The product structure of Newton strata in the good reduction of Shimura varieties of Hodge type
Author:
Paul Hamacher
Journal:
J. Algebraic Geom. 28 (2019), 721-749
DOI:
https://doi.org/10.1090/jag/732
Published electronically:
July 11, 2019
MathSciNet review:
3994311
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Abstract |
References |
Additional Information
Abstract: We construct a generalisation of Mantovan’s almost product structure to Shimura varieties of Hodge type with hyperspecial level structure at $p$ and deduce that the perfection of the Newton strata are proétale locally isomorphic to the perfection of the product of a central leaf and a Rapoport-Zink space. The almost product formula can be extended to obtain an analogue of Caraiani and Scholze’s generalisation of the almost product structure for Shimura varieties of Hodge type.
References
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- Benjamin Howard and Georgios Pappas, Rapoport-Zink spaces for spinor groups, Compos. Math. 153 (2017), no. 5, 1050–1118. MR 3705249, DOI https://doi.org/10.1112/S0010437X17007011
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- Mark Kisin, ${\rm mod}\,p$ points on Shimura varieties of abelian type, J. Amer. Math. Soc. 30 (2017), no. 3, 819–914. MR 3630089, DOI https://doi.org/10.1090/jams/867
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- Thomas Lovering, Filtered $F$-crystals on Shimura varieties of abelian type, preprint, 2017, https://arxiv.org/abs/1702.06611v1.
- Elena Mantovan, On the cohomology of certain PEL-type Shimura varieties, Duke Math. J. 129 (2005), no. 3, 573–610. MR 2169874, DOI https://doi.org/10.1215/S0012-7094-05-12935-0
- Elena Mantovan, $l$-adic étale cohomology of PEL type Shimura varieties with non-trivial coefficients, WIN—women in numbers, Fields Inst. Commun., vol. 60, Amer. Math. Soc., Providence, RI, 2011, pp. 61–83. MR 2777800
- David Mumford, The red book of varieties and schemes, Lecture Notes in Mathematics, vol. 1358, Springer-Verlag, Berlin, 1988. MR 971985
- 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
- Richard Pink, Arithmetical compactification of mixed Shimura varieties, Bonner Mathematische Schriften [Bonn Mathematical Publications], vol. 209, Universität Bonn, Mathematisches Institut, Bonn, 1990. Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 1989. MR 1128753
- M. Rapoport and M. Richartz, On the classification and specialization of $F$-isocrystals with additional structure, Compositio Math. 103 (1996), no. 2, 153–181. MR 1411570
- 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
- Peter Scholze, The Langlands-Kottwitz approach for the modular curve, Int. Math. Res. Not. IMRN 15 (2011), 3368–3425. MR 2822177, DOI https://doi.org/10.1093/imrn/rnq225
- The stacks project, published under GFDL, available at math.columbia.edu/ algebraic_geometry/stacks-git/browse.html.
- Chao Zhang, Stratifications and foliations for good reductions of Shimura varieties of Hodge type, preprint, 2015, http://arxiv.org/abs/1512.08102v1.
References
- Pierre Berthelot, Lawrence Breen, and William Messing, Théorie de Dieudonné cristalline. II, Lecture Notes in Mathematics, vol. 930, Springer-Verlag, Berlin, 1982 (French). MR 667344
- Bhargav Bhatt and Peter Scholze, Projectivity of the Witt vector affine Grassmannian, Invent. Math. 209 (2017), no. 2, 329–423. MR 3674218, DOI https://doi.org/10.1007/s00222-016-0710-4
- Ana Caraiani and Peter Scholze, On the generic part of the cohomology of compact unitary Shimura varieties, Ann. of Math. (2) 186 (2017), no. 3, 649–766. MR 3702677, DOI https://doi.org/10.4007/annals.2017.186.3.1
- Paul Hamacher, The almost product structure of Newton strata in the deformation space of a Barsotti-Tate group with crystalline Tate tensors, Math. Z. 287 (2017), no. 3-4, 1255–1277. MR 3719535, DOI https://doi.org/10.1007/s00209-017-1867-2
- Michael Harris and Richard Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton, NJ, 2001. With an appendix by Vladimir G. Berkovich. MR 1876802
- Benjamin Howard and Georgios Pappas, Rapoport-Zink spaces for spinor groups, Compos. Math. 153 (2017), no. 5, 1050–1118. MR 3705249, DOI https://doi.org/10.1112/S0010437X17007011
- Wansu Kim, Rapoport-Zink spaces of Hodge type, Forum Math. Sigma 6 (2018), e8, 110. MR 3812116, DOI https://doi.org/10.1017/fms.2018.6
- Mark Kisin, Integral models for Shimura varieties of abelian type, J. Amer. Math. Soc. 23 (2010), no. 4, 967–1012. MR 2669706, DOI https://doi.org/10.1090/S0894-0347-10-00667-3
- Mark Kisin, $\textrm {mod} p$ points on Shimura varieties of abelian type, J. Amer. Math. Soc. 30 (2017), no. 3, 819–914. MR 3630089, DOI https://doi.org/10.1090/jams/867
- Robert E. Kottwitz, Isocrystals with additional structure, Compositio Math. 56 (1985), no. 2, 201–220. MR 809866
- Robert E. Kottwitz, Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), no. 2, 373–444. MR 1124982, DOI https://doi.org/10.2307/2152772
- Thomas Lovering, Filtered $F$-crystals on Shimura varieties of abelian type, preprint, 2017, https://arxiv.org/abs/1702.06611v1.
- Elena Mantovan, On the cohomology of certain PEL-type Shimura varieties, Duke Math. J. 129 (2005), no. 3, 573–610. MR 2169874, DOI https://doi.org/10.1215/S0012-7094-05-12935-0
- Elena Mantovan, $l$-adic étale cohomology of PEL type Shimura varieties with non-trivial coefficients, WIN—women in numbers, Fields Inst. Commun., vol. 60, Amer. Math. Soc., Providence, RI, 2011, pp. 61–83. MR 2777800
- David Mumford, The red book of varieties and schemes, Lecture Notes in Mathematics, vol. 1358, Springer-Verlag, Berlin, 1988. MR 971985
- 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
- Richard Pink, Arithmetical compactification of mixed Shimura varieties, Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 1989, Bonner Mathematische Schriften [Bonn Mathematical Publications], vol. 209, Universität Bonn, Mathematisches Institut, Bonn, 1990. MR 1128753
- M. Rapoport and M. Richartz, On the classification and specialization of $F$-isocrystals with additional structure, Compositio Math. 103 (1996), no. 2, 153–181. MR 1411570
- 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
- Peter Scholze, The Langlands-Kottwitz approach for the modular curve, Int. Math. Res. Not. IMRN 15 (2011), 3368–3425. MR 2822177, DOI https://doi.org/10.1093/imrn/rnq225
- The stacks project, published under GFDL, available at math.columbia.edu/ algebraic_geometry/stacks-git/browse.html.
- Chao Zhang, Stratifications and foliations for good reductions of Shimura varieties of Hodge type, preprint, 2015, http://arxiv.org/abs/1512.08102v1.
Additional Information
Paul Hamacher
Affiliation:
Zentrum Mathematik - M11, Technische Universiät München, Boltzmannstraße 3, 85748 Garching bei München, Germany
MR Author ID:
1048654
Email:
hamacher@ma.tum.de
Received by editor(s):
October 6, 2017
Received by editor(s) in revised form:
May 18, 2018, and August 30, 2018
Published electronically:
July 11, 2019
Additional Notes:
The author was partially supported by the ERC starting grant 277889 “Moduli spaces of local $G$-shtukas”.
Article copyright:
© Copyright 2019
University Press, Inc.