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Local models in the ramified case. III Unitary groups

Part of: Special varieties Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  26 March 2009

G. Pappas  and
M. Rapoport
G. Pappas
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA, (pappas@math.msu.edu).
M. Rapoport
Affiliation:
Mathematisches Institut der Universität Bonn, Beringstrasse 1, 53115 Bonn, Germany, (rapoport@math.uni-bonn.de).
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Abstract

We continue our study of the reduction of PEL Shimura varieties with parahoric level structure at primes p at which the group defining the Shimura variety ramifies. We describe ‘good’ p-adic integral models of these Shimura varieties and study their étale local structure. In the present paper we mainly concentrate on the case of unitary groups for a ramified quadratic extension. Some of our results are applications of the theory of twisted affine flag varieties that we developed in a previous paper.

Keywords

Shimura varietyunitary groupparahoric subgrouplocal model

MSC classification

Secondary: 14G35: Modular and Shimura varieties 11G18: Arithmetic aspects of modular and Shimura varieties 14M15: Grassmannians, Schubert varieties, flag manifolds
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 8 , Issue 3 , July 2009 , pp. 507 - 564
Copyright
Copyright © Cambridge University Press 2009

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References

1.Arzdorf, K., On local models with special parahoric level structure, preprint arXiv: 0804.1886.Google Scholar
2.Bruhat, F. and Tits, J., Groupes réductifs sur un corps local, Publ. Math. IHES 41 (1972), 5–251.CrossRefGoogle Scholar
3.Bruhat, F. and Tits, J., Groupes réductifs sur un corps local, II, Schémas en groupes, Existence d'une donnée radicielle valuée, Publ. Math. IHES 60 (1984), 197–376.Google Scholar
4.Chai, C. and Norman, P., Singularities of the Λ0(p)-level structure, J. Alg. Geom. 1 (1992), 251–278.Google Scholar
5.de Jong, J., The moduli spaces of principally polarized abelian varieties with λ0(p)-level structure, J. Alg. Geom. 2 (1993), 667–688.Google Scholar
6.Deligne, P. and Pappas, G., Singularités des espaces de modules de Hilbert, en les caractéristiques divisant le discriminant, Compositio Math. 90 (1994), 59–79.Google Scholar
7.Faltings, G., Explicit resolution of local singularities of moduli-spaces, J. Reine Angew. Math. 483 (1997), 183–196.Google Scholar
8.Faltings, G., Toroidal resolutions for some matrix singularities, in Moduli of Abelian Varieties, Texel Island, 1999, pp. 157–184, Progress in Mathematics, Volume 195 (Birkhäuser, Basel, 2001).Google Scholar
9.Genestier, A., Un modèle semi-stable de la variété de Siegel de genre 3 avec structures de niveau de type λ0(p), Compositio Math. 123 (2000), 303–328.CrossRefGoogle Scholar
10.Görtz, U., On the flatness of models of certain Shimura varieties of PEL-type, Math. Annalen 321 (2001), 689–727.CrossRefGoogle Scholar
11.Görtz, U., On the flatness of local models for the symplectic group, Adv. Math. 176 (2003), 89–115.CrossRefGoogle Scholar
12.Görtz, U., Computing the alternating trace of Frobenius on the sheaves of nearby cycles on local models for GL4 and GL5, J. Alg. 278 (2004), 148–172.CrossRefGoogle Scholar
13.Görtz, U., Topological flatness of local models in the ramified case, Math. Z. 250 (2005), 775–790.Google Scholar
14.Haines, T. and Ngô, B. C., Nearby cycles for local models of some Shimura varieties, Compositio Math. 133 (2002), 117–150.CrossRefGoogle Scholar
15.Haines, T. and Ngô, B. C., Alcoves associated to special fibers of local models, Am. J. Math. 124 (2002), 1125–1152.CrossRefGoogle Scholar
16.Haines, T. and Rapoport, M., On parahoric subgroups (Appendix to [27]), Adv. Math. 219 (2008), 188–198.Google Scholar
17.Kostant, B. and Rallis, S., On orbits associated with symmetric spaces, Bull. Am. Math. Soc. 75 (1969), 879–883.CrossRefGoogle Scholar
18.Kostant, B. and Rallis, S., Orbits and representations associated with symmetric spaces, Am. J. Math. 93 (1971), 753–809.CrossRefGoogle Scholar
19.Kottwitz, R., Points on some Shimura varieties over finite fields, J. Am. Math. Soc. 5 (1992), 373–444.CrossRefGoogle Scholar
20.Krämer, N., Local models for ramified unitary groups, Ab. Math. Sem. Univ. Hamburg 73 (2003), 67–80.CrossRefGoogle Scholar
21.Milne, J., Introduction to Shimura varieties, in Harmonic analysis, the trace formula, and Shimura varieties, pp. 265–378, Clay Mathematical Proceedings, Volume 4 (American Mathematical Society, Providence, RI, 2005).Google Scholar
22.Ohta, T., The singularities of the closures of nilpotent orbits in certain symmetric pairs, Tohoku Math. J. 38 (1986), 441–468.CrossRefGoogle Scholar
23.Pappas, G., On the arithmetic moduli schemes of PEL Shimura varieties, J. Alg. Geom. 9 (2000), 577–605.Google Scholar
24.Pappas, G., Local models and wonderful completions, Notes.Google Scholar
25.Pappas, G. and Rapoport, M., Local models in the ramified case, I, The EL-case, J. Alg. Geom. 12 (2003), 107–145.CrossRefGoogle Scholar
26.Pappas, G. and Rapoport, M., Local models in the ramified case, II, Splitting models, Duke Math. J. 127 (2005), 193–250.CrossRefGoogle Scholar
27.Pappas, G. and Rapoport, M., Twisted loop groups and their affine flag varieties, Adv. Math. 219 (2008), 118–188.CrossRefGoogle Scholar
28.Rapoport, M., A guide to the reduction modulo p of Shimura varieties, Astérisque 298 (2005), 271–318.Google Scholar
29.Rapoport, M. and Zink, Th., Period spaces for p-divisible groups, Annals of Mathematics Studies, Volume 141 (Princeton University Press, 1996).Google Scholar
30.Sekiguchi, J., The nilpotent subvariety of the vector space associated to a symmetric pair, Publ. RIMS Kyoto 20 (1984), 155–212.CrossRefGoogle Scholar
31.Sekiguchi, J., Remarks on real nilpotent orbits of a symmetric pair, J. Math. Soc. Jpn 39 (1987), 127–138.Google Scholar
32.Smithling, B., Topological flatness of orthogonal local models in the split, even case, in preparation.Google Scholar
33.Tits, J., Reductive groups over local fields, in Automorphic Forms, Representations and L-Functions: Proc. Symp. Pure Mathematics, Oregon State University, Corvallis, OR, 1977, Part 1, pp. 29–69, Proceedings of Symposia in Pure Mathematics, Volume 33 (American Mathematical Society, Providence, RI, 1979).Google Scholar