On the computation of Mordell-Weil and 2-Selmer groups of elliptic curves.
type: Item
instance of: scholarly article
title: On the computation of Mordell-Weil and 2-Selmer groups of elliptic curves.
zbMATH Open document ID: 1042.11032
published in: Rocky Mountain Journal of Mathematics
publication date: 2003-12-03T00:00:00Z
full work available at URL: CONT32-3.html
review text: Let \(E\) be an elliptic curve defined over \(\mathbb Q\) with discriminant \(\Delta\); if \(E\) has three rational points \((e_j,0)\) (\(j = 1, 2, 3\)) of order \(2\), then the map sending a point \(P = (x,y)\) of order \(> 2\) to the triple of elements \((x-e_j)\mathbb Q^{\times\,2}\) induces an injective homomorphism \(\alpha: E(\mathbb Q)/2E(\mathbb Q) \longrightarrow (\mathbb Q^\times/\mathbb Q^{\times\,2})^3\). This monomorphism can be used to verify the independence of rational points in \(E(\mathbb Q)\). For general elliptic curves, Brumer suggested to compose \(\alpha\) with reduction modulo (prime ideals above) \(p\) for primes not dividing \(6 \Delta\). Using sufficiently many such primes \(p\), one gets an injective homomorphism \(\varepsilon: E(\mathbb Q)/2E(\mathbb Q) \longrightarrow (\mathbb Z/2\mathbb Z)^M\) for some \(M\) which can be used to check whether a given list of rational points is independent.
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 2012041
zbMATH Keywords: elliptic curve
zbMATH Keywords: Mordell-Weil rank
zbMATH Keywords: 2-descent
author: John E. Cremona
describes a project that uses: ecdata
MaRDI profile type: Publication
OpenAlex ID: W2088424215
cites work: Notes on elliptic curves. I.
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DOI: 10.1216/RMJM/1034968425
description: scientific article; zbMATH DE number 2012041