Matrix Kloosterman sums and product-trace estimates for semisimple algebras

2026/07/25 by Xuejun Guo, Chen Lin, Chenhao Tang
Mathematics · #math.NT #math.AG #math.RT

paper · pdf

Abstract

Let k=\mathbbF_q, E=\mathbbF_qⁿ and Tr=Tr_E/k. For r≥ 2, a∈ k^× and x∈ E^×, let N(E,r,x,a) be the number of r-tuples (x₁,⋯,x_r) in (E^×)^r satisfying x₁⋯ x_r=x and Tr(x₁+⋯+x_r)=a. We prove |N(E,r,x,a)-((qⁿ-1)^r-1+(-1)^r)/q|≤ (rⁿ-1) q^{((r-1)n-1)/(2)}. This proves the square-root estimate predicted in Wan's conjecture and generalizes a previous result of Moisio and Wan. For a finite semisimple algebra B=∏_i=1^s M_di(\mathbbF_qⁿⁱ) over k and a regular element x∈ B^×, the same method combined with Zelingher's formula leads to analogous square-root estimates.

Citations

• Gauss sums over some matrix groups
• On Katz's bound for number of elements with given trace and norm
• Lectures on zeta functions over finite fields
• Moment Zeta Functions for Toric Calabi-Yau Hypersurfaces

Related

• Equations and character sums with matrix powers, Kloosterman sums over small subgroups and quantum ergodicity
• A Matrix Analogue of Schur-Siegel-Smyth Trace Problem
• Sums of reciprocals of fractional parts
• A lower bound for classical Kloosterman sums and an application
• Product of exponentials concentrates around the exponential of the sum