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# Mathematics > Representation Theory

# Title: Rigid local systems and finite general linear groups

(Submitted on 14 Feb 2020 (v1), last revised 3 Aug 2020 (this version, v2))

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 $n \ge 2$ and 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. [Gr], [KT1], [KT2], [KT3] for (certain) finite symplectic and unitary groups, or certain sporadic groups, cf. [KRL], [KRLT1], [KRLT2], [KRLT3]. 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.

## Submission history

From: Pham Tiep [view email]**[v1]**Fri, 14 Feb 2020 03:57:22 GMT (33kb)

**[v2]**Mon, 3 Aug 2020 17:28:51 GMT (33kb)

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