Moduli of Fontaine--Laffaille representations and a mod-p local-global compatibility result

Abstract

Let F/F + be a CM field and let ve be a finite unramified place of F above the prime p. Let r : Gal(Q/F) → GLn(Fp) be a continuous representation which we assume to be modular for a unitary group over F + which is compact at all real places. We prove, under Taylor–Wiles hypotheses, that the smooth GLn(Fve)-action on the corresponding Hecke isotypical part of the modp cohomology with infinite level above ve|F + determines r|Gal(Qp/Fve), when this latter restriction is Fontaine–Laffaille and has a suitably generic semisimplification.