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Shimura varieties and twisted orbital integrals. (English) Zbl 0533.14009
Langlands’ work on the Shimura varieties attached to quaternion algebras over totally real number fields led him to outline an attack on the problem of expressing the zeta function of a general Shimura variety in terms of automorphic L-functions. He divided his approach into parts, one of them being a combinatorial problem which arises during the comparison of the Selberg trace formula with the number of points on the Shimura variety over a finite field. The author of this paper shows that Langlands’ combinatorial problem can be reduced to a standard problem in local harmonic analysis: that of establishing the spherical function identities needed for base change.

##### MSC:
 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14G25 Global ground fields in algebraic geometry 11R80 Totally real fields 14K15 Arithmetic ground fields for abelian varieties 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols
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##### References:
 [1] Bourbaki, N.: Groupes et algebrès de Lie. Chap. IV, V, VI. Paris: Hermann 1968 [2] Bruhat, F., Tits, J.: Groupes réductifs sur un corps local. I. Publ. Math. IHES41, 5-251 (1972) · Zbl 0254.14017 [3] Deligne, P.: Variétés de Shimura. In: Automorphic forms, representations andL-functions. Proc. Sympos. Pure Math., Vol. 33, Part 2, pp. 247-289. Providence, R.I.: Am. Math. Soc. 1979 [4] Greenberg, M.: Schemata over local rings. II. Ann. Math.78, 256-266 (1963) · Zbl 0126.16704 [5] Kottwitz, R.: Orbital integrals on GL3. Am. J. Math.102, 327-384 (1980) · Zbl 0437.22011 [6] Kottwitz, R.: Rational conjugacy classes in reductive groups. Duke Math. J.49, 785-806 (1982) · Zbl 0506.20017 [7] Kottwitz, R.: Stable trace formula: cuspidal tempered terms. Duke Math. J. (to appear) · Zbl 0576.22020 [8] Langlands, R.P.: Some contemporary problems with origins in the Jugendtraum. In: Mathematical developments arising from Hilbert problems. Proc. Sympos. Pure Math., Vol. 28, pp. 401-418. Providence, R.I.: 1976, Am. Math. Soc. 1976 · Zbl 0345.14006 [9] Langlands, R.P.: Shimura varieties and the Selberg trace formula. Can. J. Math.29, 1292-1299 (1977) · Zbl 0385.14005 [10] Langlands, R.P.: On the zeta-functions of some simple Shimura varieties. Can. J. Math.31, 1121-1216 (1979) · Zbl 0444.14016 [11] Langlands, R.P.: Base change for GL(2). Princeton: Princeton University Press 1980 · Zbl 0444.22007 [12] Langlands, R.P.: Les débuts d’une formule des traces stable. Publ. Math. Univ. Paris VII,13 (1983) [13] Milne, J.S.: The action of an automorphism of ? on a Shimura variety and its special points. Prog. Math.35, Boston: Birkhäuser 1983 · Zbl 0527.14035 [14] Satake, I.: Theory of spherical functions on reductive algebraic groups overp-adic fields. Publ. Math. IHES18, 1-69 (1963) [15] Serre, J.-P.: Algèbres de Lie Semi-simples complexes. New York: Benjamin 1966 · Zbl 0144.02105 [16] Tits, J.: Reductive groups over local fields. In: Automorphic forms, representations andL-functions. Proc. Sympos. Pure Math., Vol. 33, Part 1, pp. 29-69. Providence, R.I.: Am. Math. Soc. 1979
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