The p-rank strata of the moduli space of hyperelliptic curves

Abstract

We prove results about the intersection of the p-rank strata and the boundary of the moduli space of hyperelliptic curves in characteristic p > 2. Using this, we prove that the Z/\ell-monodromy of every irreducible component of the stratum H_g^f of hyperelliptic curves of genus g and p-rank f is the symplectic group Sp_{2g}(Z/\ell) if g > 2, f > 0 and \ell is an odd prime distinct from p. These results yield applications about the generic behavior of hyperelliptic curves of given genus and p-rank. The first application is that a generic hyperelliptic curve of genus g > 2 and p-rank 0 is not supersingular. Other applications are about absolutely simple Jacobians and the generic behavior of class groups and zeta functions of hyperelliptic curves of given genus and $p$-rank over finite fields.

Slight reorganization, improved results for p-rank zero locus