Acta Mathematica Vietnamica

, Volume 43, Issue 1, pp 45–66 | Cite as

Langlands Parameterization over Function Fields Following V. Lafforgue

  • Jochen HeinlothEmail author


These are a slightly expanded notes of an expository talk on V. Lafforgue’s construction of one direction (called automorphic to Galois) of the Langlands correspondence for function fields.


Langlands correspondence Drinfeld shtukas 

Mathematics Subject Classification (2010)

11R39 14D20 



I would like to thank B. C. Ngô for the invitation to VIASM and many discussions. Also I would like to thank the participants of the Forschungsseminar on [8] in Essen for all of their comments and discussions.


  1. 1.
    Beilinson, A., Drinfeld, V.: Quantization of Hitchin’s integrable system and Hecke eigensheaves. Preprint.
  2. 2.
    Drinfeld, V.: Langlands’ conjecture for GL(2) over function fields. Proc. Int. Congress Math. (Helsinki) 1978, 565–574 (1980)zbMATHGoogle Scholar
  3. 3.
    Gaitsgory, D.: Construction of central elements in the affine Hecke algebra via nearby cycles. Invent. Math. 144, 253–280 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Genestier, A., Lafforgue, V: Chtoukas restreints pour les groupes réductifs et paramétrisation de Langlands locale. arXiv:
  5. 5.
    Lafforgue, L.: Chtoucas de Drinfeld et correspondance de Langlands. Invent. Math. 147, 1–241 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Lafforgue, L.: Chtoukas de Drinfeld et conjecture de Ramanujan-Petersson. Astérisque 243 ii+ 329 pp (1997)Google Scholar
  7. 7.
    Lau, E.: On generalized D-shtukas. (Dissertation) Bonner Mathematische Schriften 369 xii+ 110 pp (2004)Google Scholar
  8. 8.
    Lafforgue, V.: Chtoukas pour les groupes réductifs et paramétrisation de Langlands globale. arXiv:
  9. 9.
    Lafforgue, V.: Introduction to chtoucas for reductive groups and to the global Langlands parameterization. arXiv:
  10. 10.
    Mirković, I., Vilonen, K.: Geometric Langlands duality and representations of algebraic groups over commutative rings. Ann. of Math. (2) 166(1), 95–143 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ngô, B.C.: Introduction to Drinfeld’s shtukas (Informal lecture notes)Google Scholar
  12. 12.
    Reich, R.: Twisted geometric Satake equivalence via gerbes on the factorizable grassmannian. Represent. Theory 16, 345–449 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Richardson, R. E.: Conjugacy classes of n-tuples in Lie algebras and algebraic groups. Duke Math. J. 57, 1–35 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Richarz, T.: A new approach to the geometric Satake equivalence. Doc. Math. 19, 209–246 (2014)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Scholze, P.: Lectures on p-adic geometry (Notes by J. Weinstein).
  16. 16.
    Stroh, B.: La paramétrisation de Langlands globale sur les corps de fonctions, d’après Vincent Lafforgue. Séminaire Bourbaki (janvier 2016),
  17. 17.
    Taylor, R.: Galois representations associated to Siegel modular forms of low weight. Duke Math. J. 63, 281–332 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Varshavsky, Y.: Moduli of principal F-sheaves. Selecta Math. (N.S.) 10(1), 131–166 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Wiles, A.: On ordinary A-adic representations associated to modular forms. Invent. Math. 94, 529–573 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Zhu, X.: An introduction to affine Grassmannians and the geometric Satake equivalence. arXiv:

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Fachbereich MathematikUniversität Duisburg–EssenEssenGermany

Personalised recommendations