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

We prove automorphy lifting results for certain essentially conjugate self-dual $p$-adic Galois representations $\rho$ over CM imaginary fields $F$, which satisfy in particular that $p$ splits in $F$, and that the restriction of $\rho$ on any decomposition group above $p$ is reducible with all the Jordan-H"older 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 $R=\mathbb{T}$-type result over the ${\mathrm{G}\mathrm{L}}_2({\mathbb{Q}}_p)$-ordinary families considered by Breuil-Ding.