p-adic Hodge theory and beyond – ThéHopaD

This project lies in the area of arithmetic and algebraic geometry. It aims at advancing both arithmetic and geometric aspects of $p$-adic Hodge theory by focusing on two of the deepest and most challenging questions : the $p$-adic Langlands programme for the arithmetic side and the $p$-adic Simpson correspondence for the geometric side.

The $p$-adic Langlands programme was born at the turn of the 21th century with the aim of understanding how the $p$-adic Hodge theory on the Galois representations side could be understood on the $GL_n$ representations side (it is indeed essentially absent from the classical Langlands programme). More precisely, the point is to understand the $p$-adic representations of $GL_n$ of a finite extension of $Q_p$ that are realized on the completed $p$-adic étale cohomology of a tower of Shimura varieties. One expects these $p$-adic representations of $GL_n$ at least to "contain" the Fontaine's theory of the $p$-adic local Galois representations associated to those algebraic automorphic representations that are related to the relevant Shimura varieties. This $p$-adic programme is still in a first development phase (it is essentially complete only for $GL_2(Q_p)$), yet it has already produced deep applications to modularity theorems. Our aim is to develop it and generalize it, in particular to state (possibly conjectural) results that outweigh the current ones.

The $p$-adic Simpson correspondence, recently initiated by Faltings, aims at describing all $p$-adic representations of the fundamental group of a (proper) smooth variety over a $p$-adic field in terms of linear algebra (Higgs modules). While the theory is still in an early development stage, it already appears as an important theoretical headway, with many deep potential applications both in arithmetics and in geometry ($p$-adic Galois representations, the cohomology of Shimura varieties, p-adic uniformization,...). Our project aims at developing the $p$-adic correspondence on the model of the complex Simpson theory from which it is inspired. Our main goals are in particular to extend its range of applications and to describe as precisely as possible its image (which corresponds to the difficult part of Simpson's results in the complex case).