Skip to Main Content

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 mathematics.

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

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Compatibility of local and global Langlands correspondences
HTML articles powered by AMS MathViewer

by Richard Taylor and Teruyoshi Yoshida
J. Amer. Math. Soc. 20 (2007), 467-493
Published electronically: July 10, 2006


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 $v|l$.
  • Allen Altman and Steven Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Mathematics, Vol. 146, Springer-Verlag, Berlin-New York, 1970. MR 0274461, DOI 10.1007/BFb0060932
  • Laurent Berger, Représentations $p$-adiques et équations différentielles, Invent. Math. 148 (2002), no. 2, 219–284 (French, with English summary). MR 1906150, DOI 10.1007/s002220100202
  • [Bo]bo P. Boyer, Monodromie du faisceau pervers des cycles évanescents et quelques variétés de Shimura simples et applications (avec un appendice de L. Fargues),
  • W. Casselman, The unramified principal series of ${\mathfrak {p}}$-adic groups. I. The spherical function, Compositio Math. 40 (1980), no. 3, 387–406. MR 571057
  • Laurent Clozel, Motifs et formes automorphes: applications du principe de fonctorialité, Automorphic forms, Shimura varieties, and $L$-functions, Vol. I (Ann Arbor, MI, 1988) Perspect. Math., vol. 10, Academic Press, Boston, MA, 1990, pp. 77–159 (French). MR 1044819
  • P. Deligne, Les constantes des équations fonctionnelles des fonctions $L$, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973, pp. 501–597 (French). MR 0349635
  • P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973, pp. 143–316 (French). MR 0337993
  • V. G. Drinfel′d, Elliptic modules, Mat. Sb. (N.S.) 94(136) (1974), 594–627, 656 (Russian). MR 0384707
  • Périodes $p$-adiques, Société Mathématique de France, Paris, 1994 (French). Papers from the seminar held in Bures-sur-Yvette, 1988; Astérisque No. 223 (1994). MR 1293969
  • A. Fröhlich, Formal groups, Lecture Notes in Mathematics, No. 74, Springer-Verlag, Berlin-New York, 1968. MR 0242837, DOI 10.1007/BFb0074373
  • Ulrich Görtz, On the flatness of models of certain Shimura varieties of PEL-type, Math. Ann. 321 (2001), no. 3, 689–727. MR 1871975, DOI 10.1007/s002080100250
  • Henri Gillet and William Messing, Cycle classes and Riemann-Roch for crystalline cohomology, Duke Math. J. 55 (1987), no. 3, 501–538. MR 904940, DOI 10.1215/S0012-7094-87-05527-X
  • Michael Harris and Richard Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton, NJ, 2001. With an appendix by Vladimir G. Berkovich. MR 1876802
  • Tetsushi Ito, Weight-monodromy conjecture for $p$-adically uniformized varieties, Invent. Math. 159 (2005), no. 3, 607–656. MR 2125735, DOI 10.1007/s00222-004-0395-y
  • Nicholas M. Katz and Barry Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol. 108, Princeton University Press, Princeton, NJ, 1985. MR 772569, DOI 10.1515/9781400881710
  • Nicholas M. Katz and William Messing, Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math. 23 (1974), 73–77. MR 332791, DOI 10.1007/BF01405203
  • A. Mokrane, La suite spectrale des poids en cohomologie de Hyodo-Kato, Duke Math. J. 72 (1993), no. 2, 301–337 (French). MR 1248675, DOI 10.1215/S0012-7094-93-07211-0
  • Tadashi Ochiai, $l$-independence of the trace of monodromy, Math. Ann. 315 (1999), no. 2, 321–340. MR 1715253, DOI 10.1007/s002080050370
  • M. Rapoport and Th. Zink, Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik, Invent. Math. 68 (1982), no. 1, 21–101 (German). MR 666636, DOI 10.1007/BF01394268
  • Takeshi Saito, Modular forms and $p$-adic Hodge theory, Invent. Math. 129 (1997), no. 3, 607–620. MR 1465337, DOI 10.1007/s002220050175
  • Takeshi Saito, Weight spectral sequences and independence of $l$, J. Inst. Math. Jussieu 2 (2003), no. 4, 583–634. MR 2006800, DOI 10.1017/S1474748003000173
  • Jean-Pierre Serre, Abelian $l$-adic representations and elliptic curves, W. A. Benjamin, Inc., New York-Amsterdam, 1968. McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute. MR 0263823
  • J. Tate, Number theoretic background, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–26. MR 546607
  • Takeshi Tsuji, $p$-adic étale cohomology and crystalline cohomology in the semi-stable reduction case, Invent. Math. 137 (1999), no. 2, 233–411. MR 1705837, DOI 10.1007/s002220050330
  • [Y]y T. Yoshida, On non-abelian Lubin-Tate theory via vanishing cycles, arXiv:math-NT/0404375, to appear in Ann. de l’Institut Fourier.
Similar Articles
Bibliographic Information
  • Richard Taylor
  • Affiliation: Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, Massachusetts 02138
  • Email:
  • Teruyoshi Yoshida
  • Affiliation: Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, Massachusetts 02138
  • Email:
  • 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:
  • MathSciNet review: 2276777