Contributions to the Langlands program

Abstract : This thesis deals with two problems within the Langlands program. For the first problem, in the situation of $\GL_2$ and a non-minuscule cocharacter, we provide a counter-example (under some natural assumptions) to the Rapoport-Zink conjecture, communicated to us by Laurent Fargues.The second problem deals with a result in the $p$-adic Langlands program. Let $A$ be a $\qp$-affinoid algebra, in the sense of Tate. We develop a theory of locally convex $A$-modules parallel to the treatment in the case of a field by Schneider and Teitelbaum. We prove that there is an integration map linking a category of locally analytic representations in $A$-modules and separately continuous relative distribution modules. There is a suitable theory of locally analytic cohomology for these objects and a version of Shapiro's Lemma. In the case of a field this has been substantially developed by Kohlhaase. As an application we propose a $p$-adic Langlands correspondence in families: For a regular trianguline $(\varphi,\Gamma)$-module of dimension 2 over the relative Robba ring $\Robba_A$ we construct a locally analytic $\GL_2(\qp)$-representation in $A$-modules. This is joint work with Joaquin Rodrigues.
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Ildar Gaisin. Contributions to the Langlands program. Number Theory [math.NT]. Université Pierre et Marie Curie - Paris VI, 2017. English. ⟨NNT : 2017PA066111⟩. ⟨tel-01624238⟩

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