Gauss sums of some matrix groups over $\mathbb{Z}/n\mathbb{Z}$

Abstract

In this paper, we will explicitly calculate Gauss sums for the general linear groups and the special linear groups over ${\mathbb{Z}}_n$, where ${\mathbb{Z}}_n=\mathbb{Z}/n\mathbb{Z}$ and $n>0$ is an integer. For $r$ being a positive integer, the formulae of Gauss sums for ${\mathrm{GL}}_r({\mathbb{Z}}_n)$ can be expressed in terms of classical Gauss sums over ${\mathbb{Z}}_n$, while the formulae of Gauss sums for ${\mathrm{SL}}_r({\mathbb{Z}}_n)$ can be expressed in terms of hyper-Kloosterman sums over ${\mathbb{Z}}_n$. As an application, we count the number of $r\times r$ invertible matrices over ${\mathbb{Z}}_n$ with given trace by using the the formulae of Gauss sums for ${\mathrm{GL}}_r({\mathbb{Z}}_n)$ and the orthogonality of Ramanujan sums.

14 pages