Abbey Bourdon, Sachi Hashimoto, Timo Keller, Z. Klagsbrun, David Lowry-Duda, Travis Morrison, Filip Najman, Himanshu Shukla
{"title":"对孤立𝑗变量进行分类","authors":"Abbey Bourdon, Sachi Hashimoto, Timo Keller, Z. Klagsbrun, David Lowry-Duda, Travis Morrison, Filip Najman, Himanshu Shukla","doi":"10.1090/mcom/3956","DOIUrl":null,"url":null,"abstract":"<p>We develop an algorithm to test whether a non-complex multiplication elliptic curve <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper E slash bold upper Q\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>E</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"bold\">Q</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">E/\\mathbf {Q}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> gives rise to an isolated point of any degree on any modular curve of the form <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X 1 left-parenthesis upper N right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>X</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>N</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">X_1(N)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. This builds on prior work of Zywina which gives a method for computing the image of the adelic Galois representation associated to <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper E\">\n <mml:semantics>\n <mml:mi>E</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">E</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. Running this algorithm on all elliptic curves presently in the <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L\">\n <mml:semantics>\n <mml:mi>L</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">L</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-functions and Modular Forms Database and the Stein–Watkins Database gives strong evidence for the conjecture that <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper E\">\n <mml:semantics>\n <mml:mi>E</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">E</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> gives rise to an isolated point on <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X 1 left-parenthesis upper N right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>X</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>N</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">X_1(N)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> if and only if <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"j left-parenthesis upper E right-parenthesis equals negative 140625 slash 8 comma negative 9317\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>j</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>E</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>=</mml:mo>\n <mml:mo>−</mml:mo>\n <mml:mn>140625</mml:mn>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:mn>8</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mo>−</mml:mo>\n <mml:mn>9317</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">j(E)=-140625/8, -9317</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"351 slash 4\">\n <mml:semantics>\n <mml:mrow>\n <mml:mn>351</mml:mn>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:mn>4</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">351/4</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, or <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"negative 162677523113838677\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo>−</mml:mo>\n <mml:mn>162677523113838677</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">-162677523113838677</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>.</p>","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Towards a classification of isolated 𝑗-invariants\",\"authors\":\"Abbey Bourdon, Sachi Hashimoto, Timo Keller, Z. Klagsbrun, David Lowry-Duda, Travis Morrison, Filip Najman, Himanshu Shukla\",\"doi\":\"10.1090/mcom/3956\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We develop an algorithm to test whether a non-complex multiplication elliptic curve <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper E slash bold upper Q\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>E</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo>/</mml:mo>\\n </mml:mrow>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"bold\\\">Q</mml:mi>\\n </mml:mrow>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">E/\\\\mathbf {Q}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> gives rise to an isolated point of any degree on any modular curve of the form <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper X 1 left-parenthesis upper N right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msub>\\n <mml:mi>X</mml:mi>\\n <mml:mn>1</mml:mn>\\n </mml:msub>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>N</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">X_1(N)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. This builds on prior work of Zywina which gives a method for computing the image of the adelic Galois representation associated to <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper E\\\">\\n <mml:semantics>\\n <mml:mi>E</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">E</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. Running this algorithm on all elliptic curves presently in the <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper L\\\">\\n <mml:semantics>\\n <mml:mi>L</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">L</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-functions and Modular Forms Database and the Stein–Watkins Database gives strong evidence for the conjecture that <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper E\\\">\\n <mml:semantics>\\n <mml:mi>E</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">E</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> gives rise to an isolated point on <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper X 1 left-parenthesis upper N right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msub>\\n <mml:mi>X</mml:mi>\\n <mml:mn>1</mml:mn>\\n </mml:msub>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>N</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">X_1(N)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> if and only if <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"j left-parenthesis upper E right-parenthesis equals negative 140625 slash 8 comma negative 9317\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>j</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>E</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo>=</mml:mo>\\n <mml:mo>−</mml:mo>\\n <mml:mn>140625</mml:mn>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo>/</mml:mo>\\n </mml:mrow>\\n <mml:mn>8</mml:mn>\\n <mml:mo>,</mml:mo>\\n <mml:mo>−</mml:mo>\\n <mml:mn>9317</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">j(E)=-140625/8, -9317</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"351 slash 4\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mn>351</mml:mn>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo>/</mml:mo>\\n </mml:mrow>\\n <mml:mn>4</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">351/4</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, or <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"negative 162677523113838677\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mo>−</mml:mo>\\n <mml:mn>162677523113838677</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">-162677523113838677</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>.</p>\",\"PeriodicalId\":2,\"journal\":{\"name\":\"ACS Applied Bio Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.6000,\"publicationDate\":\"2024-04-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Bio Materials\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/mcom/3956\",\"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":"100","ListUrlMain":"https://doi.org/10.1090/mcom/3956","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: 0
摘要
我们开发了一种算法来检验非复数乘法椭圆曲线 E / Q E/\mathbf {Q} 是否会在任何形式为 X 1 ( N ) X_1(N)的模态曲线上产生一个任意度的孤立点。这建立在 Zywina 之前的工作基础上，Zywina 给出了一种计算与 E E 相关联的adelic伽罗瓦表示的映像的方法。在 L L 函数和模块形式数据库以及 Stein-Watkins 数据库中的所有椭圆曲线上运行这一算法，有力地证明了以下猜想：当且仅当 j ( E ) = - 140625 / 8 , - 9317 j(E)=-140625/8, -9317 , 351 / 4 351/4 , 或 - 162677523113838677 -162677523113838677 时，E E 在 X 1 ( N ) X_1(N)上产生孤立点。
We develop an algorithm to test whether a non-complex multiplication elliptic curve E/QE/\mathbf {Q} gives rise to an isolated point of any degree on any modular curve of the form X1(N)X_1(N). This builds on prior work of Zywina which gives a method for computing the image of the adelic Galois representation associated to EE. Running this algorithm on all elliptic curves presently in the LL-functions and Modular Forms Database and the Stein–Watkins Database gives strong evidence for the conjecture that EE gives rise to an isolated point on X1(N)X_1(N) if and only if j(E)=−140625/8,−9317j(E)=-140625/8, -9317, 351/4351/4, or −162677523113838677-162677523113838677.
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