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

Abstract

We construct explicit local systems on the affine line in characteristic $p>2$, whose geometric monodromy groups are the finite symplectic groups $S{p}_{2n}(q)$ for all $n\ge 2$, and others whose geometric monodromy groups are the special unitary groups $S{U}_n(q)$ 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 $S{p}_{2n}(q)$ for any $n\ge 2$, and others whose geometric monodromy groups are the finite general unitary groups $G{U}_n(q)$ for any odd $n\ge 3$.

56 pages

References in corpus (2)

• Symmetric powers and a problem of Kollar and Larsen
• Monodromy groups of certain Kloosterman and hypergeometric sheaves

Cited by in corpus (1)

• Gauss sums and Van der Geer--Van der Vlugt curves