{"title":"多二次方数域上椭圆曲线的扭转点","authors":"Koji Matsuda","doi":"10.1016/j.jnt.2024.03.018","DOIUrl":null,"url":null,"abstract":"<div><p>We compute the Mordell–Weil groups of the modular Jacobian varieties of hyperelliptic modular curves <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>M</mi><mo>,</mo><mi>M</mi><mi>N</mi><mo>)</mo></math></span> over every composite field of some quadratic number fields. Also we prove criteria for the existence of elliptic curves over such number fields with prescribed torsion points generalizing the results for quadratic number fields of Kamienny and Najman.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"262 ","pages":"Pages 28-43"},"PeriodicalIF":0.6000,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24000933/pdfft?md5=f88774ff4e0aef9e748abddb14237f52&pid=1-s2.0-S0022314X24000933-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Torsion points of elliptic curves over multi-quadratic number fields\",\"authors\":\"Koji Matsuda\",\"doi\":\"10.1016/j.jnt.2024.03.018\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We compute the Mordell–Weil groups of the modular Jacobian varieties of hyperelliptic modular curves <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>M</mi><mo>,</mo><mi>M</mi><mi>N</mi><mo>)</mo></math></span> over every composite field of some quadratic number fields. Also we prove criteria for the existence of elliptic curves over such number fields with prescribed torsion points generalizing the results for quadratic number fields of Kamienny and Najman.</p></div>\",\"PeriodicalId\":50110,\"journal\":{\"name\":\"Journal of Number Theory\",\"volume\":\"262 \",\"pages\":\"Pages 28-43\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-04-23\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0022314X24000933/pdfft?md5=f88774ff4e0aef9e748abddb14237f52&pid=1-s2.0-S0022314X24000933-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Number Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022314X24000933\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Number Theory","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X24000933","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Torsion points of elliptic curves over multi-quadratic number fields
We compute the Mordell–Weil groups of the modular Jacobian varieties of hyperelliptic modular curves over every composite field of some quadratic number fields. Also we prove criteria for the existence of elliptic curves over such number fields with prescribed torsion points generalizing the results for quadratic number fields of Kamienny and Najman.
期刊介绍：
The Journal of Number Theory (JNT) features selected research articles that represent the broad spectrum of interest in contemporary number theory and allied areas. A valuable resource for mathematicians, the journal provides an international forum for the publication of original research in this field.
The Journal of Number Theory is encouraging submissions of quality, long articles where most or all of the technical details are included. The journal now considers and welcomes also papers in Computational Number Theory.
Starting in May 2019, JNT will have a new format with 3 sections:
JNT Prime targets (possibly very long with complete proofs) high impact papers. Articles published in this section will be granted 1 year promotional open access.
JNT General Section is for shorter papers. We particularly encourage submission from junior researchers. Every attempt will be made to expedite the review process for such submissions.
Computational JNT . This section aims to provide a forum to disseminate contributions which make significant use of computer calculations to derive novel number theoretic results. There will be an online repository where supplementary codes and data can be stored.
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