p-adic Langlands and the Emerton-Gee stack

ongoing

Connections between automorphic forms and p-adic Galois representations are at the heart of the Langlands program and are the source of many of the most important advances in number theory. The most powerful technique for proving these connections is the use of automorphy lifting theorems. These theorems are well established in the two dimensional case, but are much weaker in higher dimensions, due to a lack of understanding of the corresponding Galois deformation rings. I propose to use a completely new way of studying p-adic Galois representations, which is known as the Emerton–Gee stack. This opens up a new horizon, because it will allow me to use global geometric techniques, rather than being limited to studying infinitesimal neighbourhoods as in all previous work over the last 30 years. I intend to completely prove the Breuil–Mézard conjecture, which is a major open problem, and implies automorphy lifting theorems for p-adic representations with optimal local conditions at p. This will put the higher-dimensional setting on an equal footing with the 2-dimensional case, opening up a new frontier. These theorems in turn have applications to problems such as the modularity of abelian surfaces, which is at the cutting edge of the Langlands program. I will completely resolve the weight part of Serre’s conjecture in arbitrary dimension; it is currently unknown in any dimension higher than 2. I also propose to use the Emerton–Gee stack to prove a geometrization of the p-adic Langlands correspondence, and to explore generalizations of the correspondence, going beyond the frontier reached 10 years ago, of 2-dimensional representations over the p-adic numbers. Finally, I will investigate a “prismatic” version of the Emerton–Gee stacks, and new connections between the p-adic Langlands correspondence and the global Langlands correspondence for function fields.