Rigid local systems on \(\mathbb G_m\) with finite monodromy
type: Item
instance of: scholarly article
title: Rigid local systems on \(\mathbb G_m\) with finite monodromy
zbMATH Open document ID: 1193.22001
published in: Advances in Mathematics
publication date: 2010-07-20T00:00:00Z
review text: Katz has constructed some rigid local systems of rank 7 on \(\mathbb G_m\) in finite characteristics, whose monodromy is a finite subgroup \(G\) of a complex Lie group \(M\) of type \(G_2\). Among the five subgroups \(G\) of \(M\) which appear in his work, four occur in their natural characteristic: \[ \mathrm{SL}_2(8) \text{ in }\operatorname{char} 2,\quad \mathrm{PU}_3(3) \text{ in } \operatorname{char} 3, \quad \mathrm{PGL}_2(7) \text{ in }\operatorname{char} 7, \quad \mathrm{PSL}_2(13) \text{ in }\operatorname{char} 13. \] In this paper, using the theory of Deligne-Lusztig curves, the author gives another construction of these four local systems on \(\mathbb G_m\), and finds similar local systems with finite monodromy in other complex Lie groups. Some examples of finite groups which occur in exceptional complex Lie groups \(M\), analogous to the ones above are: \[ \begin{alignedat}{2}2 G &= \mathrm{PSL}_2(27) \text{ in }\operatorname{char}3,\quad& M&=F_4,\\ G &= \mathrm{PSU}_3(8) \text{ in }\operatorname{char}2,\quad& M&=E_7,\\ G &=\mathrm{PGL}_2(31) \text{ in }\operatorname{char}31,\quad& M&=E_8,\\ G &= \mathrm{PSL}_2(61) \text{ in }\operatorname{char}61,\quad& M&=E_8. \end{alignedat} \] There are interesting families in the classical groups. For example, when \(q=p^f>2\), we have \[ G=\mathrm{PU}_3(q) \text{ in }\operatorname{char}p,\quad M=\mathrm{Sp}_{2n},\quad 2n=q(q-1). \] As a bonus, the author obtains simple wild parameters for the local field \(k((1/t))\), by restricting the local systems to the decomposition group at \(t=\infty\). These parameters are representations of the local Galois group into \(M\), which have no inertial invariants on the adjoint representation and have Swan conductor equal to the rank of \(M\).
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 5758035
zbMATH Keywords: rigid local systems
zbMATH Keywords: Deligne-Lusztig curves
zbMATH Keywords: finite simple subgroups of complex Lie groups
zbMATH Keywords: wild parameters
zbMATH Keywords: rigid embeddings
zbMATH Keywords: wild inertia group
zbMATH Keywords: Swan conductors
zbMATH Keywords: Galois group
author: Benedict H. Gross
MaRDI profile type: Publication
full work available at URL: j.aim.2010.02.008
OpenAlex ID: W2008775317
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DOI: 10.1016/J.AIM.2010.02.008
description: scientific article; zbMATH DE number 5758035