A twist of rank 4 of an elliptic curve of conductor 15
type: Item
instance of: scholarly article
title: A twist of rank 4 of an elliptic curve of conductor 15
zbMATH Open document ID: 0914.11031
author: Emmanuel Halberstadt
author: Alain Kraus
published in: Proceedings of the Japan Academy. Series A
publication date: 1999-06-09T00:00:00Z
review text: Let \(E\) be a given elliptic curve, defined over \(\mathbb Q\), let \(d\) be a rational integer without quadratic factor, denote by \(E_d\) the twist of \(E\) by \(\sqrt{d}\). A conjecture due to \textit{T. Honda} [Jap. J. Math. 30, 84-101 (1960; Zbl 0109.39602)] predicts that, as \(d\) varies, the rank of the \(\mathbb Z\)-module \(E_d(\mathbb Q)\) is bounded by a constant which depends only on \(E\). In order to justify this conjecture the following problem arises. Find explicitly some example of an elliptic curve \(E\) and some integer \(d\) such that the difference between the rank of \(E_d\) and the rank of \(E\) is as big as possible. Moreover it is reasonable to require that \(E\) has the smallest possible conductor among all its twists. In this article, applying techniques which were in part already used by \textit{C. L. Stewart} and \textit{J. Top} [J. Am. Math. Soc. 8, 943-973 (1995; Zbl 0857.11026)], the authors sketch the proof that the elliptic curve \(E\) denoted 15A1 by \textit{J. E. Cremona} [Algorithms for modular elliptic curves, Cambridge University Press (1997; Zbl 0872.14041)], which has rank \(0\), has a twist \(E_d\) whose rank is \(4\). A numerical verification is needed, which is possible only after some (interesting) reductions. This example is suggested by the Birch-Swinnerton-Dyer conjecture.
reviewed by: Federico Pellarin
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 11G40
zbMATH DE Number: 1192517
zbMATH Keywords: Mordell-Weil rank
zbMATH Keywords: quadratic twists
zbMATH Keywords: elliptic curve
zbMATH Keywords: Birch-Swinnerton-Dyer conjecture
describes a project that uses: ecdata
MaRDI profile type: Publication
full work available at URL: pjaa.74.36
OpenAlex ID: W2089269805
cites work: Q3139838
cites work: Q4003857
cites work: Q3847902
cites work: Q4083553
cites work: The Difference Between the Weil Height and the Canonical Height on Elliptic Curves
cites work: On Ranks of Twists of Elliptic Curves and Power-Free Values of Binary Forms
DOI: 10.3792/PJAA.74.36
description: scientific article; zbMATH DE number 1192517