Diagrammes de Diamond et (φ, Γ)-modules

Abstract

Let ρ be a 2-dimensional continuous semi-simple generic representation of Gal(ℚp/ℚp) over Fp. The modulo p Langlands correspondence for GL2(ℚp) defined in [5], as realized in [9], can be reformulated as a quite simple recipee giving back the (φ, Γ)-module of the dual of ρ starting from the “Diamond diagram” associated to ρ. Let F be a finite unramified extension of ℚp and ρ a 2-dimensional continuous semi-simple generic representation of Gal(ℚp/F) over Fp. When one formally extends this recipee to the Diamond diagrams associated to ρ in [6], we show that one essentially finds the (φ, Γ)-module of the tensor induction from F to ℚp of the dual of ρ.