On the structure of Selmer groups
type: Item
instance of: scholarly article
title: On the structure of Selmer groups
zbMATH Open document ID: 0752.11025
DOI: 10.1007/BF01445205
published in: Mathematische Annalen
publication date: 1992-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/164864
review text: Let \(E\) be a modular elliptic curve over \(\mathbb{Q}\) and \(K\) be an imaginary quadratic field of discriminant \(\equiv 0\pmod{4\times\text{conductor }E)}\). For a rational prime \(\ell\) a set \(T\subset\bigcup_ n H^ 1(K,E_{\ell^ n})\) of explicit cohomology classes was constructed earlier by the author [Euler systems, The Grothendieck Festschr., Vol. II, Prog. Math. 87, 435-483 (1990; Zbl 0742.14017)]. Let \(\ell\) be such that the image of the \(\ell\)-adic representation of \(\text{Gal}(\overline {\mathbb{Q}}/\mathbb{Q})\) corresponding to \(E\), is maximal and \(\ell\) does not divide the discriminant of \(\text{End}(E)\). All sufficiently large \(\ell\) satisfy this condition. Then, conjecturally, \(T\neq 0\). If \(T\neq 0\) we obtain some expressions for invariants of Selmer groups \(S(K,E_{\ell^ n})\) in terms of \(T\). Analogous results for the remaining primes are formulated.
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 59957
zbMATH Keywords: modular elliptic curve
zbMATH Keywords: cohomology classes
zbMATH Keywords: invariants of Selmer groups
author: Victor Kolyvagin
MaRDI profile type: Publication
cites work: Q4012045
cites work: Q3972742
cites work: Q4011129
description: scientific article; zbMATH DE number 59957