Point counting in families of hyperelliptic curves

Let E(Gamma) be a family of hyperelliptic curves defined by Y(2) = (Q) over bar (X,Gamma) where (Q) over bar is defined over a small finite field of odd characteristic. Then with (gamma) over bar in an extension degree n field over this small field, we present a deterministic algorithm for computing the zeta function of the curve E (gamma) over bar 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) over tilde( n(3)). An implementation of this last result turns out to be quite efficient for n big enough.