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

We compute the Z/ℓ-monodromy and Z ℓ-monodromy of every irreducible component of the moduli space M f g of curves of genus g and p-rank f in characteristic p. In particular, we prove that the Z/ℓ-monodromy of every component of M f g is the symplectic group Sp 2g (Z/ℓ) if g ≥ 3 and ℓ � = p is prime. We give applications to the generic behavior of automorphism groups, Jacobians, class groups, and zeta functions of curves of given genus and p-rank.