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

TL;DR

The paper explores the Mordell-Weil sieve method for proving the non-existence of rational points on curves, especially genus 2, detailing explicit local information computation and implementation efficiency.

Contribution

It provides a detailed description of the Mordell-Weil sieve technique, including explicit methods for genus 2 curves and an implementation analysis.

Findings

Effective local information computation for genus 2 curves

Implementation of the Mordell-Weil sieve algorithm

Discussion on the efficiency of the method

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.