Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T08:29:43.944Z Has data issue: false hasContentIssue false
  • English
  • Français

Cohomologie d’intersection des variétés modulaires de Siegel, suite

Part of: (Co)homology theory Arithmetic algebraic geometry Linear algebraic groups and related topics

Published online by Cambridge University Press:  28 September 2011

Sophie Morel
Sophie Morel*
Affiliation:
Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, USA (email: morel@math.harvard.edu)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this work, we study the intersection cohomology of Siegel modular varieties. The goal is to express the trace of a Hecke operator composed with a power of the Frobenius endomorphism (at a good place) on this cohomology in terms of the geometric side of Arthur’s invariant trace formula for well-chosen test functions. Our main tools are the results of Kottwitz about the contribution of the cohomology with compact support and about the stabilization of the trace formula, Arthur’s L2 trace formula and the fixed point formula of Morel [Complexes pondérés sur les compactifications de Baily–Borel. Le cas des variétés de Siegel, J. Amer. Math. Soc. 21 (2008), 23–61]. We ‘stabilize’ this last formula, i.e. express it as a sum of stable distributions on the general symplectic groups and its endoscopic groups, and obtain the formula conjectured by Kottwitz in [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]. Applications of the results of this article have already been given by Kottwitz, assuming Arthur’s conjectures. Here, we give weaker unconditional applications in the cases of the groups GSp4 and GSp6.

Keywords

Siegel modular varietiesintersection cohomologydiscrete automorphic representations of symplectic groups

MSC classification

Secondary: 11G18: Arithmetic aspects of modular and Shimura varieties 14F20: Étale and other Grothendieck topologies and (co)homologies 20G35: Linear algebraic groups over adèles and other rings and schemes
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 6 , November 2011 , pp. 1671 - 1740
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

Références

[Art89]Arthur, J., The L 2-Lefschetz numbers of Hecke operators, Invent. Math. 97 (1989), 257–290.CrossRefGoogle Scholar
[Bor77]Borel, A., Automorphic L-functions, in Automorphic forms, representations, and L-functions, Proceedings of Symposia in Pure Mathematics, vol. 33 (American Mathematical Society, Providence, RI, 1977), 26–61.Google Scholar
[Che89]Chernousov, V. I., The Hasse principle for groups of type E 8, Soviet Math. Dokl. 39 (1989), 592–596.Google Scholar
[CD85]Clozel, L. and Delorme, P., Pseudo-coefficients et cohomologie des groupes de Lie réductifs réels, C. R. Acad. Sci. Paris Sér. I 300 (1985), 385–387.Google Scholar
[Die71]Dieudonné, J., La géométrie des groupes classiques, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 5, Troisième édition (Springer, Berlin, 1971).Google Scholar
[GHM94]Goresky, M., Harder, G. and MacPherson, R., Weighted cohomology, Invent. Math. 166 (1994), 139–213.CrossRefGoogle Scholar
[GKM97]Goresky, M., Kottwitz, R. and MacPherson, R., Discrete series characters and the Lefschetz formula for Hecke operators, Duke Math. J. 89 (1997), 477–554; Duke Math. J. 92 (1998), 665–666.CrossRefGoogle Scholar
[Hal93]Hales, T., A simple definition of transfer factors for unramified groups, in Representation theory of groups and algebras, Contemporary Mathematics, vol. 145, eds Adams, J., Herb, R., Kudla, S., Li, J.-S., Lipsman, R. and Rosenberg, J. (American Mathematical Society, Providence, RI, 1993), 109–134.CrossRefGoogle Scholar
[Her79]Herb, R., Characters of averaged discrete series on semisimple real Lie groups, Pacific J. Math. 80 (1979), 169–177.CrossRefGoogle Scholar
[Kot]Kottwitz, R., non publié.Google Scholar
[Kot83]Kottwitz, R., Sign changes in harmonic analysis on reductive groups, Trans. Amer. Math. Soc. 278 (1983), 289–297.CrossRefGoogle Scholar
[Kot84a]Kottwitz, R., Shimura varieties and twisted orbital integrals, Math. Ann. 269 (1984), 287–300.CrossRefGoogle Scholar
[Kot84b]Kottwitz, R., Stable trace formula: cuspidal tempered terms, Duke Math. J. 51 (1984), 611–650.CrossRefGoogle Scholar
[Kot86a]Kottwitz, R., Base change for units of Hecke algebras, Compositio Math. 60 (1986), 237–250.Google Scholar
[Kot86b]Kottwitz, R., Stable trace formula: elliptic singular terms, Math. Ann. 275 (1986), 365–399.CrossRefGoogle Scholar
[Kot88]Kottwitz, R., Tamagawa numbers, Ann. of Math. (2) 127 (1988), 629–646.CrossRefGoogle Scholar
[Kot90]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
[Lan79]Langlands, R., Stable conjugacy: definitions and lemmas, Canad. J. Math. 31 (1979), 700–725.CrossRefGoogle Scholar
[Lan83]Langlands, R., Les débuts d’une formule des traces stable, Publications Mathématiques de l’Université Paris VII, vol. 13 (Université de Paris VII, U.E.R. de Mathématiques, Paris, 1983).Google Scholar
[LR92]Langlands, R. and Ramakrishnan, D. (eds), The zeta function of Picard modular surfaces (Université de Montréal, Centre de Recherches Mathématiques, Montreal, QC, 1992).Google Scholar
[LS87]Langlands, R. and Shelstad, D., On the definition of transfer factors, Math. Ann. 278 (1987), 219–271.CrossRefGoogle Scholar
[LS90]Langlands, R. and Shelstad, D., Descent for transfer factors, in The Grothendieck festschrift, Vol. II, Progress in Mathematics, vol. 87 (Birkhäuser, Basel, 1990), 485–563.Google Scholar
[Lau97]Laumon, G., Sur la cohomologie à supports compacts des variétés de Shimura pour GSp(4)ℚ, Compositio Math. 105 (1997), 267–359.CrossRefGoogle Scholar
[Lau05]Laumon, G., Fonctions zêta des variétés de Siegel de dimension trois, in Formes automorphes II. Le cas du groupe GSp(4), Astérisque 302 (2005), 1–66.Google Scholar
[LN08]Laumon, G. and Ngo, B.-C., Le lemme fondamental pour les groupes unitaires, Ann. of Math. (2) 168 (2008), 477–573.CrossRefGoogle Scholar
[Mor08]Morel, S., Complexes pondérés sur les compactifications de Baily–Borel. Le cas des variétés de Siegel, J. Amer. Math. Soc. 21 (2008), 23–61.CrossRefGoogle Scholar
[Mor10]Morel, S., On the cohomology of certain non-compact Shimura varieties. With an appendix by Robert Kottwitz, Annals of Mathematics Studies, vol. 173 (Princeton University Press, Princeton, NJ, 2010).Google Scholar
[Ngo10]Ngô, B. C., Le lemme fondamental pour les algèbres de Lie, Publ. Math. Inst. Hautes Études Sci. 111 (2010), 1–169.CrossRefGoogle Scholar
[Ono66]Ono, T., On Tamagawa numbers, in Algebraic groups and discontinuous subgroups, Proceedings of Symposia in Pure Mathematics, vol. 9 eds Borel, A. and Mostow, G. (American Mathematical Society, Providence, RI, 1966).Google Scholar
[Pin89]Pink, R., Arithmetical compactification of mixed Shimura varieties, Bonner Mathematische Schriften, vol. 209 (University of Bonn, 1989).Google Scholar
[Pin92]Pink, R., On ℓ-adic sheaves on Shimura varieties and their higher direct images in the Baily–Borel compactification, Math. Ann. 292 (1992), 197–240.CrossRefGoogle Scholar
[Rog90]Rogawski, J., Automorphic representations of unitary groups in three variables, Annals of Mathematics Study, vol. 123 (Princeton University Press, Princeton, NJ, 1990).CrossRefGoogle Scholar
[Wal97]Waldspurger, J.-L., Le lemme fondamental implique le transfert, Compositio Math. 105 (1997), 153–236.CrossRefGoogle Scholar
[Wal06]Waldspurger, J.-L., Endoscopie et changement de caractéristique, J. Inst. Math. Jussieu 5 (2006), 423–525.CrossRefGoogle Scholar
[Wal08]Waldspurger, J.-L., L’endoscopie tordue n’est pas si tordue, Mem. Amer. Math. Soc. 194 (2008).Google Scholar