Abstract. In this article we show how to generalize the CM-method for elliptic curves to genus two. We describe the algorithm in detail and discuss the results of our implementation. 1.

In [3], a reduction theory for binary forms of degree 3 and 4 with integer coefficients was developed in detail, the motivation in the case of quartics being to improve 2-descent algorithms for elliptic curves over Q. In this paper we extend some of these results to forms of higher degree. One application of this is to the

We introduce a new approach of computing the automorphism group and the field of moduli of points p = [C] in the moduli space of hyperelliptic curves Hg. Further, we show that for every moduli point p ∈ Hg(L) such that the reduced automorphism group of p has at least two involutions, there exists a representative C of the isomorphism class p which is defined over L. 1