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  • Corentin Perret-Gentil (a1)


We show that integral monodromy groups of Kloosterman $\ell$ -adic sheaves of rank $n\geqslant 2$ 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 hyper-Kloosterman sums and their reductions in forthcoming work.



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Current address: Centre de recherches mathématiques, Université de Montréal, Case postale 6128, Montréal QC H3C 3J7, Canada



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