Integral canonical models of Shimura varieties
Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 2, p. 301-312

The aim of these notes is to provide an introduction to the subject of integral canonical models of Shimura varieties, and then to sketch a proof of the existence of such models for Shimura varieties of Hodge and, more generally, abelian type. For full details the reader is refered to [Ki 3].

Le but de cette note est de fournir une introduction à la théorie des modèles entiers canoniques des variétés de Shimura, et de donner une esquisse de la preuve d’existence de tels modèles pour les variétés de Shimura de type Hodge, et plus généralement, de type abélien. Pour plus de détails, le lecteur est renvoyé à [Ki 3].

@article{JTNB_2009__21_2_301_0,
     author = {Kisin, Mark},
     title = {Integral canonical models of Shimura varieties},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {2},
     year = {2009},
     pages = {301-312},
     doi = {10.5802/jtnb.672},
     mrnumber = {2541427},
     zbl = {pre05620652},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2009__21_2_301_0}
}
Kisin, Mark. Integral canonical models of Shimura varieties. Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 2, pp. 301-312. doi : 10.5802/jtnb.672. http://www.numdam.org/item/JTNB_2009__21_2_301_0/

[CS] J-L. Colliot-Thélène, J-J Sansuc, Fibrés quadratiques et composantes connexes réelles. Math. Ann. 244 (1979), 105–134. | MR 550842 | Zbl 0418.14016

[De 1] P. Deligne, Variètés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques. Automorphic forms, representations and L-functions (Corvallis 1977), Proc. Sympos. Pure Math XXXIII, 247–289, AMS, 1979. | MR 546620 | Zbl 0437.14012

[De 2] P. Deligne, Hodge cycles on abelian varieties Hodge cycles motives and Shimura varieties, Lecture notes in math. 900, 9–100, Springer, 1982. | Zbl 0537.14006

[DP] P. Deligne, G. Pappas, Singularités des espaces de modules de Hilbert, en les caractéristiques divisant le discriminant. Compositio Math. 90 (1994), 59–79. | Numdam | MR 1266495 | Zbl 0826.14027

[Ki 1] M. Kisin, Crystalline representations and F-crystals. Algebraic geometry and number theory, Progr. Math 253, 459–496, Birkhäuser, Boston, 2006. | MR 2263197 | Zbl pre05234060

[Ki 2] M. Kisin, Modularity of 2-adic Barsotti-Tate representations. Preprint 2007.

[Ki 3] M. Kisin, Integral models for Shimura varieties of abelian type. Preprint, 2008.

[La] R. Langlands, Some contemporary problems with origins in the Jugendtraum. Mathematical developments arising from Hilbert problems (De Kalb 1974), Proc. Sympos. Pure Math. XXVIII, 401–418, AMS, 1976. | MR 437500 | Zbl 0345.14006

[LR] R. Langlands, M. Rapoport, Shimuravarietäten und Gerben. J. Reine Angew. Math 378 (1987), 113–220. | MR 895287 | Zbl 0615.14014

[Mi 1] J. Milne, The points on a Shimura variety modulo a prime of good reduction. The zeta functions of Picard modular surfaces, 151–253, CRM 1992. | MR 1155229 | Zbl 0821.14016

[Mi 2] J. Milne, Canonical models of (mixed) Shimura varieties and automorphic vector bundles. Automorphic forms, Shimura varieties. and L-functions I (Ann Arbor 1988), Perspectives in Math. 10, 284–414, Academic Press, 1990. | MR 1044823 | Zbl 0704.14016

[Mi 3] J. Milne, On the conjecture of Langlands and Rapoport. Available at arxiv.org, 1995, 31 pages.

[Mo] B. Moonen, Models of Shimura varieties in mixed characteristics. Galois representations in arithmetic algebraic geometry (Durham, 1996), London Math. Soc. Lecture Note Ser. 254, 257–350, CUP 1998. | MR 1696489 | Zbl 0962.14017

[Sp] T. Springer, Reductive groups. Automorphic forms, representations and L-functions (Corvallis 1977), Proc. Sympos. Pure Math XXXIII, 3–27, AMS 1979. | MR 546587 | Zbl 0416.20034

[PY] G. Prasad; J-K. Yu, On quasi-reductive group schemes. With an appendix by Brian Conrad. J. Algebraic Geom. 15 (2006), 507–549. | MR 2219847 | Zbl 1112.14053

[Va 1] A. Vasiu, Integral Canonical Models of Shimura Varieties of Preabelian Type. Asian J. Math, 1999, 401–518. | MR 1796512 | Zbl 1002.11052

[Va 2] A. Vasiu, Integral Canonical Models of Shimura Varieties of Preabelian Type (Fully corrected version). Available at arxiv.org, 2003, 135 pages. | MR 1796512

[Va 3] A. Vasiu, A motivic conjecture of Milne. Available at arxiv.org, 2003, 46 pages. | MR 1796512

[Va 4] A. Vasiu, Good reduction of Shimura varieties in arbitrary unramified mixed characteristic I. Available at arxiv.org, 2007, 48 pages.

[Va 5] A. Vasiu, Good reduction of Shimura varieties in arbitrary unramified mixed characteristic II. Available at arxiv.org, 2007.