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

Let K be a finite extension of Qp. It is believed that one can attach a smooth Fp-representation of GLn(K) (or a packet of such representations) to a continuous Galois representation of Gal(K/Qp) with coefficients in GLn(Fp) in a natural way, that is called mod p Langlands program for GLn(K). This is known only for GL2(Qp): one of the main difficulties is that there is no classification of such smooth representations of GLn(K) unless K = Qp and n = 2. However, for a given continuous Galois representation R of Gal(K/Qp) with coefficients in GLn(Fp), one can define a smooth Fp-representation P of GLn(K) by a space of mod p automorphic forms on a compact unitary group, which is believed to be a candidate on the automorphic side corresponding to R for mod p Langlands correspondence in the spirit of Emerton. The structure of P is very mysterious as a representation of GLn(K), but it is conjectured that P determines R, which is called mod-p local-global compatibility. In this talk, we discuss a way to prove this conjecture in the case that R is Fontaine--Laffaille. More precisely, we prove that the tamely ramified part of R is determined by the Serre weights attached to R, and the wildly ramied part of R is obtained in terms of refined Hecke actions on P. This is based on a joint work with Daniel Le, Bao Le Hung, Stefano Morra, and Zicheng Qian