#### Date of Award

2016

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Graduate Group

Mathematics

#### First Advisor

Ted Chinburg

#### 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 $\pi_1^{\text{\'et}}(A_g)$ 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 $q$. 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. As a corollary, we deduce that when $\lambda=0$,

\[\frac{\#\Hom(\pi_1^{\text{\'et}}(A_g),GL_n(\mathbb F_q))}{\#GL_n(\mathbb F_q)}\]

is a Laurent polynomial in $q$.

#### Recommended Citation

Frankel, Brett, "Representations of Fundamental Groups of Abelian Varieties in Characteristic P" (2016). *Publicly Accessible Penn Dissertations*. 1723.

https://repository.upenn.edu/edissertations/1723