The Number of Monodromy Representations of Abelian Varieties of Low $p$-Rank

Abstract

Let $A_g$ be an abelian variety of dimension $g$ and $p$-rank $\lambda \le 1$ over an algebraically closed field of characteristic $p>0$. We compute the number of homomorphisms from ${\pi }_1^{\text{eˊt}}(A_g,a)$ to $G{L}_n({\mathbb{F}}_q)$, where $q$ is any power of $p$. We show that for fixed $g$, $\lambda$, 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 $\lambda =0$, \[\frac{\#\operatorname{Hom}(π_1^{\text{ét}}(A_g,a),GL_n(\mathbb F_q))}{\#GL_n(\mathbb F_q)}\] is a Laurent polynomial in $q$.

Final version, now 17 pages. Corrected a sign error in the application of Theorem 1 to the case $n=k=2$. This computation has also been worked out explicitly, as an example of how to apply the theorem. Other minor changes. To appear in J. of Algebra

References in corpus (1)

• Arithmetic of singular character varieties and their $E$-polynomials