Date of Award:

8-2025

Document Type:

Thesis

Degree Name:

Master of Science (MS)

Department:

Mathematics and Statistics

Committee Chair(s)

Zhaohu Nie

Committee

Zhaohu Nie

Committee

Andreas Malmendier

Committee

David Brown

Abstract

This thesis consists of two sections. The first section is an introductory survey of number theory discussing the reciprocity laws with a focus on accessibility. Number Theory has always been a fundamental area of mathematical study, with Gauss calling it “the queen of mathematics”. The reciprocity laws are a classical set of results from number theory which have driven number theory for quite a long time. Unfortunately, these results, while important, have always been very inaccessible to undergraduate students, making it hard to start studying the field. This survey attempts to help bridge that gap, giving a resource for novices in the subject to pull from. This section then is essentially a literature review. The second section is a presentation of a concrete result in number theory from my advisor Zhaohu Nie and myself and will be a presentation of the content covered in our preprint paper at [5] with further commentary and explanation for context. This concrete result gives a set of coset representatives for the action of particular congruence subgroups of the modular group on the hyperbolic upper half plane. In particular, we find a set such that the fundamental domain is connected and can be written down without a brute force approach.

Creative Commons License

Creative Commons Attribution 4.0 License
This work is licensed under a Creative Commons Attribution 4.0 License.

Download

Included in

Mathematics Commons

Share

COinS

Copyright for this work is retained by the student. If you have any questions regarding the inclusion of this work in the Digital Commons, please email us at .