Rigid local systems and finite general linear groups

Abstract

We use hypergeometric sheaves on ${{\mathbb {G}}}_m/{{\mathbb {F}}}_q$, 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 $\mathrm {GL}_n(q)$ for any $n \ge 2$ 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 $\mathrm {GL}_n(q)$ in this hypergeometric way. A pullback construction then yields local systems on ${{\mathbb {A}}}^1/{{\mathbb {F}}}_q$ whose geometric monodromy groups are $\mathrm {SL}_n(q)$. These turn out to recover a construction of Abhyankar.