Computing rational points on rank 0 genus 3 hyperelliptic curves

TL;DR

This paper develops and implements an algorithm using the Chabauty-Coleman method to compute all rational points on genus 3 hyperelliptic curves with Mordell-Weil rank zero, demonstrating its effectiveness on a large dataset.

Contribution

The authors adapt and implement the Chabauty-Coleman method specifically for genus 3 hyperelliptic curves with rank 0 Jacobians, enabling systematic computation of rational points.

Findings

Successfully computed rational points on 5870 curves.

Validated the effectiveness of the Chabauty-Coleman method for genus 3 hyperelliptic curves.

Provided a practical algorithm for future research in rational point computation.

Abstract

We compute rational points on genus $3$ odd degree hyperelliptic curves $C$ over $\mathbb{Q}$ that have Jacobians of Mordell-Weil rank $0$. The computation applies the Chabauty-Coleman method to find the zero set of a certain system of $p$-adic integrals, which is known to be finite and include the set of rational points $C(\mathbb{Q})$. We implemented an algorithm in Sage to carry out the Chabauty-Coleman method on a database of $5870$ curves.