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Independence of $\boldsymbol{\ell}$ of monodromy groups

Author: CheeWhye Chin
Journal: J. Amer. Math. Soc. 17 (2004), 723-747
MSC (2000): Primary 14G10; Secondary 11G40, 14F20
Published electronically: March 30, 2004
MathSciNet review: 2053954
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Abstract: Let $X$ be a smooth curve over a finite field of characteristic $p$, let $E$ be a number field, and let $\mathbf{L} = \{\mathcal{L}_\lambda\}$ be an $E$-compatible system of lisse sheaves on the curve $X$. For each place $\lambda$ of $E$not lying over $p$, the $\lambda$-component of the system $\mathbf{L}$is a lisse $E_\lambda$-sheaf $\mathcal{L}_\lambda$ on $X$, whose associated arithmetic monodromy group is an algebraic group over the local field $E_\lambda$. We use Serre's theory of Frobenius tori and Lafforgue's proof of Deligne's conjecture to show that when the $E$-compatible system $\mathbf{L}$ is semisimple and pure of some integer weight, the isomorphism type of the identity component of these monodromy groups is ``independent of $\lambda$''. More precisely, after replacing $E$ by a finite extension, there exists a connected split reductive algebraic group $G_0$ over the number field $E$such that for every place $\lambda$ of $E$not lying over $p$, the identity component of the arithmetic monodromy group of $\mathcal{L}_\lambda$is isomorphic to the group $G_0$ with coefficients extended to the local field $E_\lambda$.

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Additional Information

CheeWhye Chin
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Address at time of publication: The Broad Institute – MIT, 320 Charles Street, Cambridge, Massachusetts 02141

Keywords: Independence of $\ell$, monodromy groups, compatible systems
Received by editor(s): May 18, 2003
Published electronically: March 30, 2004
Dedicated: Dedicated to Nicholas M. Katz on his 60th birthday
Article copyright: © Copyright 2004 CheeWhye Chin