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 ≥ 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 Xs(13), completing the classification of non-CM elliptic curves over Q with split Cartan level structure due to Bilu-Parent and Bilu-Parent-Rebolledo.

We give the first explicit examples beyond the Chabauty-Coleman method where Kim's nonabelian Chabauty program determines the set of rational points of a curve defined over Q or a quadratic number field. We accomplish this by studying the role of p-adic heights in explicit nonabelian Chabauty.

The Chabauty-Kim method is a method for finding rational points on curves under certain technical conditions, generalising Chabauty's proof of the Mordell conjecture for curves with Mordell-Weil rank less than their genus. We show how the Chabauty-Kim method, when these technical conditions are satisfied in depth 2, may be applied to bound the number of rational points on a curve of higher rank. This provides a nonabelian generalisation of Coleman's effective Chabauty theorem.

We give new instances where Chabauty-Kim sets can be proved to be finite, by developing a notion of "generalised height functions" on Selmer varieties. We also explain how to compute these generalised heights in terms of iterated integrals and give the first explicit nonabelian Chabauty result for a curve X/Q whose Jacobian has Mordell-Weil rank larger than its genus.

A. 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.