The Langlands program, initiated in the 1960s and successfully developed in the last 15 years, is a set of conjectures predicting a unification of number theory and of representation theory of groups: the Langlands correspondence provides a way to interpret results in number theory in terms of group theory, and vice versa. A decade ago, the question of a p-adic/mod p component of this program was raised, motivated by natural questions of p-adic arithmetic geometry. As of now, only very few cases are understood, but they have already had spectacular consequences such as the proof of most cases of the Fontaine-Mazur conjecture. Because of unexpected and poorly understood phenomena, statements of a general p-adic/mod p local Langlands conjecture remain elusive. The proposal aims to study the mod p representations of p-adic reductive groups in order to understand the right terms of a mod p Langlands correspondence. The PI's previous work has shown that studying mod p representations of Hecke algebras is a promising approach, because it suggests that a mod p Langlands correspondence does exist. The proposal outlines strategies to study families and complexes of mod p representations of p-adic reductive groups and their associated Hecke algebras. At the heart of this proposal is the wish to go beyond the (so far most investigated) focus on irreducible objects and give a homological approach to the representations. It goes along with exploring the possibility of a mod p principle of functoriality and shedding a geometric light on the potential mod p Langlands correspondence.
The p-adic/mod p Langlands program holds profound prospects for modern number theory with deep ramifications in arithmetic algebraic geometry. It is a new, fertile area involving tools and objects that were still completely abstruse a few years ago. The proposal aims to give a geometric incarnation to the objects that appear naturally in the mod p framework. The outcome of this work will most likely be naturally connected to other areas of mathematics influenced by the Langlands conjectures, such as the geometric Langlands program, and its links with geometric representation theory. The proposal describes strategies towards such connections.
