TY - JOUR

T1 - Rigid local systems and finite general linear groups

AU - Katz, Nicholas M.

AU - Tiep, Pham Huu

N1 - Funding Information:
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.
Publisher Copyright:
© 2020, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2021/8

Y1 - 2021/8

N2 - 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 GL n(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 GL n(q) in this hypergeometric way. A pullback construction then yields local systems on A1/ Fq whose geometric monodromy groups are SL n(q). These turn out to recover a construction of Abhyankar.

AB - 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 GL n(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 GL n(q) in this hypergeometric way. A pullback construction then yields local systems on A1/ Fq whose geometric monodromy groups are SL n(q). These turn out to recover a construction of Abhyankar.

KW - Finite general linear groups

KW - Monodromy groups

KW - Rigid local systems

KW - Weil representations

UR - http://www.scopus.com/inward/record.url?scp=85095984361&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85095984361&partnerID=8YFLogxK

U2 - 10.1007/s00209-020-02617-2

DO - 10.1007/s00209-020-02617-2

M3 - Article

AN - SCOPUS:85095984361

SN - 0025-5874

VL - 298

SP - 1293

EP - 1321

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 3-4

ER -