{"title":"构建由3-Selmer同伴组成的家庭。","authors":"Harry Spencer","doi":"10.1007/s40993-025-00647-5","DOIUrl":null,"url":null,"abstract":"<p><p>Mazur and Rubin introduced the notion of <i>n</i>-Selmer companion elliptic curves and gave several examples of pairs of non-isogenous Selmer companions. We construct several pairs of families of elliptic curves, each parameterised by <math><mrow><mi>t</mi> <mo>∈</mo> <mi>Z</mi></mrow> </math> , such that the two curves in a pair corresponding to a given <i>t</i> are non-isogenous 3-Selmer companions, possibly provided that <i>t</i> satisfies a simple congruence condition.</p>","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"11 3","pages":"67"},"PeriodicalIF":0.8000,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12227473/pdf/","citationCount":"0","resultStr":"{\"title\":\"Constructing families of 3-Selmer companions.\",\"authors\":\"Harry Spencer\",\"doi\":\"10.1007/s40993-025-00647-5\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p><p>Mazur and Rubin introduced the notion of <i>n</i>-Selmer companion elliptic curves and gave several examples of pairs of non-isogenous Selmer companions. We construct several pairs of families of elliptic curves, each parameterised by <math><mrow><mi>t</mi> <mo>∈</mo> <mi>Z</mi></mrow> </math> , such that the two curves in a pair corresponding to a given <i>t</i> are non-isogenous 3-Selmer companions, possibly provided that <i>t</i> satisfies a simple congruence condition.</p>\",\"PeriodicalId\":43826,\"journal\":{\"name\":\"Research in Number Theory\",\"volume\":\"11 3\",\"pages\":\"67\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2025-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12227473/pdf/\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Research in Number Theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s40993-025-00647-5\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"2025/7/4 0:00:00\",\"PubModel\":\"Epub\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Research in Number Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s40993-025-00647-5","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2025/7/4 0:00:00","PubModel":"Epub","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Mazur and Rubin introduced the notion of n-Selmer companion elliptic curves and gave several examples of pairs of non-isogenous Selmer companions. We construct several pairs of families of elliptic curves, each parameterised by , such that the two curves in a pair corresponding to a given t are non-isogenous 3-Selmer companions, possibly provided that t satisfies a simple congruence condition.
期刊介绍:
Research in Number Theory is an international, peer-reviewed Hybrid Journal covering the scope of the mathematical disciplines of Number Theory and Arithmetic Geometry. The Mission of the Journal is to publish high-quality original articles that make a significant contribution to these research areas. It will also publish shorter research communications (Letters) covering nascent research in some of the burgeoning areas of number theory research. This journal publishes the highest quality papers in all of the traditional areas of number theory research, and it actively seeks to publish seminal papers in the most emerging and interdisciplinary areas here as well. Research in Number Theory also publishes comprehensive reviews.