The Breuil-Mezard conjecture for non-scalar split residual representations

Abstract

We prove the Breuil-Mezard conjecture for split non-scalar residual representations of Gal(Qp/Qp) by local methods. Combined with the cases previously proved in [18] and [24], this completes the proof of the conjecture (when p>3). As a consequence, the local restriction in the proof of the Fontaine-Mazur conjecture in [18] is removed.