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

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

Contributed by Luis Dieulefait ↗

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

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.

Provenance statement

This work was produced with the assistace of Claude (Anthropic), an AI assistant. It gives a generalization to totally real number fields in which 2 is totally split of a result obtained by OpenAI on the Fontaine-Mazur conjecture for residually reducible 2-adic Galois representations. It is first proved under the assumption that the the representation is non-split at ordinary places, and then via a combination with ideas from a previous paper, using fully ordinary subfamilies and results of Hida and Zhao on the local indecomposability of ordinary representations attached to non-CM abelian varieties, it is extended to the split ordinary case assuming the existence of a Steinberg place.

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Version history

  1. v1Submitted by Luis DieulefaitInitial depositCurrentNov 03, 2026