The exotic inverted Kloosterman sum

Abstract

Let $B$ be a product of finitely many finite fields containing ${\mathbb{F}}_q$, $\ddot{I}ˆ:{\mathbb{F}}_q\to {\overline{\mathbb{Q}}}_{\ell }^{\ast }$ a nontrivial additive character, and $\ddot{I}‡:B^{\ast }\to {\overline{\mathbb{Q}}}_{\ell }^{\ast }$ a multiplicative character. Katz introduced the so-called exotic inverted Kloosterman sum \begin{eqnarray*} \mathrm{EIK}(\mathbb F_q, a):=\sum_{\substack{x\in B^* \\ \mathrm{Tr}_{B/\mathbb F_q}(x)\not =0\\ \mathrm{N}_{B/\mathbb F_q}(x)=a}} χ(x)ψ\Big(\frac{1}{\mathrm{Tr}_{B/\mathbb F_q}(x)}\Big), \ \ a\in \mathbb F_q^*. \end{eqnarray*} We estimate this sum using $\ell$-adic cohomology theory. Our main result is that, up to a trivial term, the associated exotic inverted Kloosterman sheaf is lisse of rank at most $2(n+1)$ and mixed of weight at most $n$, where $n+1={\dim }_{{\mathbb{F}}_q}B$. Up to a trivial main term, this gives the expected square root cancellation.

Revised version