Kottwitz-Rapoport and $p$-rank strata in the reduction of Shimura varieties of PEL type

Hartwig, Philipp

Annales de l'Institut Fourier, Tome 65 (2015) no. 3, pp. 1031-1103.

Nous étudions la réduction de certains modèles entiers des variétés de Shimura de type PEL à structure de niveau Iwahori. Sur ces espaces on a la stratification de Kottwitz-Rapoport et la stratification de $p$-rang. Nous montrons que le $p$-rang est constant sur un strate de Kottwitz-Rapoport, généralisant un résultat de Ngô et Genestier. Nous montrons une formule abstraite, uniforme pour le $p$-rang sur un strate de Kottwitz-Rapoport. Dans les cas symplectique et unitaire nous trouvons des formules explicites pour sa valeur. Nous appliquons ces formules à la question de la densité du lieu ordinaire et à la question de la dimension du lieu de $p$-rang 0.

We study the reduction of certain integral models of Shimura varieties of PEL type with Iwahori level structure. On these spaces we have the Kottwitz-Rapoport and the $p$-rank stratification. We show that the $p$-rank is constant on a KR stratum, generalizing a result of Ngô and Genestier. We prove an abstract, uniform formula for the $p$-rank on a KR stratum. In the symplectic and in the unitary case we derive explicit formulas for its value. We apply these formulas to the question of the density of the ordinary locus and to the question of the dimension of the $p$-rank 0 locus.

Classification : 14G35, 14K10
Keywords: Abelian varieties, $p$-rank stratification, Kottwitz-Rapoport stratification, Iwahori decomposition, ordinary locus, Hilbert-Blumenthal modular varieties, affine Deligne-Lusztig varieties
Mot clés : Variétés abéliennes, stratification de $p$-rang, stratification de Kottwitz-Rapoport, décomposition d’Iwahori, lieu ordinaire, espaces de modules de Hilbert-Blumenthal, variétés de Deligne-Lusztig affines

[1] Amitsur, S. A. On the characteristic polynomial of a sum of matrices, Linear and Multilinear Algebra, Volume 8 (1979/80) no. 3, pp. 177-182 | DOI | MR | Zbl

[2] Demazure, M. Lectures on $p$-divisible groups, Lecture Notes in Mathematics, Vol. 302, Springer-Verlag, Berlin, 1972, pp. v+98 | MR | Zbl

[3] Genestier, A. Un modèle semi-stable de la variété de Siegel de genre 3 avec structures de niveau de type ${\Gamma }_0\left(p\right)$, Compositio Math., Volume 123 (2000) no. 3, pp. 303-328 | DOI | MR | Zbl

[4] Goren, E.; Kassaei, P. Canonical subgroups over Hilbert modular varieties, J. Reine Angew. Math., Volume 670 (2012), pp. 1-63 | DOI | MR | Zbl

[5] Görtz, U.; Yu, C.-F. Supersingular Kottwitz-Rapoport strata and Deligne-Lusztig varieties, J. Inst. Math. Jussieu, Volume 9 (2010) no. 2, pp. 357-390 | DOI | MR | Zbl

[6] Görtz, U.; Yu, C.-F. The supersingular locus of Siegel modular varieties with Iwahori level structure, Math. Ann., Volume 353 (2012) no. 2, pp. 465-498 | DOI | MR | Zbl

[7] Haines, T. Introduction to Shimura varieties with bad reduction of parahoric type, Harmonic analysis, the trace formula, and Shimura varieties (Clay Math. Proc.), Volume 4, Amer. Math. Soc., Providence, RI, 2005, pp. 583-642 | MR | Zbl

[8] Haines, Thomas J.; Châu, Ngô Bao Alcoves associated to special fibers of local models, Amer. J. Math., Volume 124 (2002) no. 6, pp. 1125-1152 http://muse.jhu.edu/journals/american_journal_of_mathematics/v124/124.6haines.pdf | DOI | MR | Zbl

[9] Hamacher, P. The $p$-rank stratification on the Siegel moduli space with Iwahori level structure, Manuscripta Math., Volume 143 (2014) no. 1-2, pp. 51-80 | DOI | MR

[10] Hartwig, P. Kottwitz-Rapoport and p-rank strata in the reduction of Shimura varieties of PEL type, Universität Duisburg-Essen, Germany (2012) (Masters thesis http://duepublico.uni-duisburg-essen.de/servlets/DocumentServlet?id=29122)

[11] Hoeve, M. Stratifications on moduli spaces of abelian varieties and Deligne-Lusztig varieties, Faculteit der Natuurwetenschappen, Wiskunde en Informatica, The Netherlands (2010) (Masters thesis http://dare.uva.nl/en/record/341590)

[12] Kottwitz, R. Isocrystals with additional structure, Compositio Math., Volume 56 (1985) no. 2, pp. 201-220 | Numdam | MR | Zbl

[13] Kottwitz, R. Points on some Shimura varieties over finite fields, J. Amer. Math. Soc., Volume 5 (1992) no. 2, pp. 373-444 | DOI | MR | Zbl

[14] Kottwitz, R.; Rapoport, M. Minuscule alcoves for ${\mathrm{GL}}_n$ and $G{\mathrm{Sp}}_{2n}$, Manuscripta Math., Volume 102 (2000) no. 4, pp. 403-428 | DOI | MR | Zbl

[15] Mumford, D. Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, Bombay, 1970, pp. viii+242 | MR | Zbl

[16] Ngô, B.C.; Genestier, A. Alcôves et $p$-rang des variétés abéliennes, Ann. Inst. Fourier (Grenoble), Volume 52 (2002) no. 6, pp. 1665-1680 | DOI | Numdam | MR | Zbl

[17] Oda, Tadao The first de Rham cohomology group and Dieudonné modules, Ann. Sci. École Norm. Sup. (4), Volume 2 (1969), pp. 63-135 | Numdam | MR | Zbl

[18] Pappas, G. On the arithmetic moduli schemes of PEL Shimura varieties, J. Algebraic Geom., Volume 9 (2000) no. 3, pp. 577-605 | MR | Zbl

[19] Pappas, G.; Rapoport, M. Local models in the ramified case. II. Splitting models, Duke Math. J., Volume 127 (2005) no. 2, pp. 193-250 | DOI | MR | Zbl

[20] Pappas, G.; Rapoport, M. Twisted loop groups and their affine flag varieties, Adv. Math., Volume 219 (2008) no. 1, pp. 118-198 (With an appendix by T. Haines and M. Rapoport) | DOI | MR | Zbl

[21] Pappas, G.; Rapoport, M. Local models in the ramified case. III. Unitary groups, J. Inst. Math. Jussieu, Volume 8 (2009) no. 3, pp. 507-564 | DOI | MR | Zbl

[22] Pappas, G.; Rapoport, M.; Smithling, B. Local models of Shimura varieties, I. Geometry and combinatorics, Handbook of moduli. Vol. III (Adv. Lect. Math. (ALM)), Volume 26, Int. Press, Somerville, MA, 2013, pp. 135-217 | MR

[23] Pappas, G.; Zhu, X. Local models of Shimura varieties and a conjecture of Kottwitz, Invent. Math., Volume 194 (2013) no. 1, pp. 147-254 | DOI | MR | Zbl

[24] Rapoport, M.; Zink, Th. Period spaces for $p$-divisible groups, Annals of Mathematics Studies, 141, Princeton University Press, Princeton, NJ, 1996, pp. xxii+324 | MR | Zbl

[25] Reiner, I. Maximal orders, London Mathematical Society Monographs. New Series, 28, The Clarendon Press Oxford University Press, Oxford, 2003, pp. xiv+395 (Corrected reprint of the 1975 original, With a foreword by M. J. Taylor) | MR | Zbl

[26] Smithling, B. Topological flatness of local models for ramified unitary groups. I. The odd dimensional case, Adv. Math., Volume 226 (2011) no. 4, pp. 3160-3190 | DOI | MR | Zbl

[27] Smithling, B. Topological flatness of orthogonal local models in the split, even case. I, Math. Ann., Volume 350 (2011) no. 2, pp. 381-416 | DOI | MR | Zbl

[28] Smithling, B. Topological flatness of local models for ramified unitary groups. II. The even dimensional case, J. Inst. Math. Jussieu, Volume 13 (2014) no. 2, pp. 303-393 | DOI | MR

[29] Stamm, H. On the reduction of the Hilbert-Blumenthal-moduli scheme with ${\Gamma }_0\left(p\right)$-level structure, Forum Math., Volume 9 (1997) no. 4, pp. 405-455 | DOI | MR | Zbl

[30] Viehmann, E.; Wedhorn, T. Ekedahl-Oort and Newton strata for Shimura varieties of PEL type, Math. Ann., Volume 356 (2013) no. 4, pp. 1493-1550 | DOI | MR

[31] Wedhorn, Torsten Ordinariness in good reductions of Shimura varieties of PEL-type, Ann. Sci. École Norm. Sup. (4), Volume 32 (1999) no. 5, pp. 575-618 | DOI | Numdam | MR | Zbl

[32] Yu, C.-F. (Personal communication)