On the Fontaine–Mazur conjecture for $2$-adic representations

Let $F$ be a totally real field, abelian over \Q, in which $2$ splits completely, and let \rho:G_F\to\GL_2(\overline\Q_2) be an odd irreducible representation, de Rham with distinct Hodge–Tate weights at the places above $2$, whose residual representation is reducible with distinct characters at the places above $2$. We prove that $\rho$ is modular in the following cases: if $\rho$ is non-split at every place above $2$ at which it is reducible, which includes the case where $\rho$ is irreducible at all places above $2$, and, without any condition on the splitness, if $\rho$ is of Steinberg type at some finite place not dividing $2$, using an analogue at $2$, which we prove, of Pan's finiteness theorem for fully ordinary deformation rings. Both results include the mixed case, in which $\rho$ is reducible at some places above $2$ and irreducible at others. The pro-modularity input is obtained by extending to these fields the method of \cite{OAI}, which treats representations of G_\Q; this is where we use that the residual characters are distinct at $2$. For F=\Q(\sqrt q) with $q\equiv1\pmod8$ prime, we remove the condition at $2$ from the first result in most cases; for F=\Q(\sqrt{17}) this includes every case where the residual semisimplification is scalar. The first result also holds when the residual representation is irreducible with solvable image, where residual modularity follows from an odd Artin lift. For the second result we combine the method of \cite{Die26b} with Thorne's potential modularity theorem for ordinary representations of weight two and the local indecomposability theorem of Hida and Zhao.