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

Abstract

We establish an analogue of the classical Polya–Vinogradov inequality for $$GL(2, {\mathbbm {F}}_p)$$ , where p is a prime. In the process, we compute the ‘singular’ Gauss sums for $$GL(2, {\mathbbm {F}}_p)$$ . As an application, we show that the collection of elements in $$GL(2,{\mathbbm {Z}})$$ whose reduction modulo p are of maximal order in $$GL(2, {\mathbbm {F}}_p)$$ and whose matrix entries are bounded by x has the expected size as soon as $$x\gg p^{1/2+\varepsilon }$$ for any $$\varepsilon >0$$ .