Ted ChinburgFrankel, Brett2023-05-222016-04-212016-11-292016-01-012016-11-29https://repository.upenn.edu/handle/20.500.14332/28565Let $A_g$ 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 $q$.37 p.application/pdfBrett Frankelabelian varietiescharacter varietiesmonodromy representationsprofinite groupsMathematicsDissertation/Thesis