Stefano Morra: Looking for a mod p local Langlands correspondence

The mod p local Langlands program aims at a parametrization of smooth mod p representations of a p-adic GLn in terms of mod p continuous n-dimensional Galois representations of the p-adic field.
This was realized in the case of GL2(Qp) (and continuous 2-dimensional representations of Gal_{Qp}) by work of Colmez, following work of Breuil, and its compatibility with the mod p cohomology of (a tower above p of) modular curves was established by Emerton, all in the early 2000.
The situation for GL2 over an extension of Qp, even an unramified extension, remains unclear, even though many partial results show that such a local correspondence should exist. In particular, one of the main questions is whether the mod p Hecke eigenspaces of a tower of Shimura curves at p are "purely local", i.e. only depend on the underlying local Galois representations and not on the global context. Indeed, such a locality would give strong evidence for the existence of a mod p local Langlands correspondence outside of GL2(Qp) compatible with global constructions.

In this talk I will report on a joint work in progress, where we aim at constructing natural candidates for such a correspondence using tools from perfectoid geometry.

This is joint work in progress with C. Breuil, F. Herzig, Y. Hu, K. Koziol, B. Schraen and S-W. Shin.