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.

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A. We compute the diagonal restriction of the rst derivative with respect to the weight of a p-adic family of Hilbert modular Eisenstein series a ached to a general (odd) character of the narrow class group of a real quadratic eld, and express the Fourier coe cients of its ordinary projection in terms of the values of a distinguished rigid analytic cocycle in the sense of [DV1] at appropriate real quadratic points of Drinfeld's p-adic upper half-plane. is can be viewed as the p-adic counterpart of a seminal calculation of Gross and Zagier [GZ, §7] which arose in their "analytic proof" of the factorisation of di erences of singular moduli, and whose inspiration can be traced to Siegel's proof of the rationality of the values at negative integers of the Dedekind zeta function of a totally real eld. Our main identity enriches the dictionary between the classical theory of complex multiplication and its extension to real quadratic elds based on RM values of rigid meromorphic cocycles, and leads to an expression for the p-adic logarithms of Gross-Stark units and Stark-Heegner points in terms of the rst derivatives of certain twisted Rankin triple product p-adic Lfunctions.

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We construct explicit bases for spaces of overconvergent p-adic modular forms when p = 2, 3 and study their interaction with the Atkin operator. This results in an extension of Lauder's algorithms for overconvergent modular forms. We illustrate these algorithms with computations of slope sequences of some 2-adic eigencurves and the construction of Chow-Heegner points on elliptic curves via special values of Rankin triple product L-functions.