Programme de Langlands p-adique, invariants L et catégories dérivées

Abstract : The results of this thesis have for background the p-adic Langlands program. When V is a two dimensional p-adic representation of the Galois group of Qp, we know how to associate to V a continuous p-adic representation of GL_2(Qp). In a first chapter, we consider the case where V is semi-stable non crystalline and construct a functor which gives the Fontaine module of V, when it is applied to a locally analytic subrepresentation Sigma(V) of B(V) which was constructed by Breuil. This method, which is inspired by works of Carayol and Dat in the l-adic setting, uses the de Rham complex of Drinfel'd's Half space. When L is a finite extension of Qp, we extend this construction to some families of semi-stable non crystalline two dimensional representations of G_Qp parametrized by [L:Qp]-uples of elements of the coefficient field. We propose, in analogy with Breuil's constructions, a locally analytic representation of GL_2(L) associated to V and show that we can retrieve the Fontaine module of V by the precedent functor. In a second chapter, we are intersesting by some families of semi-stable three dimensional representations of G_Qp. In this case, the situation is much more complicated and we construct, for such a representation V, not a representation but a complex Sigma(V) of locally analytic representations of GL_3(Qp). Then we show that an analog of the functor of the first chapter, but using the two dimensional Drinfel'd's space, associates to Sigma(V) the Fontaine module of V.
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Benjamin Schraen. Programme de Langlands p-adique, invariants L et catégories dérivées. Mathématiques [math]. Université Paris Sud - Paris XI, 2009. Français. ⟨tel-00421058⟩

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