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On the Zeta-Functions of Some Simple Shimura Varieties

Published online by Cambridge University Press:  20 November 2018

R. P. Langlands
R. P. Langlands*
Affiliation:
Tehran University of Technology, Tehran, Iran
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In an earlier paper [14] I have adumbrated a method for establishing that the zeta-function of a Shimura variety associated to a quaternion algebra over a totally real field can be expressed as a product of L-functions associated to automorphic forms. Now I want to add some body to that sketch. The representation-theoretic and combinatorial aspects of the proof will be given in detail, but it will simply be assumed that the set of geometric points has the structure suggested in [13]. This is so at least when the algebra is totally indefinite, but it is proved by algebraic-geometric methods that are somewhat provisional in the context of Shimura varieties. However, contrary to the suggestion in [13] the general moduli problem has yet to be treated fully. There are unresolved difficulties, but they do not arise for the problem attached to a totally indefinite quaternion algebra, which is discussed in detail in [17].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 6 , 01 December 1979 , pp. 1121 - 1216
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Artin, E. and Tate, J., Class field theory (Harvard, 1967).Google Scholar
2. Borel, A. and Wallach, N., Seminar on continuous cohomology, IAS (1977).Google Scholar
3. Borel, A., Formes automorphes et séries de Dirichlet, Sém. Bourbaki (1975).Google Scholar
4. Deligne, P., Travaux de Shimura, Sém. Bourbaki (1971), 129–154.Google Scholar
5. Hecht, H. and Schmid, W., A proof of Blattner s conjecture, Inv. Math. 31 (1975).Google Scholar
6. Jacquet, H. and Langlands, R. P., Automorphic Forms on GL(2), Springer Lecture Notes 114 (1970).Google Scholar
7. Labesse, J.-P. and Langlands, R. P., L-indistinguishability for SL(2), Can. J. Math. 31 (1979), 726–785.Google Scholar
8. Langlands, R. P., The dimension of spaces of automorphic forms, Amer. J. Math. 85 (1963).Google Scholar
9. Langlands, R. P., Problems in the theory of automorphic forms, in Modern Analysis and Applications, Springer Lecture Notes 170 (1970).Google Scholar
10. Langlands, R. P., On Artin s L-functions, Rice University Studies. 56 (1970), 23–28.Google Scholar
11. Langlands, R. P., On the classification of irreducible representations of real algebraic groups, Notes, IAS (1973).Google Scholar
12. Langlands, R. P., Modular forms and Vadic representations, in Modular functions of one variable II, Springer Lecture Notes 349 (1973).Google Scholar
13. Langlands, R. P., Some contemporary problems with origins in the Jugendtraum in Mathematical developments arising from Hilbert problems, A.M.S. (1976), 401–418.Google Scholar
14. Langlands, R. P., Shimura varieties and the Selberg trace formula, Can. J. Math., vol. XXIX (1977).Google Scholar
15. Langlands, R. P., Base change for GL(2), Notes, IAS (1975).Google Scholar
16. Langlands, R. P., Stable conjugacy: definitions and lemmas, Can. J. Math. 31 (1979), 700–725.Google Scholar
17. Milne, J. S., Points on Shimura varieties mod p, in Automorphic Forms, Representations, and L-functions (1979), II, 165–184.Google Scholar
18. Serre, J.-P., Corps locaux, Hermann et Cie (1962).Google Scholar
19. Serre, J.-P., Cohomologie Galoisienne, Springer Lecture Notes 5 (1965).Google Scholar
20. Serre, J.-P., Facteurs locaux des fonctions zêta des variétés algébriques ﹛Définitions et conjectures), Sém. Delange-Pisot-Poitou (1969-70).Google Scholar
21. Shelstad, D., Characters and inner forms of a quasi-split group over R, Com p. Math.,. 39 (1979), 11–46.Google Scholar
22. Shelstad, D., Notes on L-indistinguishability in Automorphic forms, Representations and Lfunctions (1979), II, 193–204.Google Scholar
23. Springer, T. A. and Steinberg, R., Conjugacy classes, in Springer Lecture Notes 131 (1970).Google Scholar