Journal of the American Mathematical Society

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



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
Published electronically: July 10, 2006
MathSciNet review: 2276777
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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$.

References [Enhancements On Off] (What's this?)

  • [AK] Allen Altman and Steven Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Mathematics, Vol. 146, Springer-Verlag, Berlin-New York, 1970. MR 0274461
  • [B] Laurent Berger, Représentations 𝑝-adiques et équations différentielles, Invent. Math. 148 (2002), no. 2, 219–284 (French, with English summary). MR 1906150, 10.1007/s002220100202
  • [Bo] P. Boyer, Monodromie du faisceau pervers des cycles évanescents et quelques variétés de Shimura simples et applications (avec un appendice de L. Fargues), boyer/.
  • [Ca] W. Casselman, The unramified principal series of 𝔭-adic groups. I. The spherical function, Compositio Math. 40 (1980), no. 3, 387–406. MR 571057
  • [Cl] 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
  • [D] P. Deligne, Les constantes des équations fonctionnelles des fonctions 𝐿, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1973, pp. 501–597. Lecture Notes in Math., Vol. 349 (French). MR 0349635
  • [DR] 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) Springer, Berlin, 1973, pp. 143–316. Lecture Notes in Math., Vol. 349 (French). MR 0337993
  • [Dr] V. G. Drinfel′d, Elliptic modules, Mat. Sb. (N.S.) 94(136) (1974), 594–627, 656 (Russian). MR 0384707
  • [Fo] 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
  • [Fr] A. Fröhlich, Formal groups, Lecture Notes in Mathematics, No. 74, Springer-Verlag, Berlin-New York, 1968. MR 0242837
  • [G] Ulrich Görtz, On the flatness of models of certain Shimura varieties of PEL-type, Math. Ann. 321 (2001), no. 3, 689–727. MR 1871975, 10.1007/s002080100250
  • [GM] Henri Gillet and William Messing, Cycle classes and Riemann-Roch for crystalline cohomology, Duke Math. J. 55 (1987), no. 3, 501–538. MR 904940, 10.1215/S0012-7094-87-05527-X
  • [HT] 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
  • [I] Tetsushi Ito, Weight-monodromy conjecture for 𝑝-adically uniformized varieties, Invent. Math. 159 (2005), no. 3, 607–656. MR 2125735, 10.1007/s00222-004-0395-y
  • [KMa] 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
  • [KMe] Nicholas M. Katz and William Messing, Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math. 23 (1974), 73–77. MR 0332791
  • [M] A. Mokrane, La suite spectrale des poids en cohomologie de Hyodo-Kato, Duke Math. J. 72 (1993), no. 2, 301–337 (French). MR 1248675, 10.1215/S0012-7094-93-07211-0
  • [O] Tadashi Ochiai, 𝑙-independence of the trace of monodromy, Math. Ann. 315 (1999), no. 2, 321–340. MR 1715253, 10.1007/s002080050370
  • [RZ] 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, 10.1007/BF01394268
  • [Sa1] Takeshi Saito, Modular forms and 𝑝-adic Hodge theory, Invent. Math. 129 (1997), no. 3, 607–620. MR 1465337, 10.1007/s002220050175
  • [Sa2] Takeshi Saito, Weight spectral sequences and independence of 𝑙, J. Inst. Math. Jussieu 2 (2003), no. 4, 583–634. MR 2006800, 10.1017/S1474748003000173
  • [Se] Jean-Pierre Serre, Abelian 𝑙-adic representations and elliptic curves, McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0263823
  • [Ta] 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
  • [Ts] 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, 10.1007/s002220050330
  • [Y] T. Yoshida, On non-abelian Lubin-Tate theory via vanishing cycles, math-NT/0404375, to appear in Ann. de l'Institut Fourier.

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 11R39, 11F70, 11F80, 14G35

Retrieve articles in all journals with MSC (2000): 11R39, 11F70, 11F80, 14G35

Additional Information

Richard Taylor
Affiliation: Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, Massachusetts 02138

Teruyoshi Yoshida
Affiliation: Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, Massachusetts 02138

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.