Rigid local systems and finite general linear groups
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Mathematische Zeitschrift
Rigid local systems and finite general linear groups Nicholas M. Katz1 · Pham Huu Tiep2 Received: 17 January 2020 / Accepted: 9 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We use hypergeometric sheaves on Gm /Fq , which are particular sorts of rigid local systems, to construct explicit local systems whose arithmetic and geometric monodromy groups are the finite general linear groups GLn (q) for any n ≥ 2 and any prime power q, so long as q > 3 when n = 2. This paper continues a program of finding simple (in the sense of simple to remember) families of exponential sums whose monodromy groups are certain finite groups of Lie type, cf. Gross (Adv Math 224:2531–2543, 2010), Katz (Mathematika 64:785–846, 2018) and Katz and Tiep (Finite Fields Appl 59:134–174, 2019; Adv Math 358:106859, 2019; Proc Lond Math Soc, 2020) for (certain) finite symplectic and unitary groups, or certain sporadic groups, cf. Katz and Rojas-León (Finite Fields Appl 57:276–286, 2019) and Katz et al. (J Number Theory 206:1–23, 2020; Int J Number Theory 16:341–360, 2020; Trans Am Math Soc 373:2007–2044, 2020). The novelty of this paper is obtaining GLn (q) in this hypergeometric way. A pullback construction then yields local systems on A1 /Fq whose geometric monodromy groups are SLn (q). These turn out to recover a construction of Abhyankar. Keywords Rigid local systems · Monodromy groups · Weil representations · Finite general linear groups Mathematics Subject Classification 11T23 · 20C33 · 20G40
P. H. Tiep gratefully acknowledges the support of the NSF (Grant DMS-1840702), and the Joshua Barlaz Chair in Mathematics. The authors are grateful to the referee for careful reading of the paper and many comments and suggestions that help greatly improve the exposition of the paper.
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Pham Huu Tiep [email protected] Nicholas M. Katz [email protected]
1
Department of Mathematics, Princeton University, Princeton, NJ 08544, USA
2
Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA
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N. M. Katz, P. H. Tiep
Contents Introduction . . . . . . . . . . . . . . . . . . . . . . 1 The set up . . . . . . . . . . . . . . . . . . . . . 2 The trace function of H1 . . . . . . . . . . . . . 3 The trace function of Hχ for χ = 1 . . . . . . . 4 Putting it all together . . . . . . . . . . . . . . . . 5 Galois groups in this context . . . . . . . . . . . . 6 Weil-type representations of special linear groups . 7 Weil representations of SL2 (q) . . . . . . . . . . 8 The structure of monodromy groups . . . . . . . . 9 Relation to work of Abhyankar . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .
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