Practical p-adic Langlands – PPAL

The Langlands program has made great progress since the 2000s, notably with the proof of the local Langlands correspondence, the global Langlands correspondence for function fields, the fundamental lemma and in particular the p-adic local Langlands correspondence for GL(2,Qp), started by C. Breuil, which is a major achievement in number theory. One of the most striking applications of this correspondence is the proof (independently by M. Emerton and M. Kisin) of numerous cases of the Fontaine-Mazur conjecture, a modularity conjecture for certain 2-dimensional p-adic representations of the absolute Galois group of Q.

The so-called categorical program, along with the Fargues-Scholze conjecture, also recently emerged as a fundamental framework to formulate classical results and obtain new ones. Some generic cases of this conjecture have already been proved, but the heart of the conjecture is still the subject of active research and promises to have applications to the mod p Langlands program beyond GL(2,Qp).

Recent progress combines both ideas and techniques from different aspects of the Langlands program: local representation theory, mod p and automorphic representations, geometry and cohomology of Rapoport-Zink spaces as well as Shimura varieties, computation of deformation spaces, Emerton-Gee stacks and the geometric program inspired by the work of Drinfeld-Gaitsgory and their collaborators.

The goal of the Practical p-adic Langlands ANR project is to bring together a broad spectrum of mathematicians based in France, covering the expertise needed to make progress in these two intertwined directions, each bringing into play their own knowledge and ideas. The novelty of the project resides on the one hand in the diverse new areas of number theory and arithmetic geometry that will be included, and on the other hand on an emphasis on the use of computational methods to verify conjectures and suggest new directions.