Integral Points on Hyperelliptic Curves

TL;DR

This paper provides explicit upper bounds for integral points on hyperelliptic curves using known rational points and Jacobian bases, and introduces a refined Mordell-Weil sieve to find all such points.

Contribution

It introduces a new explicit upper bound for integral points on hyperelliptic curves and a refined Mordell-Weil sieve method for complete determination.

Findings

Successfully determined integral points on two genus 2 hyperelliptic curves

Bounded integral points using known rational points and Jacobian bases

Enhanced Mordell-Weil sieve improves point-finding capabilities

Abstract

We give a completely explicit upper bound for integral points on (standard) affine models of hyperelliptic curves, provided we know at least one rational point and a Mordell-Weil basis of the Jacobian. We also explain a powerful refinement of the Mordell--Weil sieve which, combined with the upper bound, is capable of determining all the integral points. Our method is illustrated by determining the integral points on a two genus 2 hyperelliptic curves with Mordell--Weil Jacobian ranks of 3 and 6.