Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-23T18:42:00.210Z Has data issue: false hasContentIssue false
  • English
  • Français

A stable trace formula for Igusa varieties

Part of: Lie groups

Published online by Cambridge University Press:  23 March 2010

Sug Woo Shin
Sug Woo Shin
Affiliation:
Department of Mathematics, University of Chicago, 5734 South University Avenue, Chicago, IL 60637, USA, (swshin@uchicago.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

Igusa varieties are smooth varieties in positive characteristic p which are closely related to Shimura varieties and Rapoport–Zink spaces. One motivation for studying Igusa varieties is to analyse the representations in the cohomology of Shimura varieties which may be ramified at p. The main purpose of this work is to stabilize the trace formula for the cohomology of Igusa varieties arising from a PEL datum of type (A) or (C). Our proof is unconditional thanks to the recent proof of the fundamental lemma by Ngô, Waldspurger and many others.

An earlier work of Kottwitz, which inspired our work and proves the stable trace formula for the special fibres of PEL Shimura varieties with good reduction, provides an explicit way to stabilize terms at ∞. Stabilization away from p and ∞ is carried out by the usual Langlands–Shelstad transfer as in work of Kottwitz. The key point of our work is to develop an explicit method to handle the orbital integrals at p. Our approach has the technical advantage that we do not need to deal with twisted orbital integrals or the twisted fundamental lemma.

One application of our formula, among others, is the computation of the arithmetic cohomology of some compact PEL-type Shimura varieties of type (A) with non-trivial endoscopy. This is worked out in a preprint of the author's entitled ‘Galois representations arising from some compact Shimura varieties’.

Keywords

trace formulaendoscopyorbital integralsShimura varietiesIgusa varieties

MSC classification

Secondary: 22E55: Representations of Lie and linear algebraic groups over global fields and adèle rings
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 9 , Issue 4 , October 2010 , pp. 847 - 895
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Arthur, J., The L 2-Leftschetz numbers of Hecke operators, Invent. Math. 97 (1989), 257–290.CrossRefGoogle Scholar
2.Arthur, J., An introduction to the trace formula, Clay Mathematics Proceedings, Volume 4, pp. 1–263 (Clay Mathematics Institute/American Mathematical Society, Providence, RI, 2005).Google Scholar
3.Badulescu, A. I., Jacquet-Langlands et unitarisabilité, J. Inst. Math. Jussieu 6 (2007), 349–379.CrossRefGoogle Scholar
4.Borel, A., Automorphic L-functions, in Automorphic forms, representations, and L-functions (ed. Borel, A. and Casselman, W.), Proceedings of Symposia in Pure Mathematics, Volume 33.2, pp. 27–61 (American Mathematical Society, Providence, RI, 1979).Google Scholar
5.Casselman, W., Characters and Jacquet modules, Math. Annalen 230 (1977), 101–105.CrossRefGoogle Scholar
6.Deligne, P., Kazhdan, D. and Vigneras, M.-F., Représentations des algèbres centrales simples p-adiques, pp. 33–117 (Hermann, Paris, 1984).Google Scholar
7.Fargues, L., Cohomologie des espaces de modules de groupes p-divisibles et correspondances de Langlands locales, Astérisque 291 (2004), 1–200.Google Scholar
8.Flath, D., Decomposition of representations into tensor products, Proceedings of Symposia in Pure Mathematics, Volume 33.1, pp. 179–183 (American Mathematical Society, Providence, RI, 1979).Google Scholar
9.Goresky, M., Kottwitz, R. and MacPherson, R., Discrete series characters and the Lefschetz formula for Hecke operators, Duke Math. J. 89 (1997), 477–554.CrossRefGoogle Scholar
10.Hales, T., On the fundamental lemma for standard endoscopy: reduction to unit elements, Can. J. Math. 47(5)(1995), 974–994.CrossRefGoogle Scholar
11.Harris, M. and Taylor, R., The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, No. 151 (Princeton University Press, 2001).Google Scholar
12.Kottwitz, R., Rational conjugacy classes in reductive groups, Duke Math. J. 49 (1982), 785–806.CrossRefGoogle Scholar
13.Kottwitz, R., Sign changes in harmonic analysis on reductive groups, Trans. Am. Math. Soc. 278 (1983), 289–297.CrossRefGoogle Scholar
14.Kottwitz, R., Shimura varieties and twisted orbital integrals, Math. Annalen 269 (1984), 287–300.CrossRefGoogle Scholar
15.Kottwitz, R., Stable trace formula: cuspidal tempered terms, Duke Math. J. 51 (1984), 611–650.CrossRefGoogle Scholar
16.Kottwitz, R., Isocrystals with additional structure, Compositio Math. 56 (1985), 201–220.Google Scholar
17.Kottwitz, R., Stable trace formula: elliptic singular terms, Math. Annalen 275 (1986), 365–399.CrossRefGoogle Scholar
18.Kottwitz, R., Tamagawa numbers, Annals Math. 127 (1988), 629–646.CrossRefGoogle Scholar
19.Kottwitz, R., Shimura varieties and λ-adic representations, Volume I, Perspectives in Mathematics, Volume 10, pp. 161–209 (Academic Press, 1990).Google Scholar
20.Kottwitz, R., On the λ-adic representations associated to some simple Shimura variaties, Invent. Math. 108 (1992), 653–665.CrossRefGoogle Scholar
21.Kottwitz, R., Points on some Shimura varieties over finite fields, J. Am. Math. Soc. 5 (1992), 373–444.CrossRefGoogle Scholar
22.Kottwitz, R., Isocrystals with additional structure, II, Compositio Math. 109 (1997), 255–339.CrossRefGoogle Scholar
23.Kottwitz, R., Comparison of two versions of twisted transfer factors, Appendix appearing in Étude de la cohomologie de certaines varietés de Shimura non compactes, Annals of Mathematics Studies, in press.Google Scholar
24.Kottwitz, R., unpublished article.Google Scholar
25.Langlands, R., Stable conjugacy: definitions and lemmas, Can. J. Math. 31 (1979), 700–725.CrossRefGoogle Scholar
26.Langlands, R., Les d'ebuts d'une formule des traces stable, Publications Mathématiques de l'Université Paris VII, Volume 13 (l'Universite Paris VII, 1983).Google Scholar
27.Langlands, R. and Shelstad, D., On the definition of transfer factors, Math. Annalen 278 (1987), 219–271.CrossRefGoogle Scholar
28.Langlands, R. and Shelstad, D., Descent for transfer factors, in The Grothendieck Festschrift, Volume II, pp. 486–563 (Birkhäuser, Basel, 1990).Google Scholar
29.Mantovan, E., On the cohomology of certain PEL type Shimura varieties, Duke Math. J. 129 (2005), 573–610.CrossRefGoogle Scholar
30.Morel, S., Cohomologie d'intersection des variétés modulaires de Siegel, suite, preprint.Google Scholar
31.Morel, S., Étude de la cohomologie de certaines varietés de Shimura non compactes (with an appendix by R. Kottwitz), Annals of Mathematics Studies, Volume 173 (Princeton University Press, 2010).Google Scholar
32.NGô, B. C., Le lemme fondamental pour les algèbres de Lie, preprint (arXiv:0801.0446v1 [math.AG]).Google Scholar
33.Rapoport, M. and Richartz, M., On the classification and specialization of F-isocrystals with additional structure, Compositio Math. 103 (1996), 153–181.Google Scholar
34.Rapoport, M. and Zink, T., Period spaces for p-divisible groups, Annals of Mathematics Studies, No. 141 (Princeton University Press, 1996).Google Scholar
35.Serre, J.-P., Galois cohomology (Springer, 2002).Google Scholar
36.Shin, S. W., Counting points on Igusa varieties, Duke Math. J. 146 (2009), 509–568.CrossRefGoogle Scholar
37.Shin, S. W., Galois representations arising from some compact Shimura varieties, Adv. Math., in press.Google Scholar
38.Tate, J., Number theoretic background, in Automorphic forms, representations, and L-functions (ed. Borel, A. and Casselman, W.), Proceedings of Symposia in Pure Mathematics, Volume 33.2, pp. 3–26 (American Mathematical Society, Providence, RI, 1979).Google Scholar
39.Vigneras, M.-F., Caractérisation des intégrates orbitales sur un groupe réductif p-adique, J. Fac. Sci. Univ. Tokyo 28 (1982), 945–961.Google Scholar
40.Waldspurger, J.-L., Le lemme fondamental implique le transfert, Compositio Math. 105(2)(1997), 153–236.CrossRefGoogle Scholar
41.Waldspurger, J.-L., Endoscopie et changement de caractéristique, J. Inst. Math. Jussieu 5(3) (2006), 423–525.CrossRefGoogle Scholar
42.Waldspurger, J.-L., A propos du lemme fondamental tordu, Math. Annalen 343(1)(2009), 103–174.CrossRefGoogle Scholar