Fully ordinary subfamilies and the Fontaine--Mazur conjecture in the residually reducible mixed case
Source: hexagonmath.org/abs/2611.00096v1
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…
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