{"title":"Ideal Class Groups of Number Fields and Bloch-Kato's Tate-Shafarevich Groups for Symmetric Powers of Elliptic Curves","authors":"Naoto Dainobu","doi":"10.3836/tjm/1502179361","DOIUrl":null,"url":null,"abstract":". For an elliptic curve E over Q , putting K = Q ( E [ p ]) which is the p -th division field of E for an odd prime p , we study the ideal class group Cl K of K as a Gal( K/ Q ) -module. More precisely, for any j with 1 6 j 6 p − 2 , we give a condition that Cl K ⊗ F p has the symmetric power Sym j E [ p ] of E [ p ] as its quotient Gal( K/ Q ) -module, in terms of Bloch-Kato’s Tate-Shafarevich group of Sym j V p E . Here V p E denotes the rational p -adic Tate module of E . This is a partial generalization of a result of Prasad and Shekhar for the case j = 1 .","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3836/tjm/1502179361","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 2
Abstract
. For an elliptic curve E over Q , putting K = Q ( E [ p ]) which is the p -th division field of E for an odd prime p , we study the ideal class group Cl K of K as a Gal( K/ Q ) -module. More precisely, for any j with 1 6 j 6 p − 2 , we give a condition that Cl K ⊗ F p has the symmetric power Sym j E [ p ] of E [ p ] as its quotient Gal( K/ Q ) -module, in terms of Bloch-Kato’s Tate-Shafarevich group of Sym j V p E . Here V p E denotes the rational p -adic Tate module of E . This is a partial generalization of a result of Prasad and Shekhar for the case j = 1 .