We introduce a set of invariant functions on the moduli of Fontaine—Laffaille modules and prove that they separate points on the moduli in a suitable sense. Consequently, we prove the following local-global compatibility result for suitable global set up and under standard Kisin-Taylor-Wiles conditions: the Hecke eigenspace attached to a modular mod p global Galois representation determines its restriction at a place unramified over p, if the restriction is Fontaine-Laffaille and has a generic semisimplification. The genericity assumption is mild and explicit. This is a joint work with D. Le, B.V.Le Hung, S. Morra and C. Park.

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