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Compatibility of local and global Langlands correspondences


Authors: Richard Taylor and Teruyoshi Yoshida
Journal: J. Amer. Math. Soc. 20 (2007), 467-493
MSC (2000): Primary 11R39; Secondary 11F70, 11F80, 14G35
DOI: https://doi.org/10.1090/S0894-0347-06-00542-X
Published electronically: July 10, 2006
MathSciNet review: 2276777
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Abstract: We prove the compatibility of local and global Langlands correspondences for $ GL_n$, which was proved up to semisimplification in M. Harris and R. Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties, Ann. of Math. Studies 151, Princeton Univ. Press, Princeton-Oxford, 2001. More precisely, for the $ n$-dimensional $ l$-adic representation $ R_l(\Pi)$ of the Galois group of an imaginary CM-field $ L$ attached to a conjugate self-dual regular algebraic cuspidal automorphic representation $ \Pi$ of $ GL_n(\mathbb{A}_L)$, which is square integrable at some finite place, we show that Frobenius semisimplification of the restriction of $ R_l(\Pi)$ to the decomposition group of a place $ v$ of $ L$ not dividing $ l$ corresponds to $ \Pi_v$ by the local Langlands correspondence. If $ \Pi_v$ is square integrable for some finite place $ v \not\vert l$ we deduce that $ R_l(\Pi)$ is irreducible. We also obtain conditional results in the case $ v\vert l$.


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Additional Information

Richard Taylor
Affiliation: Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, Massachusetts 02138
Email: rtaylor@math.harvard.edu

Teruyoshi Yoshida
Affiliation: Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, Massachusetts 02138
Email: yoshida@math.harvard.edu

DOI: https://doi.org/10.1090/S0894-0347-06-00542-X
Received by editor(s): April 8, 2005
Published electronically: July 10, 2006
Additional Notes: This material is based upon work supported by the National Science Foundation under Grant No. 0100090. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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