Exponential sums and total Weil representations of finite symplectic and unitary groups

TL;DR

This paper constructs explicit local systems with geometric monodromy groups as finite symplectic and unitary groups, providing simple trace functions related to exponential sums and hypergeometric sheaves.

Contribution

It introduces explicit local systems on the affine line with prescribed monodromy groups in their Weil representations, and describes associated simple exponential sum trace functions.

Findings

Local systems with monodromy groups as $Sp_{2n}(q)$ and $SU_n(q)$

Trace functions form simple exponential sums

Hypergeometric sheaves with symplectic and unitary monodromy groups

Abstract

We construct explicit local systems on the affine line in characteristic $p>2$, whose geometric monodromy groups are the finite symplectic groups $Sp_{2n}(q)$ for all $n \ge 2$, and others whose geometric monodromy groups are the special unitary groups $SU_n(q)$ for all odd $n \ge 3$, and $q$ any power of $p$, in their total Weil representations. One principal merit of these local systems is that their associated trace functions are one-parameter families of exponential sums of a very simple, i.e., easy to remember, form. We also exhibit hypergeometric sheaves on $G_m$, whose geometric monodromy groups are the finite symplectic groups $Sp_{2n}(q)$ for any $n \ge 2$, and others whose geometric monodromy groups are the finite general unitary groups $GU_n(q)$ for any odd $n \geq 3$.