Date of Award

5-2012

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Legacy Department

Mathematical Science

Advisor

James, Kevin

Committee Member

Brown , James

Committee Member

Maharaj , Hiren

Abstract

In this thesis we explore three different subfields in the area of number theory. The first topic we investigate involves modular forms, specifically nearly holomorphic eigenforms. In Chapter 3, we show the product of two nearly holomorphic eigenforms is an eigenform for only a finite list of examples. The second type of problem we analyze is related to the rank of elliptic curves. Specifically in Chapter 5 we give a graph theoretical approach to calculating the size of 3-Selmer groups for a given family of elliptic curves. By calculating the size of the 3-Selmer groups, we give an upper bound for the rank of an elliptic curve. Finally, in Chapter 7, we conclude with an exposition of work from Goss, Thakur and Diaz-Vargas related to Drinfeld modules. We discuss how to build a zeta function for Drinfeld modules and introduce a symmetric group discovered by Thakur and Diaz-Vargas. An element in the symmetric group is essentially a set permutation of the p-adic integers. It is suspected that there is a relationship between this group and the zeros of certain special zeta functions. We give a specific example of this suspected connection and make a conjecture about this action.

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