Fully ordinary subfamilies and the Fontaine–Mazur conjecture in the residually reducible mixed case

Let $F$ be a totally real field, abelian over \Q, in which an odd prime $p$ splits completely, and let \rho:G_F\to\GL_2(\overline\Q_p) be a continuous, absolutely irreducible, totally odd representation, unramified outside finitely many places and de Rham with distinct Hodge–Tate weights at the places above $p$, with reducible residual representation. We prove that $\rho$ is modular in the mixed case, in which $\rho$ is reducible at some places above $p$ and irreducible at the others, provided that the ratio of the two residual characters extends to G_\Q and its restriction to G_{\Q_p} is neither $1$ nor $\omega^{\pm1}$. No assumption is made on the splitness of $\rho$ at the reducible places, on potential modularity, or on the local behaviour of $\rho$ away from $p$. A suitable irreducible component through the point of $\rho$ in a partially ordinary deformation space contains a fully ordinary subfamily of maximal dimension. A classical point of this subfamily is a regular point of the deformation space, by the vanishing of adjoint Selmer groups, and a dimension estimate for a partially ordinary Hecke algebra then identifies the component of $\rho$ with a component of the Hecke algebra.