# 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.
Keywords :
Document type :
Theses

Cited literature [21 references]

https://tel.archives-ouvertes.fr/tel-01624238
Contributor : Abes Star <>
Submitted on : Thursday, October 26, 2017 - 10:05:38 AM
Last modification on : Sunday, March 31, 2019 - 1:28:59 AM
Document(s) archivé(s) le : Saturday, January 27, 2018 - 12:34:37 PM

### File

2017PA066111.pdf
Version validated by the jury (STAR)

### Identifiers

• HAL Id : tel-01624238, version 1

### Citation

Ildar Gaisin. Contributions to the Langlands program. Number Theory [math.NT]. Université Pierre et Marie Curie - Paris VI, 2017. English. ⟨NNT : 2017PA066111⟩. ⟨tel-01624238⟩

Record views