The exotic inverted Kloosterman sum

TL;DR

This paper studies the exotic inverted Kloosterman sum, providing estimates using $\, ext{l}$-adic cohomology, and shows the associated sheaf has controlled rank and weight, leading to square root cancellation.

Contribution

It introduces a new approach to estimate exotic inverted Kloosterman sums using $\, ext{l}$-adic cohomology, establishing bounds on the associated sheaf's rank and weight.

Findings

The exotic inverted Kloosterman sheaf is lisse of rank at most 2(n+1).

The sheaf is mixed of weight at most n.

The sum exhibits the expected square root cancellation.

Abstract

Let $B$ be a product of finitely many finite fields containing $\mathbb F_q$, $\psi:\mathbb F_q\to \overline{\mathbb Q}_\ell^*$ a nontrivial additive character, and $\chi: B^*\to \overline{\mathbb Q}_\ell^*$ 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}} \chi(x)\psi\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.