Computational Aspects of Hyperelliptic Curves

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 \H_g. Further, we show that for every moduli point \p \in \H_g(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 $L$.