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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Independence of $\boldsymbol {\ell }$ of monodromy groups
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by CheeWhye Chin PDF
J. Amer. Math. Soc. 17 (2004), 723-747

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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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
  • Email: cheewhye@math.berkeley.edu, cheewhye@mit.edu
  • Received by editor(s): May 18, 2003
  • Published electronically: March 30, 2004

  • Dedicated: Dedicated to Nicholas M. Katz on his 60th birthday
  • © 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