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Unrestricted pro-modularity at the prime two
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Abstract
Every continuous, odd, absolutely irreducible two-dimensional 2-adic representation of $G_{\mathbb Q}$ that is unramified outside finitely many finite primes occurs in the full completed Hecke algebra at some odd tame level. The level may contain auxiliary tame primes, and scalar and reducible residual representations are included. This is a completed-Hecke occurrence result, with no de Rham hypothesis; it does not assert classical modularity.
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Possibly relevant literature (found by the agent, not checked)
- Przemyslaw Chojecki, Weak local-global compatibility and ordinary representations (arXiv:1306.3496, 2013). Proposes a general formalism for proving the "pro-modular Fontaine–Mazur conjecture" and checks it in the ordinary case, the same kind of occurrence statement as Theorem 1.1.
- Xinyao Zhang, On the pro-modularity in the residually reducible case for some totally real fields (arXiv:2411.18661, v1 Nov 2024, v5 7 October 2026). Proves componentwise pro-modularity at odd primes over abelian totally real fields split at the coefficient prime, following Skinner–Wiles and Pan (precedents listed in the introduction), giving a comparison for this paper's potential pro-modularity step, with additional local and degree hypotheses.
- Xinyao Zhang, Modular points and dimensions of Eisenstein deformation spaces (arXiv:2512.21249, v1 Dec 2025, v3 7 October 2026). Proves that components have dimension 4 and modular points are Zariski dense for Eisenstein deformation spaces at odd primes, implying Emerton's equidimensionality conjecture there; this is the odd-prime counterpart of the dimension input taken from the companion paper, and its use of modular points, connectedness and adjoint Selmer vanishing relates to this paper's propagation of Hecke support.
Digests and expositions
- Ying Xie, What the Machine Invented: Mining New Mathematical Instruments from OpenAI's October 2026 Release of AI-Generated Mathematics (arXiv:2610.17090, 2026-10-12). The family 010 card (Section 9.1.3) is centred on the Fontaine–Mazur companion paper and describes this pro-modularity paper as "doubly dependent", i.e. resting on both companion manuscripts.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 010
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Unrestricted-pro-modularity-at-the-prime-two-October-6-2026,
author = {{OpenAI}},
title = {{Unrestricted pro-modularity at the prime two}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-6-2026/two-adic-promodularity.pdf}{OAI:Unrestricted-pro-modularity-at-the-prime-two-October-6-2026}},
year = {2026}
}Versions
- Earlier edition: October 4, 2026
This version updates citations to revised manuscripts in this collection; see the previous version.
Family 010: Unrestricted pro-modularity at the prime two
Every continuous odd absolutely irreducible two-dimensional 2-adic representation of $G_{\mathbb Q}$ unramified outside finitely many primes occurs in a completed Hecke algebra at some odd tame level. Also proves classical modularity up to Tate twist for irreducible odd representations with these finiteness conditions that are de Rham at 2 with distinct Hodge–Tate weights, resolving the dyadic Fontaine–Mazur case without residual restrictions.
Other papers in this family:
- The Dimension of the Two-Adic Hecke Algebra at Odd Level · October 5, 2026
- Fontaine–Mazur modularity at the prime 2 · October 6, 2026