Moments of Gaussian Periods and Modified Fermat Curves

TL;DR

This work uses supercharacter theory to connect moments of Gaussian periods with arithmetic geometry. By expressing $V_4(p)$ in terms of counts of $\,\mathbb{F}_p$-rational points on modified Fermat curves, the authors obtain explicit formulas for fixed $k$ and derive exact results for small fixed $d$ (notably $d=3$ and $d=4$) as well as general bounds for all $d>2$ via Hasse–Weil. The circularity condition provides practical exact results for almost all primes when $k$ is fixed, while the general $d$ analysis leverages the eigenstructure of the associated matrices to reduce moment computations to curve-point counts. The results illuminate a deep link between Gaussian periods, supercharacters, and the arithmetic of Fermat-type curves, with explicit expressions and bounds that sharpen our understanding of power moments in this cyclotomic setting.

Abstract

We use supercharacter theory to study moments of Gaussian periods. For $p-1=dk$ and fixed $k$, we compute the fourth absolute moments for all but finitely many primes $p$. For $d$ fixed, we relate the fourth absolute moments to the number of rational points on modified Fermat curves. For small $d$, this relation is in terms of a single curve. For larger $d$, we provide both exact formulas using families of modified Fermat curves and bounds via Hasse--Weil.

Figures (1)

• Figure 1: (Left) $V_4(p)$ for $d = 8$ with the bounds from Theorem \ref{['thm:all_d_intro']}. (Right) $V_4(p)$ for $d = 4$. Lower bounds are from Theorem \ref{['thm:d4p1p8']} via \ref{['eq:LowerSpecial']} and upper bounds are from Theorem \ref{['thm:all_d_intro']} (upper band is the case $p \equiv 1 \,\,(\operatorname{mod} 8)$ and lower band is the case $p \equiv 5 \,\,(\operatorname{mod} 8)$).

Theorems & Definitions (30)

• Theorem 2
• Theorem 3
• Theorem 6
• Theorem 8
• Remark 9
• Lemma 13
• Example 17
• Lemma 20
• proof
• Lemma 21