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

Daqing Wan

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.

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.
Paper Structure (4 sections, 11 theorems, 94 equations)

This paper contains 4 sections, 11 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem $\ref{['thm']}$
  3. Proof of Theorem \ref{['K']}
  4. Product-trace version

Key Result

Theorem 1.1

Let $B$ be a finite semi-simple algebra over $\mathbb{F}_q$ of degree $n=\sum_{i=1}^k d_i^2 n_i \geq 2$ as given in equation (B). For $a \in \mathbb{F}_q$ and $b\in\mathbb{F}_q^*$, we have

Theorems & Definitions (24)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Proposition 1.4
  • Lemma 1.5
  • Theorem 1.6
  • Remark 1.7
  • Proposition 2.1
  • Lemma 2.2
  • Theorem 2.3
  • ...and 14 more