Cambridge Journal of Mathematics

Volume 9 (2021)

Number 3

Hypergeometric sheaves and finite symplectic and unitary groups

Pages: 577 – 691

DOI:  https://dx.doi.org/10.4310/CJM.2021.v9.n3.a2

Authors

Nicholas M. Katz (Department of Mathematics, Princeton University, Princeton, New Jersey, U.S.A.)

Pham Huu Tiep (Department of Mathematics, Rutgers University, Piscataway, New Jersey, U.S.A.)

Abstract

We construct hypergeometric sheaves whose geometric monodromy groups are the finite symplectic groups $\mathrm{Sp}_{2n} (q)$ for any odd $n \geq 3$, for $q$ any power of an odd prime $p$. We construct other hypergeometric sheaves whose geometric monodromy groups are the finite unitary groups $\mathrm{GU}_n (q)$, for any even $n \geq 2$, for $q$ any power of any prime $p$. Suitable Kummer pullbacks of these sheaves yield local systems on $\mathbb{A}^1$, whose geometric monodromy groups are $\mathrm{Sp}_{2n} (q)$, respectively $\mathrm{SU}_n (q)$, in their total Weil representation of degree $q^n$, and whose trace functions are simple-to-remember one-parameter families of two-variable exponential sums. The main novelty of this paper is three-fold. First, it treats unitary groups $\mathrm{GU}_n (q)$ with $n$ even via hypergeometric sheaves for the first time. Second, in both the symplectic and the unitary cases, it uses a maximal torus which is a product of two sub-tori to furnish a generator of local monodromy at $0$. Third, this is the first natural occurrence of families of two-variable exponential sums in the context of finite classical groups.

Keywords

local systems, hypergeometric sheaves, monodromy groups, finite simple groups, Weil representation

2010 Mathematics Subject Classification

Primary 11T23, 14D05, 20C33. Secondary 20C15, 20D06, 20G40.

The second author gratefully acknowledges the support of the NSF (grant DMS-1840702), the Joshua Barlaz Chair in Mathematics, and the Charles Simonyi Endowment at the Institute for Advanced Study (Princeton).

Received 30 July 2021

Published 7 December 2021