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hexagon:2611.00096v1

math.NT — Number Theory

Fully ordinary subfamilies and the Fontaine--Mazur conjecture in the residually reducible mixed case

Contributed by Luis Dieulefait

Let FF be a totally real field, abelian over $\Q$, in which an odd prime pp 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 pp, with reducible residual representation. We prove that ρ\rho is modular in the mixed case, in which ρ\rho is reducible at some places above pp and irreducible at the others, provided that the ratio of the two residual characters extends to $G_\Q$ and its restriction to $G_{\Q_p}$ is neither 11 nor ω±1\omega^{\pm1}. No assumption is made on the splitness of ρ\rho at the reducible places, on potential modularity, or on the local behaviour of ρ\rho away from pp. A suitable irreducible component through the point of ρ\rho in a partially ordinary deformation space contains a fully ordinary subfamily of maximal dimension. A classical point of this subfamily is a regular point of the deformation space, by the vanishing of adjoint Selmer groups, and a dimension estimate for a partially ordinary Hecke algebra then identifies the component of ρ\rho with a component of the Hecke algebra.

Primarily AI-generated textHuman understanding: some partsGalois representationsmodular forms
Nov 04, 2026PDF ↗View work →
hexagon:2611.00067v1

math.NT — Number Theory

On the Fontaine--Mazur conjecture for 22-adic representations

Contributed by Luis Dieulefait

Let FF be a totally real field, abelian over $\Q$, in which 22 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 22, whose residual representation is reducible with distinct characters at the places above 22. We prove that ρ\rho is modular in the following cases: if ρ\rho is non-split at every place above 22 at which it is reducible, which includes the case where ρ\rho is irreducible at all places above 22, and, without any condition on the splitness, if ρ\rho is of Steinberg type at some finite place not dividing 22, using an analogue at 22, which we prove, of Pan's finiteness theorem for fully ordinary deformation rings. Both results include the mixed case, in which ρ\rho is reducible at some places above 22 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 22. For $F=\Q(\sqrt q)$ with q≡1(mod8)q\equiv1\pmod8 prime, we remove the condition at 22 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.

Primarily AI-generated textHuman understanding: some partsGalois representationsmodular forms
Nov 04, 2026PDF ↗View work →