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Visual Aspects of Gaussian Periods and Analogues

Samantha Platt

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.

Visual Aspects of Gaussian Periods and Analogues

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.
Paper Structure (18 sections, 10 theorems, 71 equations, 11 figures)

This paper contains 18 sections, 10 theorems, 71 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Definitions and Motivations
  4. Supercharacter Theory
  5. Class Field Theory
  6. Observations and Generalizations: Supercharacter Theory Perspective
  7. Gaussian Periods as Traces of Special Unitary Matrices
  8. Generalization of Duke--Garcia--Lutz
  9. Observations and Generalizations: Class Field Theory Perspective
  10. Elliptic Curves and Complex Multiplication
  11. The Galois Group and Its Action
  12. Computation and Examples
  13. Observations
  14. Obstacles
  15. Questions and Strategies
  16. ...and 3 more sections

Key Result

Theorem 2

Let $n = p^a$, where $p$ is an odd prime. Choose $\omega \in (\mathbb{Z}/p^a\mathbb{Z})^\times$ so that it has multiplicative order $d$ dividing $p - 1$. Let $\Phi_d(x)$ denote the $d$-th cyclotomic polynomial, and let $\mathbb{T}$ denote the complex unit circle. Then the Gaussian period plot $\text where the constants $c_{mj}$ are defined by the following relations: Moreover, for a fixed $d$, as

Figures (11)

  • Figure 1: Examples of Gaussian period plots for various choices of $n$ and $\omega$
  • Figure 2: Examples of Duke--Garcia--Lutz Theorem for various values of $d$
  • Figure 3: A 4-sided hypocycloid rolling along the inside of a 5-sided hypocycloid when $n = 11^5$ and $\omega = 37107$
  • Figure 4: Examples of cyclic supercharacter plots for various $n$, $m$, and $A$.
  • Figure 5: Examples of Theorem \ref{['thm:DGLgeneralize']}
  • ...and 6 more figures

Theorems & Definitions (40)

  • Definition 1
  • Theorem 2: Theorem 6.3 of DGL
  • Definition 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Definition 7
  • Example 1
  • Example 2
  • Proposition 8
  • ...and 30 more