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ON THE GEOMETRY OF THE PAPPAS–RAPOPORT MODELS FOR PEL SHIMURA VARIETIES

Part of: Algebraic groups Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry

Published online by Cambridge University Press:  18 February 2022

Stéphane Bijakowski  and
Valentin Hernandez Open the ORCID record for Valentin Hernandez [Opens in a new window]
Stéphane Bijakowski
Affiliation:
Centre de mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau, France (stephane.bijakowski@polytechnique.edu)
Valentin Hernandez*
Affiliation:
Laboratoire Mathématiques d’Orsay, Bâtiment 307, Université Paris-Sud, 91405 Orsay, France
*
valentin.hernandez@math.cnrs.fr
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Abstract

In this article we study integral models of Shimura varieties, called Pappas–Rapoport splitting model, for ramified P.E.L. Shimira data. We study the special fiber and some stratification of these models, in particular we show that these are smooth and the Rapoport locus and the $\mu $-ordinary locus are dense, under some condition on the ramification.

Keywords

Shimura varietiesintegral models of Shimura varietiesp-divisible groupsNewton stratification

MSC classification

Primary: 11G18: Arithmetic aspects of modular and Shimura varieties
Secondary: 14G35: Modular and Shimura varieties 14L05: Formal groups, $p$-divisible groups
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 22 , Issue 5 , September 2023 , pp. 2403 - 2445
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

Langlands, R. P. and Ramakrishnan, D. (eds.), The Zeta Functions of Picard Modular Surfaces: Based on Lectures, Delivered at a CRM Workshop in the Spring of 1988, Montréal, Canada (Centre de Recherches Mathématiques, Université de Montréal, Montréal, 1992), xiv+492.Google Scholar
Berthelot, P., Breen, L. and Messing, W., Théorie de Dieudonné cristalline. II, Lecture Notes in Mathematics, 930 (Springer-Verlag, Berlin, 1982).CrossRefGoogle Scholar
Bijakowski, S. and Hernandez, V., ‘Groupes $p$ -divisibles avec condition de Pappas-Rapoport et invariants de Hasse’, J. Éc. polytech. Math. 4 (2017), 935–972.CrossRefGoogle Scholar
Clozel, L., Harris, M., Labesse, J.-P. and Ngô, B.-C. (eds.), Stabilization of the Trace Formula, Shimura Varieties, and Arithmetic Applications, Vol. 1: On the Stabilization of the Trace Formula (International Press, Somerville, MA, 2011).Google Scholar
Deligne, P., ‘Travaux de Shimura’, in Séminaire Bourbaki, 23ème année (1970/71), Exp. No. 389, Lecture Notes in Mathematics, 244, pp. 123–165 (Springer, Berlin, 1971).CrossRefGoogle Scholar
Deligne, P. and Rapoport, M., ‘Les schemas de modules de courbes elliptiques’, in Modular Functions of one Variable II, Proc. internat. Summer School, Univ. Antwerp 1972, Lecture Notes in Mathematics, 349, pp. 143–316 (Springer, 1973).CrossRefGoogle Scholar
Hamacher, P., ‘The geometry of Newton strata in the reduction modulo $p$ of Shimura varieties of PEL type’, Duke Math. J. 164(15) (2015), 2809–2895.CrossRefGoogle Scholar
Harris, M. and Taylor, R., The Geometry and Cohomology of Some Simple Shimura Varieties, Annals of Mathematics Studies, 151 (Princeton University Press, Princeton, NJ, 2001).Google Scholar
He, X. and Nie, S., ‘On the $\mu$ -ordinary locus of a Shimura variety’, Adv. Math. 321 (2017), 513–528.CrossRefGoogle Scholar
He, X. and Rapoport, M., ‘Stratifications in the reduction of Shimura varieties’, Manuscripta Math. 152(3-4) (2017), 317–343.CrossRefGoogle Scholar
Katz, N. M. and Mazur, B., Arithmetic Moduli of Elliptic Curves, Annals of Mathematics Studies, 108 (Princeton University Press, Princeton, NJ, 1985).CrossRefGoogle Scholar
Kottwitz, R. E., ‘Points on some Shimura varieties over finite fields’, J. Amer. Math. Soc. 5(2) (1992), 373–444.10.1090/S0894-0347-1992-1124982-1CrossRefGoogle Scholar
Kramer, N., ‘Local models for ramified unitary groups’, Abh. Math. Semin. Univ. Hambg. 73 (2003), 67–80.CrossRefGoogle Scholar
Lan, K.-W., Arithmetic Compactifications of PEL-Type Shimura Varieties, London Mathematical Society Monographs Series, 36 (Princeton University Press, Princeton, NJ, 2013).Google Scholar
Lan, K.-W., ‘Compactifications of PEL-type Shimura varieties in ramified characteristics’, Forum Math. Sigma 4 (2016), 98 pp.CrossRefGoogle Scholar
Lan, K.-W., ‘Compactifications of splitting models of PEL-type Shimura varieties’, Trans. Amer. Math. Soc. 370(4) (2018), 2463–2515.CrossRefGoogle Scholar
Lau, E., ‘Dieudonné theory over semiperfect rings and perfectoid rings’, Compos. Math. 154(9) (2018), 1974–2004.CrossRefGoogle Scholar
Milne, J. S., ‘Introduction to Shimura varieties’, in Harmonic Analysis, the Trace Formula, and Shimura Varieties: Proceedings of the Clay Mathematics Institute 2003 Summer School, Toronto, Canada, June 2–27, 2003, pp. 265–378 (American Mathematical Society, Providence, RI, 2005).Google Scholar
Pappas, G. and Rapoport, M., ‘Local models in the ramified case. II: Splitting models’, Duke Math. J. 127(2) (2005), 193–250.CrossRefGoogle Scholar
Rapoport, M. and Richartz, M., ‘On the classification and specialization of $F$ -isocrystals with additional structure’, Compos. Math. 103(2) (1996), 153–181.Google Scholar
Reduzzi, D. A. and Xiao, L., ‘Invariants de Hasse partiels sur les modèles de décomposition des variétés de Hilbert modulaires’, Ann. Sci. Éc. Norm. Supér. (4) 50(3) (2017), 579–607.Google Scholar
Sasaki, S., ‘Integral models of Hilbert modular varieties in the ramified case, deformations of modular Galois representations, and weight one forms’, Invent. Math. 215(1) (2019), 171–264.10.1007/s00222-018-0825-xCrossRefGoogle Scholar
Shin, S. W., ‘Galois representations arising from some compact Shimura varieties’, Ann. of Math. (2) 173(3) (2011), 1645–1741.CrossRefGoogle Scholar
Stamm, H., ‘On the reduction of the Hilbert-Blumenthal-moduli scheme with ${\varGamma}_0(p)$ -level structure’, Forum Math. 9(4) (1997), 405–455.10.1515/form.1997.9.405CrossRefGoogle Scholar
Viehmann, E. and Wedhorn, T., ‘Ekedahl-Oort and Newton strata for Shimura varieties of PEL type’, Math. Ann. 356(4) (2013), 1493–1550.CrossRefGoogle Scholar
Wedhorn, T., ‘Ordinariness in good reductions of Shimura varieties of PEL-type’, Ann. Sci. Éc. Norm. Supér. (4) 32(5) (1999), 575–618.CrossRefGoogle Scholar
Wortmann, D., ‘The $\mu$ -ordinary locus for Shimura varieties of Hodge type’, Preprint, 2013, https://arxiv.org/abs/1310.6444.Google Scholar
Zink, T., ‘The display of a formal $p$ -divisible group’, in Cohomologies $p$ -adiques et applications arithmétiques (I), pp. 127–248 (Société Mathématique de France, Paris, 2002).Google Scholar