计算模曲线上的二次点𝑋0(二进制)

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS

ACS Applied Bio Materials Pub Date : 2023-10-03 DOI:10.1090/mcom/3902

Nikola Adžaga, Timo Keller, Philippe Michaud-Jacobs, Filip Najman, Ekin Ozman, Borna Vukorepa

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The values of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N\"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding=\"application/x-tex\">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we consider are contained in the set <disp-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper L equals StartSet 58 comma 68 comma 74 comma 76 comma 80 comma 85 comma 97 comma 98 comma 100 comma 103 comma 107 comma 109 comma 113 comma 121 comma 127 EndSet period\"> <mml:semantics> <mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">L</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:mn>58</mml:mn> <mml:mo>,</mml:mo> <mml:mn>68</mml:mn> <mml:mo>,</mml:mo> <mml:mn>74</mml:mn> <mml:mo>,</mml:mo> <mml:mn>76</mml:mn> <mml:mo>,</mml:mo> <mml:mn>80</mml:mn> <mml:mo>,</mml:mo> <mml:mn>85</mml:mn> <mml:mo>,</mml:mo> <mml:mn>97</mml:mn> <mml:mo>,</mml:mo> <mml:mn>98</mml:mn> <mml:mo>,</mml:mo> <mml:mn>100</mml:mn> <mml:mo>,</mml:mo> <mml:mn>103</mml:mn> <mml:mo>,</mml:mo> <mml:mn>107</mml:mn> <mml:mo>,</mml:mo> <mml:mn>109</mml:mn> <mml:mo>,</mml:mo> <mml:mn>113</mml:mn> <mml:mo>,</mml:mo> <mml:mn>121</mml:mn> <mml:mo>,</mml:mo> <mml:mn>127</mml:mn> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\begin{equation*} \\mathcal {L}=\\{58, 68, 74, 76, 80, 85, 97, 98, 100, 103, 107, 109, 113, 121, 127 \\}. \\end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> We obtain that all the non-cuspidal quadratic points on <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X 0 left-parenthesis upper N right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">X_0(N)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N element-of script upper L\"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">L</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">N\\in \\mathcal {L}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are complex multiplication (CM) points, except for one pair of Galois conjugate points on <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X 0 left-parenthesis 103 right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mn>103</mml:mn> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">X_0(103)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> defined over <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q left-parenthesis StartRoot 2885 EndRoot right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">Q</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msqrt> <mml:mn>2885</mml:mn> </mml:msqrt> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mathbb {Q}(\\sqrt {2885})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also compute the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"j\"> <mml:semantics> <mml:mi>j</mml:mi> <mml:annotation encoding=\"application/x-tex\">j</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-invariants of the elliptic curves parametrised by these points, and for the CM points determine their geometric endomorphism rings.","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":"{\"title\":\"Computing quadratic points on modular curves 𝑋₀(𝑁)\",\"authors\":\"Nikola Adžaga, Timo Keller, Philippe Michaud-Jacobs, Filip Najman, Ekin Ozman, Borna Vukorepa\",\"doi\":\"10.1090/mcom/3902\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curves <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper X 0 left-parenthesis upper N right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">X_0(N)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of genus up to <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"8\\\"> <mml:semantics> <mml:mn>8</mml:mn> <mml:annotation encoding=\\\"application/x-tex\\\">8</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and genus up to <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"10\\\"> <mml:semantics> <mml:mn>10</mml:mn> <mml:annotation encoding=\\\"application/x-tex\\\">10</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper N\\\"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> prime, for which they were previously unknown. The values of <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper N\\\"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we consider are contained in the set <disp-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script upper L equals StartSet 58 comma 68 comma 74 comma 76 comma 80 comma 85 comma 97 comma 98 comma 100 comma 103 comma 107 comma 109 comma 113 comma 121 comma 127 EndSet period\\\"> <mml:semantics> <mml:mrow> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">L</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">{</mml:mo> <mml:mn>58</mml:mn> <mml:mo>,</mml:mo> <mml:mn>68</mml:mn> <mml:mo>,</mml:mo> <mml:mn>74</mml:mn> <mml:mo>,</mml:mo> <mml:mn>76</mml:mn> <mml:mo>,</mml:mo> <mml:mn>80</mml:mn> <mml:mo>,</mml:mo> <mml:mn>85</mml:mn> <mml:mo>,</mml:mo> <mml:mn>97</mml:mn> <mml:mo>,</mml:mo> <mml:mn>98</mml:mn> <mml:mo>,</mml:mo> <mml:mn>100</mml:mn> <mml:mo>,</mml:mo> <mml:mn>103</mml:mn> <mml:mo>,</mml:mo> <mml:mn>107</mml:mn> <mml:mo>,</mml:mo> <mml:mn>109</mml:mn> <mml:mo>,</mml:mo> <mml:mn>113</mml:mn> <mml:mo>,</mml:mo> <mml:mn>121</mml:mn> <mml:mo>,</mml:mo> <mml:mn>127</mml:mn> <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">}</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\begin{equation*} \\\\mathcal {L}=\\\\{58, 68, 74, 76, 80, 85, 97, 98, 100, 103, 107, 109, 113, 121, 127 \\\\}. \\\\end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> We obtain that all the non-cuspidal quadratic points on <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper X 0 left-parenthesis upper N right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">X_0(N)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper N element-of script upper L\\\"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">L</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">N\\\\in \\\\mathcal {L}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are complex multiplication (CM) points, except for one pair of Galois conjugate points on <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper X 0 left-parenthesis 103 right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mn>103</mml:mn> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">X_0(103)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> defined over <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper Q left-parenthesis StartRoot 2885 EndRoot right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi mathvariant=\\\"double-struck\\\">Q</mml:mi> </mml:mrow> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:msqrt> <mml:mn>2885</mml:mn> </mml:msqrt> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbb {Q}(\\\\sqrt {2885})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also compute the <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"j\\\"> <mml:semantics> <mml:mi>j</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">j</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-invariants of the elliptic curves parametrised by these points, and for the CM points determine their geometric endomorphism rings.\",\"PeriodicalId\":2,\"journal\":{\"name\":\"ACS Applied Bio Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.6000,\"publicationDate\":\"2023-10-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"5\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Bio Materials\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/mcom/3902\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATERIALS SCIENCE, BIOMATERIALS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/mcom/3902","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}

引用次数: 5

摘要

本文改进了现有的求模曲线上二次点的方法，并应用这些方法成功地求出了在所有模曲线x0 (N) X＿0(N)上的格数不超过8 8和格数不超过10 10的所有N N素数的二次点，这些二次点以前是未知的。我们所考虑的N N的值包含在集合L = {58、68、74、76、80、85、97、98、100、103、107、109、113、121、127}中。\begin{equation*} \mathcal {L}=\{58, 68, 74, 76, 80, 85, 97, 98, 100, 103, 107, 109, 113, 121, 127 \}. \end{equation*}我们得到了除X 0(103) X＿0(103)上定义在Q(2885) \mathbb Q(\sqrt 2885)上的一对伽罗瓦共轭点外，对于N∈{L} N, X 0(N) X＿0(N)上的所有非尖次二次点\in{}\mathcal L都是复乘法(CM)点。我们还计算了由这些点参数化的椭圆曲线的j j不变量，并确定了CM点的几何自同态环。{}

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Computing quadratic points on modular curves 𝑋₀(𝑁)

In this paper we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curves X 0 ( N ) X_0(N) of genus up to 8 8 , and genus up to 10 10 with N N prime, for which they were previously unknown. The values of N N we consider are contained in the set L = { 58 , 68 , 74 , 76 , 80 , 85 , 97 , 98 , 100 , 103 , 107 , 109 , 113 , 121 , 127 } . \begin{equation*} \mathcal {L}=\{58, 68, 74, 76, 80, 85, 97, 98, 100, 103, 107, 109, 113, 121, 127 \}. \end{equation*} We obtain that all the non-cuspidal quadratic points on X 0 ( N ) X_0(N) for N ∈ L N\in \mathcal {L} are complex multiplication (CM) points, except for one pair of Galois conjugate points on X 0 ( 103 ) X_0(103) defined over Q ( 2885 ) \mathbb {Q}(\sqrt {2885}) . We also compute the j j -invariants of the elliptic curves parametrised by these points, and for the CM points determine their geometric endomorphism rings.

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来源期刊

ACS Applied Bio Materials Chemistry-Chemistry (all)

CiteScore

9.40

自引率

2.10%

发文量

464