Visual Aspects of Gaussian Periods and Analogues

TL;DR

This paper explores the visual patterns of Gaussian periods in the complex plane, revealing new insights and conjectures, and extends the study to related theories with practical visualization tools.

Contribution

It introduces a visual perspective on Gaussian periods, presents structural examples, and proposes generalizations to supercharacter and class field theories with accompanying visualization code.

Findings

Emergence of interesting patterns in Gaussian periods when plotted in the complex plane

Proposals for extending the visual study to supercharacter and class field theories

Provision of code for generating Gaussian period visualizations

Abstract

Gaussian periods have been studied for centuries in the realms of number theory, field theory, cryptography, and elsewhere. However, it was only within the last decade or so that they began to be studied from a visual perspective. By plotting Gaussian periods in the complex plane, various interesting and insightful patterns emerge, leading to various conjectures and theorems about their properties. In this paper, we offer a description of Gaussian periods, along with examples of the structure that can occur when plotting them in the complex plane. In addition to this, we offer two ways in which this study can be generalized to other situations -- one relating to supercharacter theory, the other relating to class field theory -- along with discussions and visual examples of each. We end the paper by including some code for readers to generate images on their own.