Some instances of modularity for residually reducible representations in the mixed case

Abstract

Let $F$ be a totally real field, abelian over $\Q$, in which an odd prime $p$ splits completely, and let $ρ:G_F\to\GL_2(\overline\Q_p)$ be an odd irreducible representation which is de Rham with distinct Hodge--Tate weights at the places above $p$ and whose residual representation is reducible. We consider the \emph{mixed} case of the Fontaine--Mazur conjecture, in which $ρ$ is reducible at some places above $p$ and irreducible at the others. This case is not covered by the work of Pan and Zhang. We prove some instances of modularity in this situation. Our main result is that $ρ$ is modular if it is non-split at every place above $p$ where it is reducible, by combining Zhang's pro-modularity theorem with a classicality criterion based on Emerton's ordinary parts. Together with a theorem of Zhao on local indecomposability, this gives the modularity of the $λ$-adic representations attached to abelian varieties of $\GL_2$-type in the mixed case, and, using a result of Snowden, of potentially modular representations of parallel weight two under a mild additional condition. It also gives the modularity of representations whose local type at every place above $p$ is of discrete series type, and, through a member of the compatible system with interchanged Hodge-Tate weights, of potentially modular representations of weight $(2,k)$ over real quadratic fields with Steinberg type at one place above $p$, under conditions on that member.