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 xidx/y3xidx/y3 rather than the xidx/yxidx/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.

► Explicit extension of Kedlaya’s algorithm to even degree Weierstrass models. ► Use of a new differential pseudo-basis for computations. ► Full integrality analysis for the reduction of pseudo-basis elements. ► Integrality analysis independent of Edixhoven’s criterion.