Chabauty-Coleman experiments for genus 3 hyperelliptic curves

TL;DR

This paper implements a Chabauty-Coleman based algorithm in Sage to compute rational points on genus 3 hyperelliptic curves with Jacobians of rank 1, analyzing thousands of cases and discovering some unexpected rational points.

Contribution

It provides a practical computational method and implementation for finding rational points on certain genus 3 hyperelliptic curves using Chabauty-Coleman theory.

Findings

Successfully computed rational points on ~17,000 curves.

Identified cases with global points not initially known.

Demonstrated the effectiveness of the algorithm on large datasets.

Abstract

We describe a computation of rational points on genus 3 hyperelliptic curves $C$ defined over $\mathbb{Q}$ whose Jacobians have Mordell-Weil rank 1. Using the method of Chabauty and Coleman, we present and implement an algorithm in Sage to compute the zero locus of two Coleman integrals and analyze the finite set of points cut out by the vanishing of these integrals. We run the algorithm on approximately 17,000 curves from a forthcoming database of genus 3 hyperelliptic curves and discuss some interesting examples where the zero set includes global points not found in $C(\mathbb{Q})$.