Point Counting in Families of Hyperelliptic Curves

Let EΓ be a family of hyperelliptic curves defined by Y 2 = Q(X,Γ), where Q is defined over a small finite field of odd characteristic. Then with γ in an extension degree n field over this small field, we present a deterministic algorithm for computing the zeta function of the curve Eγ by using Dwork deformation in rigid cohomology. The complexity of the algorithm is O(n) and it needs O(n) bits of memory. A slight adaptation requires only O(n) space, but costs time Õ(n). An implementation of this last result turns out to be quite efficient for n big enough. AMS (MOS) Subject Classification Codes: 11G20, 11Y99, 12H25, 14F30, 14G50, 14Q05.