Infinite descent on elliptic curves
type: Item
instance of: scholarly article
title: Infinite descent on elliptic curves
zbMATH Open document ID: 0852.11028
author: Samir Siksek
published in: Rocky Mountain Journal of Mathematics
publication date: 1996-12-05T00:00:00Z
review text: In order to compute the Modell-Weil group \(E(K)\) of an elliptic curve \(E\) defined over an algebraic number field \(K\) (often \(K= \mathbb{Q}\)), one first needs to calculate the rank of the curve. This is done by carrying out a 2-descent which, with some luck, gives a basis for the quotient group \(E(K)/ 2E (K)\). From \(|E(K)/ 2E (K) |= 2^r\), one immediately reads off the rank \(r\). In most `reasonable' cases -- the discriminant of \(E\) should not be too large -- this part of the process is relatively straightforward. Once the rank is known, it usually is much harder to calculate a finite basis for \(E(K)\). For this second part of the computation, the standard procedure is to use infinite descent, which is the process of obtaining a basis for \(E(K)\), given a basis for \(E(K)/ mE (K)\) for some \(m\geq 2\). To this end one applies Zagier's theorem in conjunction with explicit bounds for the difference of the logarithmic height \(h\) and the canonical height \(\widehat {h}\) of points \(P\) on the curve [see: \textit{J. H. Silverman}, Math. Comput. 55, 723-743 (1990; Zbl 0729.14026)]. In the present paper, the author describes a method by which the existing upper bound for \(h(P)- \widehat {h} (P)\), established by Silverman in the paper cited above, can often be considerably improved. In many nontrivial examples, well-chosen from the existing literature, the author very convincingly illustrates the superiority of his approach.
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11Y16
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14Q05
zbMATH DE Number: 894664
zbMATH Keywords: Modell-Weil group
zbMATH Keywords: elliptic curve
zbMATH Keywords: infinite descent
zbMATH Keywords: logarithmic height
reviewed by: Roelof J. Stroeker
describes a project that uses: PARI/GP
describes a project that uses: ecdata
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072159
OpenAlex ID: W2006264362
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DOI: 10.1216/RMJM/1181072159
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description: scientific article; zbMATH DE number 894664