Norm-trace and Kloosterman sums in finite semi-simple algebras

Abstract

An asymptotic formula with a square root error term is obtained for the number of elements with given trace and norm in a finite semisimple algebra over a finite field. This extends previous results from finite etale algebras (commutative case) to finite semi-simple algebras (non-commutative case). The main idea is to apply the Eichler formula for Gauss sums over the general linear group and the Hasse-Davenport relation to reduce the problem to the classical geometric case where the result is known to be true. As an application of this reduction, we also obtain a square root estimate for Kloosterman sums over semi-simple algebras. Similar square root estimates are discussed when norm-trace is replaced by product-trace, leading to a new conjecture on product-trace counting over finite semi-simple algebras.