Integral models for Shimura varieties of abelian type

Kisin, Mark

doi:10.1090/S0894-0347-10-00667-3

Integral models for Shimura varieties of abelian type
HTML articles powered by AMS MathViewer

by Mark Kisin

J. Amer. Math. Soc. 23 (2010), 967-1012

DOI: https://doi.org/10.1090/S0894-0347-10-00667-3

Published electronically: April 21, 2010

PDF | Request permission

Abstract:

We construct smooth integral models of Shimura varieties of abelian type at primes where the level structure is hyperspecial.

References

Don Blasius, A $p$-adic property of Hodge classes on abelian varieties, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 293–308. MR 1265557, DOI 10.1090/pspum/055.2/1265557
P. Berthelot and A. Ogus, $F$-isocrystals and de Rham cohomology. I, Invent. Math. 72 (1983), no. 2, 159–199. MR 700767, DOI 10.1007/BF01389319
C. Breuil, Schémas en groupes et corps des normes (unpublished), 1998, 13 pages.
F. Bruhat and J. Tits, Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée, Inst. Hautes Études Sci. Publ. Math. 60 (1984), 197–376 (French). MR 756316
J.-L. Colliot-Thélène and J.-J. Sansuc, Fibrés quadratiques et composantes connexes réelles, Math. Ann. 244 (1979), no. 2, 105–134 (French). MR 550842, DOI 10.1007/BF01420486
Claude Chevalley, Deux théorèmes d’arithmétique, J. Math. Soc. Japan 3 (1951), 36–44 (French). MR 44570, DOI 10.2969/jmsj/00310036
Jean-Louis Colliot-Thélène and Venapally Suresh, Quelques questions d’approximation faible pour les tores algébriques, Ann. Inst. Fourier (Grenoble) 57 (2007), no. 1, 273–288 (French, with English and French summaries). MR 2316239, DOI 10.5802/aif.2258
Pierre Deligne, Travaux de Shimura, Séminaire Bourbaki, 23ème année (1970/1971), Lecture Notes in Math., Vol. 244, Springer, Berlin, 1971, pp. Exp. No. 389, pp. 123–165 (French). MR 0498581
Pierre Deligne, James S. Milne, Arthur Ogus, and Kuang-yen Shih, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, vol. 900, Springer-Verlag, Berlin-New York, 1982. MR 654325, DOI 10.1007/978-3-540-38955-2
Pierre Deligne, Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 247–289 (French). MR 546620
Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin-New York, 1970 (French). MR 0417174, DOI 10.1007/BFb0061194
Schémas en groupes. I: Propriétés générales des schémas en groupes, Lecture Notes in Mathematics, Vol. 151, Springer-Verlag, Berlin-New York, 1970 (French). Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3); Dirigé par M. Demazure et A. Grothendieck. MR 0274458
Gerd Faltings, Integral crystalline cohomology over very ramified valuation rings, J. Amer. Math. Soc. 12 (1999), no. 1, 117–144. MR 1618483, DOI 10.1090/S0894-0347-99-00273-8
Gerd Faltings and Ching-Li Chai, Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 22, Springer-Verlag, Berlin, 1990. With an appendix by David Mumford. MR 1083353, DOI 10.1007/978-3-662-02632-8
Jean-Marc Fontaine, Représentations $p$-adiques des corps locaux. I, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 249–309 (French). MR 1106901
Groupes de monodromie en géométrie algébrique. I, Lecture Notes in Mathematics, Vol. 288, Springer-Verlag, Berlin-New York, 1972 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I); Dirigé par A. Grothendieck. Avec la collaboration de M. Raynaud et D. S. Rim. MR 0354656
Mark Kisin, Crystalline representations and $F$-crystals, Algebraic geometry and number theory, Progr. Math., vol. 253, Birkhäuser Boston, Boston, MA, 2006, pp. 459–496. MR 2263197, DOI 10.1007/978-0-8176-4532-8_{7}
Mark Kisin, Potentially semi-stable deformation rings, J. Amer. Math. Soc. 21 (2008), no. 2, 513–546. MR 2373358, DOI 10.1090/S0894-0347-07-00576-0
Mark Kisin, Modularity of 2-adic Barsotti-Tate representations, Invent. Math. 178 (2009), no. 3, 587–634. MR 2551765, DOI 10.1007/s00222-009-0207-5
Mark Kisin, Integral canonical models of Shimura varieties, J. Théor. Nombres Bordeaux 21 (2009), no. 2, 301–312 (English, with English and French summaries). MR 2541427, DOI 10.5802/jtnb.672
Robert E. Kottwitz, Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), no. 2, 373–444. MR 1124982, DOI 10.1090/S0894-0347-1992-1124982-1
R. P. Langlands, Some contemporary problems with origins in the Jugendtraum, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974) Amer. Math. Soc., Providence, R.I., 1976, pp. 401–418. MR 0437500
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, DOI 10.1007/BFb0058301
D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906, DOI 10.1007/978-3-642-57916-5
James S. Milne, The points on a Shimura variety modulo a prime of good reduction, The zeta functions of Picard modular surfaces, Univ. Montréal, Montreal, QC, 1992, pp. 151–253. MR 1155229
J. S. Milne, Canonical models of (mixed) Shimura varieties and automorphic vector bundles, Automorphic forms, Shimura varieties, and $L$-functions, Vol. I (Ann Arbor, MI, 1988) Perspect. Math., vol. 10, Academic Press, Boston, MA, 1990, pp. 283–414. MR 1044823
J. Milne, On the conjecture of Langlands and Rapoport, available at www.arxiv.org (2007).
J. S. Milne, Shimura varieties and motives, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 447–523. MR 1265562, DOI 10.1090/pspum/055.2/1265562
Ben Moonen, Models of Shimura varieties in mixed characteristics, Galois representations in arithmetic algebraic geometry (Durham, 1996) London Math. Soc. Lecture Note Ser., vol. 254, Cambridge Univ. Press, Cambridge, 1998, pp. 267–350. MR 1696489, DOI 10.1017/CBO9780511662010.008
Gopal Prasad and Jiu-Kang Yu, On quasi-reductive group schemes, J. Algebraic Geom. 15 (2006), no. 3, 507–549. With an appendix by Brian Conrad. MR 2219847, DOI 10.1090/S1056-3911-06-00422-X
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, DOI 10.1515/9781400882601
Neantro Saavedra Rivano, Catégories Tannakiennes, Lecture Notes in Mathematics, Vol. 265, Springer-Verlag, Berlin-New York, 1972 (French). MR 0338002
T. A. Springer, Reductive groups, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–27. MR 546587
J. Tits, Reductive groups over local fields, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 29–69. MR 546588
Adrian Vasiu, Integral canonical models of Shimura varieties of preabelian type, Asian J. Math. 3 (1999), no. 2, 401–518. MR 1796512, DOI 10.4310/AJM.1999.v3.n2.a8
A. Vasiu, Integral Canonical Models of Shimura Varieties of Preabelian Type. (Fully corrected version.), available at www.arxiv.org (2003), 135 pages.
A. Vasiu, A motivic conjecture of Milne, available at www.arxiv.org (2003), 46 pages.
A. Vasiu, Good reductions of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic I, available at www.arxiv.org (2007), 48 pages.
A. Vasiu, Good reductions of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic II, available at www.arxiv.org (2007), 33 pages.
William C. Waterhouse, Introduction to affine group schemes, Graduate Texts in Mathematics, vol. 66, Springer-Verlag, New York-Berlin, 1979. MR 547117, DOI 10.1007/978-1-4612-6217-6
Thomas Zink, The display of a formal $p$-divisible group, Astérisque 278 (2002), 127–248. Cohomologies $p$-adiques et applications arithmétiques, I. MR 1922825
Thomas Zink, Windows for displays of $p$-divisible groups, Moduli of abelian varieties (Texel Island, 1999) Progr. Math., vol. 195, Birkhäuser, Basel, 2001, pp. 491–518. MR 1827031