$\mathrm{GL}_2(\mathbb{Q}_p)$-ordinary families and automorphy lifting

Abstract

We prove automorphy lifting results for certain essentially conjugate self-dual p-adic Galois representations ρ over CM imaginary fields F , which satisfy in particular that p splits in F , and that the restriction of ρ on any decomposition group above p is reducible with all the Jordan-Hölder factors of dimension at most 2. We also show some results on Breuil's locally analytic socle conjecture in certain non-trianguline case. The main results are obtained by establishing an R = T-type result over the GL 2 (Q p )-ordinary families considered in [7].1 Where we use the convention that the Hodge-Tate weight of the cyclotomic character is 1. 2 I.e. the eigenvalues (φ1, φ2, φ3) of the crystalline Frobeinus satisfy φiφ −1 j / ∈ {1, p} for i = j. 3 We need some more technical assumptions when p = 3, that we ignore in the introduction.