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.