Curves of given p-rank with trivial automorphism group

Abstract

Let k be an algebraically closed field of characteristic p > 0. Suppose g � 3 and 0 � fg. We prove there is a smooth projective k-curve of genus g and p-rank f with no non-trivial automorphisms. In addition, we prove there is a smooth projective hyperelliptic k-curve of genus g and p-rank f whose only non-trivial automorphism is the hyperelliptic involution. The proof involves computations about the dimension of the moduli space of (hyperelliptic) k-curves of genus g and p-rank f with extra automorphisms.