A refined Chabauty--Coleman bound for surfaces

TL;DR

This paper refines the Chabauty--Coleman bound for rational points on certain hyperbolic surfaces, specifically for the surface $W_2$ derived from a genus 3 hyperelliptic curve, enabling explicit determination of rational points in examples.

Contribution

It adapts Caro and Pasten's method to produce a sharper bound for $W_2$ on genus 3 hyperelliptic curves, facilitating explicit rational point computations.

Findings

Refined upper bounds for rational points on $W_2$

Explicit determination of $W_2(Q)$ in selected cases

Extension of Chabauty--Coleman techniques to new surface classes

Abstract

Caro and Pasten gave an explicit upper bound on the number of rational points on a hyperbolic surface that is embedded in an abelian variety of rank at most one. We show how to use their method to produce a refined bound on the number of rational points on the surface $W_2 := C+C$ in the case of a hyperelliptic curve $C$ of genus $3$ over $\mathbb{Q}$. Combining this with work of Siksek, we use this to determine $W_2(\mathbb{Q})$ in a selection of examples.