Quadratic Chabauty for modular curves: algorithms and examples

Abstract

We describe how the quadratic Chabauty method may be applied to 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 non-trivial 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 compute the set of rational points on the genus 6 non-split Cartan modular curve $X_{\scriptstyle \mathrm { ns}} ^+ (17)$.