Rigid local systems and finite general linear groups

2020/02/14 by Katz, Nicholas M., Tiep, Pham Huu
#11T23 #20C33 #20G40 #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2002.05863

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 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 A¹/F_q whose geometric monodromy groups are SL_n(q). These turn out to recover a construction of Abhyankar.

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