Independence of $\boldsymbol {\ell }$ of monodromy groups
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- by CheeWhye Chin
- J. Amer. Math. Soc. 17 (2004), 723-747
- DOI: https://doi.org/10.1090/S0894-0347-04-00456-4
- Published electronically: March 30, 2004
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 $E_\lambda$.References
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Bibliographic 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
- Email: cheewhye@math.berkeley.edu, cheewhye@mit.edu
- Received by editor(s): May 18, 2003
- Published electronically: March 30, 2004
- © Copyright 2004 CheeWhye Chin
- Journal: J. Amer. Math. Soc. 17 (2004), 723-747
- MSC (2000): Primary 14G10; Secondary 11G40, 14F20
- DOI: https://doi.org/10.1090/S0894-0347-04-00456-4
- MathSciNet review: 2053954
Dedicated: Dedicated to Nicholas M. Katz on his 60th birthday