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For certain mod p Galois representations ρ̄, arising from modular forms on definite unitary groups in two variables, we express the ρ̄-part of completed cohomology Ĥ ρ̄ (away from Σ = Σp ∪ Σ0) as a tensor product Πp⊗ΠΣ0 . Here Πp is attached to the universal deformation ρ via the p-adic local Langlands correspondence for GL2(Qp), and ΠΣ0 is given by the local Langlands correspondence in families, of Emerton and Helm. 1 2

Strong local-global compatibility in the $p$-adic Langlands program for $U(2)$