{"id":"hexagon/2611.00096","pdf":"https://hexagonmath.org/pdf/2611.00096v1","url":"https://hexagonmath.org/abs/2611.00096v1","tags":["msc:11F80","Galois representations","modular forms"],"score":0,"title":"Fully ordinary subfamilies and the Fontaine--Mazur conjecture in the residually reducible mixed case","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Luis Dieulefait"],"reviews":0,"subject":"math.NT","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-11-04","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}