Gauss sums of some matrix groups over $\Bbb Z/n\Bbb Z$

TL;DR

This paper explicitly calculates Gauss sums for general and special linear groups over modular integers, expressing them in terms of classical and hyper-Kloosterman sums, and applies these results to count matrices with a given trace.

Contribution

It provides explicit formulas for Gauss sums of ${ m GL}_r(Z_n)$ and ${ m SL}_r(Z_n)$, linking them to classical and hyper-Kloosterman sums, and applies these to matrix counting problems.

Findings

Formulas for Gauss sums of ${ m GL}_r(Z_n)$ in terms of classical Gauss sums.

Formulas for Gauss sums of ${ m SL}_r(Z_n)$ in terms of hyper-Kloosterman sums.

Application to counting invertible matrices with specified trace.

Abstract

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