Linear quadratic Chabauty

TL;DR

This paper introduces a simplified and faster quadratic Chabauty method for computing integral points on certain hyperelliptic curves, extending to number fields, with practical examples.

Contribution

A new quadratic Chabauty approach that simplifies and accelerates the computation of integral points on hyperelliptic curves, including over number fields.

Findings

Method is significantly simpler than existing approaches.

Approach is faster and applicable to a broader class of curves.

Successful examples over  and () demonstrate effectiveness.

Abstract

We present a new quadratic Chabauty method to compute the integral points on certain even degree hyperelliptic curves. Our approach relies on a nontrivial degree zero divisor supported at the two points at infinity to restrict the $p$-adic height to a linear function; we can then express this restriction in terms of holomorphic Coleman integrals under the standard quadratic Chabauty assumption. Then we use this linear relation to extract the integral points on the curve. We also generalize our method to integral points over number fields. Our method is significantly simpler and faster than all other existing versions of the quadratic Chabauty method. We give examples over $\Q$ and $\Q(\sqrt{7})$.