A Computational Approach to Ramanujan-Type Congruences for Products of Level 13

Author

Aiden Michael Lalonde, The University of Texas Rio Grande Valley

Date of Award

12-1-2025

Document Type

Thesis

Degree Name

Master of Science (MS)

Department

Mathematics

First Advisor

Timothy Huber

Second Advisor

Brandt Kronholm

Third Advisor

Josef Sifuentes

Abstract

In this thesis, we find products that are modular forms of level 13 whose Fourier coefficients satisfy certain congruences. These products are constructed in terms of integer powers of Klein forms and the Dedekind eta function. The products that are modular forms for Γ1 (13) have vectors of exponents that are elements of a bounded polytope. A parameterization for this polytope allows us to find a complete set of products that are modular forms for Γ1 (13). The corresponding graded ring of modular forms containing these products is constructed from a finite set of generators. These generators satisfy a finite number of algebraic relations, which we find and reduce to a minimal set using Groebner bases. This completely characterizes the graded ring of modular forms for Γ1(13) in terms of the generators and relations. This thesis lays out a framework for the techniques used to find modular forms of level 13 whose Fourier coefficients satisfy new congruences modulo certain integers N. The techniques used can be generalized to find more congruences for different weight modular forms and different moduli.

Comments

Copyright 2025 Aiden M. Lalonde. All Rights Reserved. https://proquest.com/docview/3292512149

Recommended Citation

Lalonde, A. M. (2025). A Computational Approach to Ramanujan-Type Congruences for Products of Level 13 [Master's thesis, The University of Texas Rio Grande Valley]. ScholarWorks @ UTRGV. https://scholarworks.utrgv.edu/etd/1851