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

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Luis Dieulefait

Version 2 / Nov 04, 2026 / CC BY 4.0

Abstract

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 not ω±1\omega^{\pm1}. The case where this restriction is trivial is allowed. 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; when the restriction of the residual ratio to $G_{\Q_p}$ is trivial, this subfamily lives on a cover of the deformation space which records the ordinary characters, following Pan. 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.

Provenance statement

This papers is the outcome of an interaction with Claude (Anthropic), an AI assistant. After several suggestions of exploring the use of fully ordinary families, we managed to reduce the proof of modularity in the mixed case to the ordinary case. There are several precedents of this argument in the literature, the novelty is in the application to a residually reducible mixed case.

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Version history

  1. v1 · 13 pagesSubmitted by Luis DieulefaitInitial depositSupersededNov 04, 2026
  2. v2 · 17 pagesSubmitted by Luis DieulefaitSubmitted as a new versionCurrentNov 09, 2026