Orbits of Galois Invariant n-Sets of [formula omitted]1 under the Action of PGL2

Abstract

For any finite field k we count the number of orbits of galois invariant n-sets of P1(k̄) under the action of PGL2(k). For k of odd characteristic, this counts the number of k-points of the moduli space of hyperelliptic curves of genus g over k. We get in this way an explicit formula for the number of hyperelliptic curves over k of genus g, up to k-isomorphism and quadratic twist.