Singular Gauss sums, Polya-Vinogradov inequality for $GL(2)$ and growth of primitive elements

TL;DR

This paper develops an analogue of the Polya-Vinogradov inequality for $GL(2, _p)$, computes singular Gauss sums, and investigates the growth of primitive elements in $GL(2,Z)$ with bounded entries.

Contribution

It introduces a Polya-Vinogradov inequality analogue for $GL(2, _p)$ and computes singular Gauss sums, linking these to the growth of primitive elements in $GL(2, Z)$.

Findings

Established the Polya-Vinogradov inequality for $GL(2, _p)$.

Computed singular Gauss sums for $GL(2, _p)$.

Proved existence of elements in $GL(2, Z)$ with bounded entries that are primitive modulo $p$.

Abstract

We establish an analogue of the classical Polya-Vinogradov inequality for $GL(2, \F_p)$, where $p$ is a prime. In the process, we compute the `singular' Gauss sums for $GL(2, \F_p)$. As an application, we show that the collection of elements in $GL(2,\Z)$ whose reduction modulo $p$ are of maximal order in $GL(2, \F_p)$ and whose matrix entries are bounded by $x$ has the expected size as soon as $x\gg p^{1/2+\ep}$ for any $\ep>0$. In particular, there exist elements in $GL(2,\Z)$ with matrix entries that are of the order $O(p^{1/2+\ep})$ whose reduction modulo $p$ are primitive elements.