Representations of fundamental groups of abelian varieties in characteristic p

Let Ag be an abelian variety of dimension g and p-rank λ ≤ 1 over an algebraically closed field of characteristic p \u3e 0 . We compute the number of homomorphisms from π1ét(A g) to GLn(Fq), where q is any power of p. We show that for fixed g, λ, and n, the number of such representations is polynomial in q. We show that the set of such homomorphisms forms a constructable set, and use the geometry of this space to deduce information about the coefficients and degree of the polynomial. In the last chapter we prove a divisibility theorem about the number of homomorphisms from certain semidirect products of profinite groups into finite groups. (Abstract shortened by ProQuest.