On Connected Components of Shimura Varieties

Haines, Thomas J.

Canadian journal of mathematics, Tome 54 (2002) no. 2, pp. 352-395

Résumé

We study the cohomology of connected components of Shimura varieties ${{S}_{Kp}}$ coming from the group $\text{GS}{{\text{p}}_{2g}}$ , by an approach modeled on the stabilization of the twisted trace formula, due to Kottwitz and Shelstad. More precisely, for each character $\bar{\omega }$ on the group of connected components of ${{S}_{Kp}}$ we define an operator $L(\omega )$ on the cohomology groups with compact supports $H_{c}^{i}\left( {{S}_{Kp,}}{{\overset{-}{\mathop{\mathbb{Q}}}\,}_{\ell }} \right)$ , and then we prove that the virtual trace of the composition of $L(\omega )$ with a Hecke operator $f$ away from $p$ and a sufficiently high power of a geometric Frobenius $\Phi _{p}^{r}$ , can be expressed as a sum of $\omega$ -weighted (twisted) orbital integrals (where $\omega$ -weighted means that the orbital integrals and twisted orbital integrals occuring here each have a weighting factor coming from the character $\bar{\omega }$ ). As the crucial step, we define and study a new invariant ${{\alpha }_{1}}\left( {{\gamma }_{0}};\gamma ,\delta\right)$ which is a refinement of the invariant $\alpha \left( {{\gamma }_{0}},\,\gamma ,\,\delta\right)$ defined by Kottwitz. This is done by using a theorem of Reimann and Zink.

Détail

DOI : 10.4153/CJM-2002-012-x

Mots-clés : 14G35, 11F70

Bibliographie

[1] [1] Deligne, P., Travaux de Shimura. Sém. Bourbaki Février 1971, Exposé 389, Lecture Notes in Math. 244, Springer, Heidelberg, 1971. Google Scholar

[2] [2] Flicker, Y. and Kazhdan, D., Geometric Ramanujan Conjecture and Drinfeld Reciprocity Law. In: Number Theory, Trace Formulas, and Discrete Groups: Symposium in Honor of Atle Selberg (eds. K. Aubert, E. Bombieri, D. Goldfeld), Academic Press, 1989, 201–218. Google Scholar

[3] [3] Fujiwara, K., Rigid Geometry, Lefschetz-Verdier trace formula and Deligne's conjecture. Invent. Math. 127 (1997), 489–533. Google Scholar

[4] [4] Hales, T., Shalika Germs on GSp(4). Astérisque 171–172 (1989), 195–256. Google Scholar

[5] [5] Ihara, Y., Hecke Polynomials as congruence ζ-functions in elliptic modular case. Ann. of Math. (2) 85, 1967, 267–295. Google Scholar

[6] [6] Kottwitz, R., Stable trace formula: cuspidal tempered terms. Duke Math. J. (3) 51 (1984), 611–650. Google Scholar

[7] [7] Kottwitz, R., Isocrystals with additional structure. Compositio Math. (2) 56 (1985), 201–220 Google Scholar

[8] [8] Kottwitz, R., Stable trace formula: elliptic singular terms. Math. Ann. (3) 275 (1986), 365–399. Google Scholar

[9] [9] Kottwitz, R., Shimura varieties and λ-adic representations. In: Automorphic Forms, Shimura Varieties and L-functions, Part I, Perspectives in Mathematics Vol. 10, Academic Press, San Diego, CA, 1990, 161–209. Google Scholar

[10] [10] Kottwitz, R., On the λ-adic representations associated to some simple Shimura varieties. Invent. Math. 108 (1992), 653–665. Google Scholar

[11] [11] Kottwitz, R., Points on some Shimura varieties over finite fields. J. Amer. Math. Soc. 5 (1992), 373–444. Google Scholar

[12] [12] Kottwitz, R., Isocrystals with additional structure II. Compositio Math. (3) 109 (1997), 255–339. Google Scholar

[13] [13] Kottwitz, R. and Shelstad, D., Foundations of twisted endoscopy. Astérisque 255, 1999. Google Scholar

[14] [14] Langlands, R. P., Some contemporary problems with origins in the Jugendtraum. In: Mathematical Developments Arising from Hilbert Problems, Proc. Symp. Pure Math. 28, Amer. Math. Soc., Providence, RI, 1976, 401–418. Google Scholar

[15] [15] Langlands, R. P., Shimura varieties and the Selberg trace formula. Canad. J. Math. (5) 29 (1977), 1292–1299. Google Scholar

[16] [16] Langlands, R. P., On the zeta-functions of some simple Shimura varieties. Canad. J. Math. (6) 31 (1979), 1121–1216. Google Scholar

[17] [17] Langlands, R. P. and Ramakrishnan, D. (eds.), The Zeta Function of Picard Modular Surfaces. Univ. Montréal, Montréal, QC, 1992. Google Scholar

[18] [18] Langlands, R. P. and Rapoport, M., Shimuravarietäten und Gerben. J. Reine Angew. Math. 378 (1987), 113–220. Google Scholar

[19] [19] Laumon, G., Sur la cohomologie à supports compacts des variétés de Shimura pour GSp(4)Q. Compositio Math. (3) 105 (1997), 267–359. Google Scholar

[20] [20] Milne, J. S., Étale Cohomology. Princeton Math. Series 33, Princeton Univ. Press, 1980. Google Scholar

[21] [21] Milne, J. S., The points on a Shimura variety modulo a prime of good reduction. In: The zeta functions of Picard modular surfaces, Univ. Montréal, Montréal, PQ, 1992, 151–253. Google Scholar

[22] [22] Mumford, D., Abelian varieties. Tata Institute of Fundamental Research Studies in Mathematics 5. Published for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London 1970. Google Scholar

[23] [23] Mumford, D., Geometric invariant theory. Springer, Heidelberg, 1965. Google Scholar

[24] [24] Pfau, M., The reduction of connected Shimura varieties at primes of good reduction. Dissertation, University of Michigan, 1993. Google Scholar

[25] [25] Platonov, V. and Rapinchuk, A., Algebraic Groups and Number Theory. Pure Appl. Math. 139, Academic Press, Inc., 1994. Google Scholar

[26] [26] Reimann, H., The semi-simple zeta function of quaternionic Shimura varieties. Lecture Notes in Math. 1657, Springer-Verlag, 1997. Google Scholar

[27] [27] Reimann, H. and Zink, Th., Der Dieudonńemodul einer polarisierten abelschen Mannigfaltigkeit vom CM-Typ. Ann. of Math. (2) 128 (1988), 461–482. Google Scholar

[28] [28] Serre, J. P., Corps locaux. Hermann, Paris, 1962. Google Scholar

[29] [29] Shimura, G., Correspondances modulaires et les fonctions ζ de courbes algébriques. J. Math. Soc. Japan 10 (1958), 1–28. Google Scholar

[30] [30] Shimura, G., On the zeta-functions of the algebraic curves uniformized by certain automorphic functions. J. Math. Soc. Japan 13 (1961), 275–331. Google Scholar

[31] [31] Shimura, G., Construction of class fields and zeta functions of algebraic curves. Ann. of Math. 85 (1967), 58–159. Google Scholar

[32] [32] Tate, J., Classes d’isogénie des variétés abéliennes sur un corps fini (d’après T. Honda). Sém. Bourbaki Nov. 1968, Exposé 352. Google Scholar

[33] [33] Waldspurger, J.-L., Quelques résultats de finitude concernant les distributions invariantes sur les algébres de Lie p-adiques. Preprint. Google Scholar

[34] [34] Waldspurger, J.-L., Homogéńeité de certaines distributions sur les groupes p-adiques. Preprint. Google Scholar

[35] [35] Weissauer, R., A special case of the fundamental lemma. Preprints, Parts I, II, III, IV. Google Scholar