Representations of Fundamental Groups of Abelian Varieties in Characteristic P

Loading...
Thumbnail Image
Degree type
Doctor of Philosophy (PhD)
Graduate group
Mathematics
Discipline
Subject
abelian varieties
character varieties
monodromy representations
profinite groups
Mathematics
Funder
Grant number
License
Copyright date
2016-11-29T00:00:00-08:00
Distributor
Related resources
Contributor
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 $q$.

Advisor
Ted Chinburg
Date of degree
2016-01-01
Date Range for Data Collection (Start Date)
Date Range for Data Collection (End Date)
Digital Object Identifier
Series name and number
Volume number
Issue number
Publisher
Publisher DOI
Journal Issue
Comments
Recommended citation