Moduli spaces of $p$-divisible groups
Author:
Eva Viehmann
Journal:
J. Algebraic Geom. 17 (2008), 341-374
DOI:
https://doi.org/10.1090/S1056-3911-07-00480-8
Published electronically:
December 5, 2007
MathSciNet review:
2369090
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Abstract |
References |
Additional Information
Abstract: We study the global structure of moduli spaces of quasi-isogenies of $p$-divisible groups introduced by Rapoport and Zink. We determine their dimensions and their sets of connected components and of irreducible components. If the isocrystals of the $p$-divisible groups are simple, we compute the cohomology of the moduli space. As an application we determine which moduli spaces are smooth.
References
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- Ulrich Görtz, Thomas J. Haines, Robert E. Kottwitz, and Daniel C. Reuman, Dimensions of some affine Deligne-Lusztig varieties, Ann. Sci. École Norm. Sup. (4) 39 (2006), no. 3, 467–511 (English, with English and French summaries). MR 2265676, DOI https://doi.org/10.1016/j.ansens.2005.12.004
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- William Messing, The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, Lecture Notes in Mathematics, Vol. 264, Springer-Verlag, Berlin-New York, 1972. MR 0347836
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- Frans Oort, Foliations in moduli spaces of abelian varieties, J. Amer. Math. Soc. 17 (2004), no. 2, 267–296. MR 2051612, DOI https://doi.org/10.1090/S0894-0347-04-00449-7
- Frans Oort and Thomas Zink, Families of $p$-divisible groups with constant Newton polygon, Doc. Math. 7 (2002), 183–201. MR 1938119
- Michael Rapoport, A guide to the reduction modulo $p$ of Shimura varieties, Astérisque 298 (2005), 271–318 (English, with English and French summaries). Automorphic forms. I. MR 2141705
- M. Rapoport and Th. Zink, Period spaces for $p$-divisible groups, Annals of Mathematics Studies, vol. 141, Princeton University Press, Princeton, NJ, 1996. MR 1393439
- Eva Viehmann, The dimension of some affine Deligne-Lusztig varieties, Ann. Sci. École Norm. Sup. (4) 39 (2006), no. 3, 513–526 (English, with English and French summaries). MR 2265677, DOI https://doi.org/10.1016/j.ansens.2006.04.001
- I. Vollaard, The supersingular locus of the Shimura variety of $GU(1,s)$, preprint, 2005, math.AG/0509067.
- Thomas Zink, The display of a formal $p$-divisible group, Astérisque 278 (2002), 127–248. Cohomologies $p$-adiques et applications arithmétiques, I. MR 1922825
References
- L. Fargues, Cohomologie des espaces de modules de groupes $p$-divisibles et correspondances de Langlands locales, in Variétés de Shimura, espaces de Rapoport-Zink et correspondances de Langlands locales, Astérisque 291 (2004), 1–199. MR 2074714 (2005g:11110b)
- U. Görtz, Th. Haines, R. Kottwitz, D. Reuman, Dimensions of some affine Deligne-Lusztig varieties, Ann. Sci. École Norm. Sup. 39 (2006), 467–511. MR 2265676
- A. J. de Jong, F. Oort, Purity of the stratification by Newton polygons, J. Amer. Math. Soc. 13 (2000), 209–241. MR 1703336 (2000m:14050)
- E. Mantovan, On certain unitary group Shimura varieties, in Variétés de Shimura, espaces de Rapoport-Zink et correspondances de Langlands locales, Astérisque 291 (2004), 201–331. MR 2074715 (2005g:11110c)
- W. Messing, The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, Lecture Notes in Math. 264, Springer, 1972. MR 0347836 (50:337)
- F. Oort, Newton polygon strata in the moduli space of abelian varieties in Moduli of abelian varieties (Texel Island, 1999), 417–440, Progr. Math., 195, Birkhäuser, Basel, 2001. MR 1827028 (2002c:14069)
- F. Oort, Minimal $p$-divisible groups, Ann. of Math. (2) 161 (2005), 1021–1036. MR 2153405 (2006i:14042)
- F. Oort, Foliations in moduli spaces of abelian varieties, J. Amer. Math. Soc. 17 (2004), no. 2, 267–296. MR 2051612 (2005c:14051)
- F. Oort, Th. Zink, Families of $p$-divisible groups with constant Newton polygon, Documenta Math. 7 (2002), 183–201. MR 1938119 (2003m:14066)
- M. Rapoport, A guide to the reduction modulo $p$ of Shimura varieties, Astérisque 298 (2005), 271–318. MR 2141705 (2006c:11071)
- M. Rapoport, Th. Zink, Period spaces for $p$-divisible groups, Princeton Univ. Press, 1996. MR 1393439 (97f:14023)
- E. Viehmann, The dimension of some affine Deligne-Lusztig varieties, Ann. Sci. École Norm. Sup. 39 (2006), 513–526. MR 2265677
- I. Vollaard, The supersingular locus of the Shimura variety of $GU(1,s)$, preprint, 2005, math.AG/0509067.
- Th. Zink, The display of a formal $p$-divisible group, in Cohomologies $p$-adiques et applications arithmétiques, I, Astérisque 278 (2002), 127–248. MR 1922825 (2004b:14083)
Additional Information
Eva Viehmann
Affiliation:
Mathematisches Institut der Universität Bonn, Beringstrasse 1, 53115 Bonn, Germany
Email:
viehmann@math.uni-bonn.de
Received by editor(s):
March 31, 2006
Received by editor(s) in revised form:
February 8, 2007
Published electronically:
December 5, 2007
