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Document Type

Open Access Dissertation

Degree Name

Doctor of Philosophy (PhD)

Degree Program

Mathematics

Year Degree Awarded

2017

Month Degree Awarded

May

First Advisor

Siman Wong

Subject Categories

Number Theory

Abstract

This doctoral dissertation concerns two problems in number theory. First, we examine a family of discrete dynamical systems in F_2[t] analogous to the 3x + 1 system on the positive integers. We prove a statistical result about the large-scale dynamics of these systems that is stronger than the analogous theorem in Z. We also investigate mx + 1 systems in rings of functions over a family of algebraic curves over F_2 and prove a similar result there.

Second, we describe some interesting properties of zeta functions of algebraic curves. Generally L-functions vanish only to the order required by their root number. However, we demonstrate that for a certain class of quaternion extensions of F_p(t), the zeta function vanishes at a higher order than the root number demands, indicating some other phenomenon at work.

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Number Theory Commons

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