Point counting in families of hyperelliptic curves

Abstract

Let E_G be a family of hyperelliptic curves defined by Y^2=Q(X,G), where Q is defined over a small finite field of odd characteristic. Then with g in an extension degree n field over this small field, we present a deterministic algorithm for computing the zeta function of the curve E_g by using Dwork deformation in rigid cohomology. The time complexity of the algorithm is O(n^(2.667)) and it needs O(n^(2.5)) bits of memory. A slight adaptation requires only O(n^2) space, but costs time O(n^3). An implementation of this last result turns out to be quite efficient for n big enough.