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.
All Science Journal Classification (ASJC) codes
- Finite general linear groups
- Monodromy groups
- Rigid local systems
- Weil representations