\(S\)-integral points on elliptic curves -- notes on a paper of B. M. M. de Weger.
type: Item
instance of: scholarly article
title: \(S\)-integral points on elliptic curves -- notes on a paper of B. M. M. de Weger.
zbMATH Open document ID: 1065.11014
author: Emanuel Herrmann
author: Attila Pethő
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2002-07-24T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93821
full work available at URL: jourelec
review text: \textit{B. M. M. de Weger} solved the Diophantine equation \(y^2=x^3-228x+848\) (*) [J. Théor. Nombres Bordeaux 9, 281--301 (1997; Zbl 0898.11008)] in \(S\)-integers with \(S=\{2,\infty\}\), by using tools from Algebraic Number Theory and lower estimates for linear forms in complex and \(q\)-adic logarithms of algebraic numbers. In the present paper a shorter solution for the \(S\)-solutions to (*) is given, for a larger set \(S\), namely \(S=\{2,3,5,7,\infty\}\). Now, linear forms in elliptic logarithms are used and the basic tool for computing lower bounds for such linear forms is a theorem of \textit{G. Rémond} and \textit{F. Urfels} [J. Number Theory 57, No. 1, 133--160 (1996; Zbl 0853.11055)], which applies to elliptic curves of rank at most 2, hence to the elliptic curve defined by (*). It should be noted that the same authors in cooperation with \textit{J. Gebel} and \textit{H. G. Zimmer} have given an alternative approach to (*), which avoids linear forms in \(q\)-adic elliptic logarithms [Math. Proc. Camb. Philos. Soc. 127, No. 3, 383--402 (1999; Zbl 0949.11033)], but the bounds resulting there are much larger than those of the present paper which result from the theorem of Rémond and Urfels.
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 11J86
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 1770780
zbMATH Keywords: \(S\)-integral point
zbMATH Keywords: elliptic curve linear form in elliptic logarithms
zbMATH Keywords: \(q\)-adic elliptic logarithm
reviewed by: Nikos Tzanakis
describes a project that uses: Magma
describes a project that uses: SIMATH
MaRDI profile type: Publication
OpenAlex ID: W2319934208
cites work: The Magma algebra system. I: The user language
cites work: Minorations de formes linéaires de logarithmes elliptiques
cites work: Computing integral points on elliptic curves
cites work: Q4375609
cites work: Computing all S-integral points on elliptic curves
cites work: Diophantine approximation of \(p\)-adic elliptic logarithms
cites work: Q3710637
cites work: S-integral points on elliptic curves
cites work: Algorithm for determining the type of a singular fiber in an elliptic pencil
cites work: Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms. The case of quartic equations
cites work: Q3993308
cites work: \(S\)-integral solutions to a Weierstrass equation
DOI: 10.5802/JTNB.332
description: scientific article; zbMATH DE number 1770780