Integral monodromy groups of Kloosterman sheaves

TL;DR

This paper proves that the integral monodromy groups of Kloosterman sheaves are maximally large for sufficiently large primes, extending Katz's results and using finite group theory to facilitate applications in number theory.

Contribution

It extends Katz's results on monodromy groups of Kloosterman sheaves to larger primes using finite group classification, with implications for hyper-Kloosterman sum reductions.

Findings

Monodromy groups are as large as possible for large enough primes.

Uses classification of maximal subgroups of finite groups of Lie type.

Results applicable to hyper-Kloosterman sum analysis.

Abstract

We show that integral monodromy groups of Kloosterman $\ell$-adic sheaves of rank $n\ge 2$ on $\mathbb{G}_m/\mathbb{F}_q$ are as large as possible when the characteristic $\ell$ is large enough, depending only on the rank. This variant of Katz's results over $\mathbb{C}$ was known by works of Gabber, Larsen, Nori and Hall under restrictions such as $\ell$ large enough depending on $\operatorname{char}(\mathbb{F}_q)$ with an ineffective constant, which is unsuitable for applications. We use the theory of finite groups of Lie type to extend Katz's ideas, in particular the classification of maximal subgroups of Aschbacher and Kleidman-Liebeck. These results will apply to study reductions of hyper-Kloosterman sums in forthcoming work.