Gauss Sums on $GL_2(\mathbb{Z}/p^l\mathbb{Z})$

TL;DR

This paper explicitly computes Gauss sums for all irreducible characters of the group $GL_2(Z/p^lZ)$, a novel contribution to understanding character sums over matrix groups on finite rings.

Contribution

It provides the first explicit determination of Gauss sums on $GL_2$ over finite rings, extending the study beyond finite fields.

Findings

Explicit formulas for Gauss sums on $GL_2(Z/p^lZ)$

First known results for matrix groups over finite rings

Broadens understanding of character sums in algebraic groups

Abstract

We determine explicitly the Gauss sums on the general linear group $GL_2(\mathbb{Z}/p^l\mathbb{Z})$ for all irreducible characters, where $p$ is an odd prime and $l$ is an integer > 1. While there are several studies of the Gauss sums on finite algebraic groups defined over a finite field, this paper seems to be the first one which determines the Gauss sums on a matrix group over a finite ring.