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 introduce a new algorithm to compute the zeta function of a curve over a finite field. This method extends Kedlaya's algorithm to a very general class of curves using a map to the projective line. We develop all the necessary bounds, analyse the complexity of the algorithm and provide some examples computed with our implementation.

Abstract. We introduce a new algorithm to compute the zeta function of a curve over a finite field. This method extends previous work of ours to all curves for which a good lift to characteristic zero is known. We develop all the necessary bounds, analyse the complexity of the algorithm and provide a complete implementation.

We present various improvements to the deformation method for computing the zeta function of smooth projective hypersurfaces over finite fields using p-adic cohomology. This includes new bounds for the p-adic and t-adic precisions required to obtain provably correct results and gains in the efficiency of the individual steps of the method. The algorithm that we thus obtain has lower time and space complexities than existing methods. Moreover, our implementation is more practical and can be applied more generally, which we illustrate with examples of generic quintic curves and quartic surfaces.

The Coleman integral is a p-adic line integral that plays a key role in computing several important invariants in arithmetic geometry. We give an algorithm for explicit Coleman integration on curves, using the algorithms of the second author [Tui16, Tui17] to compute the action of Frobenius on p-adic cohomology. We present a collection of examples computed with our implementation.