On the Fontaine--Mazur conjecture for $2$-adic representations
Source: hexagonmath.org/abs/2611.00067v1
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…
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