Monodromy of the p-rank strata of the moduli space of curves

TL;DR

This paper computes the monodromy groups of components of the moduli space of algebraic curves with fixed genus and p-rank, revealing symplectic group structures and implications for automorphisms and Jacobians.

Contribution

It provides the first comprehensive calculation of monodromy groups for all components of the p-rank strata in the moduli space of curves, establishing symplectic group structures.

Findings

Z/ell-monodromy is the symplectic group Sp_{2g}(Z/ell) for g>=3 and ell ≠ p

Results apply to automorphism groups, Jacobians, class groups, and zeta functions

Monodromy groups are explicitly computed for all irreducible components

Abstract

We compute the Z/\ell and \ell-adic monodromy of every irreducible component of the moduli space M_g^f of curves of genus and and p-rank f. In particular, we prove that the Z/\ell-monodromy of every component of M_g^f is the symplectic group Sp_{2g}(Z/\ell) if g>=3 and \ell is a prime distinct from p. We give applications to the generic behavior of automorphism groups, Jacobians, class groups, and zeta functions of curves of given genus and p-rank.