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Points on some Shimura varieties over finite fields. (English) Zbl 0796.14014
The author proves the Langlands conjecture on zeta functions of Shimura varieties of PEL-type in cases $$A$$ and $$C$$. More precisely, he computes the number of fixed points of twisted Hecke correspondences $$\Phi^ j_ \wp \circ f$$ inside a single isogeny class in terms of orbital and twisted orbital integrals.
An important rôle is played by a triple $$(\gamma_ 0, \gamma, \delta)$$ associated to points $$(A, \lambda, i, \eta)$$ on the Shimura variety in characteristic $$p$$. It is shown that the Kottwitz invariant $$\alpha (\gamma_ 0, \gamma, \delta)$$ of such a triple vanishes; conversely, if $$\alpha (\gamma_ 0, \gamma, \delta) = 1$$ and two other obvious conditions are satisfied then $$(\gamma_ 0, \gamma, \delta)$$ comes from some $$(A, \lambda,i)$$. The proof of the vanishing of $$\alpha$$ involves a consideration of the filtered Dieudonné module associated to a crystalline representation of the Galois group of a local field.
Apart from a lot of useful lemma’s, the article also includes Honda-Tate theory for abelian varieties with endomorphisms.

##### MSC:
 14G35 Modular and Shimura varieties 14G15 Finite ground fields in algebraic geometry 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 11S40 Zeta functions and $$L$$-functions 14K05 Algebraic theory of abelian varieties 11G18 Arithmetic aspects of modular and Shimura varieties
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##### References:
 [1] Armand Borel, Properties and linear representations of Chevalley groups, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) Lecture Notes in Mathematics, Vol. 131, Springer, Berlin, 1970, pp. 1 – 55. · Zbl 0793.01013 [2] A. Borel and J.-P. Serre, Théorèmes de finitude en cohomologie galoisienne, Comment. Math. Helv. 39 (1964), 111 – 164 (French). · Zbl 0143.05901 [3] Pierre Deligne, Variétés abéliennes ordinaires sur un corps fini, Invent. Math. 8 (1969), 238 – 243 (French). · Zbl 0179.26201 [4] -, Travaux de Griffiths, Sém. Bourbaki Mai 1970, Exposé 376, Lecture Notes in Math., vol. 180, Springer, Heidelberg, 1971. [5] Pierre Deligne, Travaux de Shimura, Séminaire Bourbaki, 23ème année (1970/71), Exp. No. 389, Springer, Berlin, 1971, pp. 123 – 165. Lecture Notes in Math., Vol. 244 (French). [6] -, Variétés de Shimura, Automorphic Forms, Representations and $$L$$-functions, Proc. Sympos. Pure Math., vol. 33, Amer. Math. Soc., Providence, RI, 1979, pp. 247-290. [7] Gerd Faltings, Crystalline cohomology and \?-adic Galois-representations, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988) Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 25 – 80. · Zbl 0805.14008 [8] Jean-Marc Fontaine, Sur certains types de représentations \?-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate, Ann. of Math. (2) 115 (1982), no. 3, 529 – 577 (French). · Zbl 0544.14016 [9] Jean-Marc Fontaine and William Messing, \?-adic periods and \?-adic étale cohomology, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985) Contemp. Math., vol. 67, Amer. Math. Soc., Providence, RI, 1987, pp. 179 – 207. · Zbl 0632.14016 [10] Taira Honda, Isogeny classes of abelian varieties over finite fields, J. Math. Soc. Japan 20 (1968), 83 – 95. · Zbl 0203.53302 [11] Yasutaka Ihara, The congruence monodromy problems, J. Math. Soc. Japan 20 (1968), 107 – 121. · Zbl 0188.24802 [12] -, On congruence monodromy problems I, II, Univ. of Tokyo, 1968, 1969. · Zbl 0228.14009 [13] Robert E. Kottwitz, Shimura varieties and twisted orbital integrals, Math. Ann. 269 (1984), no. 3, 287 – 300. · Zbl 0533.14009 [14] Robert E. Kottwitz, Stable trace formula: cuspidal tempered terms, Duke Math. J. 51 (1984), no. 3, 611 – 650. · Zbl 0576.22020 [15] Robert E. Kottwitz, Isocrystals with additional structure, Compositio Math. 56 (1985), no. 2, 201 – 220. · Zbl 0597.20038 [16] Robert E. Kottwitz, Stable trace formula: elliptic singular terms, Math. Ann. 275 (1986), no. 3, 365 – 399. · Zbl 0577.10028 [17] Robert E. Kottwitz, Shimura varieties and \?-adic representations, Automorphic forms, Shimura varieties, and \?-functions, Vol. I (Ann Arbor, MI, 1988) Perspect. Math., vol. 10, Academic Press, Boston, MA, 1990, pp. 161 – 209. · Zbl 0743.14019 [18] R. P. Langlands, Some contemporary problems with origins in the Jugendtraum, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Vol. XXVIII, Northern Illinois Univ., De Kalb, Ill., 1974) Amer. Math. Soc., Providence, R. I., 1976, pp. 401 – 418. [19] R. P. Langlands, Shimura varieties and the Selberg trace formula, Canad. J. Math. 29 (1977), no. 6, 1292 – 1299. · Zbl 0385.14005 [20] R. P. Langlands, On the zeta functions of some simple Shimura varieties, Canad. J. Math. 31 (1979), no. 6, 1121 – 1216. · Zbl 0444.14016 [21] -, Automorphic representations, Shimura varieties and motives, Automorphic Forms, Representations and $$L$$-functions, Proc. Sympos. Pure Math., vol. 33, 1979, Amer. Math. Soc., Providence, RI, pp. 205-246. [22] -, Les débuts d’une formule des traces stable, Publ. Math. Univ. Paris VII, vol. 13, Paris, 1983. [23] R. P. Langlands and M. Rapoport, Shimuravarietäten und Gerben, J. Reine Angew. Math. 378 (1987), 113 – 220 (German). · Zbl 0615.14014 [24] William Messing, The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, Lecture Notes in Mathematics, Vol. 264, Springer-Verlag, Berlin-New York, 1972. · Zbl 0243.14013 [25] J. S. Milne, Points on Shimura varieties mod \?, Automorphic forms, representations and \?-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 165 – 184. · Zbl 0418.14022 [26] J. S. Milne, The conjecture of Langlands and Rapoport for Siegel modular varieties, Bull. Amer. Math. Soc. (N.S.) 24 (1991), no. 2, 335 – 341. · Zbl 0743.11031 [27] David Mumford, Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 34, Springer-Verlag, Berlin-New York, 1965. · Zbl 0147.39304 [28] -, Abelian varieties, Oxford Univ. Press, London, 1974. [29] Harry Reimann and Thomas Zink, Der Dieudonnémodul einer polarisierten abelschen Mannigfaltigkeit vom CM-Typ, Ann. of Math. (2) 128 (1988), no. 3, 461 – 482 (German). · Zbl 0674.14030 [30] -, The good reduction of Shimura varieties associated to quaternion algebras over a totally real number field, preprint. [31] Jean-Pierre Serre and John Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492 – 517. · Zbl 0172.46101 [32] Goro Shimura, Moduli of abelian varieties and number theory, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 312 – 332. [33] Goro Shimura and Yutaka Taniyama, Complex multiplication of abelian varieties and its applications to number theory, Publications of the Mathematical Society of Japan, vol. 6, The Mathematical Society of Japan, Tokyo, 1961. · Zbl 0112.03502 [34] Robert Steinberg, Regular elements of semisimple algebraic groups, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 49 – 80. [35] John Tate, Duality theorems in Galois cohomology over number fields, Proc. Internat. Congr. Mathematicians (Stockholm, 1962) Inst. Mittag-Leffler, Djursholm, 1963, pp. 288 – 295. [36] John Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966), 134 – 144. · Zbl 0147.20303 [37] -, Classes d’isogénie des variétés abéliennes sur un corps fini (d’après T. Honda), Sém. Bourbaki Nov. 1968, Exposé 352. [38] Thomas Zink, Über die schlechte Reduktion einiger Shimuramannigfaltigkeiten, Compositio Math. 45 (1982), no. 1, 15 – 107 (German). · Zbl 0483.14006 [39] Thomas Zink, Isogenieklassen von Punkten von Shimuramannigfaltigkeiten mit Werten in einem endlichen Körper, Math. Nachr. 112 (1983), 103 – 124 (German). · Zbl 0604.14029
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