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.
Monsieur Benjamin Schraen (Centre National de la Recherche Scientifique (CNRS) - Délégation Régionale Ile-de-France Sud)
The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.
CNRS Centre National de la Recherche Scientifique (CNRS) - Délégation Régionale Ile-de-France Sud
Help of the ANR 87,035 euros
Beginning and duration of the scientific project: December 2016 - 18 Months