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

TL;DR

This paper investigates the structure and monodromy of hyperelliptic curves in characteristic p, revealing generic properties and applications to Jacobians, class groups, and zeta functions.

Contribution

It proves the symplectic monodromy for hyperelliptic p-rank strata and explores their generic geometric and arithmetic properties.

Findings

The p-rank strata intersect boundary components in specific ways.

Monodromy groups are the full symplectic group for certain hyperelliptic strata.

Generic hyperelliptic curves of genus > 2 and p-rank 0 are not supersingular.

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.