Quasi-quadratic elliptic curve point counting using rigid cohomology

We present a deterministic algorithm that computes the zeta function of a nonsupersingular elliptic curve E over a finite field with p^n elements in time quasi-quadratic in n. An older algorithm having the same time complexity uses the canonical lift of E, whereas our algorithm uses rigid cohomology combined with a deformation approach. An implementation in small odd characteristic turns out to give very good results.Comment: 14 page

An extension of Kedlaya's algorithm for hyperelliptic curves

In this paper we describe a generalisation and adaptation of Kedlaya's algorithm for computing the zeta function of a hyperelliptic curve over a finite field of odd characteristic that the author used for the implementation of the algorithm in the Magma library. We generalise the algorithm to the case of an even degree model. We also analyse the adaptation of working with the $x^idx/y^3$ rather than the $x^idx/y$ differential basis. This basis has the computational advantage of always leading to an integral transformation matrix whereas the latter fails to in small genus cases. There are some theoretical subtleties that arise in the even degree case where the two differential bases actually lead to different redundant eigenvalues that must be discarded.Comment: v3: some minor changes and addition of a reference to a paper by Theo van den Bogaar