Abbey Bourdon, Sachi Hashimoto, Timo Keller, Z. Klagsbrun, David Lowry-Duda, Travis Morrison, Filip Najman, Himanshu Shukla
{"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":"52 29","pages":""},"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}
引用次数: 0
Abstract
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.
期刊介绍:
ACS Applied Bio Materials is an interdisciplinary journal publishing original research covering all aspects of biomaterials and biointerfaces including and beyond the traditional biosensing, biomedical and therapeutic applications.
The journal is devoted to reports of new and original experimental and theoretical research of an applied nature that integrates knowledge in the areas of materials, engineering, physics, bioscience, and chemistry into important bio applications. The journal is specifically interested in work that addresses the relationship between structure and function and assesses the stability and degradation of materials under relevant environmental and biological conditions.