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Monodromy groups of Kloosterman and hypergeometric sheaves

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Abstract

We study the possible structures of monodromy groups of Kloosterman and hypergeometric sheaves on \({{\mathbb {G}}}_m\) in characteristic p. We show that most such sheaves satisfy a certain condition \(\mathrm {(\mathbf{S+})}\), which has very strong consequences on their monodromy groups. We also classify the finite, almost quasisimple, groups that can occur as monodromy groups of Kloosterman and hypergeometric sheaves.

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  1. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Nicholas M. Katz

  2. Department of Mathematics, Rutgers University, Piscataway, NJ, 08854, USA

    Pham Huu Tiep

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  1. Nicholas M. Katz
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  2. Pham Huu Tiep
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Correspondence to Pham Huu Tiep.

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The second author gratefully acknowledges the support of the NSF (Grants DMS-1839351 and DMS-1840702), the Joshua Barlaz Chair in Mathematics, and the Charles Simonyi Endowment at the Institute for Advanced Study (Princeton).

Part of this work was done while the second author visited the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK. It is a pleasure to thank the Institute for hospitality and support.

The authors are grateful to Richard Lyons and Will Sawin for helpful discussions, and to the referee for very careful reading of the manuscript.

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Katz, N.M., Tiep, P.H. Monodromy groups of Kloosterman and hypergeometric sheaves. Geom. Funct. Anal. 31, 562–662 (2021). https://doi.org/10.1007/s00039-021-00578-0

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