Monodromy of the p-Rank Strata of the Moduli Space of Curves

We determine the Z/l-monodromy and Z(l)-monodromy of every irreducible component of the stratum M-g(f) 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-g(f) is the symplectic group Sp(2g)(Z/l) if g >= 3 and if l is a prime distinct from p. The method involves results on the intersection of (M) over bar (f)(g) with the boundary of (M) over bar (g). We give applications to the generic behavior of automorphism groups, Jacobians, class groups, and zeta functions of curves of given genus and p-rank.