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2-ADIC INTEGRAL CANONICAL MODELS

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  05 October 2016

WANSU KIM  and
KEERTHI MADAPUSI PERA
WANSU KIM
Affiliation:
Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, UK; wansu.kim@kcl.ac.uk
KEERTHI MADAPUSI PERA
Affiliation:
Department of Mathematics, University of Chicago, 5734 S University Ave, Chicago, IL, USA; keerthi@math.uchicago.edu

Abstract

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We use Lau’s classification of 2-divisible groups using Dieudonné displays to construct integral canonical models for Shimura varieties of abelian type at 2-adic places where the level is hyperspecial.

MSC classification

Primary: 14G35: Modular and Shimura varieties
Secondary: 11G18: Arithmetic aspects of modular and Shimura varieties
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 4 , 2016 , e28
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2016

References

André, Y., ‘On the Shafarevich and Tate conjectures for hyperkähler varieties’, Math. Ann. 305(1) (1996), 205–248.Google Scholar
Andreatta, F., Goren, E., Howard, B. and Madapusi Pera, K., ‘Faltings heights of abelian varieties with complex multiplication’, Preprint, 2015, available at http://www.math.uchicago.edu/∼keerthi/papers/colmez.pdf.Google Scholar
Artin, M., ‘Supersingular K3 surfaces’, Ann. Sci. Éc. Norm. Supér. (4) 7 (1974), 543–567; 1975.CrossRefGoogle Scholar
Berthelot, P., Breen, L. and Messing, W., Théorie de Dieudonné cristalline. II, Lecture Notes in Mathematics, 930 (Springer, Berlin, 1982).Google Scholar
Blasius, D., ‘A p-adic property of Hodge classes on abelian varieties’, inMotives, Proceedings of Symposia in Pure Mathematics, 55 (American Mathematical Society, Providence, RI, 1994), 293–308.CrossRefGoogle Scholar
Bloch, S. and Kato, K., ‘ p-adic étale cohomology’, Publ. Math. Inst. Hautes Études Sci. 63 (1986), 107–152.Google Scholar
Charles, F., ‘The Tate conjecture for K3 surfaces over finite fields’, Invent. Math. 194(1) (2013), 119–145.Google Scholar
Deligne, P., ‘Variétés de Shimura: interprétation modulaire, et techniques de construction de modeles canoniques’, inAutomorphic Forms, Representations and L-Functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII (American Mathematical Society, Providence, RI, 1979), 247–289.Google Scholar
Faltings, G., ‘Integral crystalline cohomology over very ramified valuation rings’, J. Amer. Math. Soc. 12(1) (1999), 117–144.CrossRefGoogle Scholar
Grothendieck, A., ‘Groupes de type multiplicatif: homomorphismes dans un schéma en groupes’, in Schémas en Groupes (Sém. Géométrie Algébrique, Inst. Hautes Études Sci., 1963/64), pages Fasc. 3, Exposé 9, 37 (Inst. Hautes Études Sci., Paris, 1964).Google Scholar
Kim, W., ‘The classification of p-divisible groups over 2-adic discrete valuation rings’, Math. Res. Lett. 19(1) (2012), 121–141.CrossRefGoogle Scholar
Kisin, M., ‘Crystalline representations and F-crystals’, inAlgebraic Geometry and Number Theory, Progress in Mathematics, 253 (Birkhäuser Boston, Boston, MA, 2006), 459–496.Google Scholar
Kisin, M., ‘Integral models for Shimura varieties of abelian type’, J. Amer. Math. Soc. 23(4) (2010), 967–1012.Google Scholar
Kisin, M., ‘Mod p points on Shimura varieties of abelian type’, J. Amer. Math. Soc. (2013) (to appear).Google Scholar
Lau, E., ‘Displayed equations for Galois representations’, Preprint, 2012, available at http://arxiv.org/abs/1012.4436.Google Scholar
Lau, E., ‘Relations between Dieudonné displays and crystalline Dieudonné theory’, Algebra Number Theory 8(9) (2014), 2201–2262.Google Scholar
Madapusi Pera, K., ‘The Tate conjecture for K3 surfaces in odd characteristic’, Invent. Math. 201(2) (2015), 625–668.Google Scholar
Madapusi Pera, K., ‘Toroidal compactifications of integral models of Shimura varieties of hodge type’, Preprint, 2015, available at http://www.math.uchicago.edu/∼keerthi/papers/toroidal_new.pdf.Google Scholar
Madapusi Pera, K., ‘Integral canonical models for Spin Shimura varieties’, Compos. Math. 152(4) (2016), 769–824.Google Scholar
Madapusi Sampath, K. S., ‘Toroidal compactifications of integral models of Shimura varieties of Hodge type’, PhD thesis, University of Chicago (2011).Google Scholar
Maulik, D., ‘Supersingular K3 surfaces for large primes’, Duke Math. J. 163(13) (2014), 2357–2425.Google Scholar
Moonen, B., ‘Models of Shimura varieties in mixed characteristics’, inGalois Representations in Arithmetic Algebraic Geoemety (Durham, 1996), London Mathematical Society Lecture Note Series, 254 (Cambridge University Press, Cambridge, 1998), 267–350.CrossRefGoogle Scholar
Ogus, A., ‘Supersingular K3 crystals’, inJournées de Géométrie Algébrique de Rennes (Rennes, 1978), Vol. II, Astérisque, 64 (Société Mathématique de France, Paris, 1979), 3–86.Google Scholar
Ogus, A., ‘A crystalline Torelli theorem for supersingular K3 surfaces’, inArithmetic and Geometry, Vol. II, Progress in Mathematics, 36 (Birkhäuser Boston, Boston, MA, 1983), 361–394.Google Scholar
Ogus, A., ‘Singularities of the height strata in the moduli of K3 surfaces’, inModuli of Abelian Varieties, Progress in Mathematics, 195 (Birkhäuser, Basel, 2001), 325–343.Google Scholar
Prasad, G. and Yu, J.-K., ‘On quasi-reductive group schemes’, J. Algebraic Geom. 15(3) (2006), 507–549.CrossRefGoogle Scholar
Tate, J. T., ‘ p-divisible groups’, inProc. Conf. Local Fields (Driebergen, 1966) (Springer, Berlin, 1967), 158–183.Google Scholar
Vasiu, A. and Zink, T., ‘Purity results for p-divisible groups and abelian schemes over regular bases of mixed characteristic’, Doc. Math. 15 (2010), 571–599.Google Scholar
Zarhin, J. G., ‘Abelian varieties in characteristic p ’, Mat. Zametki 19(3) (1976), 393–400.Google Scholar