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.
Madame Wieslawa Niziol (Unité de mathématiques pures et appliquées de l'ENS de Lyon)
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.
UMPA/ENSL Unité de mathématiques pures et appliquées de l'ENS de Lyon
IMJ-PRG Institut de mathématiques de Jussieu - Paris Rive Gauche
LAGA - Université Paris 13 Laboratoire Analyse, Géométrie et Applications
Help of the ANR 255,096 euros
Beginning and duration of the scientific project: October 2019 - 48 Months