## Abstract

We use hypergeometric sheaves on G_{m}/ F_{q}, 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 A^{1}/ F_{q} whose geometric monodromy groups are SL _{n}(q). These turn out to recover a construction of Abhyankar.

Original language | English (US) |
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Pages (from-to) | 1293-1321 |

Number of pages | 29 |

Journal | Mathematische Zeitschrift |

Volume | 298 |

Issue number | 3-4 |

DOIs | |

State | Published - Aug 2021 |

## All Science Journal Classification (ASJC) codes

- General Mathematics

## Keywords

- Finite general linear groups
- Monodromy groups
- Rigid local systems
- Weil representations