Explicit Evaluation of Double Gauss Sums

Abstract

We present an explicit evaluation of the double Gauss sum $G(a,b,c;S;p^n):=\sum _{x,y=0}^{p^n-1}{e}^{2\pi iS(a{x}^2+bxy+c{y}^2)/p^n}$, 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 $p$.