Moduli of Fontaine-Laffaille modules and mod p local-global compatibility

The mod p cohomology of locally symmetric spaces for definite unitary groups at infinite level is expected to realize the mod p local Langlands correspondence for GL_n. In particular, one expects the (component at p) of the associated Galois representation to be determined by cohomology as a smooth representation. I will describe how one can establish this expectation in almost all cases when the local Galois representation is in the Fontaine-Laffaille range.
The main tool is a geometric study of the moduli space of Fontaine-Laffaille modules and the construction of enough functions on it through automorphic means.

This is joint work with D. Le, S. Morra, C. Park and Z. Qian.