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The Mordell–Weil sieve: proving non-existence of rational points on curves

Part of: Arithmetic problems. Diophantine geometry Curves Diophantine equations Arithmetic algebraic geometry Computational number theory Computational aspects in algebraic geometry

Published online by Cambridge University Press:  01 August 2010

Nils Bruin  and
Michael Stoll
Nils Bruin
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada (email: nbruin@cecm.sfu.ca)
Michael Stoll
Affiliation:
Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany (email: Michael.Stoll@uni-bayreuth.de)

Abstract

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We discuss the Mordell–Weil sieve as a general technique for proving results concerning rational points on a given curve. In the special case of curves of genus 2, we describe quite explicitly how the relevant local information can be obtained if one does not want to restrict to mod p information at primes of good reduction. We describe our implementation of the Mordell–Weil sieve algorithm and discuss its efficiency.

Supplementary materials are available with this article.

MSC classification

Secondary: 11D41: Higher degree equations; Fermat's equation 11G30: Curves of arbitrary genus or genus $ne 1$ over global fields 11Y50: Computer solution of Diophantine equations 14G05: Rational points 14G25: Global ground fields 14H25: Arithmetic ground fields 14H45: Special curves and curves of low genus 14Q05: Curves
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 13 , January 2010 , pp. 272 - 306
Copyright
Copyright © London Mathematical Society 2010

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