Selmer groups of quadratic twists of elliptic curves
type: Item
instance of: scholarly article
title: Selmer groups of quadratic twists of elliptic curves
zbMATH Open document ID: 0934.11031
author: Ken Ono
author: Kevin James
published in: Mathematische Annalen
publication date: 2000-04-03T00:00:00Z
review text: Let \(E/\mathbb{Q}\) be an elliptic curve and \(D\) a square-free integer. \(E(D)\) will denote the elliptic curve over \(\mathbb{Q}\) which is the \(D\)-quadratic twist of \(E\) and, for an odd prime \(l\), \(S(E(D))_l\) the \(l\)-part of its Selmer group. The authors of the paper under review are focused on the frequency of the triviality of \(S(E(D))_l\). Numerical evidence suggests that \[ \#\{|D|< X\mid D\text{ square-free and }S(E(D))_l= \{1\}\}\gg_{E,l} X. \] For every elliptic curve \(E/\mathbb{Q}\) and for every prime \(l\gg_E 1\) then holds: \[ \#\{|D|< X\mid D\text{ square-free and }S(E(D))_l= \{1\}\}\gg_{E,l} \frac{\sqrt{X}} {\log X}. \] This result does not cover the case of an odd prime \(l\) for which \(E\) has rational \(l\)-torsion, i.e. \(l=3,5,7\). In this case by invoking a theorem of Frey the authors obtain analogous results if \(l=5\) or 7 and a better bound for \(l=3\).
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11G40
zbMATH DE Number: 1337022
zbMATH Keywords: quadratic twist
zbMATH Keywords: elliptic curve
zbMATH Keywords: Selmer group
reviewed by: Jannis A. Antoniadis
MaRDI profile type: Publication
full work available at URL: s002080050283
OpenAlex ID: W2089622504
DOI: 10.1007/S002080050283
description: scientific article; zbMATH DE number 1337022