Studies in Number Theory: Reciprocity Laws and Fundamental Domains

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.