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

## TL;DR

This paper calculates the number of monodromy representations of low p-rank abelian varieties over fields of characteristic p, showing it is polynomial in q and providing explicit formulas, with applications to divisibility properties of these counts.

## Contribution

It provides explicit polynomial formulas for the number of monodromy representations of abelian varieties of low p-rank, and establishes geometric and divisibility properties of these counts.

## Key findings

- Number of homomorphisms is polynomial in q for fixed g, λ, n.
- Explicit formulas for these polynomials are derived.
- Divisibility theorem for counts of homomorphisms from certain profinite groups.

## Abstract

Let $A_g$ be an abelian variety of dimension $g$ and $p$-rank $\lambda \leq 1$ over an algebraically closed field of characteristic $p>0$. We compute the number of homomorphisms from $\pi_1^{\text{\'et}}(A_g,a)$ to $GL_n(\mathbb F_q)$, 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 $\lambda=0$, \[\frac{\#\operatorname{Hom}(\pi_1^{\text{\'et}}(A_g,a),GL_n(\mathbb F_q))}{\#GL_n(\mathbb F_q)}\] is a Laurent polynomial in $q$.