{"id":"hexagon/2611.00067","pdf":"https://hexagonmath.org/pdf/2611.00067v1","url":"https://hexagonmath.org/abs/2611.00067v1","tags":["msc:11F80","Galois representations","modular forms"],"score":0,"title":"On the Fontaine--Mazur conjecture for $2$-adic representations","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 $2$ 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…","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withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}