p-adic Langlands Programme and p-adic Banach Representations – ProBaRe

This project aims to investigate the p-adic Langlands Program, where p is a prime number. The ultimate goal of this program is to establish a correspondence between certain p-adic Banach space representations of some p-adic reductive groups and certain p-adic representations of the absolute Galois group of a p-adic local field. It is consequently linked to p-adic Hodge theory. Especially, this correspondence should describe integral p-adic properties of automorphic forms and Galois representations as for example p-adic interpolation. It began under the impulsion of Christophe Breuil and the case of the group of 2 by 2 invertible matrices in p-adic numbers was completely elucidated between years 2000 and 2010. It was a great achievement since it permitted to prove important conjectures on 2-dimensional Galois representations as the Fontaine-Mazur and Breuil-M\'ezard conjectures. The extension of this program to other groups is consequently a big challenge in Number Theory and the key toward a full understanding of p-adic properties of Galois representations and automorphic forms. However, this extension encounters very numerous difficulties : even the formulation of the expected correspondence is not clear. This project has for purpose to investigate this generalization with a focus on the methods of representation theory. The main objectifve of this project is to understand the structure and properties of locally analytic representations associated to p-adic Banach space representations going beyond the case of finite slope representations. A very important tool in this study is the action of the Harish-Chandra center on these representations together with their canonical dimension.