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

Abstract

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.

47 pages, 4 figures. Revised version, following suggestions/questions by the referee. Main changes: (1) We fixed a gap in the proof of Lemma 4.3 and added a discussion of the case r < g-1. (2) We added a heuristic discussion of the success probability at a single prime in Section 8