Explicit Evaluation of Double Gauss Sums

TL;DR

This paper provides an explicit formula for evaluating double Gauss sums involving quadratic forms over modular arithmetic, which are important in number theory and related fields.

Contribution

It introduces a new explicit evaluation method for double Gauss sums with quadratic forms over prime power moduli, extending previous results.

Findings

Derived explicit formulas for double Gauss sums

Extended evaluation techniques to prime power moduli

Facilitated computation of quadratic exponential sums

Abstract

We present an explicit evaluation of the double Gauss sum $\displaystyle G(a,b,c;S;p^n):=\sum_{x,y=0}^{p^n-1} e^{2\pi i S(ax^2+bxy+cy^2)/p^n}$, where $a, b, c$ are integers such that $\gcd(a,b,c)=1$, $p$ is a prime, $n$ is a positive integer, and $S$ is an integer coprime to $p$.