Hostname: page-component-f7d5f74f5-dcd55 Total loading time: 0 Render date: 2023-10-03T04:10:15.636Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "coreDisableSocialShare": false, "coreDisableEcommerceForArticlePurchase": false, "coreDisableEcommerceForBookPurchase": false, "coreDisableEcommerceForElementPurchase": false, "coreUseNewShare": true, "useRatesEcommerce": true } hasContentIssue false

Supersingular Kottwitz–Rapoport strata and Deligne–Lusztig varieties

Published online by Cambridge University Press:  11 August 2009

Ulrich Görtz
Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany, (
Chia-Fu Yu
Institute of Mathematics, Academia Sinica, 128 Academia Road, Section 2, Nankang, Taipei, Taiwan and National Center for Theoretical Sciences (Taipei Office), National Taiwan University, Taipei 10617, Taiwan, (


We investigate the special fibres of Siegel modular varieties with Iwahori level structure. On these spaces, we have the Newton stratification, and the Kottwitz–Rapoport (KR) stratification; one would like to understand how these stratifications are related to each other. We give a simple description of all KR strata which are entirely contained in the supersingular locus as disjoint unions of Deligne–Lusztig varieties. We also give an explicit numerical description of the KR stratification in terms of abelian varieties.

Research Article
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.)


1.Beauville, A. and Laszlo, Y., Conformal blocks and generalized theta functions, Commun. Math. Phys. 164(2) (1994), 385419.CrossRefGoogle Scholar
2.Bonnafé, C. and Rouquier, R., On the irreducibility of Deligne–Lusztig varieties, C. R. Acad. Sci. Paris Sér. I 343 (2006), 3739.CrossRefGoogle Scholar
3.Bonnafé, C. and Rouquier, R., Affineness of Deligne–Lusztig varieties for minimal length elements, J. Alg. 320 (2008), 12001206.CrossRefGoogle Scholar
4.Borel, A., Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup, Invent. Math. 35 (1976), 233259.CrossRefGoogle Scholar
5.Boyer, P., Mauvaise réduction des variétés de Drinfeld et correspondance de Langlands locale, Invent. Math. 138 (1999), 573629.CrossRefGoogle Scholar
6.Carter, R., Simple groups of Lie type (Wiley, 1972).Google Scholar Jong, A. J., The moduli spaces of polarized abelian varieties, Math. Annalen 295 (1993), 485503.CrossRefGoogle Scholar Jong, A. J., The moduli spaces of principally polarized abelian varieties with Γ0(p)-level structure, J. Alg. Geom. 2 (1993), 667688.Google Scholar
9.Deligne, P. and Lusztig, G., Representations of reductive groups over finite fields, Annals Math. 103 (1976), 103161.CrossRefGoogle Scholar
10.Digne, F. and Michel, J., Endomorphisms of Deligne–Lusztig varieties, Nagoya Math. J. 183 (2006), 35103.CrossRefGoogle Scholar
11.Ekedahl, T. and van der Geer, G., Cycle classes of the E–O stratification on the moduli of abelian varieties, preprint arXiv:math.AG/0412272v2 (to appear in Arithmetic, algebra and geometry—Manin-Festschrift (Birkhäuser, Basel)).Google Scholar
12.Faltings, G., Algebraic loop groups and moduli spaces of bundles, J. Eur. Math. Soc. 5 (2003), 4168.CrossRefGoogle Scholar
13.Fargues, L., Cohomologie des espaces de modules de groupes p-divisibles et correspondances de Langlands locales, Astérisque 291 (2004), 1199.Google Scholar
14.Görtz, U., On the flatness of local models for the symplectic group, Adv. Math. 176 (2003), 89115.CrossRefGoogle Scholar
15.Görtz, U. and Hoeve, M., Ekedahl–Oort strata and Kottwitz–Rapoport strata, preprint arXiv:0808.2537 (2008).Google Scholar
16.Görtz, U. and Yu, C.-F., The supersingular locus of Siegel modular varieties with Iwahori level structure, preprint arXiv:0807.1229 (2008).Google Scholar
17.Görtz, U. and Yu, C.-F., Components of supersingular Kottwitz–Rapoport strata, in preparation.Google Scholar
18.Görtz, U., Haines, T., Kottwitz, R. and Reuman, D., Dimensions of some affine Deligne–Lusztig varieties, Annales Scient. Éc. Norm. Sup. 39 (2006), 467511CrossRefGoogle Scholar
19.Haastert, B., Die Quasiaffinität der Deligne–Lusztig-Varietäten, J. Alg. 102 (1986), 186193.CrossRefGoogle Scholar
20.Haines, T., The combinatorics of Bernstein functions, Trans. Am. Math. Soc. 353(3) (2001), 12511278.CrossRefGoogle Scholar
21.Haines, T., Introduction to Shimura varieties with bad reduction of parahoric type, in Harmonic analysis, the trace formula, and Shimura varieties, Clay Mathematics Proceedings, Volume 4, pp. 583642 (American Mathematical Society, Providence, RI, 2005).Google Scholar
22.Haines, T. and Ngô, B. C., Nearby cycles for local models of some Shimura varieties, Compositio Math. 133 (2002), 117150.CrossRefGoogle Scholar
23.Hansen, S., The geometry of Deligne–Lusztig varieties; higher-dimensional AG codes, PhD thesis, University of Aarhus, Denmark (available at; 1999).Google Scholar
24.Harashita, S., Ekedahl–Oort strata contained in the supersingular locus and Deligne–Lusztig varieties, preprint (available at; 2007; to appear in J. Alg. Geom.).Google Scholar
25.Harris, M., Local Langlands correspondences and vanishing cycles on Shimura varieties, in European Congress of Mathematics, Barcelona, 2000, Volume I, pp. 407427, Progress in Mathematics, Volume 201 (Birkhäuser, Basel, 2001).Google Scholar
26.Harris, M., The local Langlands correspondence: notes of (half) a course at the IHP Spring 2000, in Automorphic forms, Volume I, Astérique 298 (2005), 17145.Google Scholar
27.Harris, M. and Taylor, R., The geometry and cohomology of some simple Shimura varieties (with an appendix by V. G. Berkovich), Annals of Mathematics Studies, Volume 151 (Princeton University Press, 2001).Google Scholar
28.He, X., On the affineness of Deligne–Lusztig varieties, J. Alg. 320 (2008), 12071219.CrossRefGoogle Scholar
29.Hoeve, M., Ekedahl–Oort strata in the supersingular locus, preprint arXiv:0802.4012 (2008).Google Scholar
30.Koblitz, N., p-adic variant of the zeta function of families of varieties defined over finite fields, Compositio Math. 31 (1975), 119218.Google Scholar
31.Kottwitz, R. E. and Rapoport, M., Minuscule alcoves for GLn and GSp2n, Manuscr. Math. 102 (2000), 403428.CrossRefGoogle Scholar
32.Li, K.-Z. and Oort, F., Moduli of supersingular Abelian varieties, Lecture Notes in Mathematics, Volume 1680 (Springer, 1998).CrossRefGoogle Scholar
33.Ngô, B. C. and Genestier, A., Alcôves et p-rang des variétés abéliennes, Annales Inst. Fourier 52 (2002), 16651680.CrossRefGoogle Scholar
34.Oort, F., A stratification of a moduli space of abelian varieties, in Moduli of Abelian Varieties, Texel Island, 1999, Progress in Mathematics, Volume 195, pp. 255298 (Birkhäuser, Basel, 2001).Google Scholar
35.Orlik, S. and Rapoport, M., Deligne–Lusztig varieties and period domains over finite fields, J. Alg. 320 (2008), 12201234.CrossRefGoogle Scholar
36.Rapoport, M., A guide to the reduction modulo p of Shimura varieties, Astérisque 298 (2005), 271318.Google Scholar
37.Rapoport, M. and Zink, Th., Period spaces for p-divisible groups, Annals of Mathematics Studies, Volume 141 (Princeton University Press, 1996).Google Scholar
38.Tits, J., Reductive groups over local fields, in Automorphic Forms, Representations and L-Functions, 1977, pp. 2969, Proceedings of Symposia in Pure Mathematics, Volume 33, Part 1 (American Mathematical Society, Providence, RI, 1979).Google Scholar
39.Vollaard, I., The supersingular locus of the Shimura variety of GU(1, s), Can. J. Math., to appear.Google Scholar
40.Vollaard, I. and Wedhorn, T., The supersingular locus of the Shimura variety of GU(1, s), II, preprint arXiv:0804.1522 (2008).Google Scholar
41.Yoshida, T., On non-abelian Lubin–Tate theory via vanishing cycles, Annales Inst. Fourier, to appear.Google Scholar
42.Yu, C.-F., Irreducibility of the Siegel moduli spaces with parahoric level structure, Int. Math. Res. Not. 2004(48) (2004), 25932597.CrossRefGoogle Scholar
43.Yu, C.-F., The supersingular loci and mass formulas on Siegel modular varieties, Documenta Math. 11 (2006), 449468.Google Scholar
44.Yu, C.-F., Irreducibility and p-adic monodromies on the Siegel moduli spaces, Adv. Math. 218 (2008), 12531285.CrossRefGoogle Scholar
45.Yu, C.-F., Kottwitz–Rapoport strata in Siegel moduli spaces, Taiwanese J. Math., to appear.Google Scholar