Quadratic Chabauty: p-adic height pairings and integral points on hyperelliptic curves

TL;DR

This paper develops a p-adic height pairing formula for hyperelliptic curves and introduces a Chabauty-like method to find p-integral points, providing explicit bounds and computational techniques when the Mordell-Weil rank equals the genus.

Contribution

It presents a new explicit formula for p-adic height pairings and a novel method for computing p-integral points on hyperelliptic curves with rank equal to genus.

Findings

Derived a formula for p-adic height pairing components.

Established a Chabauty-like method for p-integral points.

Provided explicit bounds and computational procedures.

Abstract

We give a formula for the component at p of the p-adic height pairing of a divisor of degree 0 on a hyperelliptic curve. We use this to give a Chabauty-like method for finding p-adic approximations to p-integral points on such curves when the Mordell-Weil rank of the Jacobian equals the genus. In this case we get an explicit bound for the number of such p-integral points, and we are able to use the method in explicit computation. An important aspect of the method is that it only requires a basis of the Mordell-Weil group tensored with the rationals.