Explicit Chabauty-Kim for the Split Cartan Modular Curve of Level 13

TL;DR

This paper extends explicit quadratic Chabauty methods to non-hyperelliptic curves, enabling the computation of rational points on certain genus g curves, and applies this to classify non-CM elliptic curves with split Cartan level structure.

Contribution

It develops an algorithm for non-hyperelliptic curves with specific Jacobian properties and applies it to the modular curve X_s(13) for classification purposes.

Findings

Algorithm successfully computes rational points on the specified curve.

Complete classification of non-CM elliptic curves with split Cartan level structure.

Extended Chabauty methods to non-hyperelliptic cases.

Abstract

We extend the explicit quadratic Chabauty methods developed in previous work by the first two authors to the case of non-hyperelliptic curves. This results in an algorithm to compute the rational points on a curve of genus $g \ge 2$ over the rationals whose Jacobian has Mordell-Weil rank $g$ and Picard number greater than one, and which satisfies some additional conditions. This algorithm is then applied to the modular curve $X_{s}(13)$, completing the classification of non-CM elliptic curves over $\mathbf{Q}$ with split Cartan level structure due to Bilu-Parent and Bilu-Parent-Rebolledo.