Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 010-03. Fontaine–Mazur modularity at the prime 2
Manuscript map · lines 2635–2644
OpenAI@2026-10-07#rem:oai-p-010-03
Authors: OpenAI. Family 010.
Title: Fontaine–Mazur modularity at the prime 2.
Abstract: We prove that every continuous, irreducible, odd two-dimensional 2-adic representation of $G_{{}mathbb Q}$, unramified outside finitely many primes and de Rham at 2 with distinct Hodge–Tate weights, is modular up to Tate twist. This resolves the odd, regular two-dimensional Fontaine–Mazur conjecture over ℚ at 2, including all residual representations.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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