PerCoLaTor Project

 

With the $p$-adic Langlands program, we are now in the similar situation as we were in the 80's with the classical Langlands program, with an increasing number of questions and projects. One has to define the right objects to consider and to connect them with the older, more familiar theory. The technical and theoretical difficulties are huge and each question is a real challenge, appealing to a large variety of expertises.

Thanks to works of Berger, Breuil, Colmez , Emerton, Kisin and Paskunas, the theory of $p$-adic representations of $GL_2(\Qm_p)$, in the context of $p$-adic Langlands program, is rather well understood. Going beyond this case, by considering for example a finite extension of $\Qm_p$, or more challenging reductive groups other than $GL_2$, seems to call for completely new techniques and ideas. Let us explain some of the fields which are expected to interact.

- From a purely point of view, the first goal is to generalize the now classical theory of $(\varphi,\Gamma)$-modules, either for the simplest groups of rank $1$ such as $SL_{2}(F)$ or $U(2,1)(E/F)$ for $E/F$ a quadratic extension, or for the theory of overconvergent $(\varphi,\Gamma)$-modules of Lubin-Tate type in several variables.

- From a geometric viewpoint, one idea is to use the characterisation by Scholze of the $p$-adic Langlands correspondence in terms of some kind of a trace formula over the stack of $p$-divisible groups with fixed height. Passing from this characterisation to Colmez's construction seems to be a good starting point toward the generalization of the correspondence to other groups.

- Globally, the Hodge-Tate map from the infinite level Harris-Taylor-Kottwitz Shimura varieties to projective space, seen as an adic space. Via the notion of Drinfeld sheaf, we should be able to obtain a new realization of the $l$-adic Langlands correspondence.

Going beyond the case of $GL_2(\Qm_p)$ in the $p$-adic Langlands program is a highly challenging project which, as was the case with the classical Langlands program for many years, should interact with questions around the cohomology of Shimura varieties. One of these questions concerns the eigenvarieties and modularity theorems for Galois representations.

The heart of the Langlands program deals with the construction of global Galois representation attached to algebraic automorphic representations of reductive groups. For a PEL reductive group and for cohomological automorphic representations, huge advances have been made. The next step is the construction of Galois representations for some algebraic non cohomological Galois representations. One possibility is to investigate the coherent cohomology of Shimura varieties.

Recent works of P. Scholze on perfectoids, infinte level Shimura varieties and finiteness of de Rham cohomology on adic spaces have generated a lot of interest among the community. Everyone is quite convinced that these tools announce a new revolution in the Langlands program. The area of expertise of the members of our project covers the vast field of knowledge needed to make major breakthroughs and to carry with us a great number of students and young researchers.