Moments of Gaussian Periods and Modified Fermat Curves

TL;DR

This paper employs supercharacter theory to analyze moments of Gaussian periods, deriving explicit formulas and bounds for the fourth absolute moments in relation to rational points on modified Fermat curves, advancing understanding in algebraic number theory.

Contribution

It introduces a novel application of supercharacter theory to compute moments of Gaussian periods and connects these moments to rational points on modified Fermat curves for various parameters.

Findings

Computed fourth absolute moments for most primes p with fixed d and k.

Established a relationship between moments and rational points on Fermat curves.

Provided exact formulas and bounds for moments using algebraic curves.

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.