Inventiones mathematicae

, Volume 192, Issue 3, pp 663–715 | Cite as

The Local Langlands Correspondence for GL n over p-adic fields



We extend our methods from Scholze (Invent. Math. 2012, doi: 10.1007/s00222-012-0419-y) to reprove the Local Langlands Correspondence for GL n over p-adic fields as well as the existence of -adic Galois representations attached to (most) regular algebraic conjugate self-dual cuspidal automorphic representations, for which we prove a local-global compatibility statement as in the book of Harris-Taylor (The Geometry and Cohomology of Some Simple Shimura Varieties, 2001).

In contrast to the proofs of the Local Langlands Correspondence given by Henniart (Invent. Math. 139(2), 439–455, 2000), and Harris-Taylor (The Geometry and Cohomology of Some Simple Shimura Varieties, 2001), our proof completely by-passes the numerical Local Langlands Correspondence of Henniart (Ann. Sci. Éc. Norm. Super. 21(4), 497–544, 1988). Instead, we make use of a previous result from Scholze (Invent. Math. 2012, doi: 10.1007/s00222-012-0419-y) describing the inertia-invariant nearby cycles in certain regular situations.



First of all, I thank my advisor M. Rapoport for explaining me the Langlands-Kottwitz method of counting points, which plays a crucial role in this article, for his encouragement to work on this topic, and for the many other things he taught me. Furthermore, my thanks go to Guy Henniart and Vincent Sécherre for their advice in type theory, among other things. Moreover, I am grateful for the financial support of the Hausdorff Center for Mathematics in Bonn, and the hospitality of the Institut Henri Poincaré and Harvard University, where part of this work was carried out.


  1. 1.
    Arthur, J., Clozel, L.: Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula. Annals of Mathematics Studies, vol. 120. Princeton University Press, Princeton (1989) MATHGoogle Scholar
  2. 2.
    Artin, M.: Algebraization of formal moduli. I. In: Global Analysis (Papers in Honor of K. Kodaira), pp. 21–71. Univ. Tokyo Press, Tokyo (1969) Google Scholar
  3. 3.
    Berkovich, V.G.: Vanishing cycles for formal schemes. II. Invent. Math. 125(2), 367–390 (1996) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bernstein, I.N., Zelevinsky, A.V.: Induced representations of reductive \({\mathfrak{p}}\)-adic groups. I. Ann. Sci. Éc. Norm. Super. 10(4), 441–472 (1977) MathSciNetMATHGoogle Scholar
  5. 5.
    Buzzard, K., Gee, T.: The conjectural connections between automorphic representations and Galois representations. arXiv:1009.0785. Proceedings of the LMS Durham Symposium (2011, to appear)
  6. 6.
    Drinfel’d, V.G.: Elliptic modules. Mat. Sb. 94(136), 594–627 (1974) MathSciNetGoogle Scholar
  7. 7.
    Faltings, G.: Group schemes with strict \(\mathcal{O}\)-action. Mosc. Math. J. 2(2), 249–279 (2002). Dedicated to Yuri I. Manin on the occasion of his 65th birthday MathSciNetMATHGoogle Scholar
  8. 8.
    Fargues, L.: Motives and automorphic forms: the Abelian case
  9. 9.
    Harris, M.: The local Langlands conjecture for GL(n) over a p-adic field, n<p. Invent. Math. 134(1), 177–210 (1998) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Harris, M., Taylor, R.: The Geometry and Cohomology of Some Simple Shimura Varieties. Annals of Mathematics Studies, vol. 151. Princeton University Press, Princeton (2001). With an appendix by Vladimir G. Berkovich MATHGoogle Scholar
  11. 11.
    Henniart, G.: La conjecture de Langlands locale numérique pour GL(n). Ann. Sci. Éc. Norm. Super. 21(4), 497–544 (1988) MathSciNetMATHGoogle Scholar
  12. 12.
    Henniart, G.: Caractérisation de la correspondance de Langlands locale par les facteurs ϵ de paires. Invent. Math. 113(2), 339–350 (1993) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Henniart, G.: Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique. Invent. Math. 139(2), 439–455 (2000) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Jacquet, H., Piatetski-Shapiro, I.I., Shalika, J.: Conducteur des représentations du groupe linéaire. Math. Ann. 256(2), 199–214 (1981) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Kazhdan, D.: Cuspidal geometry of p-adic groups. J. Anal. Math. 47, 1–36 (1986) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Kottwitz, R.E.: On the λ-adic representations associated to some simple Shimura varieties. Invent. Math. 108(3), 653–665 (1992) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Kottwitz, R.E.: Points on some Shimura varieties over finite fields. J. Am. Math. Soc. 5(2), 373–444 (1992) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Langlands, R.P.: Representations of abelian algebraic groups. Pac. J. Math. (Special Issue), 231–250 (1997). Olga Taussky-Todd: in memoriam Google Scholar
  19. 19.
    Mantovan, E.: On certain unitary group Shimura varieties. Astérisque 291, 201–331 (2004) Variétés de Shimura, espaces de Rapoport-Zink et correspondances de Langlands locales MathSciNetGoogle Scholar
  20. 20.
    Mantovan, E.: On the cohomology of certain PEL-type Shimura varieties. Duke Math. J. 129(3), 573–610 (2005) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Mieda, Y.: On l-independence for the étale cohomology of rigid spaces over local fields. Compos. Math. 143(2), 393–422 (2007) MathSciNetMATHGoogle Scholar
  22. 22.
    Schneider, P., Zink, E.-W.: K-types for the tempered components of a p-adic general linear group. J. Reine Angew. Math. 517, 161–208 (1999). With an appendix by Schneider and U. Stuhler MathSciNetMATHGoogle Scholar
  23. 23.
    Scholze, P.: The Langlands-Kottwitz approach for some simple Shimura varieties. Invent. Math. (2012). doi: 10.1007/s00222-012-0419-y MATHGoogle Scholar
  24. 24.
    Scholze, P.: The Langlands-Kottwitz approach for the modular curve. Int. Math. Res. Not. 15, 3368–3425 (2011) MathSciNetGoogle Scholar
  25. 25.
    Scholze, P.: The Langlands-Kottwitz method and deformation spaces of p-divisible groups. arXiv:1110.0230. Am. Math. Soc. (2011, to appear)
  26. 26.
    Scholze, P., Shin, S.W.: On the cohomology of compact unitary group Shimura varieties at ramified split places. arXiv:1110.0232. Am. Math. Soc. (2011, to appear)
  27. 27.
    Shin, S.W.: Counting points on Igusa varieties. Duke Math. J. 146(3), 509–568 (2009) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Strauch, M.: On the Jacquet-Langlands correspondence in the cohomology of the Lubin-Tate deformation tower. Astérisque 298, 391–410 (2005). Automorphic forms. I MathSciNetGoogle Scholar
  29. 29.
    Varshavsky, Y.: Lefschetz-Verdier trace formula and a generalization of a theorem of Fujiwara. Geom. Funct. Anal. 17(1), 271–319 (2007) MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Zelevinsky, A.V.: Induced representations of reductive \({\mathfrak{p}}\)-adic groups. II. On irreducible representations of GL(n). Ann. Sci. Éc. Norm. Super. 13(2), 165–210 (1980) MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Mathematisches Institut der Universität BonnBonnGermany

Personalised recommendations