": "text"}], "index": 14}, {"bbox": [110, 345, 500, 358], "spans": [{"bbox": [110, 349, 122, 357], "score": 0.91, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [123, 345, 150, 358], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [150, 349, 164, 357], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [164, 345, 500, 358], "score": 1.0, "content": ", otherwise it would violates the calibration condition. Consider", "type": "text"}], "index": 15}, {"bbox": [109, 359, 344, 372], "spans": [{"bbox": [109, 360, 302, 372], "score": 1.0, "content": "now the following smooth sections of", "type": "text"}, {"bbox": [302, 359, 339, 371], "score": 0.93, "content": "\\Lambda^{2}T^{}\\Lambda", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [339, 360, 344, 372], "score": 1.0, "content": ":", "type": "text"}], "index": 16}], "index": 13.5}, {"type": "interline_equation", "bbox": [244, 377, 365, 411], "lines": [{"bbox": [244, 377, 365, 411], "spans": [{"bbox": [244, 377, 365, 411], "score": 0.91, "content": "\\begin{array}{c c c c}{{\\alpha_{I,J}:\\ \\Lambda}}&{{\\to}}&{{\\Lambda^{2}\\,T^{}\\Lambda}}\\\\ {{p}}&{{\\mapsto}}&{{\\omega_{I,J}\|T_{p}\\Lambda}}\\end{array}", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [109, 413, 501, 487], "lines": [{"bbox": [109, 416, 500, 432], "spans": [{"bbox": [109, 416, 217, 432], "score": 1.0, "content": "and the zero section ", "type": "text"}, {"bbox": [217, 417, 301, 429], "score": 0.94, "content": "s_{0}:\\Lambda\\to\\Lambda^{2}\\,T^{}\\Lambda", "type": "inline_equation", "height": 12, "width": 84}, {"bbox": [302, 416, 367, 432], "score": 1.0, "content": ". Obviously, ", "type": "text"}, {"bbox": [367, 418, 395, 430], "score": 0.95, "content": "s_{0}(\\Lambda)", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [395, 416, 458, 432], "score": 1.0, "content": " is closed in", "type": "text"}, {"bbox": [459, 417, 496, 429], "score": 0.92, "content": "\\Lambda^{2}T^{}\\Lambda", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [497, 416, 500, 432], "score": 1.0, "content": ",", "type": "text"}], "index": 18}, {"bbox": [110, 431, 500, 446], "spans": [{"bbox": [110, 431, 272, 446], "score": 1.0, "content": "and by the previous reasoning ", "type": "text"}, {"bbox": [272, 433, 281, 442], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [281, 431, 405, 446], "score": 1.0, "content": " can be decomposed as ", "type": "text"}, {"bbox": [405, 432, 500, 445], "score": 0.93, "content": "\\Lambda\\,=\\,\\alpha_{I}^{-1}(s_{0}(\\Lambda))\\cup", "type": "inline_equation", "height": 13, "width": 95}], "index": 19}, {"bbox": [110, 444, 501, 461], "spans": [{"bbox": [110, 446, 166, 460], "score": 0.97, "content": "\\alpha_{J}^{-1}(s_{0}(\\Lambda))", "type": "inline_equation", "height": 14, "width": 56}, {"bbox": [166, 444, 501, 461], "score": 1.0, "content": ", that is as the disjoint union of two proper closed subsets. But", "type": "text"}], "index": 20}, {"bbox": [110, 461, 501, 474], "spans": [{"bbox": [110, 461, 272, 474], "score": 1.0, "content": "this is clearly impossible, since ", "type": "text"}, {"bbox": [273, 462, 281, 471], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [281, 461, 501, 474], "score": 1.0, "content": " is connected, and this implies that one of", "type": "text"}], "index": 21}, {"bbox": [109, 474, 502, 489], "spans": [{"bbox": [109, 474, 286, 489], "score": 1.0, "content": "the two closed subset is empty, so ", "type": "text"}, {"bbox": [287, 477, 295, 485], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [296, 474, 385, 489], "score": 1.0, "content": " is bi-Lagrangian.", "type": "text"}, {"bbox": [491, 477, 502, 487], "score": 0.9912325739860535, "content": "\u53e3", "type": "text"}], "index": 22}], "index": 20}, {"type": "text", "bbox": [126, 492, 435, 506], "lines": [{"bbox": [127, 493, 434, 509], "spans": [{"bbox": [127, 493, 434, 509], "score": 1.0, "content": "The previous theorem is important in view of the following:", "type": "text"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [110, 507, 501, 563], "lines": [{"bbox": [126, 507, 501, 524], "spans": [{"bbox": [126, 507, 501, 524], "score": 1.0, "content": "Corollary 2.1: Every (connected, compact and without border) special", "type": "text"}], "index": 24}, {"bbox": [111, 523, 500, 537], "spans": [{"bbox": [111, 523, 379, 537], "score": 1.0, "content": "Lagrangian submanifold \u039b of a hyperkaehler 4-fold ", "type": "text"}, {"bbox": [380, 524, 391, 533], "score": 0.89, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [391, 523, 493, 537], "score": 1.0, "content": " can be realized as ", "type": "text"}, {"bbox": [493, 528, 500, 533], "score": 0.26, "content": "a", "type": "inline_equation", "height": 5, "width": 7}], "index": 25}, {"bbox": [110, 537, 502, 552], "spans": [{"bbox": [110, 537, 502, 552], "score": 1.0, "content": "complex submanifold, via hyperkaehler rotation of the complex structure of", "type": "text"}], "index": 26}, {"bbox": [110, 552, 126, 565], "spans": [{"bbox": [110, 554, 121, 563], "score": 0.89, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [122, 552, 126, 565], "score": 1.0, "content": ".", "type": "text"}], "index": 27}], "index": 25.5}, {"type": "text", "bbox": [109, 564, 500, 607], "lines": [{"bbox": [127, 566, 498, 580], "spans": [{"bbox": [127, 566, 190, 580], "score": 1.0, "content": "Proof: Let ", "type": "text"}, {"bbox": [190, 568, 199, 577], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [199, 566, 407, 580], "score": 1.0, "content": " be a special Lagrangian submanifold of ", "type": "text"}, {"bbox": [408, 568, 419, 577], "score": 0.91, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [419, 566, 498, 580], "score": 1.0, "content": " in the complex", "type": "text"}], "index": 28}, {"bbox": [109, 581, 500, 596], "spans": [{"bbox": [109, 581, 160, 596], "score": 1.0, "content": "structure ", "type": "text"}, {"bbox": [161, 582, 172, 591], "score": 0.91, "content": "K", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [172, 581, 282, 596], "score": 1.0, "content": ". Then by definition ", "type": "text"}, {"bbox": [283, 582, 388, 595], "score": 0.93, "content": "\\mathrm{Re}(\\Omega_{K})_{\|\\Lambda}=\\mathrm{Vol}_{g}(\\Lambda)", "type": "inline_equation", "height": 13, "width": 105}, {"bbox": [389, 581, 500, 596], "score": 1.0, "content": ", but by the previous", "type": "text"}], "index": 29}, {"bbox": [110, 595, 296, 610], "spans": [{"bbox": [110, 595, 187, 610], "score": 1.0, "content": "theorem, since ", "type": "text"}, {"bbox": [187, 596, 232, 608], "score": 0.95, "content": "\\omega_{J}\|_{\\Lambda}=0", "type": "inline_equation", "height": 12, "width": 45}, {"bbox": [233, 595, 296, 610], "score": 1.0, "content": " this means:", "type": "text"}], "index": 30}], "index": 29}, {"type": "interline_equation", "bbox": [257, 614, 353, 642], "lines": [{"bbox": [257, 614, 353, 642], "spans": [{"bbox": [257, 614, 353, 642], "score": 0.95, "content": "\\mathrm{Vol}_{g}(\\Lambda)=\\frac{1}{2}\\int_{\\Lambda}\\omega_{I}^{2}.", "type": "interline_equation"}], "index": 31}], "index": 31}, {"type": "text", "bbox": [110, 645, 501, 675], "lines": [{"bbox": [111, 648, 500, 661], "spans": [{"bbox": [111, 648, 275, 661], "score": 1.0, "content": "By Wirtinger\u2019s theorem, since ", "type": "text"}, {"bbox": [275, 649, 284, 658], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [284, 648, 500, 661], "score": 1.0, "content": " is assumed to be compact and without", "type": "text"}], "index": 32}, {"bbox": [110, 662, 501, 675], "spans": [{"bbox": [110, 662, 345, 675], "score": 1.0, "content": "border, condition (3) is equivalent to say that ", "type": "text"}, {"bbox": [345, 664, 354, 672], "score": 0.91, "content": "\\Lambda", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [354, 662, 501, 675], "score": 1.0, "content": " is a complex submanifold of", "type": "text"}], "index": 33}], "index": 32.5}], "layout_bboxes": [], "page_idx": 4, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [244, 377, 365, 411], "lines": [{"bbox": [244, 377, 365, 411], "spans": [{"bbox": [244, 377, 365, 411], "score": 0.91, "content": "\\begin{array}{c c c c}{{\\alpha_{I,J}:\\ \\Lambda}}&{{\\to}}&{{\\Lambda^{2}\\,T^{}\\Lambda}}\\\\ {{p}}&{{\\mapsto}}&{{\\omega_{I,J}\|T_{p}\\Lambda}}\\end{array}", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "interline_equation", "bbox": [257, 614, 353, 642], "lines": [{"bbox": [257, 614, 353, 642], "spans": [{"bbox": [257, 614, 353, 642], "score": 0.95, "content": "\\mathrm{Vol}_{g}(\\Lambda)=\\frac{1}{2}\\int_{\\Lambda}\\omega_{I}^{2}.", "type": "interline_equation"}], "index": 31}], "index": 31}], "discarded_blocks": [{"type": "discarded", "bbox": [300, 691, 309, 702], "lines": [{"bbox": [301, 693, 309, 704], "spans": [{"bbox": [301, 693, 309, 704], "score": 1.0, "content": "5", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [109, 125, 500, 197], "lines": [{"bbox": [127, 127, 501, 142], "spans": [{"bbox": [127, 127, 214, 142], "score": 1.0, "content": "Second subcase: ", "type": "text"}, {"bbox": [214, 133, 222, 138], "score": 0.89, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [222, 127, 248, 142], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [248, 133, 254, 141], "score": 0.9, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [255, 127, 501, 142], "score": 1.0, "content": " have a 1-dimensional intersection spanned by a", "type": "text"}], "index": 0}, {"bbox": [110, 142, 499, 156], "spans": [{"bbox": [110, 142, 144, 156], "score": 1.0, "content": "vector ", "type": "text"}, {"bbox": [145, 147, 155, 154], "score": 0.91, "content": "v_{1}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [155, 142, 348, 156], "score": 1.0, "content": ". In this case we can choose a basis of ", "type": "text"}, {"bbox": [348, 144, 358, 153], "score": 0.87, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [358, 142, 420, 156], "score": 1.0, "content": " of the form ", "type": "text"}, {"bbox": [420, 143, 499, 155], "score": 0.92, "content": "(v_{1},I v_{1},J v_{1},w)", "type": "inline_equation", "height": 12, "width": 79}], "index": 1}, {"bbox": [110, 157, 500, 171], "spans": [{"bbox": [110, 157, 114, 171], "score": 1.0, "content": "(", "type": "text"}, {"bbox": [115, 162, 122, 167], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [122, 157, 203, 171], "score": 1.0, "content": " is spanned by ", "type": "text"}, {"bbox": [203, 158, 244, 170], "score": 0.94, "content": "(v_{1},I v_{1})", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [245, 157, 284, 171], "score": 1.0, "content": ", while ", "type": "text"}, {"bbox": [284, 162, 290, 169], "score": 0.9, "content": "\\rho", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [291, 157, 371, 171], "score": 1.0, "content": " is spanned by ", "type": "text"}, {"bbox": [372, 158, 418, 170], "score": 0.92, "content": "\\left(v_{1},J v_{1}\\right))", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [418, 157, 500, 171], "score": 1.0, "content": ". Again by the", "type": "text"}], "index": 2}, {"bbox": [110, 172, 499, 185], "spans": [{"bbox": [110, 172, 499, 185], "score": 1.0, "content": "same computation of the previous subcase one shows that this configuration", "type": "text"}], "index": 3}, {"bbox": [109, 186, 307, 198], "spans": [{"bbox": [109, 186, 307, 198], "score": 1.0, "content": "is not compatible with the calibration.", "type": "text"}], "index": 4}], "index": 2, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [109, 127, 501, 198]}, {"type": "text", "bbox": [109, 198, 500, 255], "lines": [{"bbox": [127, 199, 500, 215], "spans": [{"bbox": [127, 199, 248, 215], "score": 1.0, "content": "Third subcase: Finally ", "type": "text"}, {"bbox": [248, 204, 271, 213], "score": 0.88, "content": "\\pi{=}\\rho", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [271, 199, 500, 215], "score": 1.0, "content": " can not clearly happen, since otherwise one", "type": "text"}], "index": 5}, {"bbox": [110, 214, 499, 228], "spans": [{"bbox": [110, 214, 221, 228], "score": 1.0, "content": "can choose a basis of ", "type": "text"}, {"bbox": [221, 219, 228, 225], "score": 0.89, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [229, 214, 278, 228], "score": 1.0, "content": " equal to ", "type": "text"}, {"bbox": [278, 215, 320, 228], "score": 0.94, "content": "(v_{1},I v_{1})", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [320, 214, 444, 228], "score": 1.0, "content": ", but then, in this basis ", "type": "text"}, {"bbox": [444, 219, 457, 227], "score": 0.91, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [458, 214, 499, 228], "score": 1.0, "content": " is iden-", "type": "text"}], "index": 6}, {"bbox": [109, 227, 500, 245], "spans": [{"bbox": [109, 227, 380, 245], "score": 1.0, "content": "tically vanishing, contrary to the assumption that ", "type": "text"}, {"bbox": [380, 234, 415, 242], "score": 0.91, "content": "\\rho~=~\\pi", "type": "inline_equation", "height": 8, "width": 35}, {"bbox": [415, 227, 500, 245], "score": 1.0, "content": " is a symplectic", "type": "text"}], "index": 7}, {"bbox": [110, 241, 211, 259], "spans": [{"bbox": [110, 241, 192, 259], "score": 1.0, "content": "2-plane also for ", "type": "text"}, {"bbox": [192, 248, 206, 256], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [206, 241, 211, 259], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 6.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [109, 199, 500, 259]}, {"type": "text", "bbox": [109, 255, 499, 284], "lines": [{"bbox": [126, 257, 499, 271], "spans": [{"bbox": [126, 257, 330, 271], "score": 1.0, "content": "Since the second case can never happen ", "type": "text"}, {"bbox": [331, 259, 340, 268], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [340, 257, 499, 271], "score": 1.0, "content": " has to be Lagrangian also with", "type": "text"}], "index": 9}, {"bbox": [109, 271, 182, 287], "spans": [{"bbox": [109, 271, 164, 287], "score": 1.0, "content": "respect to ", "type": "text"}, {"bbox": [164, 277, 178, 285], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [178, 271, 182, 287], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 9.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [109, 257, 499, 287]}, {"type": "text", "bbox": [109, 284, 500, 371], "lines": [{"bbox": [128, 286, 501, 301], "spans": [{"bbox": [128, 286, 501, 301], "score": 1.0, "content": "Up to now, we have worked only locally; to conclude the proof it is", "type": "text"}], "index": 11}, {"bbox": [110, 302, 501, 315], "spans": [{"bbox": [110, 302, 235, 315], "score": 1.0, "content": "necessary to show that if ", "type": "text"}, {"bbox": [235, 303, 255, 315], "score": 0.93, "content": "T_{p}\\Lambda", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [256, 302, 406, 315], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [406, 306, 420, 313], "score": 0.89, "content": "\\omega_{J}", "type": "inline_equation", "height": 7, "width": 14}, {"bbox": [420, 302, 501, 315], "score": 1.0, "content": ", then it can not", "type": "text"}], "index": 12}, {"bbox": [110, 315, 500, 330], "spans": [{"bbox": [110, 315, 195, 330], "score": 1.0, "content": "be possible that ", "type": "text"}, {"bbox": [195, 317, 216, 329], "score": 0.94, "content": "T_{q}\\Lambda", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [216, 315, 373, 330], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [374, 320, 386, 327], "score": 0.89, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [386, 315, 466, 330], "score": 1.0, "content": ", for a different ", "type": "text"}, {"bbox": [467, 317, 496, 328], "score": 0.93, "content": "q\\in\\Lambda", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [496, 315, 500, 330], "score": 1.0, "content": ".", "type": "text"}], "index": 13}, {"bbox": [109, 330, 500, 344], "spans": [{"bbox": [109, 330, 275, 344], "score": 1.0, "content": "Notice that any tangent space to ", "type": "text"}, {"bbox": [275, 331, 284, 340], "score": 0.89, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [284, 330, 500, 344], "score": 1.0, "content": " can not be Lagrangian with respect to both", "type": "text"}], "index": 14}, {"bbox": [110, 345, 500, 358], "spans": [{"bbox": [110, 349, 122, 357], "score": 0.91, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [123, 345, 150, 358], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [150, 349, 164, 357], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [164, 345, 500, 358], "score": 1.0, "content": ", otherwise it would violates the calibration condition. Consider", "type": "text"}], "index": 15}, {"bbox": [109, 359, 344, 372], "spans": [{"bbox": [109, 360, 302, 372], "score": 1.0, "content": "now the following smooth sections of", "type": "text"}, {"bbox": [302, 359, 339, 371], "score": 0.93, "content": "\\Lambda^{2}T^{}\\Lambda", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [339, 360, 344, 372], "score": 1.0, "content": ":", "type": "text"}], "index": 16}], "index": 13.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [109, 286, 501, 372]}, {"type": "interline_equation", "bbox": [244, 377, 365, 411], "lines": [{"bbox": [244, 377, 365, 411], "spans": [{"bbox": [244, 377, 365, 411], "score": 0.91, "content": "\\begin{array}{c c c c}{{\\alpha_{I,J}:\\ \\Lambda}}&{{\\to}}&{{\\Lambda^{2}\\,T^{}\\Lambda}}\\\\ {{p}}&{{\\mapsto}}&{{\\omega_{I,J}\|T_{p}\\Lambda}}\\end{array}", "type": "interline_equation"}], "index": 17}], "index": 17, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [109, 413, 501, 487], "lines": [{"bbox": [109, 416, 500, 432], "spans": [{"bbox": [109, 416, 217, 432], "score": 1.0, "content": "and the zero section ", "type": "text"}, {"bbox": [217, 417, 301, 429], "score": 0.94, "content": "s_{0}:\\Lambda\\to\\Lambda^{2}\\,T^{}\\Lambda", "type": "inline_equation", "height": 12, "width": 84}, {"bbox": [302, 416, 367, 432], "score": 1.0, "content": ". Obviously, ", "type": "text"}, {"bbox": [367, 418, 395, 430], "score": 0.95, "content": "s_{0}(\\Lambda)", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [395, 416, 458, 432], "score": 1.0, "content": " is closed in", "type": "text"}, {"bbox": [459, 417, 496, 429], "score": 0.92, "content": "\\Lambda^{2}T^{*}\\Lambda", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [497, 416, 500, 432], "score": 1.0, "content": ",", "type": "text"}], "index": 18}, {"bbox": [110, 431, 500, 446], "spans": [{"bbox": [110, 431, 272, 446], "score": 1.0, "content": "and by the previous reasoning ", "type": "text"}, {"bbox": [272, 433, 281, 442], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [281, 431, 405, 446], "score": 1.0, "content": " can be decomposed as ", "type": "text"}, {"bbox": [405, 432, 500, 445], "score": 0.93, "content": "\\Lambda\\,=\\,\\alpha_{I}^{-1}(s_{0}(\\Lambda))\\cup", "type": "inline_equation", "height": 13, "width": 95}], "index": 19}, {"bbox": [110, 444, 501, 461], "spans": [{"bbox": [110, 446, 166, 460], "score": 0.97, "content": "\\alpha_{J}^{-1}(s_{0}(\\Lambda))", "type": "inline_equation", "height": 14, "width": 56}, {"bbox": [166, 444, 501, 461], "score": 1.0, "content": ", that is as the disjoint union of two proper closed subsets. But", "type": "text"}], "index": 20}, {"bbox": [110, 461, 501, 474], "spans": [{"bbox": [110, 461, 272, 474], "score": 1.0, "content": "this is clearly impossible, since ", "type": "text"}, {"bbox": [273, 462, 281, 471], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [281, 461, 501, 474], "score": 1.0, "content": " is connected, and this implies that one of", "type": "text"}], "index": 21}, {"bbox": [109, 474, 502, 489], "spans": [{"bbox": [109, 474, 286, 489], "score": 1.0, "content": "the two closed subset is empty, so ", "type": "text"}, {"bbox": [287, 477, 295, 485], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [296, 474, 385, 489], "score": 1.0, "content": " is bi-Lagrangian.", "type": "text"}, {"bbox": [491, 477, 502, 487], "score": 0.9912325739860535, "content": "\u53e3", "type": "text"}], "index": 22}], "index": 20, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [109, 416, 502, 489]}, {"type": "text", "bbox": [126, 492, 435, 506], "lines": [{"bbox": [127, 493, 434, 509], "spans": [{"bbox": [127, 493, 434, 509], "score": 1.0, "content": "The previous theorem is important in view of the following:", "type": "text"}], "index": 23}], "index": 23, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [127, 493, 434, 509]}, {"type": "text", "bbox": [110, 507, 501, 563], "lines": [{"bbox": [126, 507, 501, 524], "spans": [{"bbox": [126, 507, 501, 524], "score": 1.0, "content": "Corollary 2.1: Every (connected, compact and without border) special", "type": "text"}], "index": 24}, {"bbox": [111, 523, 500, 537], "spans": [{"bbox": [111, 523, 379, 537], "score": 1.0, "content": "Lagrangian submanifold \u039b of a hyperkaehler 4-fold ", "type": "text"}, {"bbox": [380, 524, 391, 533], "score": 0.89, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [391, 523, 493, 537], "score": 1.0, "content": " can be realized as ", "type": "text"}, {"bbox": [493, 528, 500, 533], "score": 0.26, "content": "a", "type": "inline_equation", "height": 5, "width": 7}], "index": 25}, {"bbox": [110, 537, 502, 552], "spans": [{"bbox": [110, 537, 502, 552], "score": 1.0, "content": "complex submanifold, via hyperkaehler rotation of the complex structure of", "type": "text"}], "index": 26}, {"bbox": [110, 552, 126, 565], "spans": [{"bbox": [110, 554, 121, 563], "score": 0.89, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [122, 552, 126, 565], "score": 1.0, "content": ".", "type": "text"}], "index": 27}], "index": 25.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [110, 507, 502, 565]}, {"type": "text", "bbox": [109, 564, 500, 607], "lines": [{"bbox": [127, 566, 498, 580], "spans": [{"bbox": [127, 566, 190, 580], "score": 1.0, "content": "Proof: Let ", "type": "text"}, {"bbox": [190, 568, 199, 577], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [199, 566, 407, 580], "score": 1.0, "content": " be a special Lagrangian submanifold of ", "type": "text"}, {"bbox": [408, 568, 419, 577], "score": 0.91, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [419, 566, 498, 580], "score": 1.0, "content": " in the complex", "type": "text"}], "index": 28}, {"bbox": [109, 581, 500, 596], "spans": [{"bbox": [109, 581, 160, 596], "score": 1.0, "content": "structure ", "type": "text"}, {"bbox": [161, 582, 172, 591], "score": 0.91, "content": "K", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [172, 581, 282, 596], "score": 1.0, "content": ". 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In this case we can choose a basis of of the form ( is spanned by , while is spanned by . Again by the same computation of the previous subcase one shows that this configuration is not compatible with the calibration. Third subcase: Finally can not clearly happen, since otherwise one can choose a basis of equal to , but then, in this basis is iden- tically vanishing, contrary to the assumption that is a symplectic 2-plane also for . Since the second case can never happen has to be Lagrangian also with respect to . Up to now, we have worked only locally; to conclude the proof it is necessary to show that if is Lagrangian with respect to , then it can not be possible that is Lagrangian with respect to , for a different . Notice that any tangent space to can not be Lagrangian with respect to both and , otherwise it would violates the calibration condition. Consider now the following smooth sections of : $$ \begin{array}{c c c c}{{\alpha_{I,J}:\ \Lambda}}&{{\to}}&{{\Lambda^{2}\,T^{*}\Lambda}}\\ {{p}}&{{\mapsto}}&{{\omega_{I,J}\|T_{p}\Lambda}}\end{array} $$ and the zero section . Obviously, is closed in , and by the previous reasoning can be decomposed as , that is as the disjoint union of two proper closed subsets. But this is clearly impossible, since is connected, and this implies that one of the two closed subset is empty, so is bi-Lagrangian. 口 The previous theorem is important in view of the following: Corollary 2.1: Every (connected, compact and without border) special Lagrangian submanifold Λ of a hyperkaehler 4-fold can be realized as complex submanifold, via hyperkaehler rotation of the complex structure of . Proof: Let be a special Lagrangian submanifold of in the complex structure . Then by definition , but by the previous theorem, since this means: $$ \mathrm{Vol}_{g}(\Lambda)=\frac{1}{2}\int_{\Lambda}\omega_{I}^{2}. $$ By Wirtinger’s theorem, since is assumed to be compact and without border, condition (3) is equivalent to say that is a complex submanifold of 5	<div class="pdf-page"> <p>Second subcase: and have a 1-dimensional intersection spanned by a vector . In this case we can choose a basis of of the form ( is spanned by , while is spanned by . Again by the same computation of the previous subcase one shows that this configuration is not compatible with the calibration.</p> <p>Third subcase: Finally can not clearly happen, since otherwise one can choose a basis of equal to , but then, in this basis is iden- tically vanishing, contrary to the assumption that is a symplectic 2-plane also for .</p> <p>Since the second case can never happen has to be Lagrangian also with respect to .</p> <p>Up to now, we have worked only locally; to conclude the proof it is necessary to show that if is Lagrangian with respect to , then it can not be possible that is Lagrangian with respect to , for a different . Notice that any tangent space to can not be Lagrangian with respect to both and , otherwise it would violates the calibration condition. Consider now the following smooth sections of :</p> <p>and the zero section . Obviously, is closed in , and by the previous reasoning can be decomposed as , that is as the disjoint union of two proper closed subsets. But this is clearly impossible, since is connected, and this implies that one of the two closed subset is empty, so is bi-Lagrangian. 口</p> <p>The previous theorem is important in view of the following:</p> <p>Corollary 2.1: Every (connected, compact and without border) special Lagrangian submanifold Λ of a hyperkaehler 4-fold can be realized as complex submanifold, via hyperkaehler rotation of the complex structure of .</p> <p>Proof: Let be a special Lagrangian submanifold of in the complex structure . Then by definition , but by the previous theorem, since this means:</p> <p>By Wirtinger’s theorem, since is assumed to be compact and without border, condition (3) is equivalent to say that is a complex submanifold of</p> </div>	<div class="pdf-page"> <p class="pdf-text" data-x="182" data-y="161" data-width="654" data-height="93">Second subcase: and have a 1-dimensional intersection spanned by a vector . In this case we can choose a basis of of the form ( is spanned by , while is spanned by . Again by the same computation of the previous subcase one shows that this configuration is not compatible with the calibration.</p> <p class="pdf-text" data-x="182" data-y="256" data-width="654" data-height="73">Third subcase: Finally can not clearly happen, since otherwise one can choose a basis of equal to , but then, in this basis is iden- tically vanishing, contrary to the assumption that is a symplectic 2-plane also for .</p> <p class="pdf-text" data-x="182" data-y="329" data-width="652" data-height="38">Since the second case can never happen has to be Lagrangian also with respect to .</p> <p class="pdf-text" data-x="182" data-y="367" data-width="654" data-height="112">Up to now, we have worked only locally; to conclude the proof it is necessary to show that if is Lagrangian with respect to , then it can not be possible that is Lagrangian with respect to , for a different . Notice that any tangent space to can not be Lagrangian with respect to both and , otherwise it would violates the calibration condition. Consider now the following smooth sections of :</p> <p class="pdf-text" data-x="182" data-y="533" data-width="656" data-height="96">and the zero section . Obviously, is closed in , and by the previous reasoning can be decomposed as , that is as the disjoint union of two proper closed subsets. But this is clearly impossible, since is connected, and this implies that one of the two closed subset is empty, so is bi-Lagrangian. 口</p> <p class="pdf-text" data-x="210" data-y="636" data-width="517" data-height="18">The previous theorem is important in view of the following:</p> <p class="pdf-text" data-x="184" data-y="655" data-width="654" data-height="72">Corollary 2.1: Every (connected, compact and without border) special Lagrangian submanifold Λ of a hyperkaehler 4-fold can be realized as complex submanifold, via hyperkaehler rotation of the complex structure of .</p> <p class="pdf-text" data-x="182" data-y="729" data-width="654" data-height="55">Proof: Let be a special Lagrangian submanifold of in the complex structure . Then by definition , but by the previous theorem, since this means:</p> <p class="pdf-text" data-x="184" data-y="833" data-width="654" data-height="39">By Wirtinger’s theorem, since is assumed to be compact and without border, condition (3) is equivalent to say that is a complex submanifold of</p> <div class="pdf-discarded" data-x="501" data-y="893" data-width="16" data-height="14" style="opacity: 0.5;">5</div> </div>	{ "type": [ "inline_equation", "inline_equation", "inline_equation", "text", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "interline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "text", "inline_equation", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "interline_equation", "inline_equation", "inline_equation" ], "coordinates": [ [ 212, 164, 838, 183 ], [ 184, 183, 834, 201 ], [ 184, 202, 836, 221 ], [ 184, 222, 834, 239 ], [ 182, 240, 513, 256 ], [ 212, 257, 836, 277 ], [ 184, 276, 834, 294 ], [ 182, 293, 836, 316 ], [ 184, 311, 353, 334 ], [ 210, 332, 834, 350 ], [ 182, 350, 304, 371 ], [ 214, 369, 838, 389 ], [ 184, 390, 838, 407 ], [ 184, 407, 836, 426 ], [ 182, 426, 836, 444 ], [ 184, 446, 836, 462 ], [ 182, 464, 575, 480 ], [ 408, 487, 610, 531 ], [ 182, 537, 836, 558 ], [ 184, 557, 836, 576 ], [ 184, 574, 838, 596 ], [ 184, 596, 838, 612 ], [ 182, 612, 839, 632 ], [ 212, 637, 726, 658 ], [ 210, 655, 838, 677 ], [ 185, 676, 836, 694 ], [ 184, 694, 839, 713 ], [ 184, 713, 210, 730 ], [ 212, 731, 833, 749 ], [ 182, 751, 836, 770 ], [ 184, 769, 495, 788 ], [ 430, 793, 590, 830 ], [ 185, 837, 836, 854 ], [ 184, 855, 838, 872 ] ], "content": [ "Second subcase: \\pi and \\rho have a 1-dimensional intersection spanned by a", "vector v_{1} . In this case we can choose a basis of V of the form (v_{1},I v_{1},J v_{1},w)", "( \\pi is spanned by (v_{1},I v_{1}) , while \\rho is spanned by \\left(v_{1},J v_{1}\\right)) . Again by the", "same computation of the previous subcase one shows that this configuration", "is not compatible with the calibration.", "Third subcase: Finally \\pi{=}\\rho can not clearly happen, since otherwise one", "can choose a basis of \\pi equal to (v_{1},I v_{1}) , but then, in this basis \\omega_{J} is iden-", "tically vanishing, contrary to the assumption that \\rho~=~\\pi is a symplectic", "2-plane also for \\omega_{J} .", "Since the second case can never happen V has to be Lagrangian also with", "respect to \\omega_{J} .", "Up to now, we have worked only locally; to conclude the proof it is", "necessary to show that if T_{p}\\Lambda is Lagrangian with respect to \\omega_{J} , then it can not", "be possible that T_{q}\\Lambda is Lagrangian with respect to \\omega_{I} , for a different q\\in\\Lambda .", "Notice that any tangent space to \\Lambda can not be Lagrangian with respect to both", "\\omega_{I} and \\omega_{J} , otherwise it would violates the calibration condition. Consider", "now the following smooth sections of \\Lambda^{2}T^{}\\Lambda :", "\\begin{array}{c c c c}{{\\alpha_{I,J}:\\ \\Lambda}}&{{\\to}}&{{\\Lambda^{2}\\,T^{}\\Lambda}}\\\\ {{p}}&{{\\mapsto}}&{{\\omega_{I,J}\|T_{p}\\Lambda}}\\end{array}", "and the zero section s_{0}:\\Lambda\\to\\Lambda^{2}\\,T^{}\\Lambda . Obviously, s_{0}(\\Lambda) is closed in \\Lambda^{2}T^{}\\Lambda ,", "and by the previous reasoning \\Lambda can be decomposed as \\Lambda\\,=\\,\\alpha_{I}^{-1}(s_{0}(\\Lambda))\\cup", "\\alpha_{J}^{-1}(s_{0}(\\Lambda)) , that is as the disjoint union of two proper closed subsets. But", "this is clearly impossible, since \\Lambda is connected, and this implies that one of", "the two closed subset is empty, so \\Lambda is bi-Lagrangian. 口", "The previous theorem is important in view of the following:", "Corollary 2.1: Every (connected, compact and without border) special", "Lagrangian submanifold Λ of a hyperkaehler 4-fold X can be realized as a", "complex submanifold, via hyperkaehler rotation of the complex structure of", "X .", "Proof: Let \\Lambda be a special Lagrangian submanifold of X in the complex", "structure K . Then by definition \\mathrm{Re}(\\Omega_{K})_{\|\\Lambda}=\\mathrm{Vol}_{g}(\\Lambda) , but by the previous", "theorem, since \\omega_{J}\|_{\\Lambda}=0 this means:", "\\mathrm{Vol}_{g}(\\Lambda)=\\frac{1}{2}\\int_{\\Lambda}\\omega_{I}^{2}.", "By Wirtinger’s theorem, since \\Lambda is assumed to be compact and without", "border, condition (3) is equivalent to say that \\Lambda is a complex submanifold of" ], "index": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 14, 15, 25, 26, 27, 28, 29, 30, 42, 60, 61, 62, 63, 64, 83, 107, 108, 109, 110, 135, 136, 137, 166, 198, 199 ] }	{ "coordinates": [], "index": [], "caption": [], "caption_coordinates": [] }	{ "type": [ "interline_equation", "interline_equation", "interline_equation", "interline_equation" ], "coordinates": [ [ 408, 487, 610, 531 ], [ 430, 793, 590, 830 ], [ 408, 487, 610, 531 ], [ 430, 793, 590, 830 ] ], "content": [ "", "", "\\begin{array}{c c c c}{{\\alpha_{I,J}:\\ \\Lambda}}&{{\\to}}&{{\\Lambda^{2}\\,T^{*}\\Lambda}}\\\\ {{p}}&{{\\mapsto}}&{{\\omega_{I,J}\|T_{p}\\Lambda}}\\end{array}", "\\mathrm{Vol}_{g}(\\Lambda)=\\frac{1}{2}\\int_{\\Lambda}\\omega_{I}^{2}." ], "index": [ 0, 1, 2, 3 ] }	{ "type": [], "coordinates": [], "content": [], "index": [] }	{"document_info": {"total_pages": 9, "parse_type": "txt", "version": "1.0.1", "total_content_list_items": 72}, "content_statistics": {"total_blocks": 81, "total_lines": 269, "total_images": 0, "total_equations": 18, "total_tables": 0, "total_characters": 16112, "total_words": 2578, "block_type_distribution": {"title": 6, "text": 55, "discarded": 9, "interline_equation": 9, "list": 2}}, "model_statistics": {"total_layout_detections": 813, "avg_confidence": 0.9488487084870848, "min_confidence": 0.26, "max_confidence": 1.0}, "page_statistics": [{"page_number": 0, "page_size": [612.0, 792.0], "blocks": 10, "lines": 26, "images": 0, "equations": 0, "tables": 0, "characters": 1419, "words": 204, "discarded_blocks": 2}, {"page_number": 1, "page_size": [612.0, 792.0], "blocks": 15, "lines": 37, "images": 0, "equations": 0, "tables": 0, "characters": 2328, "words": 359, "discarded_blocks": 1}, {"page_number": 2, "page_size": [612.0, 792.0], "blocks": 12, "lines": 35, "images": 0, "equations": 4, "tables": 0, "characters": 1843, "words": 306, "discarded_blocks": 1}, {"page_number": 3, "page_size": [612.0, 792.0], "blocks": 8, "lines": 30, "images": 0, "equations": 6, "tables": 0, "characters": 1797, "words": 312, "discarded_blocks": 0}, {"page_number": 4, "page_size": [612.0, 792.0], "blocks": 12, "lines": 34, "images": 0, "equations": 4, "tables": 0, "characters": 1905, "words": 328, "discarded_blocks": 1}, {"page_number": 5, "page_size": [612.0, 792.0], "blocks": 11, "lines": 32, "images": 0, "equations": 4, "tables": 0, "characters": 1846, "words": 308, "discarded_blocks": 1}, {"page_number": 6, "page_size": [612.0, 792.0], "blocks": 6, "lines": 46, "images": 0, "equations": 0, "tables": 0, "characters": 3307, "words": 521, "discarded_blocks": 1}, {"page_number": 7, "page_size": [612.0, 792.0], "blocks": 5, "lines": 29, "images": 0, "equations": 0, "tables": 0, "characters": 1666, "words": 239, "discarded_blocks": 1}, {"page_number": 8, "page_size": [612.0, 792.0], "blocks": 2, "lines": 0, "images": 0, "equations": 0, "tables": 0, "characters": 1, "words": 1, "discarded_blocks": 1}]}
0001060v1	5	[ 612, 792 ]	{ "type": [ "text", "text", "text", "text", "interline_equation", "text", "interline_equation", "text", "text", "text", "discarded" ], "coordinates": [ [ 184, 161, 838, 236 ], [ 182, 244, 836, 281 ], [ 184, 283, 839, 318 ], [ 182, 319, 836, 394 ], [ 324, 412, 694, 430 ], [ 182, 442, 836, 514 ], [ 354, 518, 662, 537 ], [ 182, 544, 838, 601 ], [ 182, 608, 836, 720 ], [ 184, 721, 838, 815 ], [ 501, 893, 517, 907 ] ], "content": [ ", in the complex structure , that is performing a hyperkaehler rotation. Notice that in the complex structure , is still a Lagrangian submanifold with respect to and , so it is Lagrangian with respect to the holomorphic (in the structure ) 2-form . 口", "Collecting the results so far proved, we can show that special Lagrangian submanifolds of are particularly rigid:", "Proposition 2.1: Any (connected, compact and without border) special Lagrangian submanifold Λ of a hyperkaehler 4-fold is real analytic.", "Proof: Let be a special Lagrangian submanifold of , having fixed some complex structure on , let us say ; then, by Corollary 2.1 there exists a new complex structure, let us say , in which is holomorphic, that is, it is locally given by:", "", "Now observe that coming back to the original complex structure , we in- duce an analytic change of coordinates from the holomorphic coordinates to new holomorphic coordinates ( i ∂∂wi ) such that locally:", "", "for some complex constants . Thus in the complex structure the special Lagrangian submanifold is given by which is again the zero locus of a set of functions analytic in . 口", "Quite naturally, the action of the hyperkaehler rotation can be extended also to the holomorphic functions defined on complex submanifolds of ; in particular we have an action of the hyperkaehler rotation on the structure sheaf (here, as always, we identify with its direct image , where is the holomorphic embedding). We are thus led to give the following:", "Definition 2.2: Let Λ be a special Lagrangian submanifold of a hyper- kaehler 4-fold (in the complex structure ). Then we define the special Lagrangian structure sheaf as the sheaf obtained by the action of the hy- perkaehler rotation on the structure sheaf of , as a complex Lagrangian submanifold of , (in the structure ).", "6" ], "index": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ] }	[{"type": "text", "text": "$X$ , in the complex structure $I$ , that is performing a hyperkaehler rotation. Notice that in the complex structure $I$ , $\\Lambda$ is still a Lagrangian submanifold with respect to $\\omega_{K}$ and $\\omega_{I}$ , so it is Lagrangian with respect to the holomorphic (in the structure $I$ ) 2-form $\\Omega_{I}:=\\omega_{J}+i\\omega_{K}$ . \u53e3 ", "page_idx": 5}, {"type": "text", "text": "Collecting the results so far proved, we can show that special Lagrangian submanifolds of $X$ are particularly rigid: ", "page_idx": 5}, {"type": "text", "text": "Proposition 2.1: Any (connected, compact and without border) special Lagrangian submanifold \u039b of a hyperkaehler 4-fold $X$ is real analytic. ", "page_idx": 5}, {"type": "text", "text": "Proof: Let $\\Lambda$ be a special Lagrangian submanifold of $X$ , having fixed some complex structure on $X$ , let us say $K$ ; then, by Corollary 2.1 there exists a new complex structure, let us say $I$ , in which $\\Lambda$ is holomorphic, that is, it is locally given by: ", "page_idx": 5}, {"type": "equation", "text": "$$\nf_{1}(z_{1},\\ldots,z_{4})=0\\quad\\mathrm{and}\\quad f_{2}(z_{1},\\ldots,z_{4})=0.\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "Now observe that coming back to the original complex structure $K$ , we induce an analytic change of coordinates from the holomorphic coordinates $z^{i}$ $\\begin{array}{r}{\\int\\!\\frac{\\partial}{\\partial z^{i}}\\,=\\,i\\frac{\\partial}{\\partial z^{i}}\\rangle}\\end{array}$ to new holomorphic coordinates $w^{i}$ ( $\\begin{array}{r}{K\\frac{\\partial}{\\partial w^{i}}\\,=\\,i\\frac{\\partial}{\\partial w^{i}}\\Big)}\\end{array}$ i \u2202\u2202wi ) such that locally: ", "page_idx": 5}, {"type": "equation", "text": "$$\nz^{i}=c_{1}w^{i}+c_{2}\\bar{w}^{i}\\;\\;\\;\\;\\bar{z}^{i}=d_{1}w^{i}+d_{2}\\bar{w}^{i},\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "for some complex constants $c_{j},d_{j}$ . Thus in the complex structure $K$ the special Lagrangian submanifold $\\Lambda$ is given by $f_{j}(c_{1}w^{i}\\!+\\!c_{2}\\bar{w}^{i},d_{1}w^{i}\\!+\\!d_{2}\\bar{w}^{i})=0$ which is again the zero locus of a set of functions analytic in $w^{i},\\bar{w}^{i}$ . \u53e3 ", "page_idx": 5}, {"type": "text", "text": "Quite naturally, the action of the hyperkaehler rotation can be extended also to the holomorphic functions defined on complex submanifolds $S$ of $X$ ; in particular we have an action of the hyperkaehler rotation on the structure sheaf $O_{S}$ (here, as always, we identify $O_{S}$ with its direct image $j_{*}O_{S}$ , where $j\\,:\\,S\\,\\rightarrow\\,X$ is the holomorphic embedding). We are thus led to give the following: ", "page_idx": 5}, {"type": "text", "text": "Definition 2.2: Let \u039b be a special Lagrangian submanifold of a hyperkaehler 4-fold $X$ (in the complex structure $K$ ). Then we define the special Lagrangian structure sheaf ${\\mathcal{L}}_{\\Lambda}$ as the sheaf obtained by the action of the hyperkaehler rotation on the structure sheaf $O_{\\Lambda}$ of $\\Lambda$ , as a complex Lagrangian submanifold of $X$ , (in the structure $I$ ). 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then, by Corollary 2.1 there", "type": "text"}], "index": 9}, {"bbox": [110, 279, 499, 291], "spans": [{"bbox": [110, 279, 325, 291], "score": 1.0, "content": "exists a new complex structure, let us say ", "type": "text"}, {"bbox": [325, 280, 331, 289], "score": 0.89, "content": "I", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [332, 279, 384, 291], "score": 1.0, "content": ", in which ", "type": "text"}, {"bbox": [384, 280, 393, 289], "score": 0.91, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [393, 279, 499, 291], "score": 1.0, "content": " is holomorphic, that", "type": "text"}], "index": 10}, {"bbox": [110, 294, 232, 306], "spans": [{"bbox": [110, 294, 232, 306], "score": 1.0, "content": "is, it is locally given by:", "type": "text"}], "index": 11}], "index": 9.5}, {"type": "interline_equation", "bbox": [194, 319, 415, 333], "lines": [{"bbox": [194, 319, 415, 333], "spans": [{"bbox": [194, 319, 415, 333], "score": 0.88, "content": "f_{1}(z_{1},\\ldots,z_{4})=0\\quad\\mathrm{and}\\quad f_{2}(z_{1},\\ldots,z_{4})=0.", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [109, 342, 500, 398], "lines": [{"bbox": [109, 344, 499, 359], "spans": [{"bbox": [109, 344, 448, 359], "score": 1.0, "content": "Now observe that coming back to the original complex structure ", "type": "text"}, {"bbox": [449, 347, 460, 356], "score": 0.89, "content": "K", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [460, 344, 499, 359], "score": 1.0, "content": ", we in-", "type": "text"}], "index": 13}, {"bbox": [109, 359, 499, 374], "spans": [{"bbox": [109, 359, 489, 374], "score": 1.0, "content": "duce an analytic change of coordinates from the holomorphic coordinates ", "type": "text"}, {"bbox": [489, 360, 499, 370], "score": 0.91, "content": "z^{i}", "type": "inline_equation", "height": 10, "width": 10}], "index": 14}, {"bbox": [113, 369, 502, 395], "spans": [{"bbox": [113, 374, 175, 389], "score": 0.9, "content": "\\begin{array}{r}{\\int\\!\\frac{\\partial}{\\partial z^{i}}\\,=\\,i\\frac{\\partial}{\\partial z^{i}}\\rangle}\\end{array}", "type": "inline_equation", "height": 15, "width": 62}, {"bbox": [176, 369, 352, 395], "score": 1.0, "content": " to new holomorphic coordinates ", "type": "text"}, {"bbox": [352, 375, 365, 385], "score": 0.9, "content": "w^{i}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [365, 369, 373, 395], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [374, 374, 444, 389], "score": 0.92, "content": "\\begin{array}{r}{K\\frac{\\partial}{\\partial w^{i}}\\,=\\,i\\frac{\\partial}{\\partial w^{i}}\\Big)}\\end{array}", "type": "inline_equation", "height": 15, "width": 70}, {"bbox": [421, 372, 502, 391], "score": 1.0, "content": "i \u2202\u2202wi ) such that", "type": "text"}], "index": 15}, {"bbox": [109, 389, 148, 402], "spans": [{"bbox": [109, 389, 148, 402], "score": 1.0, "content": "locally:", "type": "text"}], "index": 16}], "index": 14.5}, {"type": "interline_equation", "bbox": [212, 401, 396, 416], "lines": [{"bbox": [212, 401, 396, 416], "spans": [{"bbox": [212, 401, 396, 416], "score": 0.92, "content": "z^{i}=c_{1}w^{i}+c_{2}\\bar{w}^{i}\\;\\;\\;\\;\\bar{z}^{i}=d_{1}w^{i}+d_{2}\\bar{w}^{i},", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [109, 421, 501, 465], "lines": [{"bbox": [109, 424, 501, 439], "spans": [{"bbox": [109, 424, 261, 439], "score": 1.0, "content": "for some complex constants ", "type": "text"}, {"bbox": [261, 426, 286, 438], "score": 0.94, "content": "c_{j},d_{j}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [287, 424, 466, 439], "score": 1.0, "content": ". Thus in the complex structure ", "type": "text"}, {"bbox": [466, 426, 477, 434], "score": 0.88, "content": "K", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [478, 424, 501, 439], "score": 1.0, "content": " the", "type": "text"}], "index": 18}, {"bbox": [109, 439, 499, 453], "spans": [{"bbox": [109, 439, 273, 453], "score": 1.0, "content": "special Lagrangian submanifold ", "type": "text"}, {"bbox": [273, 440, 281, 449], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [282, 439, 340, 453], "score": 1.0, "content": " is given by ", "type": "text"}, {"bbox": [341, 439, 499, 452], "score": 0.92, "content": "f_{j}(c_{1}w^{i}\\!+\\!c_{2}\\bar{w}^{i},d_{1}w^{i}\\!+\\!d_{2}\\bar{w}^{i})=0", "type": "inline_equation", "height": 13, "width": 158}], "index": 19}, {"bbox": [110, 453, 501, 467], "spans": [{"bbox": [110, 453, 424, 467], "score": 1.0, "content": "which is again the zero locus of a set of functions analytic in ", "type": "text"}, {"bbox": [424, 453, 454, 466], "score": 0.81, "content": "w^{i},\\bar{w}^{i}", "type": "inline_equation", "height": 13, "width": 30}, {"bbox": [454, 453, 458, 467], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [492, 456, 501, 465], "score": 0.9820061922073364, "content": "\u53e3", "type": "text"}], "index": 20}], "index": 19}, {"type": "text", "bbox": [109, 471, 500, 557], "lines": [{"bbox": [128, 474, 500, 487], "spans": [{"bbox": [128, 474, 500, 487], "score": 1.0, "content": "Quite naturally, the action of the hyperkaehler rotation can be extended", "type": "text"}], "index": 21}, {"bbox": [109, 488, 500, 501], "spans": [{"bbox": [109, 488, 459, 501], "score": 1.0, "content": "also to the holomorphic functions defined on complex submanifolds ", "type": "text"}, {"bbox": [460, 489, 468, 498], "score": 0.89, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [468, 488, 484, 501], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [484, 489, 496, 498], "score": 0.88, "content": "X", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [496, 488, 500, 501], "score": 1.0, "content": ";", "type": "text"}], "index": 22}, {"bbox": [109, 503, 501, 516], "spans": [{"bbox": [109, 503, 501, 516], "score": 1.0, "content": "in particular we have an action of the hyperkaehler rotation on the structure", "type": "text"}], "index": 23}, {"bbox": [109, 516, 500, 531], "spans": [{"bbox": [109, 516, 140, 531], "score": 1.0, "content": "sheaf ", "type": "text"}, {"bbox": [140, 518, 155, 529], "score": 0.92, "content": "O_{S}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [156, 516, 307, 531], "score": 1.0, "content": " (here, as always, we identify ", "type": "text"}, {"bbox": [307, 518, 323, 529], "score": 0.92, "content": "O_{S}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [323, 516, 436, 531], "score": 1.0, "content": " with its direct image ", "type": "text"}, {"bbox": [436, 518, 462, 529], "score": 0.94, "content": "j_{}O_{S}", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [462, 516, 500, 531], "score": 1.0, "content": ", where", "type": "text"}], "index": 24}, {"bbox": [110, 531, 500, 545], "spans": [{"bbox": [110, 533, 173, 544], "score": 0.92, "content": "j\\,:\\,S\\,\\rightarrow\\,X", "type": "inline_equation", "height": 11, "width": 63}, {"bbox": [173, 531, 500, 545], "score": 1.0, "content": " is the holomorphic embedding). We are thus led to give the", "type": "text"}], "index": 25}, {"bbox": [109, 545, 160, 560], "spans": [{"bbox": [109, 545, 160, 560], "score": 1.0, "content": "following:", "type": "text"}], "index": 26}], "index": 23.5}, {"type": "text", "bbox": [110, 558, 501, 631], "lines": [{"bbox": [127, 559, 500, 574], "spans": [{"bbox": [127, 559, 500, 574], "score": 1.0, "content": "Definition 2.2: Let \u039b be a special Lagrangian submanifold of a hyper-", "type": "text"}], "index": 27}, {"bbox": [109, 574, 501, 589], "spans": [{"bbox": [109, 574, 184, 589], "score": 1.0, "content": "kaehler 4-fold ", "type": "text"}, {"bbox": [185, 577, 195, 585], "score": 0.82, "content": "X", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [196, 574, 335, 589], "score": 1.0, "content": " (in the complex structure ", "type": "text"}, {"bbox": [335, 576, 346, 585], "score": 0.83, "content": "K", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [347, 574, 501, 589], "score": 1.0, "content": "). Then we define the special", "type": "text"}], "index": 28}, {"bbox": [111, 589, 499, 604], "spans": [{"bbox": [111, 589, 249, 604], "score": 1.0, "content": "Lagrangian structure sheaf ", "type": "text"}, {"bbox": [249, 591, 264, 601], "score": 0.88, "content": "{\\mathcal{L}}_{\\Lambda}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [265, 589, 499, 604], "score": 1.0, "content": " as the sheaf obtained by the action of the hy-", "type": "text"}], "index": 29}, {"bbox": [110, 604, 500, 617], "spans": [{"bbox": [110, 604, 324, 617], "score": 1.0, "content": "perkaehler rotation on the structure sheaf ", "type": "text"}, {"bbox": [324, 605, 340, 616], "score": 0.86, "content": "O_{\\Lambda}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [340, 604, 357, 617], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [357, 605, 366, 614], "score": 0.61, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [366, 604, 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259], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [201, 248, 416, 263], "score": 1.0, "content": " be a special Lagrangian submanifold of ", "type": "text"}, {"bbox": [416, 251, 428, 260], "score": 0.91, "content": "X", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [428, 248, 500, 263], "score": 1.0, "content": ", having fixed", "type": "text"}], "index": 8}, {"bbox": [109, 264, 500, 278], "spans": [{"bbox": [109, 264, 256, 278], "score": 1.0, "content": "some complex structure on ", "type": "text"}, {"bbox": [256, 265, 267, 274], "score": 0.9, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [268, 264, 331, 278], "score": 1.0, "content": ", let us say ", "type": "text"}, {"bbox": [331, 265, 342, 274], "score": 0.89, "content": "K", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [343, 264, 500, 278], "score": 1.0, "content": "; then, by Corollary 2.1 there", "type": "text"}], "index": 9}, {"bbox": [110, 279, 499, 291], "spans": [{"bbox": [110, 279, 325, 291], "score": 1.0, "content": "exists a new complex structure, let us say ", "type": "text"}, {"bbox": [325, 280, 331, 289], "score": 0.89, "content": "I", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [332, 279, 384, 291], "score": 1.0, "content": ", in which ", "type": "text"}, {"bbox": [384, 280, 393, 289], "score": 0.91, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [393, 279, 499, 291], "score": 1.0, "content": " is holomorphic, that", "type": "text"}], "index": 10}, {"bbox": [110, 294, 232, 306], "spans": [{"bbox": [110, 294, 232, 306], "score": 1.0, "content": "is, it is locally given by:", "type": "text"}], "index": 11}], "index": 9.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [109, 248, 500, 306]}, {"type": "interline_equation", "bbox": [194, 319, 415, 333], "lines": [{"bbox": [194, 319, 415, 333], "spans": [{"bbox": [194, 319, 415, 333], "score": 0.88, "content": "f_{1}(z_{1},\\ldots,z_{4})=0\\quad\\mathrm{and}\\quad f_{2}(z_{1},\\ldots,z_{4})=0.", "type": "interline_equation"}], "index": 12}], "index": 12, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [109, 342, 500, 398], "lines": [{"bbox": [109, 344, 499, 359], "spans": [{"bbox": [109, 344, 448, 359], "score": 1.0, "content": "Now observe that coming back to the original complex structure ", "type": "text"}, {"bbox": [449, 347, 460, 356], "score": 0.89, "content": "K", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [460, 344, 499, 359], "score": 1.0, "content": ", we in-", "type": "text"}], "index": 13}, {"bbox": [109, 359, 499, 374], "spans": [{"bbox": [109, 359, 489, 374], "score": 1.0, "content": "duce an analytic change of coordinates from the holomorphic coordinates ", "type": "text"}, {"bbox": [489, 360, 499, 370], "score": 0.91, "content": "z^{i}", "type": "inline_equation", "height": 10, "width": 10}], "index": 14}, {"bbox": [113, 369, 502, 395], "spans": [{"bbox": [113, 374, 175, 389], "score": 0.9, "content": "\\begin{array}{r}{\\int\\!\\frac{\\partial}{\\partial z^{i}}\\,=\\,i\\frac{\\partial}{\\partial z^{i}}\\rangle}\\end{array}", "type": "inline_equation", "height": 15, "width": 62}, {"bbox": [176, 369, 352, 395], "score": 1.0, "content": " to new holomorphic coordinates ", "type": "text"}, {"bbox": [352, 375, 365, 385], "score": 0.9, "content": "w^{i}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [365, 369, 373, 395], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [374, 374, 444, 389], "score": 0.92, "content": "\\begin{array}{r}{K\\frac{\\partial}{\\partial w^{i}}\\,=\\,i\\frac{\\partial}{\\partial w^{i}}\\Big)}\\end{array}", "type": "inline_equation", "height": 15, "width": 70}, {"bbox": [421, 372, 502, 391], "score": 1.0, "content": "i \u2202\u2202wi ) such that", "type": "text"}], "index": 15}, {"bbox": [109, 389, 148, 402], "spans": [{"bbox": [109, 389, 148, 402], "score": 1.0, "content": "locally:", "type": "text"}], "index": 16}], "index": 14.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [109, 344, 502, 402]}, {"type": "interline_equation", "bbox": [212, 401, 396, 416], "lines": [{"bbox": [212, 401, 396, 416], "spans": [{"bbox": [212, 401, 396, 416], "score": 0.92, "content": "z^{i}=c_{1}w^{i}+c_{2}\\bar{w}^{i}\\;\\;\\;\\;\\bar{z}^{i}=d_{1}w^{i}+d_{2}\\bar{w}^{i},", "type": "interline_equation"}], "index": 17}], "index": 17, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [109, 421, 501, 465], "lines": [{"bbox": [109, 424, 501, 439], "spans": [{"bbox": [109, 424, 261, 439], "score": 1.0, "content": "for some complex constants ", "type": "text"}, {"bbox": [261, 426, 286, 438], "score": 0.94, "content": "c_{j},d_{j}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [287, 424, 466, 439], "score": 1.0, "content": ". Thus in the complex structure ", "type": "text"}, {"bbox": [466, 426, 477, 434], "score": 0.88, "content": "K", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [478, 424, 501, 439], "score": 1.0, "content": " the", "type": "text"}], "index": 18}, {"bbox": [109, 439, 499, 453], "spans": [{"bbox": [109, 439, 273, 453], "score": 1.0, "content": "special Lagrangian submanifold ", "type": "text"}, {"bbox": [273, 440, 281, 449], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [282, 439, 340, 453], "score": 1.0, "content": " is given by ", "type": "text"}, {"bbox": [341, 439, 499, 452], "score": 0.92, "content": "f_{j}(c_{1}w^{i}\\!+\\!c_{2}\\bar{w}^{i},d_{1}w^{i}\\!+\\!d_{2}\\bar{w}^{i})=0", "type": "inline_equation", "height": 13, "width": 158}], "index": 19}, {"bbox": [110, 453, 501, 467], "spans": [{"bbox": [110, 453, 424, 467], "score": 1.0, "content": "which is again the zero locus of a set of functions analytic in ", "type": "text"}, {"bbox": [424, 453, 454, 466], "score": 0.81, "content": "w^{i},\\bar{w}^{i}", "type": "inline_equation", "height": 13, "width": 30}, {"bbox": [454, 453, 458, 467], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [492, 456, 501, 465], "score": 0.9820061922073364, "content": "\u53e3", "type": "text"}], "index": 20}], "index": 19, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [109, 424, 501, 467]}, {"type": "text", "bbox": [109, 471, 500, 557], "lines": [{"bbox": [128, 474, 500, 487], "spans": [{"bbox": [128, 474, 500, 487], "score": 1.0, "content": "Quite naturally, the action of the hyperkaehler rotation can be extended", "type": "text"}], "index": 21}, {"bbox": [109, 488, 500, 501], "spans": [{"bbox": [109, 488, 459, 501], "score": 1.0, "content": "also to the holomorphic functions defined on complex submanifolds ", "type": "text"}, {"bbox": [460, 489, 468, 498], "score": 0.89, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [468, 488, 484, 501], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [484, 489, 496, 498], "score": 0.88, "content": "X", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [496, 488, 500, 501], "score": 1.0, "content": ";", "type": "text"}], "index": 22}, {"bbox": [109, 503, 501, 516], "spans": [{"bbox": [109, 503, 501, 516], "score": 1.0, "content": "in particular we have an action of the hyperkaehler rotation on the structure", "type": "text"}], "index": 23}, {"bbox": [109, 516, 500, 531], "spans": [{"bbox": [109, 516, 140, 531], "score": 1.0, "content": "sheaf ", "type": "text"}, {"bbox": [140, 518, 155, 529], "score": 0.92, "content": "O_{S}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [156, 516, 307, 531], "score": 1.0, "content": " (here, as always, we identify ", "type": "text"}, {"bbox": [307, 518, 323, 529], "score": 0.92, "content": "O_{S}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [323, 516, 436, 531], "score": 1.0, "content": " with its direct image ", "type": "text"}, {"bbox": [436, 518, 462, 529], "score": 0.94, "content": "j_{}O_{S}", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [462, 516, 500, 531], "score": 1.0, "content": ", where", "type": "text"}], "index": 24}, {"bbox": [110, 531, 500, 545], "spans": [{"bbox": [110, 533, 173, 544], "score": 0.92, "content": "j\\,:\\,S\\,\\rightarrow\\,X", "type": "inline_equation", "height": 11, "width": 63}, {"bbox": [173, 531, 500, 545], "score": 1.0, "content": " is the holomorphic embedding). We are thus led to give the", "type": "text"}], "index": 25}, {"bbox": [109, 545, 160, 560], "spans": [{"bbox": [109, 545, 160, 560], "score": 1.0, "content": "following:", "type": "text"}], "index": 26}], "index": 23.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [109, 474, 501, 560]}, {"type": "text", "bbox": [110, 558, 501, 631], "lines": [{"bbox": [127, 559, 500, 574], "spans": [{"bbox": [127, 559, 500, 574], "score": 1.0, "content": "Definition 2.2: Let \u039b be a special Lagrangian submanifold of a hyper-", "type": "text"}], "index": 27}, {"bbox": [109, 574, 501, 589], "spans": [{"bbox": [109, 574, 184, 589], "score": 1.0, "content": "kaehler 4-fold ", "type": "text"}, {"bbox": [185, 577, 195, 585], "score": 0.82, "content": "X", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [196, 574, 335, 589], "score": 1.0, "content": " (in the complex structure ", "type": "text"}, {"bbox": [335, 576, 346, 585], "score": 0.83, "content": "K", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [347, 574, 501, 589], "score": 1.0, "content": "). Then we define the special", "type": "text"}], "index": 28}, {"bbox": [111, 589, 499, 604], "spans": [{"bbox": [111, 589, 249, 604], "score": 1.0, "content": "Lagrangian structure sheaf ", "type": "text"}, {"bbox": [249, 591, 264, 601], "score": 0.88, "content": "{\\mathcal{L}}_{\\Lambda}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [265, 589, 499, 604], "score": 1.0, "content": " as the sheaf obtained by the action of the hy-", "type": "text"}], "index": 29}, {"bbox": [110, 604, 500, 617], "spans": [{"bbox": [110, 604, 324, 617], "score": 1.0, "content": "perkaehler rotation on the structure sheaf ", "type": "text"}, {"bbox": [324, 605, 340, 616], "score": 0.86, "content": "O_{\\Lambda}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [340, 604, 357, 617], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [357, 605, 366, 614], "score": 0.61, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [366, 604, 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1.0, "text": ""}, {"category_id": 15, "poly": [837.0, 1924.0, 859.0, 1924.0, 859.0, 1957.0, 837.0, 1957.0], "score": 1.0, "text": ""}], "page_info": {"page_no": 5, "height": 2200, "width": 1700}}	, in the complex structure , that is performing a hyperkaehler rotation. Notice that in the complex structure , is still a Lagrangian submanifold with respect to and , so it is Lagrangian with respect to the holomorphic (in the structure ) 2-form . 口 Collecting the results so far proved, we can show that special Lagrangian submanifolds of are particularly rigid: Proposition 2.1: Any (connected, compact and without border) special Lagrangian submanifold Λ of a hyperkaehler 4-fold is real analytic. Proof: Let be a special Lagrangian submanifold of , having fixed some complex structure on , let us say ; then, by Corollary 2.1 there exists a new complex structure, let us say , in which is holomorphic, that is, it is locally given by: $$ f_{1}(z_{1},\ldots,z_{4})=0\quad\mathrm{and}\quad f_{2}(z_{1},\ldots,z_{4})=0. $$ Now observe that coming back to the original complex structure , we in- duce an analytic change of coordinates from the holomorphic coordinates to new holomorphic coordinates ( i ∂∂wi ) such that locally: $$ z^{i}=c_{1}w^{i}+c_{2}\bar{w}^{i}\;\;\;\;\bar{z}^{i}=d_{1}w^{i}+d_{2}\bar{w}^{i}, $$ for some complex constants . Thus in the complex structure the special Lagrangian submanifold is given by which is again the zero locus of a set of functions analytic in . 口 Quite naturally, the action of the hyperkaehler rotation can be extended also to the holomorphic functions defined on complex submanifolds of ; in particular we have an action of the hyperkaehler rotation on the structure sheaf (here, as always, we identify with its direct image , where is the holomorphic embedding). We are thus led to give the following: Definition 2.2: Let Λ be a special Lagrangian submanifold of a hyper- kaehler 4-fold (in the complex structure ). Then we define the special Lagrangian structure sheaf as the sheaf obtained by the action of the hy- perkaehler rotation on the structure sheaf of , as a complex Lagrangian submanifold of , (in the structure ). 6	<div class="pdf-page"> <p>, in the complex structure , that is performing a hyperkaehler rotation. Notice that in the complex structure , is still a Lagrangian submanifold with respect to and , so it is Lagrangian with respect to the holomorphic (in the structure ) 2-form . 口</p> <p>Collecting the results so far proved, we can show that special Lagrangian submanifolds of are particularly rigid:</p> <p>Proposition 2.1: Any (connected, compact and without border) special Lagrangian submanifold Λ of a hyperkaehler 4-fold is real analytic.</p> <p>Proof: Let be a special Lagrangian submanifold of , having fixed some complex structure on , let us say ; then, by Corollary 2.1 there exists a new complex structure, let us say , in which is holomorphic, that is, it is locally given by:</p> <p>Now observe that coming back to the original complex structure , we in- duce an analytic change of coordinates from the holomorphic coordinates to new holomorphic coordinates ( i ∂∂wi ) such that locally:</p> <p>for some complex constants . Thus in the complex structure the special Lagrangian submanifold is given by which is again the zero locus of a set of functions analytic in . 口</p> <p>Quite naturally, the action of the hyperkaehler rotation can be extended also to the holomorphic functions defined on complex submanifolds of ; in particular we have an action of the hyperkaehler rotation on the structure sheaf (here, as always, we identify with its direct image , where is the holomorphic embedding). We are thus led to give the following:</p> <p>Definition 2.2: Let Λ be a special Lagrangian submanifold of a hyper- kaehler 4-fold (in the complex structure ). Then we define the special Lagrangian structure sheaf as the sheaf obtained by the action of the hy- perkaehler rotation on the structure sheaf of , as a complex Lagrangian submanifold of , (in the structure ).</p> </div>	<div class="pdf-page"> <p class="pdf-text" data-x="184" data-y="161" data-width="654" data-height="75">, in the complex structure , that is performing a hyperkaehler rotation. Notice that in the complex structure , is still a Lagrangian submanifold with respect to and , so it is Lagrangian with respect to the holomorphic (in the structure ) 2-form . 口</p> <p class="pdf-text" data-x="182" data-y="244" data-width="654" data-height="37">Collecting the results so far proved, we can show that special Lagrangian submanifolds of are particularly rigid:</p> <p class="pdf-text" data-x="184" data-y="283" data-width="655" data-height="35">Proposition 2.1: Any (connected, compact and without border) special Lagrangian submanifold Λ of a hyperkaehler 4-fold is real analytic.</p> <p class="pdf-text" data-x="182" data-y="319" data-width="654" data-height="75">Proof: Let be a special Lagrangian submanifold of , having fixed some complex structure on , let us say ; then, by Corollary 2.1 there exists a new complex structure, let us say , in which is holomorphic, that is, it is locally given by:</p> <p class="pdf-text" data-x="182" data-y="442" data-width="654" data-height="72">Now observe that coming back to the original complex structure , we in- duce an analytic change of coordinates from the holomorphic coordinates to new holomorphic coordinates ( i ∂∂wi ) such that locally:</p> <p class="pdf-text" data-x="182" data-y="544" data-width="656" data-height="57">for some complex constants . Thus in the complex structure the special Lagrangian submanifold is given by which is again the zero locus of a set of functions analytic in . 口</p> <p class="pdf-text" data-x="182" data-y="608" data-width="654" data-height="112">Quite naturally, the action of the hyperkaehler rotation can be extended also to the holomorphic functions defined on complex submanifolds of ; in particular we have an action of the hyperkaehler rotation on the structure sheaf (here, as always, we identify with its direct image , where is the holomorphic embedding). We are thus led to give the following:</p> <p class="pdf-text" data-x="184" data-y="721" data-width="654" data-height="94">Definition 2.2: Let Λ be a special Lagrangian submanifold of a hyper- kaehler 4-fold (in the complex structure ). Then we define the special Lagrangian structure sheaf as the sheaf obtained by the action of the hy- perkaehler rotation on the structure sheaf of , as a complex Lagrangian submanifold of , (in the structure ).</p> <div class="pdf-discarded" data-x="501" data-y="893" data-width="16" data-height="14" style="opacity: 0.5;">6</div> </div>	{ "type": [ "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "inline_equation", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "interline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "interline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "inline_equation", "text", "inline_equation", "inline_equation", "text", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation" ], "coordinates": [ [ 184, 165, 834, 183 ], [ 182, 183, 836, 201 ], [ 182, 202, 836, 221 ], [ 184, 219, 838, 240 ], [ 212, 245, 836, 267 ], [ 184, 266, 532, 283 ], [ 210, 283, 838, 306 ], [ 185, 305, 779, 323 ], [ 210, 320, 836, 340 ], [ 182, 341, 836, 359 ], [ 184, 360, 834, 376 ], [ 184, 380, 388, 395 ], [ 324, 412, 694, 430 ], [ 182, 444, 834, 464 ], [ 182, 464, 834, 483 ], [ 189, 477, 839, 510 ], [ 182, 502, 247, 519 ], [ 354, 518, 662, 537 ], [ 182, 548, 838, 567 ], [ 182, 567, 834, 585 ], [ 184, 585, 838, 603 ], [ 214, 612, 836, 629 ], [ 182, 630, 836, 647 ], [ 182, 650, 838, 667 ], [ 182, 667, 836, 686 ], [ 184, 686, 836, 704 ], [ 182, 704, 267, 724 ], [ 212, 722, 836, 742 ], [ 182, 742, 838, 761 ], [ 185, 761, 834, 780 ], [ 184, 780, 836, 797 ], [ 184, 799, 518, 817 ] ], "content": [ "X , in the complex structure I , that is performing a hyperkaehler rotation.", "Notice that in the complex structure I , \\Lambda is still a Lagrangian submanifold", "with respect to \\omega_{K} and \\omega_{I} , so it is Lagrangian with respect to the holomorphic", "(in the structure I ) 2-form \\Omega_{I}:=\\omega_{J}+i\\omega_{K} . 口", "Collecting the results so far proved, we can show that special Lagrangian", "submanifolds of X are particularly rigid:", "Proposition 2.1: Any (connected, compact and without border) special", "Lagrangian submanifold Λ of a hyperkaehler 4-fold X is real analytic.", "Proof: Let \\Lambda be a special Lagrangian submanifold of X , having fixed", "some complex structure on X , let us say K ; then, by Corollary 2.1 there", "exists a new complex structure, let us say I , in which \\Lambda is holomorphic, that", "is, it is locally given by:", "f_{1}(z_{1},\\ldots,z_{4})=0\\quad\\mathrm{and}\\quad f_{2}(z_{1},\\ldots,z_{4})=0.", "Now observe that coming back to the original complex structure K , we in-", "duce an analytic change of coordinates from the holomorphic coordinates z^{i}", "\\begin{array}{r}{\\int\\!\\frac{\\partial}{\\partial z^{i}}\\,=\\,i\\frac{\\partial}{\\partial z^{i}}\\rangle}\\end{array} to new holomorphic coordinates w^{i} ( \\begin{array}{r}{K\\frac{\\partial}{\\partial w^{i}}\\,=\\,i\\frac{\\partial}{\\partial w^{i}}\\Big)}\\end{array} i ∂∂wi ) such that", "locally:", "z^{i}=c_{1}w^{i}+c_{2}\\bar{w}^{i}\\;\\;\\;\\;\\bar{z}^{i}=d_{1}w^{i}+d_{2}\\bar{w}^{i},", "for some complex constants c_{j},d_{j} . Thus in the complex structure K the", "special Lagrangian submanifold \\Lambda is given by f_{j}(c_{1}w^{i}\\!+\\!c_{2}\\bar{w}^{i},d_{1}w^{i}\\!+\\!d_{2}\\bar{w}^{i})=0", "which is again the zero locus of a set of functions analytic in w^{i},\\bar{w}^{i} . 口", "Quite naturally, the action of the hyperkaehler rotation can be extended", "also to the holomorphic functions defined on complex submanifolds S of X ;", "in particular we have an action of the hyperkaehler rotation on the structure", "sheaf O_{S} (here, as always, we identify O_{S} with its direct image j_{*}O_{S} , where", "j\\,:\\,S\\,\\rightarrow\\,X is the holomorphic embedding). We are thus led to give the", "following:", "Definition 2.2: Let Λ be a special Lagrangian submanifold of a hyper-", "kaehler 4-fold X (in the complex structure K ). Then we define the special", "Lagrangian structure sheaf {\\mathcal{L}}_{\\Lambda} as the sheaf obtained by the action of the hy-", "perkaehler rotation on the structure sheaf O_{\\Lambda} of \\Lambda , as a complex Lagrangian", "submanifold of X , (in the structure I )." ], "index": [ 0, 1, 2, 3, 4, 5, 10, 11, 18, 19, 20, 21, 30, 43, 44, 45, 46, 60, 78, 79, 80, 99, 100, 101, 102, 103, 104, 126, 127, 128, 129, 130 ] }	{ "coordinates": [], "index": [], "caption": [], "caption_coordinates": [] }	{ "type": [ "interline_equation", "interline_equation", "interline_equation", "interline_equation" ], "coordinates": [ [ 324, 412, 694, 430 ], [ 354, 518, 662, 537 ], [ 324, 412, 694, 430 ], [ 354, 518, 662, 537 ] ], "content": [ "", "", "f_{1}(z_{1},\\ldots,z_{4})=0\\quad\\mathrm{and}\\quad f_{2}(z_{1},\\ldots,z_{4})=0.", "z^{i}=c_{1}w^{i}+c_{2}\\bar{w}^{i}\\;\\;\\;\\;\\bar{z}^{i}=d_{1}w^{i}+d_{2}\\bar{w}^{i}," ], "index": [ 0, 1, 2, 3 ] }	{ "type": [], "coordinates": [], "content": [], "index": [] }	{"document_info": {"total_pages": 9, "parse_type": "txt", "version": "1.0.1", "total_content_list_items": 72}, "content_statistics": {"total_blocks": 81, "total_lines": 269, "total_images": 0, "total_equations": 18, "total_tables": 0, "total_characters": 16112, "total_words": 2578, "block_type_distribution": {"title": 6, "text": 55, "discarded": 9, "interline_equation": 9, "list": 2}}, "model_statistics": {"total_layout_detections": 813, "avg_confidence": 0.9488487084870848, "min_confidence": 0.26, "max_confidence": 1.0}, "page_statistics": [{"page_number": 0, "page_size": [612.0, 792.0], "blocks": 10, "lines": 26, "images": 0, "equations": 0, "tables": 0, "characters": 1419, "words": 204, "discarded_blocks": 2}, {"page_number": 1, "page_size": [612.0, 792.0], "blocks": 15, "lines": 37, "images": 0, "equations": 0, "tables": 0, "characters": 2328, "words": 359, "discarded_blocks": 1}, {"page_number": 2, "page_size": [612.0, 792.0], "blocks": 12, "lines": 35, "images": 0, "equations": 4, "tables": 0, "characters": 1843, "words": 306, "discarded_blocks": 1}, {"page_number": 3, "page_size": [612.0, 792.0], "blocks": 8, "lines": 30, "images": 0, "equations": 6, "tables": 0, "characters": 1797, "words": 312, "discarded_blocks": 0}, {"page_number": 4, "page_size": [612.0, 792.0], "blocks": 12, "lines": 34, "images": 0, "equations": 4, "tables": 0, "characters": 1905, "words": 328, "discarded_blocks": 1}, {"page_number": 5, "page_size": [612.0, 792.0], "blocks": 11, "lines": 32, "images": 0, "equations": 4, "tables": 0, "characters": 1846, "words": 308, "discarded_blocks": 1}, {"page_number": 6, "page_size": [612.0, 792.0], "blocks": 6, "lines": 46, "images": 0, "equations": 0, "tables": 0, "characters": 3307, "words": 521, "discarded_blocks": 1}, {"page_number": 7, "page_size": [612.0, 792.0], "blocks": 5, "lines": 29, "images": 0, "equations": 0, "tables": 0, "characters": 1666, "words": 239, "discarded_blocks": 1}, {"page_number": 8, "page_size": [612.0, 792.0], "blocks": 2, "lines": 0, "images": 0, "equations": 0, "tables": 0, "characters": 1, "words": 1, "discarded_blocks": 1}]}
0001060v1	6	[ 612, 792 ]	{ "type": [ "title", "text", "text", "text", "text", "discarded" ], "coordinates": [ [ 182, 156, 520, 182 ], [ 182, 195, 836, 250 ], [ 182, 253, 836, 418 ], [ 182, 420, 836, 849 ], [ 210, 850, 836, 867 ], [ 501, 893, 517, 907 ] ], "content": [ "3 Concluding remarks", "It is important to remark that all previous results are true also for special Lagrangian submanifolds of K3 surfaces, but their proof is completely trivial in that case.", "Another observation is related to singular Lagrangian submanifolds: in- deed, by the previous results, it turns out that we can also give examples of special Lagrangian subvarieties, obtained via hyperkaehler rotation of La- grangian complex subvarieties. On the other hand, contrary to the case of the corresponding submanifolds, we can not expect that all special Lagrangian subvarieties are obtained in this way, and consequently we can not expect that all special Lagrangian subvarieties are real analytic. Indeed, there are examples (compare [4]) of singular special Lagrangian submanifold in which are only smooth, but not real analytic.", "The discussion about singular Lagrangian submanifolds leads us to com- ment on the mirror symmetry construction suggested in [11]. Indeed, ac- cording to the recipe of [11], any Calabi-Yau , admitting a mirror , has a peculiar fibre space structure: on a physical ground it is argued that can be realized as the total space of a fibration in special Lagrangian tori. Unfortunately, there are very few examples of such realization: in particular, as far as we know, there is only one (partial) example for Calabi-Yau 3-folds of the so called Borcea-Voisin type (see [3]). Instead, in the case of irre- ducible symplectic projective manifolds the situation is completely different. Indeed, a recent result of Matsushita (see [8] and [9]) shows that for any fibre space structure of a projective irreducible symplectic manifold , with projective base , the generic fibre is an Abelian variety (up to finite unramified cover), and it is also Lagrangian with respect to the non degenerate holomorphic 2-form ; moreover, in the case of 4-folds one can prove that the generic fibre is an Abelian surface and is equidimensional, (i.e. all irreducible components of the fibres have the same dimension). By Corollary 2.1 it turns out that this fibre space structure can also be realized as a special Lagrangian torus fibration; moreover, in this case all special La- grangian fibres, even the singular ones, are analytic, since they are obtained by performing a hyperkaehler rotation starting from Lagrangian Abelian sur- faces. So, in these cases, we have special Lagrangian torus fibration in which all fibres are analytic: one can hope to understand the degeneration types of singular special Lagrangian tori, moving from these constructions.", "Explicit examples of projective irreducible symplectic 4-folds, fibered over a projective base have been constructed by Markuschevich in [6] and [7]. One of this constructions is the following: consider a double cover of the projective plane, ramified along a smooth sextic ( is then realized as a K3 surface). Since any line in will intersect generically the sextic in six distinct point, we have that the covering deter- mines a (flat) family of hyperelliptic curves over the dual projective plane . Then the Altmann-Kleiman compactification of the relative Ja- cobian of the family turns out to be a simplectic projective irreducible 4-folds, fibered over , and in fact all fibres are Lagrangian Abelian varieties.", "7" ], "index": [ 0, 1, 2, 3, 4, 5 ] }	[{"type": "text", "text": "3 Concluding remarks ", "text_level": 1, "page_idx": 6}, {"type": "text", "text": "It is important to remark that all previous results are true also for special Lagrangian submanifolds of K3 surfaces, but their proof is completely trivial in that case. ", "page_idx": 6}, {"type": "text", "text": "Another observation is related to singular Lagrangian submanifolds: indeed, by the previous results, it turns out that we can also give examples of special Lagrangian subvarieties, obtained via hyperkaehler rotation of Lagrangian complex subvarieties. On the other hand, contrary to the case of the corresponding submanifolds, we can not expect that all special Lagrangian subvarieties are obtained in this way, and consequently we can not expect that all special Lagrangian subvarieties are real analytic. Indeed, there are examples (compare [4]) of singular special Lagrangian submanifold in $C^{n}$ which are only smooth, but not real analytic. ", "page_idx": 6}, {"type": "text", "text": "The discussion about singular Lagrangian submanifolds leads us to comment on the mirror symmetry construction suggested in [11]. Indeed, according to the recipe of [11], any Calabi-Yau $X$ , admitting a mirror $\\hat{X}$ , has a peculiar fibre space structure: on a physical ground it is argued that $X$ can be realized as the total space of a fibration in special Lagrangian tori. Unfortunately, there are very few examples of such realization: in particular, as far as we know, there is only one (partial) example for Calabi-Yau 3-folds of the so called Borcea-Voisin type (see [3]). Instead, in the case of irreducible symplectic projective manifolds the situation is completely different. Indeed, a recent result of Matsushita (see [8] and [9]) shows that for any fibre space structure $f:X\\to B$ of a projective irreducible symplectic manifold $X$ , with projective base $B$ , the generic fibre $f^{-1}(b)$ is an Abelian variety (up to finite unramified cover), and it is also Lagrangian with respect to the non degenerate holomorphic 2-form $\\Omega$ ; moreover, in the case of 4-folds one can prove that the generic fibre is an Abelian surface and $f$ is equidimensional, (i.e. all irreducible components of the fibres have the same dimension). By Corollary 2.1 it turns out that this fibre space structure can also be realized as a special Lagrangian torus fibration; moreover, in this case all special Lagrangian fibres, even the singular ones, are analytic, since they are obtained by performing a hyperkaehler rotation starting from Lagrangian Abelian surfaces. So, in these cases, we have special Lagrangian torus fibration in which all fibres are analytic: one can hope to understand the degeneration types of singular special Lagrangian tori, moving from these constructions. ", "page_idx": 6}, {"type": "text", "text": "Explicit examples of projective irreducible symplectic 4-folds, fibered over a projective base have been constructed by Markuschevich in [6] and [7]. One of this constructions is the following: consider a double cover $\\pi:S\\to P^{2}$ of the projective plane, ramified along a smooth sextic $C\\hookrightarrow P^{2}$ ( $S$ is then realized as a K3 surface). Since any line in $P^{2}$ will intersect generically the sextic $C$ in six distinct point, we have that the covering $\\pi:S\\to P^{2}$ determines a (flat) family of hyperelliptic curves over the dual projective plane $f:\\mathcal{X}\\rightarrow P^{2}$ . Then the Altmann-Kleiman compactification of the relative Jacobian of the family turns out to be a simplectic projective irreducible 4-folds, fibered over $P^{2}$ , and in fact all fibres are Lagrangian Abelian varieties. ", "page_idx": 6}]	{"preproc_blocks": [{"type": "title", "bbox": [109, 121, 311, 141], "lines": [{"bbox": [110, 123, 311, 142], "spans": [{"bbox": [110, 126, 121, 138], "score": 1.0, "content": "3", "type": "text"}, {"bbox": [138, 123, 311, 142], "score": 1.0, "content": "Concluding remarks", "type": "text"}], "index": 0}], "index": 0}, {"type": "text", "bbox": [109, 151, 500, 194], "lines": [{"bbox": [109, 154, 500, 168], "spans": [{"bbox": [109, 154, 500, 168], "score": 1.0, "content": "It is important to remark that all previous results are true also for special", "type": "text"}], "index": 1}, {"bbox": [110, 168, 499, 182], "spans": [{"bbox": [110, 168, 499, 182], "score": 1.0, "content": "Lagrangian submanifolds of K3 surfaces, but their proof is completely trivial", "type": "text"}], "index": 2}, {"bbox": [109, 183, 175, 196], "spans": [{"bbox": [109, 183, 175, 196], "score": 1.0, "content": "in that case.", "type": "text"}], "index": 3}], "index": 2}, {"type": "text", "bbox": [109, 196, 500, 324], "lines": [{"bbox": [127, 198, 498, 210], "spans": [{"bbox": [127, 198, 498, 210], "score": 1.0, "content": "Another observation is related to singular Lagrangian submanifolds: in-", "type": "text"}], "index": 4}, {"bbox": [110, 212, 499, 226], "spans": [{"bbox": [110, 212, 499, 226], "score": 1.0, "content": "deed, by the previous results, it turns out that we can also give examples", "type": "text"}], "index": 5}, {"bbox": [110, 227, 499, 240], "spans": [{"bbox": [110, 227, 499, 240], "score": 1.0, "content": "of special Lagrangian subvarieties, obtained via hyperkaehler rotation of La-", "type": "text"}], "index": 6}, {"bbox": [109, 241, 500, 254], "spans": [{"bbox": [109, 241, 500, 254], "score": 1.0, "content": "grangian complex subvarieties. 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Indeed, there are", "type": "text"}], "index": 10}, {"bbox": [109, 298, 498, 312], "spans": [{"bbox": [109, 298, 483, 312], "score": 1.0, "content": "examples (compare [4]) of singular special Lagrangian submanifold in ", "type": "text"}, {"bbox": [484, 300, 498, 309], "score": 0.91, "content": "C^{n}", "type": "inline_equation", "height": 9, "width": 14}], "index": 11}, {"bbox": [110, 312, 342, 326], "spans": [{"bbox": [110, 312, 342, 326], "score": 1.0, "content": "which are only smooth, but not real analytic.", "type": "text"}], "index": 12}], "index": 8}, {"type": "text", "bbox": [109, 325, 500, 657], "lines": [{"bbox": [127, 327, 500, 341], "spans": [{"bbox": [127, 327, 500, 341], "score": 1.0, "content": "The discussion about singular Lagrangian submanifolds leads us to com-", "type": "text"}], "index": 13}, {"bbox": [109, 342, 501, 357], "spans": [{"bbox": [109, 342, 501, 357], "score": 1.0, "content": "ment on the mirror symmetry construction suggested in [11]. Indeed, ac-", "type": "text"}], "index": 14}, {"bbox": [110, 355, 500, 369], "spans": [{"bbox": [110, 357, 344, 369], "score": 1.0, "content": "cording to the recipe of [11], any Calabi-Yau ", "type": "text"}, {"bbox": [344, 358, 355, 366], "score": 0.91, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [356, 357, 463, 369], "score": 1.0, "content": ", admitting a mirror", "type": "text"}, {"bbox": [463, 355, 474, 366], "score": 0.91, "content": "\\hat{X}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [475, 357, 500, 369], "score": 1.0, "content": ", has", "type": "text"}], "index": 15}, {"bbox": [110, 371, 499, 385], "spans": [{"bbox": [110, 371, 487, 385], "score": 1.0, "content": "a peculiar fibre space structure: on a physical ground it is argued that ", "type": "text"}, {"bbox": [488, 372, 499, 381], "score": 0.9, "content": "X", "type": "inline_equation", "height": 9, "width": 11}], "index": 16}, {"bbox": [109, 385, 500, 399], "spans": [{"bbox": [109, 385, 500, 399], "score": 1.0, "content": "can be realized as the total space of a fibration in special Lagrangian tori.", "type": "text"}], "index": 17}, {"bbox": [110, 399, 500, 414], "spans": [{"bbox": [110, 399, 500, 414], "score": 1.0, "content": "Unfortunately, there are very few examples of such realization: in particular,", "type": "text"}], "index": 18}, {"bbox": [110, 415, 500, 427], "spans": [{"bbox": [110, 415, 500, 427], "score": 1.0, "content": "as far as we know, there is only one (partial) example for Calabi-Yau 3-folds", "type": "text"}], "index": 19}, {"bbox": [110, 428, 500, 442], "spans": [{"bbox": [110, 428, 500, 442], "score": 1.0, "content": "of the so called Borcea-Voisin type (see [3]). Instead, in the case of irre-", "type": "text"}], "index": 20}, {"bbox": [109, 442, 500, 457], "spans": [{"bbox": [109, 442, 500, 457], "score": 1.0, "content": "ducible symplectic projective manifolds the situation is completely different.", "type": "text"}], "index": 21}, {"bbox": [109, 457, 500, 471], "spans": [{"bbox": [109, 457, 500, 471], "score": 1.0, "content": "Indeed, a recent result of Matsushita (see [8] and [9]) shows that for any fibre", "type": "text"}], "index": 22}, {"bbox": [110, 472, 500, 486], "spans": [{"bbox": [110, 472, 192, 486], "score": 1.0, "content": "space structure ", "type": "text"}, {"bbox": [193, 473, 253, 484], "score": 0.93, "content": "f:X\\to B", "type": "inline_equation", "height": 11, "width": 60}, {"bbox": [254, 472, 500, 486], "score": 1.0, "content": " of a projective irreducible symplectic manifold", "type": "text"}], "index": 23}, {"bbox": [110, 486, 500, 501], "spans": [{"bbox": [110, 488, 121, 497], "score": 0.89, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [121, 486, 234, 501], "score": 1.0, "content": ", with projective base ", "type": "text"}, {"bbox": [234, 488, 244, 497], "score": 0.9, "content": "B", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [244, 486, 335, 501], "score": 1.0, "content": ", the generic fibre ", "type": "text"}, {"bbox": [336, 487, 369, 500], "score": 0.94, "content": "f^{-1}(b)", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [369, 486, 500, 501], "score": 1.0, "content": " is an Abelian variety (up", "type": "text"}], "index": 24}, {"bbox": [110, 502, 499, 514], "spans": [{"bbox": [110, 502, 499, 514], "score": 1.0, "content": "to finite unramified cover), and it is also Lagrangian with respect to the non", "type": "text"}], "index": 25}, {"bbox": [110, 515, 501, 529], "spans": [{"bbox": [110, 515, 275, 529], "score": 1.0, "content": "degenerate holomorphic 2-form ", "type": "text"}, {"bbox": [275, 517, 284, 525], "score": 0.88, "content": "\\Omega", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [284, 515, 501, 529], "score": 1.0, "content": "; moreover, in the case of 4-folds one can", "type": "text"}], "index": 26}, {"bbox": [109, 530, 500, 543], "spans": [{"bbox": [109, 530, 390, 543], "score": 1.0, "content": "prove that the generic fibre is an Abelian surface and ", "type": "text"}, {"bbox": [390, 531, 397, 542], "score": 0.91, "content": "f", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [398, 530, 500, 543], "score": 1.0, "content": " is equidimensional,", "type": "text"}], "index": 27}, {"bbox": [110, 544, 499, 558], "spans": [{"bbox": [110, 544, 499, 558], "score": 1.0, "content": "(i.e. all irreducible components of the fibres have the same dimension). By", "type": "text"}], "index": 28}, {"bbox": [109, 558, 500, 572], "spans": [{"bbox": [109, 558, 500, 572], "score": 1.0, "content": "Corollary 2.1 it turns out that this fibre space structure can also be realized", "type": "text"}], "index": 29}, {"bbox": [110, 573, 499, 587], "spans": [{"bbox": [110, 573, 499, 587], "score": 1.0, "content": "as a special Lagrangian torus fibration; moreover, in this case all special La-", "type": "text"}], "index": 30}, {"bbox": [109, 587, 500, 601], "spans": [{"bbox": [109, 587, 500, 601], "score": 1.0, "content": "grangian fibres, even the singular ones, are analytic, since they are obtained", "type": "text"}], "index": 31}, {"bbox": [110, 603, 499, 615], "spans": [{"bbox": [110, 603, 499, 615], "score": 1.0, "content": "by performing a hyperkaehler rotation starting from Lagrangian Abelian sur-", "type": "text"}], "index": 32}, {"bbox": [109, 616, 500, 630], "spans": [{"bbox": [109, 616, 500, 630], "score": 1.0, "content": "faces. So, in these cases, we have special Lagrangian torus fibration in which", "type": "text"}], "index": 33}, {"bbox": [110, 631, 501, 645], "spans": [{"bbox": [110, 631, 501, 645], "score": 1.0, "content": "all fibres are analytic: one can hope to understand the degeneration types of", "type": "text"}], "index": 34}, {"bbox": [110, 645, 450, 659], "spans": [{"bbox": [110, 645, 450, 659], "score": 1.0, "content": "singular special Lagrangian tori, moving from these constructions.", "type": "text"}], "index": 35}], "index": 24}, {"type": "text", "bbox": [126, 658, 500, 671], "lines": [{"bbox": [127, 660, 500, 673], "spans": [{"bbox": [127, 660, 500, 673], "score": 1.0, "content": "Explicit examples of projective irreducible symplectic 4-folds, fibered over", "type": "text"}], "index": 36}], "index": 36}], "layout_bboxes": [], "page_idx": 6, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [300, 691, 309, 702], "lines": [{"bbox": [300, 692, 310, 705], "spans": [{"bbox": [300, 692, 310, 705], "score": 1.0, "content": "7", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "title", "bbox": [109, 121, 311, 141], "lines": [{"bbox": [110, 123, 311, 142], "spans": [{"bbox": [110, 126, 121, 138], "score": 1.0, "content": "3", "type": "text"}, {"bbox": [138, 123, 311, 142], "score": 1.0, "content": "Concluding remarks", "type": "text"}], "index": 0}], "index": 0, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [109, 151, 500, 194], "lines": [{"bbox": [109, 154, 500, 168], "spans": [{"bbox": [109, 154, 500, 168], "score": 1.0, "content": "It is important to remark that all previous results are true also for special", "type": "text"}], "index": 1}, {"bbox": [110, 168, 499, 182], "spans": [{"bbox": [110, 168, 499, 182], "score": 1.0, "content": "Lagrangian submanifolds of K3 surfaces, but their proof is completely trivial", "type": "text"}], "index": 2}, {"bbox": [109, 183, 175, 196], "spans": [{"bbox": [109, 183, 175, 196], "score": 1.0, "content": "in that case.", "type": "text"}], "index": 3}], "index": 2, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [109, 154, 500, 196]}, {"type": "text", "bbox": [109, 196, 500, 324], "lines": [{"bbox": [127, 198, 498, 210], "spans": [{"bbox": [127, 198, 498, 210], "score": 1.0, "content": "Another observation is related to singular Lagrangian submanifolds: in-", "type": "text"}], "index": 4}, {"bbox": [110, 212, 499, 226], "spans": [{"bbox": [110, 212, 499, 226], "score": 1.0, "content": "deed, by the previous results, it turns out that we can also give examples", "type": "text"}], "index": 5}, {"bbox": [110, 227, 499, 240], "spans": [{"bbox": [110, 227, 499, 240], "score": 1.0, "content": "of special Lagrangian subvarieties, obtained via hyperkaehler rotation of La-", "type": "text"}], "index": 6}, {"bbox": [109, 241, 500, 254], "spans": [{"bbox": [109, 241, 500, 254], "score": 1.0, "content": "grangian complex subvarieties. On the other hand, contrary to the case of the", "type": "text"}], "index": 7}, {"bbox": [109, 255, 500, 270], "spans": [{"bbox": [109, 255, 500, 270], "score": 1.0, "content": "corresponding submanifolds, we can not expect that all special Lagrangian", "type": "text"}], "index": 8}, {"bbox": [109, 270, 500, 284], "spans": [{"bbox": [109, 270, 500, 284], "score": 1.0, "content": "subvarieties are obtained in this way, and consequently we can not expect", "type": "text"}], "index": 9}, {"bbox": [109, 284, 500, 298], "spans": [{"bbox": [109, 284, 500, 298], "score": 1.0, "content": "that all special Lagrangian subvarieties are real analytic. Indeed, there are", "type": "text"}], "index": 10}, {"bbox": [109, 298, 498, 312], "spans": [{"bbox": [109, 298, 483, 312], "score": 1.0, "content": "examples (compare [4]) of singular special Lagrangian submanifold in ", "type": "text"}, {"bbox": [484, 300, 498, 309], "score": 0.91, "content": "C^{n}", "type": "inline_equation", "height": 9, "width": 14}], "index": 11}, {"bbox": [110, 312, 342, 326], "spans": [{"bbox": [110, 312, 342, 326], "score": 1.0, "content": "which are only smooth, but not real analytic.", "type": "text"}], "index": 12}], "index": 8, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [109, 198, 500, 326]}, {"type": "text", "bbox": [109, 325, 500, 657], "lines": [{"bbox": [127, 327, 500, 341], "spans": [{"bbox": [127, 327, 500, 341], "score": 1.0, "content": "The discussion about singular Lagrangian submanifolds leads us to com-", "type": "text"}], "index": 13}, {"bbox": [109, 342, 501, 357], "spans": [{"bbox": [109, 342, 501, 357], "score": 1.0, "content": "ment on the mirror symmetry construction suggested in [11]. Indeed, ac-", "type": "text"}], "index": 14}, {"bbox": [110, 355, 500, 369], "spans": [{"bbox": [110, 357, 344, 369], "score": 1.0, "content": "cording to the recipe of [11], any Calabi-Yau ", "type": "text"}, {"bbox": [344, 358, 355, 366], "score": 0.91, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [356, 357, 463, 369], "score": 1.0, "content": ", admitting a mirror", "type": "text"}, {"bbox": [463, 355, 474, 366], "score": 0.91, "content": "\\hat{X}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [475, 357, 500, 369], "score": 1.0, "content": ", has", "type": "text"}], "index": 15}, {"bbox": [110, 371, 499, 385], "spans": [{"bbox": [110, 371, 487, 385], "score": 1.0, "content": "a peculiar fibre space structure: on a physical ground it is argued that ", "type": "text"}, {"bbox": [488, 372, 499, 381], "score": 0.9, "content": "X", "type": "inline_equation", "height": 9, "width": 11}], "index": 16}, {"bbox": [109, 385, 500, 399], "spans": [{"bbox": [109, 385, 500, 399], "score": 1.0, "content": "can be realized as the total space of a fibration in special Lagrangian tori.", "type": "text"}], "index": 17}, {"bbox": [110, 399, 500, 414], "spans": [{"bbox": [110, 399, 500, 414], "score": 1.0, "content": "Unfortunately, there are very few examples of such realization: in particular,", "type": "text"}], "index": 18}, {"bbox": [110, 415, 500, 427], "spans": [{"bbox": [110, 415, 500, 427], "score": 1.0, "content": "as far as we know, there is only one (partial) example for Calabi-Yau 3-folds", "type": "text"}], "index": 19}, {"bbox": [110, 428, 500, 442], "spans": [{"bbox": [110, 428, 500, 442], "score": 1.0, "content": "of the so called Borcea-Voisin type (see [3]). Instead, in the case of irre-", "type": "text"}], "index": 20}, {"bbox": [109, 442, 500, 457], "spans": [{"bbox": [109, 442, 500, 457], "score": 1.0, "content": "ducible symplectic projective manifolds the situation is completely different.", "type": "text"}], "index": 21}, {"bbox": [109, 457, 500, 471], "spans": [{"bbox": [109, 457, 500, 471], "score": 1.0, "content": "Indeed, a recent result of Matsushita (see [8] and [9]) shows that for any fibre", "type": "text"}], "index": 22}, {"bbox": [110, 472, 500, 486], "spans": [{"bbox": [110, 472, 192, 486], "score": 1.0, "content": "space structure ", "type": "text"}, {"bbox": [193, 473, 253, 484], "score": 0.93, "content": "f:X\\to B", "type": "inline_equation", "height": 11, "width": 60}, {"bbox": [254, 472, 500, 486], "score": 1.0, "content": " of a projective irreducible symplectic manifold", "type": "text"}], "index": 23}, {"bbox": [110, 486, 500, 501], "spans": [{"bbox": [110, 488, 121, 497], "score": 0.89, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [121, 486, 234, 501], "score": 1.0, "content": ", with projective base ", "type": "text"}, {"bbox": [234, 488, 244, 497], "score": 0.9, "content": "B", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [244, 486, 335, 501], "score": 1.0, "content": ", the generic fibre ", "type": "text"}, {"bbox": [336, 487, 369, 500], "score": 0.94, "content": "f^{-1}(b)", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [369, 486, 500, 501], "score": 1.0, "content": " is an Abelian variety (up", "type": "text"}], "index": 24}, {"bbox": [110, 502, 499, 514], "spans": [{"bbox": [110, 502, 499, 514], "score": 1.0, "content": "to finite unramified cover), and it is also Lagrangian with respect to the non", "type": "text"}], "index": 25}, {"bbox": [110, 515, 501, 529], "spans": [{"bbox": [110, 515, 275, 529], "score": 1.0, "content": "degenerate holomorphic 2-form ", "type": "text"}, {"bbox": [275, 517, 284, 525], "score": 0.88, "content": "\\Omega", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [284, 515, 501, 529], "score": 1.0, "content": "; moreover, in the case of 4-folds one can", "type": "text"}], "index": 26}, {"bbox": [109, 530, 500, 543], "spans": [{"bbox": [109, 530, 390, 543], "score": 1.0, "content": "prove that the generic fibre is an Abelian surface and ", "type": "text"}, {"bbox": [390, 531, 397, 542], "score": 0.91, "content": "f", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [398, 530, 500, 543], "score": 1.0, "content": " is equidimensional,", "type": "text"}], "index": 27}, {"bbox": [110, 544, 499, 558], "spans": [{"bbox": [110, 544, 499, 558], "score": 1.0, "content": "(i.e. all irreducible components of the fibres have the same dimension). By", "type": "text"}], "index": 28}, {"bbox": [109, 558, 500, 572], "spans": [{"bbox": [109, 558, 500, 572], "score": 1.0, "content": "Corollary 2.1 it turns out that this fibre space structure can also be realized", "type": "text"}], "index": 29}, {"bbox": [110, 573, 499, 587], "spans": [{"bbox": [110, 573, 499, 587], "score": 1.0, "content": "as a special Lagrangian torus fibration; moreover, in this case all special La-", "type": "text"}], "index": 30}, {"bbox": [109, 587, 500, 601], "spans": [{"bbox": [109, 587, 500, 601], "score": 1.0, "content": "grangian fibres, even the singular ones, are analytic, since they are obtained", "type": "text"}], "index": 31}, {"bbox": [110, 603, 499, 615], "spans": [{"bbox": [110, 603, 499, 615], "score": 1.0, "content": "by performing a hyperkaehler rotation starting from Lagrangian Abelian sur-", "type": "text"}], "index": 32}, {"bbox": [109, 616, 500, 630], "spans": [{"bbox": [109, 616, 500, 630], "score": 1.0, "content": "faces. So, in these cases, we have special Lagrangian torus fibration in which", "type": "text"}], "index": 33}, {"bbox": [110, 631, 501, 645], "spans": [{"bbox": [110, 631, 501, 645], "score": 1.0, "content": "all fibres are analytic: one can hope to understand the degeneration types of", "type": "text"}], "index": 34}, {"bbox": [110, 645, 450, 659], "spans": [{"bbox": [110, 645, 450, 659], "score": 1.0, "content": "singular special Lagrangian tori, moving from these constructions.", "type": "text"}], "index": 35}], "index": 24, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [109, 327, 501, 659]}, {"type": "text", "bbox": [126, 658, 500, 671], "lines": [{"bbox": [127, 660, 500, 673], "spans": [{"bbox": [127, 660, 500, 673], "score": 1.0, "content": "Explicit examples of projective irreducible symplectic 4-folds, fibered over", "type": "text"}], "index": 36}, {"bbox": [109, 128, 500, 142], "spans": [{"bbox": [109, 128, 500, 142], "score": 1.0, "content": "a projective base have been constructed by Markuschevich in [6] and [7]. One", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [109, 141, 499, 156], "spans": [{"bbox": [109, 141, 436, 156], "score": 1.0, "content": "of this constructions is the following: consider a double cover ", "type": "text", "cross_page": true}, {"bbox": [436, 143, 499, 153], "score": 0.91, "content": "\\pi:S\\to P^{2}", "type": "inline_equation", "height": 10, "width": 63, "cross_page": true}], "index": 1}, {"bbox": [109, 156, 500, 171], "spans": [{"bbox": [109, 156, 397, 171], "score": 1.0, "content": "of the projective plane, ramified along a smooth sextic ", "type": "text", "cross_page": true}, {"bbox": [398, 157, 443, 167], "score": 0.93, "content": "C\\hookrightarrow P^{2}", "type": "inline_equation", "height": 10, "width": 45, "cross_page": true}, {"bbox": [443, 156, 451, 171], "score": 1.0, "content": " (", "type": "text", "cross_page": true}, {"bbox": [451, 158, 459, 167], "score": 0.88, "content": "S", "type": "inline_equation", "height": 9, "width": 8, "cross_page": true}, {"bbox": [460, 156, 500, 171], "score": 1.0, "content": " is then", "type": "text", "cross_page": true}], "index": 2}, {"bbox": [109, 171, 500, 185], "spans": [{"bbox": [109, 171, 336, 185], "score": 1.0, "content": "realized as a K3 surface). Since any line in ", "type": "text", "cross_page": true}, {"bbox": [336, 172, 350, 181], "score": 0.92, "content": "P^{2}", "type": "inline_equation", "height": 9, "width": 14, "cross_page": true}, {"bbox": [350, 171, 500, 185], "score": 1.0, "content": " will intersect generically the", "type": "text", "cross_page": true}], "index": 3}, {"bbox": [110, 185, 500, 199], "spans": [{"bbox": [110, 185, 143, 199], "score": 1.0, "content": "sextic ", "type": "text", "cross_page": true}, {"bbox": [144, 187, 153, 196], "score": 0.91, "content": "C", "type": "inline_equation", "height": 9, "width": 9, "cross_page": true}, {"bbox": [153, 185, 404, 199], "score": 1.0, "content": " in six distinct point, we have that the covering ", "type": "text", "cross_page": true}, {"bbox": [404, 186, 465, 196], "score": 0.93, "content": "\\pi:S\\to P^{2}", "type": "inline_equation", "height": 10, "width": 61, "cross_page": true}, {"bbox": [465, 185, 500, 199], "score": 1.0, "content": " deter-", "type": "text", "cross_page": true}], "index": 4}, {"bbox": [110, 200, 500, 214], "spans": [{"bbox": [110, 200, 500, 214], "score": 1.0, "content": "mines a (flat) family of hyperelliptic curves over the dual projective plane", "type": "text", "cross_page": true}], "index": 5}, {"bbox": [110, 213, 500, 228], "spans": [{"bbox": [110, 215, 170, 227], "score": 0.93, "content": "f:\\mathcal{X}\\rightarrow P^{2}", "type": "inline_equation", "height": 12, "width": 60, "cross_page": true}, {"bbox": [170, 213, 500, 228], "score": 1.0, "content": ". Then the Altmann-Kleiman compactification of the relative Ja-", "type": "text", "cross_page": true}], "index": 6}, {"bbox": [110, 230, 500, 243], "spans": [{"bbox": [110, 230, 500, 243], "score": 1.0, "content": "cobian of the family turns out to be a simplectic projective irreducible 4-folds,", "type": "text", "cross_page": true}], "index": 7}, {"bbox": [109, 243, 472, 257], "spans": [{"bbox": [109, 243, 173, 257], "score": 1.0, "content": "fibered over ", "type": "text", "cross_page": true}, {"bbox": [174, 244, 188, 254], "score": 0.91, "content": "P^{2}", "type": "inline_equation", "height": 10, "width": 14, "cross_page": true}, {"bbox": [188, 243, 472, 257], "score": 1.0, "content": ", and in fact all fibres are Lagrangian Abelian varieties.", "type": "text", "cross_page": true}], "index": 8}], "index": 36, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [127, 660, 500, 673]}]}	{"layout_dets": [{"category_id": 1, "poly": [305, 904, 1391, 904, 1391, 1827, 305, 1827], 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1391.0, 1834.0, 1391.0, 1872.0, 354.0, 1872.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [836.0, 1923.0, 862.0, 1923.0, 862.0, 1959.0, 836.0, 1959.0], "score": 1.0, "text": ""}], "page_info": {"page_no": 6, "height": 2200, "width": 1700}}	# 3 Concluding remarks It is important to remark that all previous results are true also for special Lagrangian submanifolds of K3 surfaces, but their proof is completely trivial in that case. Another observation is related to singular Lagrangian submanifolds: in- deed, by the previous results, it turns out that we can also give examples of special Lagrangian subvarieties, obtained via hyperkaehler rotation of La- grangian complex subvarieties. On the other hand, contrary to the case of the corresponding submanifolds, we can not expect that all special Lagrangian subvarieties are obtained in this way, and consequently we can not expect that all special Lagrangian subvarieties are real analytic. Indeed, there are examples (compare [4]) of singular special Lagrangian submanifold in which are only smooth, but not real analytic. The discussion about singular Lagrangian submanifolds leads us to com- ment on the mirror symmetry construction suggested in [11]. Indeed, ac- cording to the recipe of [11], any Calabi-Yau , admitting a mirror , has a peculiar fibre space structure: on a physical ground it is argued that can be realized as the total space of a fibration in special Lagrangian tori. Unfortunately, there are very few examples of such realization: in particular, as far as we know, there is only one (partial) example for Calabi-Yau 3-folds of the so called Borcea-Voisin type (see [3]). Instead, in the case of irre- ducible symplectic projective manifolds the situation is completely different. Indeed, a recent result of Matsushita (see [8] and [9]) shows that for any fibre space structure of a projective irreducible symplectic manifold , with projective base , the generic fibre is an Abelian variety (up to finite unramified cover), and it is also Lagrangian with respect to the non degenerate holomorphic 2-form ; moreover, in the case of 4-folds one can prove that the generic fibre is an Abelian surface and is equidimensional, (i.e. all irreducible components of the fibres have the same dimension). By Corollary 2.1 it turns out that this fibre space structure can also be realized as a special Lagrangian torus fibration; moreover, in this case all special La- grangian fibres, even the singular ones, are analytic, since they are obtained by performing a hyperkaehler rotation starting from Lagrangian Abelian sur- faces. So, in these cases, we have special Lagrangian torus fibration in which all fibres are analytic: one can hope to understand the degeneration types of singular special Lagrangian tori, moving from these constructions. Explicit examples of projective irreducible symplectic 4-folds, fibered over a projective base have been constructed by Markuschevich in [6] and [7]. One of this constructions is the following: consider a double cover of the projective plane, ramified along a smooth sextic ( is then realized as a K3 surface). Since any line in will intersect generically the sextic in six distinct point, we have that the covering deter- mines a (flat) family of hyperelliptic curves over the dual projective plane . Then the Altmann-Kleiman compactification of the relative Ja- cobian of the family turns out to be a simplectic projective irreducible 4-folds, fibered over , and in fact all fibres are Lagrangian Abelian varieties. 7	<div class="pdf-page"> <h1>3 Concluding remarks</h1> <p>It is important to remark that all previous results are true also for special Lagrangian submanifolds of K3 surfaces, but their proof is completely trivial in that case.</p> <p>Another observation is related to singular Lagrangian submanifolds: in- deed, by the previous results, it turns out that we can also give examples of special Lagrangian subvarieties, obtained via hyperkaehler rotation of La- grangian complex subvarieties. On the other hand, contrary to the case of the corresponding submanifolds, we can not expect that all special Lagrangian subvarieties are obtained in this way, and consequently we can not expect that all special Lagrangian subvarieties are real analytic. Indeed, there are examples (compare [4]) of singular special Lagrangian submanifold in which are only smooth, but not real analytic.</p> <p>The discussion about singular Lagrangian submanifolds leads us to com- ment on the mirror symmetry construction suggested in [11]. Indeed, ac- cording to the recipe of [11], any Calabi-Yau , admitting a mirror , has a peculiar fibre space structure: on a physical ground it is argued that can be realized as the total space of a fibration in special Lagrangian tori. Unfortunately, there are very few examples of such realization: in particular, as far as we know, there is only one (partial) example for Calabi-Yau 3-folds of the so called Borcea-Voisin type (see [3]). Instead, in the case of irre- ducible symplectic projective manifolds the situation is completely different. Indeed, a recent result of Matsushita (see [8] and [9]) shows that for any fibre space structure of a projective irreducible symplectic manifold , with projective base , the generic fibre is an Abelian variety (up to finite unramified cover), and it is also Lagrangian with respect to the non degenerate holomorphic 2-form ; moreover, in the case of 4-folds one can prove that the generic fibre is an Abelian surface and is equidimensional, (i.e. all irreducible components of the fibres have the same dimension). By Corollary 2.1 it turns out that this fibre space structure can also be realized as a special Lagrangian torus fibration; moreover, in this case all special La- grangian fibres, even the singular ones, are analytic, since they are obtained by performing a hyperkaehler rotation starting from Lagrangian Abelian sur- faces. So, in these cases, we have special Lagrangian torus fibration in which all fibres are analytic: one can hope to understand the degeneration types of singular special Lagrangian tori, moving from these constructions.</p> <p>Explicit examples of projective irreducible symplectic 4-folds, fibered over a projective base have been constructed by Markuschevich in [6] and [7]. One of this constructions is the following: consider a double cover of the projective plane, ramified along a smooth sextic ( is then realized as a K3 surface). Since any line in will intersect generically the sextic in six distinct point, we have that the covering deter- mines a (flat) family of hyperelliptic curves over the dual projective plane . Then the Altmann-Kleiman compactification of the relative Ja- cobian of the family turns out to be a simplectic projective irreducible 4-folds, fibered over , and in fact all fibres are Lagrangian Abelian varieties.</p> </div>	<div class="pdf-page"> <h1 class="pdf-title" data-x="182" data-y="156" data-width="338" data-height="26">3 Concluding remarks</h1> <p class="pdf-text" data-x="182" data-y="195" data-width="654" data-height="55">It is important to remark that all previous results are true also for special Lagrangian submanifolds of K3 surfaces, but their proof is completely trivial in that case.</p> <p class="pdf-text" data-x="182" data-y="253" data-width="654" data-height="165">Another observation is related to singular Lagrangian submanifolds: in- deed, by the previous results, it turns out that we can also give examples of special Lagrangian subvarieties, obtained via hyperkaehler rotation of La- grangian complex subvarieties. On the other hand, contrary to the case of the corresponding submanifolds, we can not expect that all special Lagrangian subvarieties are obtained in this way, and consequently we can not expect that all special Lagrangian subvarieties are real analytic. Indeed, there are examples (compare [4]) of singular special Lagrangian submanifold in which are only smooth, but not real analytic.</p> <p class="pdf-text" data-x="182" data-y="420" data-width="654" data-height="429">The discussion about singular Lagrangian submanifolds leads us to com- ment on the mirror symmetry construction suggested in [11]. Indeed, ac- cording to the recipe of [11], any Calabi-Yau , admitting a mirror , has a peculiar fibre space structure: on a physical ground it is argued that can be realized as the total space of a fibration in special Lagrangian tori. Unfortunately, there are very few examples of such realization: in particular, as far as we know, there is only one (partial) example for Calabi-Yau 3-folds of the so called Borcea-Voisin type (see [3]). Instead, in the case of irre- ducible symplectic projective manifolds the situation is completely different. Indeed, a recent result of Matsushita (see [8] and [9]) shows that for any fibre space structure of a projective irreducible symplectic manifold , with projective base , the generic fibre is an Abelian variety (up to finite unramified cover), and it is also Lagrangian with respect to the non degenerate holomorphic 2-form ; moreover, in the case of 4-folds one can prove that the generic fibre is an Abelian surface and is equidimensional, (i.e. all irreducible components of the fibres have the same dimension). By Corollary 2.1 it turns out that this fibre space structure can also be realized as a special Lagrangian torus fibration; moreover, in this case all special La- grangian fibres, even the singular ones, are analytic, since they are obtained by performing a hyperkaehler rotation starting from Lagrangian Abelian sur- faces. So, in these cases, we have special Lagrangian torus fibration in which all fibres are analytic: one can hope to understand the degeneration types of singular special Lagrangian tori, moving from these constructions.</p> <p class="pdf-text" data-x="210" data-y="850" data-width="626" data-height="17">Explicit examples of projective irreducible symplectic 4-folds, fibered over a projective base have been constructed by Markuschevich in [6] and [7]. One of this constructions is the following: consider a double cover of the projective plane, ramified along a smooth sextic ( is then realized as a K3 surface). Since any line in will intersect generically the sextic in six distinct point, we have that the covering deter- mines a (flat) family of hyperelliptic curves over the dual projective plane . Then the Altmann-Kleiman compactification of the relative Ja- cobian of the family turns out to be a simplectic projective irreducible 4-folds, fibered over , and in fact all fibres are Lagrangian Abelian varieties.</p> <div class="pdf-discarded" data-x="501" data-y="893" data-width="16" data-height="14" style="opacity: 0.5;">7</div> </div>	{ "type": [ "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "inline_equation", "text", "text", "text", "inline_equation", "inline_equation", "text", "text", "text", "text", "text", "text", "inline_equation", "inline_equation", "text", "inline_equation", "inline_equation", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "inline_equation", "inline_equation", "inline_equation", "inline_equation", "text", "inline_equation", "text", "inline_equation" ], "coordinates": [ [ 184, 159, 520, 183 ], [ 182, 199, 836, 217 ], [ 184, 217, 834, 235 ], [ 182, 236, 292, 253 ], [ 212, 256, 833, 271 ], [ 184, 274, 834, 292 ], [ 184, 293, 834, 310 ], [ 182, 311, 836, 328 ], [ 182, 329, 836, 349 ], [ 182, 349, 836, 367 ], [ 182, 367, 836, 385 ], [ 182, 385, 833, 403 ], [ 184, 403, 572, 421 ], [ 212, 422, 836, 440 ], [ 182, 442, 838, 461 ], [ 184, 458, 836, 477 ], [ 184, 479, 834, 497 ], [ 182, 497, 836, 515 ], [ 184, 515, 836, 535 ], [ 184, 536, 836, 552 ], [ 184, 553, 836, 571 ], [ 182, 571, 836, 590 ], [ 182, 590, 836, 608 ], [ 184, 610, 836, 628 ], [ 184, 628, 836, 647 ], [ 184, 649, 834, 664 ], [ 184, 665, 838, 683 ], [ 182, 685, 836, 702 ], [ 184, 703, 834, 721 ], [ 182, 721, 836, 739 ], [ 184, 740, 834, 758 ], [ 182, 758, 836, 777 ], [ 184, 779, 834, 795 ], [ 182, 796, 836, 814 ], [ 184, 815, 838, 833 ], [ 184, 833, 752, 852 ], [ 212, 853, 836, 870 ], [ 182, 165, 836, 183 ], [ 182, 182, 834, 201 ], [ 182, 201, 836, 221 ], [ 182, 221, 836, 239 ], [ 184, 239, 836, 257 ], [ 184, 258, 836, 276 ], [ 184, 275, 836, 294 ], [ 184, 297, 836, 314 ], [ 182, 314, 789, 332 ] ], "content": [ "3 Concluding remarks", "It is important to remark that all previous results are true also for special", "Lagrangian submanifolds of K3 surfaces, but their proof is completely trivial", "in that case.", "Another observation is related to singular Lagrangian submanifolds: in-", "deed, by the previous results, it turns out that we can also give examples", "of special Lagrangian subvarieties, obtained via hyperkaehler rotation of La-", "grangian complex subvarieties. On the other hand, contrary to the case of the", "corresponding submanifolds, we can not expect that all special Lagrangian", "subvarieties are obtained in this way, and consequently we can not expect", "that all special Lagrangian subvarieties are real analytic. Indeed, there are", "examples (compare [4]) of singular special Lagrangian submanifold in C^{n}", "which are only smooth, but not real analytic.", "The discussion about singular Lagrangian submanifolds leads us to com-", "ment on the mirror symmetry construction suggested in [11]. Indeed, ac-", "cording to the recipe of [11], any Calabi-Yau X , admitting a mirror \\hat{X} , has", "a peculiar fibre space structure: on a physical ground it is argued that X", "can be realized as the total space of a fibration in special Lagrangian tori.", "Unfortunately, there are very few examples of such realization: in particular,", "as far as we know, there is only one (partial) example for Calabi-Yau 3-folds", "of the so called Borcea-Voisin type (see [3]). Instead, in the case of irre-", "ducible symplectic projective manifolds the situation is completely different.", "Indeed, a recent result of Matsushita (see [8] and [9]) shows that for any fibre", "space structure f:X\\to B of a projective irreducible symplectic manifold", "X , with projective base B , the generic fibre f^{-1}(b) is an Abelian variety (up", "to finite unramified cover), and it is also Lagrangian with respect to the non", "degenerate holomorphic 2-form \\Omega ; moreover, in the case of 4-folds one can", "prove that the generic fibre is an Abelian surface and f is equidimensional,", "(i.e. all irreducible components of the fibres have the same dimension). By", "Corollary 2.1 it turns out that this fibre space structure can also be realized", "as a special Lagrangian torus fibration; moreover, in this case all special La-", "grangian fibres, even the singular ones, are analytic, since they are obtained", "by performing a hyperkaehler rotation starting from Lagrangian Abelian sur-", "faces. So, in these cases, we have special Lagrangian torus fibration in which", "all fibres are analytic: one can hope to understand the degeneration types of", "singular special Lagrangian tori, moving from these constructions.", "Explicit examples of projective irreducible symplectic 4-folds, fibered over", "a projective base have been constructed by Markuschevich in [6] and [7]. One", "of this constructions is the following: consider a double cover \\pi:S\\to P^{2}", "of the projective plane, ramified along a smooth sextic C\\hookrightarrow P^{2} ( S is then", "realized as a K3 surface). Since any line in P^{2} will intersect generically the", "sextic C in six distinct point, we have that the covering \\pi:S\\to P^{2} deter-", "mines a (flat) family of hyperelliptic curves over the dual projective plane", "f:\\mathcal{X}\\rightarrow P^{2} . Then the Altmann-Kleiman compactification of the relative Ja-", "cobian of the family turns out to be a simplectic projective irreducible 4-folds,", "fibered over P^{2} , and in fact all fibres are Lagrangian Abelian varieties." ], "index": [ 0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63 ] }	{ "coordinates": [], "index": [], "caption": [], "caption_coordinates": [] }	{ "type": [], "coordinates": [], "content": [], "index": [] }	{ "type": [], "coordinates": [], "content": [], "index": [] }	{"document_info": {"total_pages": 9, "parse_type": "txt", "version": "1.0.1", "total_content_list_items": 72}, "content_statistics": {"total_blocks": 81, "total_lines": 269, "total_images": 0, "total_equations": 18, "total_tables": 0, "total_characters": 16112, "total_words": 2578, "block_type_distribution": {"title": 6, "text": 55, "discarded": 9, "interline_equation": 9, "list": 2}}, "model_statistics": {"total_layout_detections": 813, "avg_confidence": 0.9488487084870848, "min_confidence": 0.26, "max_confidence": 1.0}, "page_statistics": [{"page_number": 0, "page_size": [612.0, 792.0], "blocks": 10, "lines": 26, "images": 0, "equations": 0, "tables": 0, "characters": 1419, "words": 204, "discarded_blocks": 2}, {"page_number": 1, "page_size": [612.0, 792.0], "blocks": 15, "lines": 37, "images": 0, "equations": 0, "tables": 0, "characters": 2328, "words": 359, "discarded_blocks": 1}, {"page_number": 2, "page_size": [612.0, 792.0], "blocks": 12, "lines": 35, "images": 0, "equations": 4, "tables": 0, "characters": 1843, "words": 306, "discarded_blocks": 1}, {"page_number": 3, "page_size": [612.0, 792.0], "blocks": 8, "lines": 30, "images": 0, "equations": 6, "tables": 0, "characters": 1797, "words": 312, "discarded_blocks": 0}, {"page_number": 4, "page_size": [612.0, 792.0], "blocks": 12, "lines": 34, "images": 0, "equations": 4, "tables": 0, "characters": 1905, "words": 328, "discarded_blocks": 1}, {"page_number": 5, "page_size": [612.0, 792.0], "blocks": 11, "lines": 32, "images": 0, "equations": 4, "tables": 0, "characters": 1846, "words": 308, "discarded_blocks": 1}, {"page_number": 6, "page_size": [612.0, 792.0], "blocks": 6, "lines": 46, "images": 0, "equations": 0, "tables": 0, "characters": 3307, "words": 521, "discarded_blocks": 1}, {"page_number": 7, "page_size": [612.0, 792.0], "blocks": 5, "lines": 29, "images": 0, "equations": 0, "tables": 0, "characters": 1666, "words": 239, "discarded_blocks": 1}, {"page_number": 8, "page_size": [612.0, 792.0], "blocks": 2, "lines": 0, "images": 0, "equations": 0, "tables": 0, "characters": 1, "words": 1, "discarded_blocks": 1}]}
0001060v1	7	[ 612, 792 ]	{ "type": [ "text", "text", "title", "list", "discarded" ], "coordinates": [ [ 182, 160, 838, 329 ], [ 184, 330, 836, 440 ], [ 182, 469, 337, 493 ], [ 192, 508, 839, 849 ], [ 501, 893, 515, 907 ] ], "content": [ "", "Finally, we believe that our characterization of special Lagrangian sub- manifolds of irreducible symplectic 4-folds can be extended also to higher dimensional irreducible symplectic manifolds: to this aim notice that the proof we have given becomes longer and longer, since one has to deal with new cases and subcases. It would be nice, instead, to find out a sort of inductive argument, which works for all dimensions.", "References", "[1] Becker K., Becker M., Strominger A., Fivebranes, membranes and non- perturbative string theory, hep-th/9507158. [2] Beauville A., Varits Kahleriennes dont la premire classe de Chern est nulle, J. Diff. Geom. 18 (1983), 755-782. [3] Gross M., Wilson P.M.H., Mirror Symmetry via 3-tori for a class of Calabi-Yau threefolds, alg-geom/9608004. [4] Harvey R., Lawson Jr., H.B., Calibrated Geometries, Acta Math., 148 (1982), 47-157. [5] Huybrechts D., Compact Hyperkahler manifolds: basic results, alg- geom/9705025. [6] Markushevich D., Completely integrable projective symplectic 4- dimensional varieties, Izvestiya: Mathematics 59 (1995), 159-187. [7] Markushevich D., Integrable symplectic structures on compact complex manifolds, Math. USSR Sb. 59 (1988), 459-469. [8] Matsushita D., On fibre space structures of a projective irreducible sym- plectic manifold, Topology 38 (1999), 79-83. [9] Matsushita D., Addendum to: On fibre space structures of a projective irreducible symplectic manifold, math.ag/9903045. [10] O’Grady K.G., Desingularized moduli spaces of sheaves on a K3, I and II, alg-geom/9708009, math.ag/9805099. [11] Strominger A., Yau S.-T., Zaslow E., Mirror Symmetry is T-duality, Nucl. Phys. B479, (1996), 243-259.", "8" ], "index": [ 0, 1, 2, 3, 4 ] }	[{"type": "text", "text": "", "page_idx": 7}, {"type": "text", "text": "Finally, we believe that our characterization of special Lagrangian submanifolds of irreducible symplectic 4-folds can be extended also to higher dimensional irreducible symplectic manifolds: to this aim notice that the proof we have given becomes longer and longer, since one has to deal with new cases and subcases. It would be nice, instead, to find out a sort of inductive argument, which works for all dimensions. ", "page_idx": 7}, {"type": "text", "text": "References ", "text_level": 1, "page_idx": 7}, {"type": "text", "text": "[1] Becker K., Becker M., Strominger A., Fivebranes, membranes and nonperturbative string theory, hep-th/9507158. \n[2] Beauville A., Varits Kahleriennes dont la premire classe de Chern est nulle, J. Diff. Geom. 18 (1983), 755-782. \n[3] Gross M., Wilson P.M.H., Mirror Symmetry via 3-tori for a class of Calabi-Yau threefolds, alg-geom/9608004. \n[4] Harvey R., Lawson Jr., H.B., Calibrated Geometries, Acta Math., 148 (1982), 47-157. \n[5] Huybrechts D., Compact Hyperkahler manifolds: basic results, alggeom/9705025. \n[6] Markushevich D., Completely integrable projective symplectic 4- dimensional varieties, Izvestiya: Mathematics 59 (1995), 159-187. \n[7] Markushevich D., Integrable symplectic structures on compact complex manifolds, Math. USSR Sb. 59 (1988), 459-469. \n[8] Matsushita D., On fibre space structures of a projective irreducible symplectic manifold, Topology 38 (1999), 79-83. \n[9] Matsushita D., Addendum to: On fibre space structures of a projective irreducible symplectic manifold, math.ag/9903045. \n[10] O\u2019Grady K.G., Desingularized moduli spaces of sheaves on a K3, I and II, alg-geom/9708009, math.ag/9805099. \n[11] Strominger A., Yau S.-T., Zaslow E., Mirror Symmetry is T-duality, Nucl. Phys. B479, (1996), 243-259. ", "page_idx": 7}]	{"preproc_blocks": [{"type": "text", "bbox": [109, 124, 501, 255], "lines": [{"bbox": [109, 128, 500, 142], "spans": [{"bbox": [109, 128, 500, 142], "score": 1.0, "content": "a projective base have been constructed by Markuschevich in [6] and [7]. One", "type": "text"}], "index": 0}, {"bbox": [109, 141, 499, 156], "spans": [{"bbox": [109, 141, 436, 156], "score": 1.0, "content": "of this constructions is the following: consider a double cover ", "type": "text"}, {"bbox": [436, 143, 499, 153], "score": 0.91, "content": "\\pi:S\\to P^{2}", "type": "inline_equation", "height": 10, "width": 63}], "index": 1}, {"bbox": [109, 156, 500, 171], "spans": [{"bbox": [109, 156, 397, 171], "score": 1.0, "content": "of the projective plane, ramified along a smooth sextic ", "type": "text"}, {"bbox": [398, 157, 443, 167], "score": 0.93, "content": "C\\hookrightarrow P^{2}", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [443, 156, 451, 171], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [451, 158, 459, 167], "score": 0.88, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [460, 156, 500, 171], "score": 1.0, "content": " is then", "type": "text"}], "index": 2}, {"bbox": [109, 171, 500, 185], "spans": [{"bbox": [109, 171, 336, 185], "score": 1.0, "content": "realized as a K3 surface). 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""}, {"category_id": 15, "poly": [837.0, 1925.0, 860.0, 1925.0, 860.0, 1958.0, 837.0, 1958.0], "score": 1.0, "text": ""}], "page_info": {"page_no": 7, "height": 2200, "width": 1700}}	Finally, we believe that our characterization of special Lagrangian sub- manifolds of irreducible symplectic 4-folds can be extended also to higher dimensional irreducible symplectic manifolds: to this aim notice that the proof we have given becomes longer and longer, since one has to deal with new cases and subcases. It would be nice, instead, to find out a sort of inductive argument, which works for all dimensions. # References - [1] Becker K., Becker M., Strominger A., Fivebranes, membranes and non- perturbative string theory, hep-th/9507158. [2] Beauville A., Varits Kahleriennes dont la premire classe de Chern est nulle, J. Diff. Geom. 18 (1983), 755-782. [3] Gross M., Wilson P.M.H., Mirror Symmetry via 3-tori for a class of Calabi-Yau threefolds, alg-geom/9608004. [4] Harvey R., Lawson Jr., H.B., Calibrated Geometries, Acta Math., 148 (1982), 47-157. [5] Huybrechts D., Compact Hyperkahler manifolds: basic results, alg- geom/9705025. [6] Markushevich D., Completely integrable projective symplectic 4- dimensional varieties, Izvestiya: Mathematics 59 (1995), 159-187. [7] Markushevich D., Integrable symplectic structures on compact complex manifolds, Math. USSR Sb. 59 (1988), 459-469. [8] Matsushita D., On fibre space structures of a projective irreducible sym- plectic manifold, Topology 38 (1999), 79-83. [9] Matsushita D., Addendum to: On fibre space structures of a projective irreducible symplectic manifold, math.ag/9903045. [10] O’Grady K.G., Desingularized moduli spaces of sheaves on a K3, I and II, alg-geom/9708009, math.ag/9805099. [11] Strominger A., Yau S.-T., Zaslow E., Mirror Symmetry is T-duality, Nucl. Phys. B479, (1996), 243-259. 8	<div class="pdf-page"> <p>Finally, we believe that our characterization of special Lagrangian sub- manifolds of irreducible symplectic 4-folds can be extended also to higher dimensional irreducible symplectic manifolds: to this aim notice that the proof we have given becomes longer and longer, since one has to deal with new cases and subcases. It would be nice, instead, to find out a sort of inductive argument, which works for all dimensions.</p> <h1>References</h1> <ul> <li>[1] Becker K., Becker M., Strominger A., Fivebranes, membranes and non- perturbative string theory, hep-th/9507158. [2] Beauville A., Varits Kahleriennes dont la premire classe de Chern est nulle, J. Diff. Geom. 18 (1983), 755-782. [3] Gross M., Wilson P.M.H., Mirror Symmetry via 3-tori for a class of Calabi-Yau threefolds, alg-geom/9608004. [4] Harvey R., Lawson Jr., H.B., Calibrated Geometries, Acta Math., 148 (1982), 47-157. [5] Huybrechts D., Compact Hyperkahler manifolds: basic results, alg- geom/9705025. [6] Markushevich D., Completely integrable projective symplectic 4- dimensional varieties, Izvestiya: Mathematics 59 (1995), 159-187. [7] Markushevich D., Integrable symplectic structures on compact complex manifolds, Math. USSR Sb. 59 (1988), 459-469. [8] Matsushita D., On fibre space structures of a projective irreducible sym- plectic manifold, Topology 38 (1999), 79-83. [9] Matsushita D., Addendum to: On fibre space structures of a projective irreducible symplectic manifold, math.ag/9903045. [10] O’Grady K.G., Desingularized moduli spaces of sheaves on a K3, I and II, alg-geom/9708009, math.ag/9805099. [11] Strominger A., Yau S.-T., Zaslow E., Mirror Symmetry is T-duality, Nucl. Phys. B479, (1996), 243-259.</li> </ul> </div>	<div class="pdf-page"> <p class="pdf-text" data-x="184" data-y="330" data-width="652" data-height="110">Finally, we believe that our characterization of special Lagrangian sub- manifolds of irreducible symplectic 4-folds can be extended also to higher dimensional irreducible symplectic manifolds: to this aim notice that the proof we have given becomes longer and longer, since one has to deal with new cases and subcases. It would be nice, instead, to find out a sort of inductive argument, which works for all dimensions.</p> <h1 class="pdf-title" data-x="182" data-y="469" data-width="155" data-height="24">References</h1> <ul class="pdf-list" data-x="192" data-y="508" data-width="647" data-height="341"> <li>[1] Becker K., Becker M., Strominger A., Fivebranes, membranes and non- perturbative string theory, hep-th/9507158. [2] Beauville A., Varits Kahleriennes dont la premire classe de Chern est nulle, J. Diff. Geom. 18 (1983), 755-782. [3] Gross M., Wilson P.M.H., Mirror Symmetry via 3-tori for a class of Calabi-Yau threefolds, alg-geom/9608004. [4] Harvey R., Lawson Jr., H.B., Calibrated Geometries, Acta Math., 148 (1982), 47-157. [5] Huybrechts D., Compact Hyperkahler manifolds: basic results, alg- geom/9705025. [6] Markushevich D., Completely integrable projective symplectic 4- dimensional varieties, Izvestiya: Mathematics 59 (1995), 159-187. [7] Markushevich D., Integrable symplectic structures on compact complex manifolds, Math. USSR Sb. 59 (1988), 459-469. [8] Matsushita D., On fibre space structures of a projective irreducible sym- plectic manifold, Topology 38 (1999), 79-83. [9] Matsushita D., Addendum to: On fibre space structures of a projective irreducible symplectic manifold, math.ag/9903045. [10] O’Grady K.G., Desingularized moduli spaces of sheaves on a K3, I and II, alg-geom/9708009, math.ag/9805099. [11] Strominger A., Yau S.-T., Zaslow E., Mirror Symmetry is T-duality, Nucl. Phys. B479, (1996), 243-259.</li> </ul> <div class="pdf-discarded" data-x="501" data-y="893" data-width="14" data-height="14" style="opacity: 0.5;">8</div> </div>	{ "type": [ "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text" ], "coordinates": [ [ 212, 333, 833, 350 ], [ 184, 352, 834, 368 ], [ 184, 372, 836, 386 ], [ 182, 390, 836, 407 ], [ 182, 408, 839, 425 ], [ 182, 426, 632, 443 ], [ 184, 473, 339, 495 ], [ 194, 510, 836, 532 ], [ 222, 531, 592, 549 ], [ 194, 561, 838, 581 ], [ 224, 581, 573, 598 ], [ 192, 611, 839, 632 ], [ 225, 630, 580, 651 ], [ 192, 663, 836, 681 ], [ 224, 682, 354, 700 ], [ 192, 712, 833, 734 ], [ 220, 733, 354, 749 ], [ 192, 762, 836, 783 ], [ 225, 782, 788, 800 ], [ 194, 813, 838, 832 ], [ 224, 832, 632, 852 ], [ 194, 164, 836, 184 ], [ 224, 184, 602, 202 ], [ 192, 213, 838, 235 ], [ 222, 234, 654, 253 ], [ 184, 265, 838, 285 ], [ 224, 284, 573, 303 ], [ 184, 315, 836, 334 ], [ 224, 334, 528, 352 ] ], "content": [ "Finally, we believe that our characterization of special Lagrangian sub-", "manifolds of irreducible symplectic 4-folds can be extended also to higher", "dimensional irreducible symplectic manifolds: to this aim notice that the", "proof we have given becomes longer and longer, since one has to deal with", "new cases and subcases. 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0002014v1	0	[ 612, 792 ]	{ "type": [ "title", "title", "text", "text", "text", "text", "text", "text", "text", "discarded", "discarded" ], "coordinates": [ [ 224, 183, 776, 202 ], [ 184, 219, 814, 266 ], [ 272, 316, 726, 337 ], [ 267, 355, 731, 393 ], [ 145, 400, 851, 418 ], [ 165, 497, 831, 572 ], [ 165, 579, 833, 729 ], [ 195, 753, 599, 771 ], [ 194, 795, 831, 927 ], [ 707, 118, 833, 134 ], [ 23, 266, 60, 674 ] ], "content": [ "Reply to A. Patrascioiu’s and E. Seiler’s comment on our paper", "Percolation properties of the 2D Heisenberg model", "B. Allés , J. J. Alonso , C. Criado , M. Pepe Dipartimento di Fisica, Universita di Milano-Bicocca and INFN Sezione di Milano, Milano, Italy", "", "Departamento de Fisica Aplicada I, Facultad de Ciencias, 29071 Malaga, Spain", "The most of the problems raised by the authors of the comment [1] about Ref. [2] are based on claims which have not been written in [2], for instance almost all the introduction and the point (1) in [1] are based on such non– existent claims.", "Instead in Ref. [2] we avoid to make claims not based on well–founded results. For instance in the abstract we write “... This result indicates how the model can avoid a previously conjectured Kosterlitz–Thouless phase transition...” and in the conclusive part we notice that “Our results exclude this massless phase for ”. Therefore it seems to us that the opening sentence in the Comment [1] “In a recent letter All´es et al. claim to show that the two dimensional classical Heisenberg model does not have a massless phase.” is strongly inadequate.", "As for the points that appear in the Comment:", "• (1) The purpose of the paper [2] is to fill a gap in the research about the critical properties of the Heisenberg model. This gap is the following one: in Ref. [3] a scenario was proposed where the 2D Heisenberg model should undergo a KT phase transition at a finite temperature. This sce- nario is based mainly on three hypotheses, the third one (which states the non–percolation of the –type or equatorial clusters) being left in [3] without a plausible justification. To back up that hypothesis a numerical test was cited in [3] but the details of the numerics (temperature, size of the lattice, etc.) and several data concerning the percolation properties of the system, were completely skipped. The only quoted result was (see beginning of section 4 in [3]) “We also tested numerically for ,... There is no indication of percolation...”. On the contrary, such inter- esting results about the critical properties should be put forward with a thorough description of the hypotheses involved. Moreover, one would like to understand how was possible to use the small value of epsilon mentioned in Ref. [3], because that value implies a really tiny temper- ature and consequently it requires a huge lattice size. If “Everybody agrees that at the standard action model has a finite correla- tion length”, see [1], also everybody would like to know details about the numerics and the computer used to simulate the model at such a small temperature.", "February 2000", "arXiv:hep-lat/0002014v1 11 Feb 2000" ], "index": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ] }	[{"type": "text", "text": "Reply to A. Patrascioiu\u2019s and E. Seiler\u2019s comment on our paper ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "Percolation properties of the 2D Heisenberg model ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "B. All\u00e9s $\\mathrm{a}$ , J. J. Alonso $\\mathrm{b}$ , C. Criado $\\mathrm{b}$ , M. Pepe $\\mathrm{c}$ $\\mathrm{a}$ Dipartimento di Fisica, Universita di Milano-Bicocca and INFN Sezione di Milano, Milano, Italy ", "page_idx": 0}, {"type": "text", "text": "", "page_idx": 0}, {"type": "text", "text": "$^\\mathrm{b}$ Departamento de Fisica Aplicada I, Facultad de Ciencias, 29071 Malaga, Spain ", "page_idx": 0}, {"type": "text", "text": "The most of the problems raised by the authors of the comment [1] about Ref. [2] are based on claims which have not been written in [2], for instance almost all the introduction and the point (1) in [1] are based on such non\u2013 existent claims. ", "page_idx": 0}, {"type": "text", "text": "Instead in Ref. [2] we avoid to make claims not based on well\u2013founded results. For instance in the abstract we write \u201c... This result indicates how the model can avoid a previously conjectured Kosterlitz\u2013Thouless $[K T]$ phase transition...\u201d and in the conclusive part we notice that \u201cOur results exclude this massless phase for $T\\,>\\,0.5$ \u201d. Therefore it seems to us that the opening sentence in the Comment [1] \u201cIn a recent letter All\u00b4es et al. claim to show that the two dimensional classical Heisenberg model does not have a massless phase.\u201d is strongly inadequate. ", "page_idx": 0}, {"type": "text", "text": "As for the points that appear in the Comment: ", "page_idx": 0}, {"type": "text", "text": "\u2022 (1) The purpose of the paper [2] is to fill a gap in the research about the critical properties of the Heisenberg model. This gap is the following one: in Ref. [3] a scenario was proposed where the 2D Heisenberg model should undergo a KT phase transition at a finite temperature. This scenario is based mainly on three hypotheses, the third one (which states the non\u2013percolation of the $\\boldsymbol{S}$ \u2013type or equatorial clusters) being left in [3] without a plausible justification. To back up that hypothesis a numerical test was cited in [3] but the details of the numerics (temperature, size of the lattice, etc.) and several data concerning the percolation properties of the system, were completely skipped. The only quoted result was (see beginning of section 4 in [3]) \u201cWe also tested numerically for $\\epsilon=1/3$ ,... There is no indication of percolation...\u201d. On the contrary, such interesting results about the critical properties should be put forward with a thorough description of the hypotheses involved. Moreover, one would like to understand how was possible to use the small value of epsilon mentioned in Ref. [3], because that value implies a really tiny temperature $T$ and consequently it requires a huge lattice size. If \u201cEverybody agrees that at $\\beta\\:=\\:2.0$ the standard action model has a finite correlation length\u201d, see [1], also everybody would like to know details about the numerics and the computer used to simulate the model at such a small temperature. ", "page_idx": 0}]	{"preproc_blocks": [{"type": "title", "bbox": [134, 142, 464, 157], "lines": [{"bbox": [135, 145, 463, 159], "spans": [{"bbox": [135, 145, 463, 159], "score": 1.0, "content": "Reply to A. Patrascioiu\u2019s and E. 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Pepe", "type": "text"}, {"bbox": [428, 252, 433, 260], "score": 0.26, "content": "\\mathrm{c}", "type": "inline_equation", "height": 8, "width": 5}], "index": 3}, {"bbox": [160, 276, 439, 293], "spans": [{"bbox": [160, 280, 165, 288], "score": 0.48, "content": "\\mathrm{a}", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [165, 276, 439, 293], "score": 0.9901946783065796, "content": "Dipartimento di Fisica, Universita di Milano-Bicocca", "type": "text"}], "index": 4}, {"bbox": [187, 293, 410, 305], "spans": [{"bbox": [187, 293, 410, 305], "score": 1.0, "content": "and INFN Sezione di Milano, Milano, Italy", "type": "text"}], "index": 5}], "index": 3, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [163, 249, 433, 263]}, {"type": "text", "bbox": [160, 275, 437, 304], "lines": [], "index": 4.5, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [160, 276, 439, 305], "lines_deleted": true}, {"type": "text", "bbox": [87, 310, 509, 324], "lines": [{"bbox": [91, 308, 508, 330], "spans": [{"bbox": [91, 312, 97, 323], "score": 0.41, "content": "^\\mathrm{b}", "type": "inline_equation", "height": 11, "width": 6}, {"bbox": [97, 308, 508, 330], "score": 0.9878029227256775, "content": "Departamento de Fisica Aplicada I, Facultad de Ciencias, 29071 Malaga, Spain", "type": "text"}], "index": 6}], "index": 6, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [91, 308, 508, 330]}, {"type": "text", "bbox": [99, 385, 497, 443], "lines": [{"bbox": [116, 387, 497, 403], "spans": [{"bbox": [116, 387, 497, 403], "score": 1.0, "content": "The most of the problems raised by the authors of the comment [1] about", "type": "text"}], "index": 7}, {"bbox": [99, 401, 498, 417], "spans": [{"bbox": [99, 401, 498, 417], "score": 1.0, "content": "Ref. [2] are based on claims which have not been written in [2], for instance", "type": "text"}], "index": 8}, {"bbox": [100, 417, 497, 431], "spans": [{"bbox": [100, 417, 497, 431], "score": 1.0, "content": "almost all the introduction and the point (1) in [1] are based on such non\u2013", "type": "text"}], "index": 9}, {"bbox": [100, 432, 179, 444], "spans": [{"bbox": [100, 432, 179, 444], "score": 1.0, "content": "existent claims.", "type": "text"}], "index": 10}], "index": 8.5, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [99, 387, 498, 444]}, {"type": "text", "bbox": [99, 448, 498, 564], "lines": [{"bbox": [117, 451, 497, 464], "spans": [{"bbox": [117, 451, 497, 464], "score": 1.0, "content": "Instead in Ref. [2] we avoid to make claims not based on well\u2013founded", "type": "text"}], "index": 11}, {"bbox": [99, 464, 497, 479], "spans": [{"bbox": [99, 464, 497, 479], "score": 1.0, "content": "results. For instance in the abstract we write \u201c... This result indicates how", "type": "text"}], "index": 12}, {"bbox": [99, 479, 498, 494], "spans": [{"bbox": [99, 479, 440, 494], "score": 1.0, "content": "the model can avoid a previously conjectured Kosterlitz\u2013Thouless ", "type": "text"}, {"bbox": [440, 481, 464, 493], "score": 0.85, "content": "[K T]", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [464, 479, 498, 494], "score": 1.0, "content": " phase", "type": "text"}], "index": 13}, {"bbox": [99, 493, 498, 509], "spans": [{"bbox": [99, 493, 498, 509], "score": 1.0, "content": "transition...\u201d and in the conclusive part we notice that \u201cOur results exclude", "type": "text"}], "index": 14}, {"bbox": [99, 507, 498, 524], "spans": [{"bbox": [99, 507, 221, 524], "score": 1.0, "content": "this massless phase for ", "type": "text"}, {"bbox": [222, 510, 263, 519], "score": 0.87, "content": "T\\,>\\,0.5", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [263, 507, 498, 524], "score": 1.0, "content": "\u201d. Therefore it seems to us that the opening", "type": "text"}], "index": 15}, {"bbox": [99, 523, 497, 537], "spans": [{"bbox": [99, 523, 497, 537], "score": 1.0, "content": "sentence in the Comment [1] \u201cIn a recent letter All\u00b4es et al. claim to show", "type": "text"}], "index": 16}, {"bbox": [98, 536, 497, 552], "spans": [{"bbox": [98, 536, 497, 552], "score": 1.0, "content": "that the two dimensional classical Heisenberg model does not have a massless", "type": "text"}], "index": 17}, {"bbox": [99, 552, 258, 567], "spans": [{"bbox": [99, 552, 258, 567], "score": 1.0, "content": "phase.\u201d is strongly inadequate.", "type": "text"}], "index": 18}], "index": 14.5, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [98, 451, 498, 567]}, {"type": "text", "bbox": [117, 583, 358, 597], "lines": [{"bbox": [118, 585, 359, 598], "spans": [{"bbox": [118, 585, 359, 598], "score": 1.0, "content": "As for the points that appear in the Comment:", "type": "text"}], "index": 19}], "index": 19, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [118, 585, 359, 598]}, {"type": "text", "bbox": [116, 615, 497, 717], "lines": [{"bbox": [118, 618, 497, 632], "spans": [{"bbox": [118, 618, 497, 632], "score": 1.0, "content": "\u2022 (1) The purpose of the paper [2] is to fill a gap in the research about", "type": "text"}], "index": 20}, {"bbox": [128, 633, 498, 647], "spans": [{"bbox": [128, 633, 498, 647], "score": 1.0, "content": "the critical properties of the Heisenberg model. This gap is the following", "type": "text"}], "index": 21}, {"bbox": [128, 648, 498, 661], "spans": [{"bbox": [128, 648, 498, 661], "score": 1.0, "content": "one: in Ref. [3] a scenario was proposed where the 2D Heisenberg model", "type": "text"}], "index": 22}, {"bbox": [128, 661, 497, 675], "spans": [{"bbox": [128, 661, 497, 675], "score": 1.0, "content": "should undergo a KT phase transition at a finite temperature. This sce-", "type": "text"}], "index": 23}, {"bbox": [129, 676, 497, 690], "spans": [{"bbox": [129, 676, 497, 690], "score": 1.0, "content": "nario is based mainly on three hypotheses, the third one (which states", "type": "text"}], "index": 24}, {"bbox": [128, 690, 497, 705], "spans": [{"bbox": [128, 690, 265, 705], "score": 1.0, "content": "the non\u2013percolation of the ", "type": "text"}, {"bbox": [266, 692, 274, 700], "score": 0.9, "content": "\\boldsymbol{S}", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [274, 690, 497, 705], "score": 1.0, "content": "\u2013type or equatorial clusters) being left in [3]", "type": "text"}], "index": 25}, {"bbox": [129, 705, 497, 718], "spans": [{"bbox": [129, 705, 497, 718], "score": 1.0, "content": "without a plausible justification. To back up that hypothesis a numerical", "type": "text"}], "index": 26}, {"bbox": [128, 94, 499, 107], "spans": [{"bbox": [128, 94, 499, 107], "score": 1.0, "content": "test was cited in [3] but the details of the numerics (temperature, size of", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [128, 108, 497, 122], "spans": [{"bbox": [128, 108, 497, 122], "score": 1.0, "content": "the lattice, etc.) and several data concerning the percolation properties", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [128, 123, 498, 137], "spans": [{"bbox": [128, 123, 498, 137], "score": 1.0, "content": "of the system, were completely skipped. The only quoted result was (see", "type": "text", "cross_page": true}], "index": 2}, {"bbox": [128, 136, 497, 152], "spans": [{"bbox": [128, 136, 445, 152], "score": 1.0, "content": "beginning of section 4 in [3]) \u201cWe also tested numerically for ", "type": "text", "cross_page": true}, {"bbox": [445, 138, 484, 151], "score": 0.81, "content": "\\epsilon=1/3", "type": "inline_equation", "height": 13, "width": 39, "cross_page": true}, {"bbox": [484, 136, 497, 152], "score": 1.0, "content": ",...", "type": "text", "cross_page": true}], "index": 3}, {"bbox": [128, 151, 496, 165], "spans": [{"bbox": [128, 151, 496, 165], "score": 1.0, "content": "There is no indication of percolation...\u201d. On the contrary, such inter-", "type": "text", "cross_page": true}], "index": 4}, {"bbox": [129, 167, 499, 180], "spans": [{"bbox": [129, 167, 499, 180], "score": 1.0, "content": "esting results about the critical properties should be put forward with a", "type": "text", "cross_page": true}], "index": 5}, {"bbox": [128, 181, 497, 194], "spans": [{"bbox": [128, 181, 497, 194], "score": 1.0, "content": "thorough description of the hypotheses involved. Moreover, one would", "type": "text", "cross_page": true}], "index": 6}, {"bbox": [128, 194, 497, 209], "spans": [{"bbox": [128, 194, 497, 209], "score": 1.0, "content": "like to understand how was possible to use the small value of epsilon", "type": "text", "cross_page": true}], "index": 7}, {"bbox": [128, 209, 497, 223], "spans": [{"bbox": [128, 209, 497, 223], "score": 1.0, "content": "mentioned in Ref. [3], because that value implies a really tiny temper-", "type": "text", "cross_page": true}], "index": 8}, {"bbox": [129, 224, 497, 237], "spans": [{"bbox": [129, 224, 159, 237], "score": 1.0, "content": "ature ", "type": "text", "cross_page": true}, {"bbox": [159, 226, 168, 234], "score": 0.91, "content": "T", "type": "inline_equation", "height": 8, "width": 9, "cross_page": true}, {"bbox": [169, 224, 497, 237], "score": 1.0, "content": " and consequently it requires a huge lattice size. If \u201cEverybody", "type": "text", "cross_page": true}], "index": 9}, {"bbox": [128, 239, 497, 251], "spans": [{"bbox": [128, 239, 206, 251], "score": 1.0, "content": "agrees that at ", "type": "text", "cross_page": true}, {"bbox": [206, 240, 248, 251], "score": 0.92, "content": "\\beta\\:=\\:2.0", "type": "inline_equation", "height": 11, "width": 42, "cross_page": true}, {"bbox": [248, 239, 497, 251], "score": 1.0, "content": " the standard action model has a finite correla-", "type": "text", "cross_page": true}], "index": 10}, {"bbox": [128, 253, 499, 268], "spans": [{"bbox": [128, 253, 499, 268], "score": 1.0, "content": "tion length\u201d, see [1], also everybody would like to know details about the", "type": "text", "cross_page": true}], "index": 11}, {"bbox": [128, 268, 498, 280], "spans": [{"bbox": [128, 268, 498, 280], "score": 1.0, "content": "numerics and the computer used to simulate the model at such a small", "type": "text", "cross_page": true}], "index": 12}, {"bbox": 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Patrascioiu’s and E. Seiler’s comment on our paper # Percolation properties of the 2D Heisenberg model arXiv:hep-lat/0002014v1 11 Feb 2000 B. Allés , J. J. Alonso , C. Criado , M. Pepe Dipartimento di Fisica, Universita di Milano-Bicocca and INFN Sezione di Milano, Milano, Italy Departamento de Fisica Aplicada I, Facultad de Ciencias, 29071 Malaga, Spain The most of the problems raised by the authors of the comment [1] about Ref. [2] are based on claims which have not been written in [2], for instance almost all the introduction and the point (1) in [1] are based on such non– existent claims. Instead in Ref. [2] we avoid to make claims not based on well–founded results. For instance in the abstract we write “... This result indicates how the model can avoid a previously conjectured Kosterlitz–Thouless phase transition...” and in the conclusive part we notice that “Our results exclude this massless phase for ”. Therefore it seems to us that the opening sentence in the Comment [1] “In a recent letter All´es et al. claim to show that the two dimensional classical Heisenberg model does not have a massless phase.” is strongly inadequate. As for the points that appear in the Comment: • (1) The purpose of the paper [2] is to fill a gap in the research about the critical properties of the Heisenberg model. This gap is the following one: in Ref. [3] a scenario was proposed where the 2D Heisenberg model should undergo a KT phase transition at a finite temperature. This sce- nario is based mainly on three hypotheses, the third one (which states the non–percolation of the –type or equatorial clusters) being left in [3] without a plausible justification. To back up that hypothesis a numerical test was cited in [3] but the details of the numerics (temperature, size of the lattice, etc.) and several data concerning the percolation properties of the system, were completely skipped. The only quoted result was (see beginning of section 4 in [3]) “We also tested numerically for ,... There is no indication of percolation...”. On the contrary, such inter- esting results about the critical properties should be put forward with a thorough description of the hypotheses involved. Moreover, one would like to understand how was possible to use the small value of epsilon mentioned in Ref. [3], because that value implies a really tiny temper- ature and consequently it requires a huge lattice size. If “Everybody agrees that at the standard action model has a finite correla- tion length”, see [1], also everybody would like to know details about the numerics and the computer used to simulate the model at such a small temperature.	<div class="pdf-page"> <h1>Reply to A. Patrascioiu’s and E. Seiler’s comment on our paper</h1> <h1>Percolation properties of the 2D Heisenberg model</h1> <p>B. Allés , J. J. Alonso , C. Criado , M. Pepe Dipartimento di Fisica, Universita di Milano-Bicocca and INFN Sezione di Milano, Milano, Italy</p> <p>Departamento de Fisica Aplicada I, Facultad de Ciencias, 29071 Malaga, Spain</p> <p>The most of the problems raised by the authors of the comment [1] about Ref. [2] are based on claims which have not been written in [2], for instance almost all the introduction and the point (1) in [1] are based on such non– existent claims.</p> <p>Instead in Ref. [2] we avoid to make claims not based on well–founded results. For instance in the abstract we write “... This result indicates how the model can avoid a previously conjectured Kosterlitz–Thouless phase transition...” and in the conclusive part we notice that “Our results exclude this massless phase for ”. Therefore it seems to us that the opening sentence in the Comment [1] “In a recent letter All´es et al. claim to show that the two dimensional classical Heisenberg model does not have a massless phase.” is strongly inadequate.</p> <p>As for the points that appear in the Comment:</p> <p>• (1) The purpose of the paper [2] is to fill a gap in the research about the critical properties of the Heisenberg model. This gap is the following one: in Ref. [3] a scenario was proposed where the 2D Heisenberg model should undergo a KT phase transition at a finite temperature. This sce- nario is based mainly on three hypotheses, the third one (which states the non–percolation of the –type or equatorial clusters) being left in [3] without a plausible justification. To back up that hypothesis a numerical test was cited in [3] but the details of the numerics (temperature, size of the lattice, etc.) and several data concerning the percolation properties of the system, were completely skipped. The only quoted result was (see beginning of section 4 in [3]) “We also tested numerically for ,... There is no indication of percolation...”. On the contrary, such inter- esting results about the critical properties should be put forward with a thorough description of the hypotheses involved. Moreover, one would like to understand how was possible to use the small value of epsilon mentioned in Ref. [3], because that value implies a really tiny temper- ature and consequently it requires a huge lattice size. If “Everybody agrees that at the standard action model has a finite correla- tion length”, see [1], also everybody would like to know details about the numerics and the computer used to simulate the model at such a small temperature.</p> </div>	<div class="pdf-page"> <div class="pdf-discarded" data-x="707" data-y="118" data-width="126" data-height="16" style="opacity: 0.5;">February 2000</div> <h1 class="pdf-title" data-x="224" data-y="183" data-width="552" data-height="19">Reply to A. Patrascioiu’s and E. Seiler’s comment on our paper</h1> <h1 class="pdf-title" data-x="184" data-y="219" data-width="630" data-height="47">Percolation properties of the 2D Heisenberg model</h1> <div class="pdf-discarded" data-x="23" data-y="266" data-width="37" data-height="408" style="opacity: 0.5;">arXiv:hep-lat/0002014v1 11 Feb 2000</div> <p class="pdf-text" data-x="272" data-y="316" data-width="454" data-height="21">B. Allés , J. J. Alonso , C. Criado , M. Pepe Dipartimento di Fisica, Universita di Milano-Bicocca and INFN Sezione di Milano, Milano, Italy</p> <p class="pdf-text" data-x="145" data-y="400" data-width="706" data-height="18">Departamento de Fisica Aplicada I, Facultad de Ciencias, 29071 Malaga, Spain</p> <p class="pdf-text" data-x="165" data-y="497" data-width="666" data-height="75">The most of the problems raised by the authors of the comment [1] about Ref. [2] are based on claims which have not been written in [2], for instance almost all the introduction and the point (1) in [1] are based on such non– existent claims.</p> <p class="pdf-text" data-x="165" data-y="579" data-width="668" data-height="150">Instead in Ref. [2] we avoid to make claims not based on well–founded results. For instance in the abstract we write “... This result indicates how the model can avoid a previously conjectured Kosterlitz–Thouless phase transition...” and in the conclusive part we notice that “Our results exclude this massless phase for ”. Therefore it seems to us that the opening sentence in the Comment [1] “In a recent letter All´es et al. claim to show that the two dimensional classical Heisenberg model does not have a massless phase.” is strongly inadequate.</p> <p class="pdf-text" data-x="195" data-y="753" data-width="404" data-height="18">As for the points that appear in the Comment:</p> <p class="pdf-text" data-x="194" data-y="795" data-width="637" data-height="132">• (1) The purpose of the paper [2] is to fill a gap in the research about the critical properties of the Heisenberg model. This gap is the following one: in Ref. [3] a scenario was proposed where the 2D Heisenberg model should undergo a KT phase transition at a finite temperature. This sce- nario is based mainly on three hypotheses, the third one (which states the non–percolation of the –type or equatorial clusters) being left in [3] without a plausible justification. To back up that hypothesis a numerical test was cited in [3] but the details of the numerics (temperature, size of the lattice, etc.) and several data concerning the percolation properties of the system, were completely skipped. The only quoted result was (see beginning of section 4 in [3]) “We also tested numerically for ,... There is no indication of percolation...”. On the contrary, such inter- esting results about the critical properties should be put forward with a thorough description of the hypotheses involved. Moreover, one would like to understand how was possible to use the small value of epsilon mentioned in Ref. [3], because that value implies a really tiny temper- ature and consequently it requires a huge lattice size. If “Everybody agrees that at the standard action model has a finite correla- tion length”, see [1], also everybody would like to know details about the numerics and the computer used to simulate the model at such a small temperature.</p> </div>	{ "type": [ "text", "text", "text", "inline_equation", "inline_equation", "text", "inline_equation", "text", "text", "text", "text", "text", "text", "inline_equation", "text", "inline_equation", "text", "text", "text", "text", "text", "text", "text", "text", "text", "inline_equation", "text", "text", "text", "text", "inline_equation", "text", "text", "text", "text", "text", "inline_equation", "inline_equation", "text", "text", "text" ], "coordinates": [ [ 225, 187, 774, 205 ], [ 182, 222, 816, 250 ], [ 455, 245, 545, 268 ], [ 272, 321, 724, 340 ], [ 267, 356, 734, 378 ], [ 312, 378, 686, 394 ], [ 152, 398, 849, 426 ], [ 194, 500, 831, 521 ], [ 165, 518, 833, 539 ], [ 167, 539, 831, 557 ], [ 167, 558, 299, 574 ], [ 195, 583, 831, 599 ], [ 165, 599, 831, 619 ], [ 165, 619, 833, 638 ], [ 165, 637, 833, 658 ], [ 165, 655, 833, 677 ], [ 165, 676, 831, 694 ], [ 163, 693, 831, 713 ], [ 165, 713, 431, 733 ], [ 197, 756, 600, 773 ], [ 197, 799, 831, 817 ], [ 214, 818, 833, 836 ], [ 214, 837, 833, 854 ], [ 214, 854, 831, 872 ], [ 215, 874, 831, 892 ], [ 214, 892, 831, 911 ], [ 215, 911, 831, 928 ], [ 214, 121, 834, 138 ], [ 214, 139, 831, 157 ], [ 214, 159, 833, 177 ], [ 214, 175, 831, 196 ], [ 214, 195, 829, 213 ], [ 215, 215, 834, 232 ], [ 214, 234, 831, 250 ], [ 214, 250, 831, 270 ], [ 214, 270, 831, 288 ], [ 215, 289, 831, 306 ], [ 214, 309, 831, 324 ], [ 214, 327, 834, 346 ], [ 214, 346, 833, 362 ], [ 214, 364, 327, 382 ] ], "content": [ "Reply to A. 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0002014v1	1	[ 612, 792 ]	{ "type": [ "text", "text", "text", "text", "text", "discarded" ], "coordinates": [ [ 214, 117, 833, 378 ], [ 195, 395, 833, 490 ], [ 212, 496, 834, 700 ], [ 195, 717, 836, 893 ], [ 197, 908, 831, 927 ], [ 491, 950, 505, 963 ] ], "content": [ "", "• (2) There is a statement in [2] which is repeated several times: all results are valid for any versor of the internal symmetry space . In par- ticular, a percolating equatorial cluster is found for every . Under these conditions, we do not see how the percolation of the equatorial cluster may lead to a breaking of the symmetry.", "On the other hand, the fractal properties of a cluster are very sensitive to the choice of parameters. By varying around the value (for 0.5), one can make the data for in Table 1 of [2] to change rather dramatically. It is important (even in the case of a high temperature regime, like ) to study this dependence. It is sensible to expect that the fractal properties of the cluster show up at the threshold of percolation. Again in [2] we do not claim that the cluster is a fractal, but just write “... [the equatorial clusters] present a high degree of roughness recalling a fractal structure”. To state any firmer claim, a deep analysis of the errors and better statistics in Table 1 should be performed. All these problems are currently investigated.", "• (3) It is true that not all flimsy clusters can avoid a KT transition. However this trivial truth proves nothing. Other kinds of lattices can hold versions of the model with no transition (see for instance [4]). On the other hand, the statement “... there should be no doubt that on such a lattice [square holes of side length the model has a KT phase transition for any finite is surprising. In Ref. [5] it is shown that for any finite the KT transition is still present but it approaches as becomes larger. The idea of a fractal as the limit of some kind of cluster should not be forgotten.", "(4) We agree with one of the sentences of this point: “It would be in- teresting to verify this [the existence of a KT transition for models on a fractal lattice]”. Yet we do not see the relevance of such an obvious claim.", "2" ], "index": [ 0, 1, 2, 3, 4, 5 ] }	[{"type": "text", "text": "", "page_idx": 1}, {"type": "text", "text": "\u2022 (2) There is a statement in [2] which is repeated several times: all results are valid for any versor $\\vec{n}$ of the internal symmetry space $O(3)$ . In particular, a percolating equatorial cluster is found for every $\\vec{n}$ . Under these conditions, we do not see how the percolation of the equatorial cluster may lead to a breaking of the $O(3)$ symmetry. ", "page_idx": 1}, {"type": "text", "text": "On the other hand, the fractal properties of a cluster are very sensitive to the choice of parameters. By varying $\\epsilon$ around the value $\\epsilon=1$ (for $T=$ 0.5), one can make the data for $\\langle M_{S}\\rangle/L^{2}$ in Table 1 of [2] to change rather dramatically. It is important (even in the case of a high temperature regime, like $T=0.5$ ) to study this dependence. It is sensible to expect that the fractal properties of the cluster show up at the threshold of percolation. Again in [2] we do not claim that the cluster is a fractal, but just write \u201c... [the equatorial clusters] present a high degree of roughness recalling a fractal structure\u201d. To state any firmer claim, a deep analysis of the errors and better statistics in Table 1 should be performed. All these problems are currently investigated. ", "page_idx": 1}, {"type": "text", "text": "\u2022 (3) It is true that not all flimsy clusters can avoid a KT transition. However this trivial truth proves nothing. Other kinds of lattices can hold versions of the $X Y$ model with no transition (see for instance [4]). On the other hand, the statement \u201c... there should be no doubt that on such a lattice [square holes of side length $L]$ the $O(2)$ model has a KT phase transition for any finite $L^{\\gamma}$ is surprising. In Ref. [5] it is shown that for any finite $L$ the KT transition is still present but it approaches $T=0$ as $L$ becomes larger. The idea of a fractal as the limit of some kind of cluster should not be forgotten. ", "page_idx": 1}, {"type": "text", "text": "(4) We agree with one of the sentences of this point: \u201cIt would be interesting to verify this [the existence of a KT transition for $X Y$ models on a fractal lattice]\u201d. Yet we do not see the relevance of such an obvious claim. 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Again in [2] we do not claim that the cluster is a fractal, but", "type": "text"}], "index": 25}, {"bbox": [127, 487, 498, 502], "spans": [{"bbox": [127, 487, 498, 502], "score": 1.0, "content": "just write \u201c... [the equatorial clusters] present a high degree of roughness", "type": "text"}], "index": 26}, {"bbox": [128, 501, 498, 516], "spans": [{"bbox": [128, 501, 498, 516], "score": 1.0, "content": "recalling a fractal structure\u201d. To state any firmer claim, a deep analysis", "type": "text"}], "index": 27}, {"bbox": [128, 516, 498, 529], "spans": [{"bbox": [128, 516, 498, 529], "score": 1.0, "content": "of the errors and better statistics in Table 1 should be performed. All", "type": "text"}], "index": 28}, {"bbox": [128, 530, 342, 544], "spans": [{"bbox": [128, 530, 342, 544], "score": 1.0, "content": "these problems are currently investigated.", "type": "text"}], "index": 29}], "index": 24}, {"type": "text", "bbox": [117, 555, 500, 691], "lines": [{"bbox": [120, 558, 497, 571], "spans": [{"bbox": [120, 558, 497, 571], "score": 1.0, "content": "\u2022 (3) It is true that not all flimsy clusters can avoid a KT transition.", "type": "text"}], "index": 30}, {"bbox": [128, 572, 498, 586], "spans": [{"bbox": [128, 572, 498, 586], "score": 1.0, "content": "However this trivial truth proves nothing. Other kinds of lattices can", "type": "text"}], "index": 31}, {"bbox": [128, 586, 494, 600], "spans": [{"bbox": [128, 586, 232, 600], "score": 1.0, "content": "hold versions of the ", "type": "text"}, {"bbox": [232, 588, 253, 597], "score": 0.9, "content": "X Y", "type": "inline_equation", "height": 9, "width": 21}, {"bbox": [253, 586, 494, 600], "score": 1.0, "content": " model with no transition (see for instance [4]).", "type": "text"}], "index": 32}, {"bbox": [129, 606, 498, 619], "spans": [{"bbox": [129, 606, 498, 619], "score": 1.0, "content": "On the other hand, the statement \u201c... there should be no doubt that on", "type": "text"}], "index": 33}, {"bbox": [128, 620, 497, 634], "spans": [{"bbox": [128, 620, 347, 634], "score": 1.0, "content": "such a lattice [square holes of side length ", "type": "text"}, {"bbox": [347, 621, 358, 633], "score": 0.5, "content": "L]", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [359, 620, 383, 634], "score": 1.0, "content": " the ", "type": "text"}, {"bbox": [383, 621, 408, 634], "score": 0.94, "content": "O(2)", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [408, 620, 497, 634], "score": 1.0, "content": " model has a KT", "type": "text"}], "index": 34}, {"bbox": [128, 635, 498, 649], "spans": [{"bbox": [128, 635, 289, 649], "score": 1.0, "content": "phase transition for any finite ", "type": "text"}, {"bbox": [289, 636, 302, 645], "score": 0.88, "content": "L^{\\gamma}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [303, 635, 498, 649], "score": 1.0, "content": " is surprising. In Ref. [5] it is shown", "type": "text"}], "index": 35}, {"bbox": [128, 649, 497, 662], "spans": [{"bbox": [128, 649, 224, 662], "score": 1.0, "content": "that for any finite ", "type": "text"}, {"bbox": [225, 651, 232, 659], "score": 0.9, "content": "L", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [233, 649, 497, 662], "score": 1.0, "content": " the KT transition is still present but it approaches", "type": "text"}], "index": 36}, {"bbox": [129, 663, 498, 677], "spans": [{"bbox": [129, 665, 162, 674], "score": 0.93, "content": "T=0", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [162, 663, 181, 677], "score": 1.0, "content": " as ", "type": "text"}, {"bbox": [181, 665, 189, 674], "score": 0.91, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [190, 663, 498, 677], "score": 1.0, "content": " becomes larger. The idea of a fractal as the limit of some", "type": "text"}], "index": 37}, {"bbox": [127, 677, 330, 692], "spans": [{"bbox": [127, 677, 330, 692], "score": 1.0, "content": "kind of cluster should not be forgotten.", "type": "text"}], "index": 38}], "index": 34}, {"type": "text", "bbox": [118, 703, 497, 717], "lines": [{"bbox": [121, 704, 497, 719], "spans": [{"bbox": [121, 704, 497, 719], "score": 1.0, "content": " (4) We agree with one of the sentences of this point: \u201cIt would be in-", "type": "text"}], "index": 39}], "index": 39}], "layout_bboxes": [], "page_idx": 1, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [294, 735, 302, 745], "lines": [{"bbox": [294, 735, 303, 748], "spans": [{"bbox": [294, 735, 303, 748], "score": 1.0, "content": "2", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [128, 91, 498, 293], "lines": [], "index": 6.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [128, 94, 499, 296], "lines_deleted": true}, {"type": "text", "bbox": [117, 306, 498, 379], "lines": [{"bbox": [118, 309, 498, 322], "spans": [{"bbox": [118, 309, 498, 322], "score": 1.0, "content": "\u2022 (2) There is a statement in [2] which is repeated several times: all results", "type": "text"}], "index": 14}, {"bbox": [128, 324, 497, 338], "spans": [{"bbox": [128, 324, 252, 338], "score": 1.0, "content": "are valid for any versor ", "type": "text"}, {"bbox": [252, 325, 259, 334], "score": 0.9, "content": "\\vec{n}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [260, 324, 427, 338], "score": 1.0, "content": " of the internal symmetry space ", "type": "text"}, {"bbox": [427, 325, 452, 337], "score": 0.95, "content": "O(3)", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [452, 324, 497, 338], "score": 1.0, "content": ". In par-", "type": "text"}], "index": 15}, {"bbox": [128, 338, 497, 352], "spans": [{"bbox": [128, 338, 420, 352], "score": 1.0, "content": "ticular, a percolating equatorial cluster is found for every ", "type": "text"}, {"bbox": [420, 339, 428, 348], "score": 0.9, "content": "\\vec{n}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [428, 338, 497, 352], "score": 1.0, "content": ". Under these", "type": "text"}], "index": 16}, {"bbox": [129, 353, 497, 366], "spans": [{"bbox": [129, 353, 497, 366], "score": 1.0, "content": "conditions, we do not see how the percolation of the equatorial cluster", "type": "text"}], "index": 17}, {"bbox": [129, 367, 366, 381], "spans": [{"bbox": [129, 367, 284, 381], "score": 1.0, "content": "may lead to a breaking of the ", "type": "text"}, {"bbox": [284, 368, 309, 380], "score": 0.94, "content": "O(3)", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [309, 367, 366, 381], "score": 1.0, "content": " symmetry.", "type": "text"}], "index": 18}], "index": 16, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [118, 309, 498, 381]}, {"type": "text", "bbox": [127, 384, 499, 542], "lines": [{"bbox": [129, 386, 498, 399], "spans": [{"bbox": [129, 386, 498, 399], "score": 1.0, "content": "On the other hand, the fractal properties of a cluster are very sensitive to", "type": "text"}], "index": 19}, {"bbox": [128, 399, 498, 416], "spans": [{"bbox": [128, 399, 323, 416], "score": 1.0, "content": "the choice of parameters. By varying ", "type": "text"}, {"bbox": [323, 405, 328, 411], "score": 0.85, "content": "\\epsilon", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [329, 399, 422, 416], "score": 1.0, "content": " around the value ", "type": "text"}, {"bbox": [422, 402, 449, 411], "score": 0.91, "content": "\\epsilon=1", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [450, 399, 475, 416], "score": 1.0, "content": " (for ", "type": "text"}, {"bbox": [475, 402, 498, 412], "score": 0.83, "content": "T=", "type": "inline_equation", "height": 10, "width": 23}], "index": 20}, {"bbox": [128, 415, 497, 429], "spans": [{"bbox": [128, 415, 286, 429], "score": 1.0, "content": "0.5), one can make the data for ", "type": "text"}, {"bbox": [286, 415, 332, 428], "score": 0.94, "content": "\\langle M_{S}\\rangle/L^{2}", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [332, 415, 497, 429], "score": 1.0, "content": " in Table 1 of [2] to change rather", "type": "text"}], "index": 21}, {"bbox": [127, 428, 498, 444], "spans": [{"bbox": [127, 428, 498, 444], "score": 1.0, "content": "dramatically. It is important (even in the case of a high temperature", "type": "text"}], "index": 22}, {"bbox": [129, 444, 498, 458], "spans": [{"bbox": [129, 444, 192, 458], "score": 1.0, "content": "regime, like ", "type": "text"}, {"bbox": [192, 445, 232, 454], "score": 0.86, "content": "T=0.5", "type": "inline_equation", "height": 9, "width": 40}, {"bbox": [233, 444, 498, 458], "score": 1.0, "content": ") to study this dependence. It is sensible to expect", "type": "text"}], "index": 23}, {"bbox": [127, 457, 500, 472], "spans": [{"bbox": [127, 457, 500, 472], "score": 1.0, "content": "that the fractal properties of the cluster show up at the threshold of", "type": "text"}], "index": 24}, {"bbox": [127, 473, 498, 486], "spans": [{"bbox": [127, 473, 498, 486], "score": 1.0, "content": "percolation. 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All", "type": "text"}], "index": 28}, {"bbox": [128, 530, 342, 544], "spans": [{"bbox": [128, 530, 342, 544], "score": 1.0, "content": "these problems are currently investigated.", "type": "text"}], "index": 29}], "index": 24, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [127, 386, 500, 544]}, {"type": "text", "bbox": [117, 555, 500, 691], "lines": [{"bbox": [120, 558, 497, 571], "spans": [{"bbox": [120, 558, 497, 571], "score": 1.0, "content": "\u2022 (3) It is true that not all flimsy clusters can avoid a KT transition.", "type": "text"}], "index": 30}, {"bbox": [128, 572, 498, 586], "spans": [{"bbox": [128, 572, 498, 586], "score": 1.0, "content": "However this trivial truth proves nothing. 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"score": 1.0, "text": ""}], "page_info": {"page_no": 1, "height": 2200, "width": 1700}}	• (2) There is a statement in [2] which is repeated several times: all results are valid for any versor of the internal symmetry space . In par- ticular, a percolating equatorial cluster is found for every . Under these conditions, we do not see how the percolation of the equatorial cluster may lead to a breaking of the symmetry. On the other hand, the fractal properties of a cluster are very sensitive to the choice of parameters. By varying around the value (for 0.5), one can make the data for in Table 1 of [2] to change rather dramatically. It is important (even in the case of a high temperature regime, like ) to study this dependence. It is sensible to expect that the fractal properties of the cluster show up at the threshold of percolation. Again in [2] we do not claim that the cluster is a fractal, but just write “... [the equatorial clusters] present a high degree of roughness recalling a fractal structure”. To state any firmer claim, a deep analysis of the errors and better statistics in Table 1 should be performed. All these problems are currently investigated. • (3) It is true that not all flimsy clusters can avoid a KT transition. However this trivial truth proves nothing. Other kinds of lattices can hold versions of the model with no transition (see for instance [4]). On the other hand, the statement “... there should be no doubt that on such a lattice [square holes of side length the model has a KT phase transition for any finite is surprising. In Ref. [5] it is shown that for any finite the KT transition is still present but it approaches as becomes larger. The idea of a fractal as the limit of some kind of cluster should not be forgotten. (4) We agree with one of the sentences of this point: “It would be in- teresting to verify this [the existence of a KT transition for models on a fractal lattice]”. Yet we do not see the relevance of such an obvious claim. 2	<div class="pdf-page"> <p>• (2) There is a statement in [2] which is repeated several times: all results are valid for any versor of the internal symmetry space . In par- ticular, a percolating equatorial cluster is found for every . Under these conditions, we do not see how the percolation of the equatorial cluster may lead to a breaking of the symmetry.</p> <p>On the other hand, the fractal properties of a cluster are very sensitive to the choice of parameters. By varying around the value (for 0.5), one can make the data for in Table 1 of [2] to change rather dramatically. It is important (even in the case of a high temperature regime, like ) to study this dependence. It is sensible to expect that the fractal properties of the cluster show up at the threshold of percolation. Again in [2] we do not claim that the cluster is a fractal, but just write “... [the equatorial clusters] present a high degree of roughness recalling a fractal structure”. To state any firmer claim, a deep analysis of the errors and better statistics in Table 1 should be performed. All these problems are currently investigated.</p> <p>• (3) It is true that not all flimsy clusters can avoid a KT transition. However this trivial truth proves nothing. Other kinds of lattices can hold versions of the model with no transition (see for instance [4]). On the other hand, the statement “... there should be no doubt that on such a lattice [square holes of side length the model has a KT phase transition for any finite is surprising. In Ref. [5] it is shown that for any finite the KT transition is still present but it approaches as becomes larger. The idea of a fractal as the limit of some kind of cluster should not be forgotten.</p> <p>(4) We agree with one of the sentences of this point: “It would be in- teresting to verify this [the existence of a KT transition for models on a fractal lattice]”. Yet we do not see the relevance of such an obvious claim.</p> </div>	<div class="pdf-page"> <p class="pdf-text" data-x="195" data-y="395" data-width="638" data-height="95">• (2) There is a statement in [2] which is repeated several times: all results are valid for any versor of the internal symmetry space . In par- ticular, a percolating equatorial cluster is found for every . Under these conditions, we do not see how the percolation of the equatorial cluster may lead to a breaking of the symmetry.</p> <p class="pdf-text" data-x="212" data-y="496" data-width="622" data-height="204">On the other hand, the fractal properties of a cluster are very sensitive to the choice of parameters. By varying around the value (for 0.5), one can make the data for in Table 1 of [2] to change rather dramatically. It is important (even in the case of a high temperature regime, like ) to study this dependence. It is sensible to expect that the fractal properties of the cluster show up at the threshold of percolation. Again in [2] we do not claim that the cluster is a fractal, but just write “... [the equatorial clusters] present a high degree of roughness recalling a fractal structure”. To state any firmer claim, a deep analysis of the errors and better statistics in Table 1 should be performed. All these problems are currently investigated.</p> <p class="pdf-text" data-x="195" data-y="717" data-width="641" data-height="176">• (3) It is true that not all flimsy clusters can avoid a KT transition. However this trivial truth proves nothing. Other kinds of lattices can hold versions of the model with no transition (see for instance [4]). On the other hand, the statement “... there should be no doubt that on such a lattice [square holes of side length the model has a KT phase transition for any finite is surprising. In Ref. [5] it is shown that for any finite the KT transition is still present but it approaches as becomes larger. The idea of a fractal as the limit of some kind of cluster should not be forgotten.</p> <p class="pdf-text" data-x="197" data-y="908" data-width="634" data-height="19">(4) We agree with one of the sentences of this point: “It would be in- teresting to verify this [the existence of a KT transition for models on a fractal lattice]”. 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To state any firmer claim, a deep analysis", "of the errors and better statistics in Table 1 should be performed. All", "these problems are currently investigated.", "• (3) It is true that not all flimsy clusters can avoid a KT transition.", "However this trivial truth proves nothing. Other kinds of lattices can", "hold versions of the X Y model with no transition (see for instance [4]).", "On the other hand, the statement “... there should be no doubt that on", "such a lattice [square holes of side length L] the O(2) model has a KT", "phase transition for any finite L^{\\gamma} is surprising. In Ref. [5] it is shown", "that for any finite L the KT transition is still present but it approaches", "T=0 as L becomes larger. The idea of a fractal as the limit of some", "kind of cluster should not be forgotten.", "(4) We agree with one of the sentences of this point: “It would be in-", "teresting to verify this [the existence of a KT transition for X Y models", "on a fractal lattice]”. 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0002014v1	2	[ 612, 792 ]	{ "type": [ "text", "text", "title", "list", "discarded" ], "coordinates": [ [ 214, 117, 833, 173 ], [ 214, 181, 833, 292 ], [ 165, 380, 319, 403 ], [ 174, 422, 833, 592 ], [ 491, 949, 506, 963 ] ], "content": [ "", "We disagree however with the authors of [1] when they say “our ar- gument does not depend on the existence of such a transition on that particular percolating cluster”. Instead, after the conclusions of Ref. [2], we think that the non–rigorous proof proposed in [3] for the case when the equatorial cluster does percolate, heavily lies on whether or not such a transition is realized.", "References", "[1] A. Patrascioiu and E. Seiler, unpublished report hep–lat/9912014 (v1). [2] B. All´es, J.J. Alonso, C. Criado and M. Pepe, Phys. Rev. Lett. 83 (1999) 3669. [3] A. Patrascioiu and E. Seiler, Nucl. Phys. B (Proc. Suppl.) 30 (1993) 184. [4] Yu.E. Lozovik and L.M. Pomirchy, Solid State Comm. 89 (1994) 145. [5] P. Minnhagen and H. Weber, Physica B152 (1988) 50.", "3" ], "index": [ 0, 1, 2, 3, 4 ] }	[{"type": "text", "text": "", "page_idx": 2}, {"type": "text", "text": "We disagree however with the authors of [1] when they say \u201cour argument does not depend on the existence of such a transition on that particular percolating cluster\u201d. Instead, after the conclusions of Ref. [2], we think that the non\u2013rigorous proof proposed in [3] for the case when the equatorial cluster does percolate, heavily lies on whether or not such a transition is realized. ", "page_idx": 2}, {"type": "text", "text": "References ", "text_level": 1, "page_idx": 2}, {"type": "text", "text": "[1] A. Patrascioiu and E. Seiler, unpublished report hep\u2013lat/9912014 (v1). \n[2] B. All\u00b4es, J.J. Alonso, C. Criado and M. Pepe, Phys. Rev. Lett. 83 (1999) 3669. \n[3] A. Patrascioiu and E. Seiler, Nucl. Phys. B (Proc. Suppl.) 30 (1993) 184. \n[4] Yu.E. Lozovik and L.M. Pomirchy, Solid State Comm. 89 (1994) 145. \n[5] P. 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Instead, after the conclusions of Ref. [2], we think that the non–rigorous proof proposed in [3] for the case when the equatorial cluster does percolate, heavily lies on whether or not such a transition is realized. # References - [1] A. Patrascioiu and E. Seiler, unpublished report hep–lat/9912014 (v1). [2] B. All´es, J.J. Alonso, C. Criado and M. Pepe, Phys. Rev. Lett. 83 (1999) 3669. [3] A. Patrascioiu and E. Seiler, Nucl. Phys. B (Proc. Suppl.) 30 (1993) 184. [4] Yu.E. Lozovik and L.M. Pomirchy, Solid State Comm. 89 (1994) 145. [5] P. Minnhagen and H. Weber, Physica B152 (1988) 50. 3	<div class="pdf-page"> <p>We disagree however with the authors of [1] when they say “our ar- gument does not depend on the existence of such a transition on that particular percolating cluster”. Instead, after the conclusions of Ref. [2], we think that the non–rigorous proof proposed in [3] for the case when the equatorial cluster does percolate, heavily lies on whether or not such a transition is realized.</p> <h1>References</h1> <ul> <li>[1] A. Patrascioiu and E. Seiler, unpublished report hep–lat/9912014 (v1). [2] B. All´es, J.J. Alonso, C. Criado and M. Pepe, Phys. Rev. Lett. 83 (1999) 3669. [3] A. Patrascioiu and E. Seiler, Nucl. Phys. B (Proc. Suppl.) 30 (1993) 184. [4] Yu.E. Lozovik and L.M. Pomirchy, Solid State Comm. 89 (1994) 145. [5] P. Minnhagen and H. Weber, Physica B152 (1988) 50.</li> </ul> </div>	<div class="pdf-page"> <p class="pdf-text" data-x="214" data-y="181" data-width="619" data-height="111">We disagree however with the authors of [1] when they say “our ar- gument does not depend on the existence of such a transition on that particular percolating cluster”. Instead, after the conclusions of Ref. [2], we think that the non–rigorous proof proposed in [3] for the case when the equatorial cluster does percolate, heavily lies on whether or not such a transition is realized.</p> <h1 class="pdf-title" data-x="165" data-y="380" data-width="154" data-height="23">References</h1> <ul class="pdf-list" data-x="174" data-y="422" data-width="659" data-height="170"> <li>[1] A. Patrascioiu and E. Seiler, unpublished report hep–lat/9912014 (v1). [2] B. All´es, J.J. Alonso, C. Criado and M. Pepe, Phys. Rev. Lett. 83 (1999) 3669. [3] A. Patrascioiu and E. Seiler, Nucl. Phys. B (Proc. Suppl.) 30 (1993) 184. [4] Yu.E. Lozovik and L.M. Pomirchy, Solid State Comm. 89 (1994) 145. [5] P. Minnhagen and H. Weber, Physica B152 (1988) 50.</li> </ul> <div class="pdf-discarded" data-x="491" data-y="949" data-width="15" data-height="14" style="opacity: 0.5;">3</div> </div>	{ "type": [ "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text", "text" ], "coordinates": [ [ 215, 183, 831, 201 ], [ 212, 202, 833, 219 ], [ 214, 222, 829, 239 ], [ 214, 240, 831, 257 ], [ 214, 258, 831, 275 ], [ 214, 277, 413, 293 ], [ 165, 384, 321, 404 ], [ 175, 429, 814, 448 ], [ 175, 458, 829, 480 ], [ 205, 480, 252, 497 ], [ 175, 510, 831, 532 ], [ 175, 543, 799, 561 ], [ 177, 574, 672, 593 ] ], "content": [ "We disagree however with the authors of [1] when they say “our ar-", "gument does not depend on the existence of such a transition on that", "particular percolating cluster”. Instead, after the conclusions of Ref. [2],", "we think that the non–rigorous proof proposed in [3] for the case when", "the equatorial cluster does percolate, heavily lies on whether or not such", "a transition is realized.", "References", "[1] A. Patrascioiu and E. Seiler, unpublished report hep–lat/9912014 (v1).", "[2] B. All´es, J.J. Alonso, C. Criado and M. Pepe, Phys. Rev. Lett. 83 (1999)", "3669.", "[3] A. Patrascioiu and E. Seiler, Nucl. Phys. B (Proc. Suppl.) 30 (1993) 184.", "[4] Yu.E. Lozovik and L.M. Pomirchy, Solid State Comm. 89 (1994) 145.", "[5] P. Minnhagen and H. Weber, Physica B152 (1988) 50." ], "index": [ 0, 1, 2, 3, 4, 5, 6, 13, 14, 15, 16, 17, 18 ] }	{ "coordinates": [], "index": [], "caption": [], "caption_coordinates": [] }	{ "type": [], "coordinates": [], "content": [], "index": [] }	{ "type": [], "coordinates": [], "content": [], "index": [] }	{"document_info": {"total_pages": 3, "parse_type": "txt", "version": "1.0.1", "total_content_list_items": 18}, "content_statistics": {"total_blocks": 22, "total_lines": 83, "total_images": 0, "total_equations": 0, "total_tables": 0, "total_characters": 5356, "total_words": 926, "block_type_distribution": {"title": 3, "text": 14, "discarded": 4, "list": 1}}, "model_statistics": {"total_layout_detections": 183, "avg_confidence": 0.936224043715847, "min_confidence": 0.26, "max_confidence": 1.0}, "page_statistics": [{"page_number": 0, "page_size": [612.0, 792.0], "blocks": 11, "lines": 41, "images": 0, "equations": 0, "tables": 0, "characters": 2670, "words": 449, "discarded_blocks": 2}, {"page_number": 1, "page_size": [612.0, 792.0], "blocks": 6, "lines": 29, "images": 0, "equations": 0, "tables": 0, "characters": 1930, "words": 348, "discarded_blocks": 1}, {"page_number": 2, "page_size": [612.0, 792.0], "blocks": 5, "lines": 13, "images": 0, "equations": 0, "tables": 0, "characters": 756, "words": 129, "discarded_blocks": 1}]}

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