A geometric linear Chabauty comparison theorem

TL;DR

This paper compares the geometric linear Chabauty method with the classical Chabauty-Coleman approach, demonstrating scenarios where the former outperforms the latter and discussing ways to enhance computational practicality.

Contribution

It introduces a comparison between geometric linear Chabauty and Chabauty-Coleman, highlighting cases where the former is superior and proposing improvements for the latter.

Findings

Geometric linear Chabauty can outperform Chabauty-Coleman in certain cases.

Chabauty-Coleman remains more practical for general computations.

Proposed methods to strengthen Chabauty-Coleman for theoretical equivalence.

Abstract

The Chabauty-Coleman method is a $p$-adic method for finding all rational points on curves of genus $g$ whose Jacobians have Mordell-Weil rank $r < g$. Recently, Edixhoven and Lido developed a geometric quadratic Chabauty method that was adapted by Spelier to cover the case of geometric linear Chabauty. We compare the geometric linear Chabauty method and the Chabauty-Coleman method and show that geometric linear Chabauty can outperform Chabauty-Coleman in certain cases. However, as Chabauty-Coleman remains more practical for general computations, we discuss how to strengthen Chabauty-Coleman to make it theoretically equivalent to geometric linear Chabauty. We apply these methods to genus 2 and genus 3 curves.