Models of some genus one curves with applications to descent
type: Item
instance of: scholarly article
title: Models of some genus one curves with applications to descent
zbMATH Open document ID: 1095.11032
author: Catherine O'Neil
published in: Journal of Number Theory
publication date: 2005-06-30T00:00:00Z
review text: The classical method for finding the rank of an elliptic curve \(E\) defined over the rationals is performing an \(n\)-descent for \(n=2\) or \(n = 3\). One important step in performing such descents is computing the Selmer group, whose elements are certain curves of genus \(1\) that become isomorphic to \(E\) over the algebraic closure of \(\mathbb Q\). For \(n = 2\), these curves of genus \(1\) have the form \(y^2 = f(x)\) for some quartic polynomial \(f\), and for \(n=3\) they are certain cubic curves (at least if we adjoin the cube roots of unity to \(\mathbb Q\)). In this article, the author derives families of genus \(1\) curves for special cases when \(n = 3\) and \(n = 5\), based on her previous work concerning the period-index obstruction [J. Number Theory 95, No. 2, 329--339 (2002); erratum ibid. 109, No. 2, 390 (2004; Zbl 1033.11029)]
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 2183816
zbMATH Keywords: period-index obstruction
zbMATH Keywords: elliptic curves
zbMATH Keywords: descent
zbMATH Keywords: norm symbol
MaRDI profile type: Publication
full work available at URL: j.jnt.2004.09.006
OpenAlex ID: W2059912670
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DOI: 10.1016/J.JNT.2004.09.006
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description: scientific article; zbMATH DE number 2183816