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G-structures entières de représentations cristallines

Abstract : Jean-Marc Fontaine showed that the tannakian category of crystalline representations of the group of Galois of a local field K is equivalent to that of the admissible filtered Phi-modules on K. Moreover, the Fontaine-Laffaille's theory, under certain restrictions, specifies this using a V_cris functor which induces an abelian equivalence between the strongly divisible lattices of the admissible filtered Phi-modules and the Galois-lattices of the corresponding Galois-representations.

The purpose of this thesis is to study the V_cris functor. Because of the restrictions related to the Fontaine-Laffaille's theory, the categories considered for the lattices are not stable by tensorial product. But we show that in spite of this problem, V_cris has good tannakian properties, which lead to interesting applications for the crystalline representations with values in the Zp-points of a smooth algebraic group over Zp.

The key point is the construction of a functor, which to a filtered Phi-module M (checking the conditions of Fountain-Laffaille) associates a (Phi, Gamma) - module of which the associated Galois-representation is fonctorially identified with V_cris (M), and who preserves the tensorial product (under certain conditions). This functor has a very strong link with the theory of the Wach modules, and that's why we can use at his best the equivalence of categories between the Galois-representations over Zp and the (Phi, Gamma) - modules.
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Submitted on : Monday, March 19, 2007 - 11:01:28 PM
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Lionel Dorat. G-structures entières de représentations cristallines. Mathématiques [math]. Université Louis Pasteur - Strasbourg I, 2006. Français. ⟨tel-00137445⟩

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