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.
Keywords
Cite
@article{arxiv.2603.18511,
title = {Norm-trace and Kloosterman sums in finite semi-simple algebras},
author = {Daqing Wan},
journal= {arXiv preprint arXiv:2603.18511},
year = {2026}
}
Comments
Norm and trace are specified to be reduced norm and reduced trace, Remark 3.5 is updated