Cohomology of locally symmetric spaces – COLOSS

The project is organized in three main axes:

1 Cohomologies of Shimura Varieties
2. p-adic Langlands program
3. theories of p-adic cohomologies

Relatively to the initial aims of the project, in all the envisioned directions, most of the aims are either already reached or deep progress have been obtained.

As environne the main obstacles focus in the p-adic Langlands program.

33 published articles
38 preprints submitted for publication

Understanding the representations of the Galois group of Q, their L-functions and the arithmetical informations hidden in the values of these L-functions at integers (conjectures of Beilinson and Bloch-Kato) is one of the main problems of modern Number Theory. The goal of the Langlands program is to establish a correspondence, preserving L-functions, between such representations and certain representations of algebraic groups (or rather their adelic points).
Alongside the classical Langlands program, a p-adic avatar has appeared and has become increasingly important following Wiles' proof of Fermat Last Theorem.

The cohomology of locally symmetric spaces (modular curves, Shimura varieties, Rapoport-Zink spaces, moduli spaces of Shtukas, etc.) encodes the Langlands correspondence in a number of cases. The project COLOSS is devoted to the study of the cohomology of these spaces and its applications to the Langlands program and its p-adic avatar. It combines foundational work on aspects of p-adic geometry and cohomology of p-adic varieties, and arithmetic applications of these theories.