Integral canonical models of Shimura varieties

@article{Kisin2009IntegralCM,
  title={Integral canonical models of Shimura varieties},
  author={Mark Kisin},
  journal={Journal de Theorie des Nombres de Bordeaux},
  year={2009},
  volume={21},
  pages={301-312}
}
  • M. Kisin
  • Published 2009
  • Mathematics
  • Journal de Theorie des Nombres de Bordeaux
Le but de cette note est de fournir une introduction a la theorie des modeles entiers canoniques des varietes de Shimura, et de donner une esquisse de la preuve d'existence de tels modeles pour les varietes de Shimura de type Hodge, et plus generalement, de type abelien. Pour plus de details, le lecteur est renvoye a [Ki 3]. 

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References

SHOWING 1-10 OF 24 REFERENCES

Singularités des espaces de modules de Hilbert, en les caractéristiques divisant le discriminant

© Foundation Compositio Mathematica, 1994, tous droits reserves. L’acces aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions

Integral Canonical Models of Shimura Varieties of Preabelian Type

We prove the existence of integral canonical models of Shimura varieties of preabelian type with respect to primes of characteristic at least 5.

A motivic conjecture of Milne

Let k be an algebraically closed field of characteristic p>0. Let W(k) be the ring of Witt vectors with coefficients in k. We prove a motivic conjecture of Milne that relates, in the case of abelian

Modularity of 2-adic Barsotti-Tate representations

We prove a modularity lifting theorem for two dimensional, 2-adic, potentially Barsotti-Tate representations. This proves hypothesis (H) of Khare-Wintenberger, and completes the proof of Serre’s

Integral models for Shimura varieties of abelian type

Introduction 967 1. Reductive groups and p-divisible groups 971 (1.1) Cocharacters and filtrations 971 (1.2) Review of S-modules 973 (1.3) Reductive groups and crystalline representations 975 (1.4)

On the conjecture of Langlands and Rapoport

FORENOTE (2007): The remarkable conjecture of Langlands and Rapoport (1987) gives a purely group-theoretic description of the points on a Shimura variety modulo a prime of good reduction. In an

Some Contemporary Problems with Origins in the Jugendtraum

The twelfth problem of Hilbert reminds us, although the reminder should be unnecessary, of the blood relationship of three subjects which have since undergone often separate developments. The first

Canonical Models of (Mixed) Shimura Varieties and Automorphic Vector Bundles

The article surveys what was known, or conjectured, about canonical models of Shimura varieties and related objects at the time it was written (1988). I Abelian varieties with complex multiplication

The points on a Shimura variety modulo a prime of good reduction

In the known process of manufacturing an amphoteric polymeric composition which process comprises firstly polymerising a mixture comprising a monomer containing a basic group and a monomer containing