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Supersingular Kottwitz–Rapoport strata and Deligne–Lusztig varieties

  • Ulrich Görtz (a1) and Chia-Fu Yu (a2)

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.

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1.Beauville A. and Laszlo Y., Conformal blocks and generalized theta functions, Commun. Math. Phys. 164(2) (1994), 385419.
2.Bonnafé C. and Rouquier R., On the irreducibility of Deligne–Lusztig varieties, C. R. Acad. Sci. Paris Sér. I 343 (2006), 3739.
3.Bonnafé C. and Rouquier R., Affineness of Deligne–Lusztig varieties for minimal length elements, J. Alg. 320 (2008), 12001206.
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.
5.Boyer P., Mauvaise réduction des variétés de Drinfeld et correspondance de Langlands locale, Invent. Math. 138 (1999), 573629.
6.Carter R., Simple groups of Lie type (Wiley, 1972). Jong A. J., The moduli spaces of polarized abelian varieties, Math. Annalen 295 (1993), 485503. Jong A. J., The moduli spaces of principally polarized abelian varieties with Γ0(p)-level structure, J. Alg. Geom. 2 (1993), 667688.
9.Deligne P. and Lusztig G., Representations of reductive groups over finite fields, Annals Math. 103 (1976), 103161.
10.Digne F. and Michel J., Endomorphisms of Deligne–Lusztig varieties, Nagoya Math. J. 183 (2006), 35103.
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)).
12.Faltings G., Algebraic loop groups and moduli spaces of bundles, J. Eur. Math. Soc. 5 (2003), 4168.
13.Fargues L., Cohomologie des espaces de modules de groupes p-divisibles et correspondances de Langlands locales, Astérisque 291 (2004), 1199.
14.Görtz U., On the flatness of local models for the symplectic group, Adv. Math. 176 (2003), 89115.
15.Görtz U. and Hoeve M., Ekedahl–Oort strata and Kottwitz–Rapoport strata, preprint arXiv:0808.2537 (2008).
16.Görtz U. and Yu C.-F., The supersingular locus of Siegel modular varieties with Iwahori level structure, preprint arXiv:0807.1229 (2008).
17.Görtz U. and Yu C.-F., Components of supersingular Kottwitz–Rapoport strata, in preparation.
18.Görtz U., Haines T., Kottwitz R. and Reuman D., Dimensions of some affine Deligne–Lusztig varieties, Annales Scient. Éc. Norm. Sup. 39 (2006), 467511
19.Haastert B., Die Quasiaffinität der Deligne–Lusztig-Varietäten, J. Alg. 102 (1986), 186193.
20.Haines T., The combinatorics of Bernstein functions, Trans. Am. Math. Soc. 353(3) (2001), 12511278.
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).
22.Haines T. and Ngô B. C., Nearby cycles for local models of some Shimura varieties, Compositio Math. 133 (2002), 117150.
23.Hansen S., The geometry of Deligne–Lusztig varieties; higher-dimensional AG codes, PhD thesis, University of Aarhus, Denmark (available at; 1999).
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.).
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).
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.
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).
28.He X., On the affineness of Deligne–Lusztig varieties, J. Alg. 320 (2008), 12071219.
29.Hoeve M., Ekedahl–Oort strata in the supersingular locus, preprint arXiv:0802.4012 (2008).
30.Koblitz N., p-adic variant of the zeta function of families of varieties defined over finite fields, Compositio Math. 31 (1975), 119218.
31.Kottwitz R. E. and Rapoport M., Minuscule alcoves for GLn and GSp2n, Manuscr. Math. 102 (2000), 403428.
32.Li K.-Z. and Oort F., Moduli of supersingular Abelian varieties, Lecture Notes in Mathematics, Volume 1680 (Springer, 1998).
33.Ngô B. C. and Genestier A., Alcôves et p-rang des variétés abéliennes, Annales Inst. Fourier 52 (2002), 16651680.
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).
35.Orlik S. and Rapoport M., Deligne–Lusztig varieties and period domains over finite fields, J. Alg. 320 (2008), 12201234.
36.Rapoport M., A guide to the reduction modulo p of Shimura varieties, Astérisque 298 (2005), 271318.
37.Rapoport M. and Zink Th., Period spaces for p-divisible groups, Annals of Mathematics Studies, Volume 141 (Princeton University Press, 1996).
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).
39.Vollaard I., The supersingular locus of the Shimura variety of GU(1, s), Can. J. Math., to appear.
40.Vollaard I. and Wedhorn T., The supersingular locus of the Shimura variety of GU(1, s), II, preprint arXiv:0804.1522 (2008).
41.Yoshida T., On non-abelian Lubin–Tate theory via vanishing cycles, Annales Inst. Fourier, to appear.
42.Yu C.-F., Irreducibility of the Siegel moduli spaces with parahoric level structure, Int. Math. Res. Not. 2004(48) (2004), 25932597.
43.Yu C.-F., The supersingular loci and mass formulas on Siegel modular varieties, Documenta Math. 11 (2006), 449468.
44.Yu C.-F., Irreducibility and p-adic monodromies on the Siegel moduli spaces, Adv. Math. 218 (2008), 12531285.
45.Yu C.-F., Kottwitz–Rapoport strata in Siegel moduli spaces, Taiwanese J. Math., to appear.
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