Abstract

We shall briefly review the Fontaine theory that provides the Grothendieck mysterious functor, relating the etale and crystalline cohomology functors. We will hopefully have a quick look at the analogous theory in the arithmetic of function fields set up. We'll further discuss the modern approach of the Breuil and Kisin to the notion of the Drinfeld shtukas. Meanwhile we'll point out the significant analogy between Breuil-Kisin local models and Rapoport-Zink local models for Shimura Varieties (by the way the comparison with the arithmetic of function fields set up might be mentioned). Finally as an application we study the geometry of the flat quotients of the Mazurs deformation rings, which generically parametrize the deformations of the crystalline Galois representations.



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