RésuméSoit p un nombre premier et F un corps local non archimédien de caractéristique p. Dans cet article, à une représentation lisse irréductible de GL2(F) sur $\smash{\bar{\mathbb{F}}_p}$ avec caractère central, nous associons un diagramme qui détermine la représentation de départ à isomorphisme près. Nous le déterminons également dans certains cas.

Abstract. We prove the Breuil-Mézard conjecture for split non-scalar residual representations of Gal(Q p /Qp) by local methods. Combined with the cases previously proved in [20] and [26], this completes the proof of the conjecture (when p ≥ 5). As a consequence, the local restriction in the proof of the Fontaine-Mazur conjecture in [20] is removed. Contents Notation• p ≥ 5 is a prime number. The p-adic valuation is normalized as v p (p) = 1.• E/Q p is a sufficiently large finite extension with ring of integers O, a (fixed) uniformizer ̟, and residue field F. Its subring of Witt vectors is denoted by W (F).• For a number field F , the completion at a place v is written as F v , for which we fix a uniformizer denoted by ̟ v . • ½ : G Qp → F × p is the trivial character. We also let ½ denote other trivial representations, if no confusion arises.• Normalize the local class field map Q × p → G ab Qp so that uniformizers correspond to geometric Frobenii. Then a character of G Qp will also be regarded as a character of Q × p .• For a ring R, m-SpecR denotes the set of maximal ideals.• For R a noetherian ring and M a finite R-module of dimension at most d, let ℓ Rp (M p ) denote the length of the R p -module M p , and letWhen the context is clear, we simply denote it by Z(M ).• For R a noetherian local ring with maximal ideal m and M a finite Rmodule, and for an m-primary ideal q of R, let e q (R, M ) denote the HilbertSamuel multiplicity of M with respect to q. We abbreviate e m (R, M ) = e(R, M ) and e q (R, R) = e q (R).• For r ≥ 0, we let Sym r E 2 (resp. Sym r F 2 ) be the usual symmetric power representation of GL 2 (Z p ) (resp. of GL 2 (F p ), but viewed as a representation of GL 2 (Z p )).

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