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 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 away from our working prime. 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_S4 (13), as well as a few other curves relevant to Mazur’s Program B.