The Action of an Automorphism of C On a Shimura Variety and its Special Points

  • J. S. Milne
Part of the Progress in Mathematics book series (PM, volume 35)


In [8, pp. 222–223] Langlands made a very precise conjecture describing how an automorphism of C acts on a Shirnura variety and its special points. The results of Milne-Shin [15], when combined with the result of Deligne [5], give a proof of the conjecture (including its supplement) for all Shirnura varieties of abelian type (this class excludes only those varieties associated with groups having factors of exceptional type and most types D). here the proof is extended to cover all Shimura varieties. As a consequence, one obtains a complete proof of Shimura’s conjecture on the existence of canonical models. The main new ingredients in the proof are the results of Kazhdan [7] and the methods of Borovoi [2].


Conjugacy Class Maximal Torus Finite Index Galois Extension Canonical Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. Borel, Density and maximality of arithmetic subgroups. J. Reine Angew. Math. 224 (1966), 78–89.Google Scholar
  2. [2]
    M. Borovoi, Canonical models of Shimura varieties. Ilandwritten notes dated 26/5/81.Google Scholar
  3. [3]
    P. Deligne, Travaux de Shimura. Sénr. Bourbaki Février 71, Exposé 389, Lecture Notes in Math., 244, Springer, Berlin, 1971.Google Scholar
  4. [4]
    P. Deligne, Variétés de Shimura: interpretation modulaire, et techniques de construction de modéles canoniques. Proc. Symp. Pure Math., A.M.S., 33 (1979), part 2, 247–290.Google Scholar
  5. [5]
    P. Deligne, Motifs et groupes de Taniyama. Lecture Notes in Math., 900, Springer, Berlin, 1982, pp. 261–279.Google Scholar
  6. [6]
    D. Kazhdan, On arithmetic varieties. Lie Groups and Their Representations. Budapest, 1971,. pp. 151–217.Google Scholar
  7. [7]
    D. Kazhdan, On arithmetic varieties, II. Preprint.Google Scholar
  8. [8]
    R. Langlands, Automorphic representations, Shimura varieties, and motives. Ein Märchen. Proc. Symp. Pure Math., A.M.S. 33 (1979), part 2, 205–246.Google Scholar
  9. [9]
    R. Langlands, Les débuts d’une formule des traces stables. Publ. Math. Univ. Paris VII. (To appear);Google Scholar
  10. [10]
    G. Margulis, Discrete groups of motions of manifolds of nonpositive curvature. Amer. Math. Soc. Transl. 109 (1977), 33–45.Google Scholar
  11. [11]
    J. Milne, The arithmetic of automorphic functions. In preparation.Google Scholar
  12. [12]
    J. Milne and K-y. Shih, The action of complex conjugation on a Shimura variety. Annals of Math., 113 (1981), 569–599.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    J. Milne and K-y. Shill, Automorphism groups of Shimura varieties and reciprocity laws. Amer. J. Math., 103 (1981), 911–935.CrossRefGoogle Scholar
  14. [14]
    J. Milne and K-y. Shih, Langlands’s construction of the Taniyama group. Lecture Notes in Math., 900, Springer, Berlin, 1982, 229260.Google Scholar
  15. [15]
    J. Milne and K-y. Shih, Conjugates of Shimura varieties. Lecture Notes in Math., 900, Springer, Berlin, 1982, pp. 280–356.Google Scholar
  16. [16]
    V. Platonov and A. Rapincuk, On the group of rational points of three-dimensional groups. Soviet Math. Dokl. 20 (1979), 693–697.zbMATHGoogle Scholar
  17. [17]
    M. Ragunathan, Discrete Subgroups of Lie Groups. Erg. Math. 68, Springer, Berlin, 1972.Google Scholar
  18. [18]
    I. Satake, Algebraic Structures of Symmetric Domains. Publ. Math. Soc. Japan 14, Princeton University Press, Princeton, 1980.Google Scholar
  19. [19]
    G. Shimura, On arithmetic automorphic functions. Actes, Congrès Intern. Math. (1970) Tom 2, 343–348.Google Scholar
  20. [20]
    G. Shirnura, Arithmetic Theory of Automorphic Functions. Publ. Math. Soc. Japan 11, Princeton University Press, Princeton, 1971.Google Scholar
  21. [21.]
    S-T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I. Comm. Pure Appl. Math., 31 (1978), 339–411.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1983

Authors and Affiliations

  • J. S. Milne
    • 1
  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA

Personalised recommendations