Number Theory Commons^™

Explicit Modularity And Moments For Certain Hypergeometric Character Sums, Brian Grove

LSU Doctoral Dissertations

A great deal of progress in number theory throughout history has been motivated by trying to solve equations. One of the most famous challenges is to show there are no positive integer solutions to $x^{n}+y^{n} = z^{n}$ for $n > 2$, posed by Fermat around 1637. Special cases, such as the $n = 3$ and $n = 4$ cases, can be established using various algebraic manipulations. However, a general solution was elusive until the late 1990s when the combined work of Wiles \cite{Wiles} and Taylor--Wiles \cite{TaylorWiles} give a full proof.

One of the key insights used in proving Fermat's conjecture involves …

Finite Monodromy And Artin Representations, Emma Lien

LSU Doctoral Dissertations

Artin representations, which are complex representations of finite Galois groups, appear in many contexts in number theory. The Langlands program predicts that Galois representations like these should arise from automorphic representations and many examples of this correspondence have been found such as in the proof of Fermat's Last Theorem. This dissertation aims to make an analysis of explicitly computable examples of Artin representations from both sides of this correspondence. On the automorphic side, certain weight 1 modular forms have been shown to be related to Artin representations and an explicit analysis of their Fourier coefficients allows us to identify the …

Congruences Between Coefficients Of A Class Of Eta-Quotients And Their Applications To Combinatorics, Shashika Petta Mestrige

LSU Doctoral Dissertations

Ramanujan in $1920$s discovered remarkable congruence properties of the partition function $p(n)$. Later, Watson and Atkin proved these congruences using the theory of modular forms. Atkin, Gordon, and Hughes extended these works to $k$-colored partition functions. In $2010$, Folsom-Kent-Ono and Boylan-Webb proved the congruences of $p(n)$ by studying a $\ell$-adic module associated with a certain sequence of modular functions which are related to $p(n)$.

Primary goal of this thesis is to generalize the work of Atkin, Gordon, Hughes, Folsom-Kent-Ono, and Boylan-Webb about the partition function to a larger class of partition functions. For this purpose we study a closely related …

Algorithms Related To Triangle Groups, Bao The Pham

LSU Doctoral Dissertations

Given a finite index subgroup of $\PSL_2(\Z)$, one can talk about the different properties of this subgroup. These properties have been studied extensively in an attempt to classify these subgroups. Tim Hsu created an algorithm to determine whether a subgroup is a congruence subgroup by using permutations \cite{hsu}. Lang, Lim, and Tan also created an algorithm to determine if a subgroup is a congruence subgroup by using Farey Symbols \cite{llt}. Sebbar classified torsion-free congruence subgroups of genus 0 \cite{sebbar}. Pauli and Cummins computed and tabulated all congruence subgroups of genus less than 24 \cite{ps}. However, there are still some problems …

On Properties Of Weil Sums Of Binomials, Liem P. Nguyen

LSU Doctoral Dissertations

This dissertation explores questions regarding the Weil sum of binomials, a finite field character sum originated from information theory. The Weil spectrum counts distinct values of the Weil sum through invertible elements in the finite field. The value of these sums and the size of the Weil spectrum are of particular interest, as they link problems in information theory, coding theory, and cryptography to other areas of math such as number theory and arithmetic geometry. In the setting of Niho exponents, we prove the Vanishing Conjecture of Helleseth ($1971$) on the presence of zero values in the Weil spectrum and …

Combinatorial And Asymptotic Statistical Properties Of Partitions And Unimodal Sequences, Walter Mcfarland Bridges

LSU Doctoral Dissertations

Our main results are asymptotic zero-one laws satisfied by the diagrams of unimodal sequences of positive integers. These diagrams consist of columns of squares in the plane; the upper boundary is called the shape. For various types of unimodal sequences, we show that, as the number of squares tends to infinity, 100% of shapes are near a certain curve---that is, there is a single limit shape. Similar phenomena have been well-studied for integer partitions, but several technical difficulties arise in the extension of such asymptotic statistical laws to unimodal sequences. We develop a widely applicable method for obtaining these limit …