Theses and Dissertations

Date of Award

8-2025

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Timothy Huber

Second Advisor

Debanjana Kundu

Third Advisor

Bingyuan Liu

Abstract

This dissertation investigates the arithmetic properties of some modular forms and eta quotients, focusing on the divisibility properties and congruence relations satisfied by these. The first part examines quotients of the Rogers-Ramanujan and Rogers-Selberg functions, defined by \[ f(\tau) = q^{r} (q^5; q^5)^{a_0} (q, q^4; q^5)^{a_1} (q^2, q^3; q^5)^{a_2}= \sum_{n=r}^{\infty} P_{a_0,a_1,a_2}(n-r) q^n, \] and \[ g(\tau) = q^{s} (q^7; q^7)^{a_0} (q, q^6; q^7)^{a_1} (q^2, q^5; q^7)^{a_2} (q^3, q^4; q^7)^{a_3}= \sum_{n=s}^{\infty} P_{a_0,a_1,a_2,a_3}(n-s) q^n, \] respectively. We establish conditions on the exponents \(a_0, a_1, a_2, a_3\) and residue classes \(r\) and $s$ modulo $p$ such that \(P_{a_0, a_1, a_2}(pn - r) \equiv 0 \pmod{p}\) and \(P_{a_0, a_1, a_2, a_3}(pn - s) \equiv 0 \pmod{p}\) for \(n \in \mathbb{N}\). Again, motivated by an algorithmic exploration, we present a class of conjectures, and conditions on exponents of the Rogers-Ramanujan quotients above such that \(P_{a_0, a_1, a_2}(5^j n - r) \equiv 0 \pmod{5^j}\) for \(n \in \mathbb{N}\) and $j\ge 1$. The second part develops \(q\)-difference equations and integral identities involving eta quotients of level \(10\), uncovering further divisibility properties and congruence relations satisfied by these quotients.

Comments

Copyright 2025 Jeffery Opoku. All Rights Reserved.

https://proquest.com/docview/3247688907

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