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Prime to p fundamental groups and tame Galois actions

Mark Kisin — 2000

Annales de l'institut Fourier

We show that for a local, discretely valued field F , with residue characteristic p , and a variety 𝒰 over F , the map ρ : Gal ( F sep / F ) Out ( π 1 , geom ( p ' ) ( 𝒰 ) ) to the outer automorphisms of the prime to p geometric étale fundamental group of 𝒰 maps the wild inertia onto a finite image. We show that under favourable conditions ρ depends only on the reduction of 𝒰 modulo a power of the maximal ideal of F . The proofs make use of the theory of logarithmic schemes.

Integral canonical models of Shimura varieties

Mark Kisin — 2009

Journal de Théorie des Nombres de Bordeaux

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].

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