The number of monodromy representations of Abelian varieties of low p-rank

Abstract

Let Ag be an abelian variety of dimension g and p-rank λ≤1 over an algebraically closed field of characteristic p>0. We compute the number of homomorphisms from π1ét(Ag,a) 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, and give an explicit formula for this polynomial. We show that the set of such homomorphisms forms a constructible set, and use the geometry of this space to deduce information about the coefficients and degree of the polynomial.In the last section we prove a divisibility theorem about the number of homomorphisms from certain semidirect products of profinite groups into finite groups. As a corollary, we deduce that when λ=0,#Hom(π1ét(Ag,a),GLn(Fq))|GLn(Fq)| is a Laurent polynomial in q.