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Journal of the American Mathematical Society

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

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Compatibility of local and global Langlands correspondences
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by Richard Taylor and Teruyoshi Yoshida PDF
J. Amer. Math. Soc. 20 (2007), 467-493 Request permission

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 $v \not | l$ we deduce that $R_l(\Pi )$ is irreducible. We also obtain conditional results in the case $v|l$.
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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
  • 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.
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2276777