Matrix Kloosterman sums and product-trace estimates for semisimple algebras

Abstract

Let $k={\mathbb{F}}_q$, $E={\mathbb{F}}_{q^n}$ and $\mathrm{Tr}={\mathrm{Tr}}_{E/k}$. For $r\ge 2$, $a\in k^{\times }$ and $x\in E^{\times }$, let $\mathrm{N}(E,r,x,a)$ be the number of $r$-tuples $(x_1,\cdots ,x_r)$ in $(E^{\times }{)}^r$ satisfying $x_1\cdots x_r=x$ and $\mathrm{Tr}(x_1+\cdots +x_r)=a$. We prove $\left|\mathrm{N}(E,r,x,a)-\left((q^n-1{)}^{r-1}+(-1{)}^r\right)/q\right|\le (r^n-1)q^{\frac{(r-1)n-1}{2}}$. 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 $B=\prod _{i=1}^s{M}_{d_i}({\mathbb{F}}_{q^{n_i}})$ over $k$ and a regular element $x\in B^{\times }$, the same method combined with Zelingher's formula leads to analogous square-root estimates.

16 pages