Quadratic Chabauty for modular curves: Algorithms and examples

TL;DR

This paper extends the quadratic Chabauty method to explicitly compute rational points on certain modular curves of genus greater than one, including examples like Atkin-Lehner quotients and non-split Cartan curves.

Contribution

It introduces algorithms for applying quadratic Chabauty to modular curves with Jacobians of rank equal to genus, broadening the scope of explicit rational point determination.

Findings

Successfully determined rational points on several genus 2 and 3 modular curves.

Extended quadratic Chabauty to curves with nontrivial local height contributions.

Computed rational points on a genus 6 non-split Cartan modular curve.

Abstract

We describe how the quadratic Chabauty method may be applied to explicitly determine the set of rational points on modular curves of genus $g>1$ whose Jacobians have Mordell--Weil rank $g$. This extends our previous work on the split Cartan curve of level 13 and allows us to consider modular curves that may have few known rational points or nontrivial local height contributions at primes of bad reduction. We illustrate our algorithms with a number of examples where we determine the set of rational points on several modular curves of genus 2 and 3: this includes Atkin--Lehner quotients $X_0^+(N)$ of prime level $N$, the curve $X_{S_4}(13)$, as well as a few other curves relevant to Mazur's Program B. We also describe the computation of rational points on the genus 6 non-split Cartan modular curve $X_{\textrm{ns}} ^+ (17)$.