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The odd regular two-dimensional Fontaine-Mazur conjecture over Q at p = 2

The Fontaine-Mazur conjecture (1995) predicts that irreducible pp-adic Galois representations unramified outside finitely many primes and de Rham at pp come from algebraic geometry. For odd two-dimensional representations of GQG_{\mathbb Q} with distinct Hodge-Tate weights it takes the form of modularity up to Tate twist. Building on Kisin and Emerton, Pan (2022) and Zhang (2025) established this for every odd prime, and at p=2p=2 Tung removed local restrictions only under a nonsolvable residual image. Let r:GQ→GL2(Q‾2)r:G_{\mathbb Q}\to\mathrm{GL}_2(\overline{\mathbb Q}_2) be continuous, irreducible, odd, unramified outside finitely many primes, and de Rham at 22 with distinct Hodge-Tate weights. Is rr a Tate twist of the Galois representation of a classical cuspidal eigenform, with no condition on the residual representation?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Galois representations; modularity lifting at p = 2
Posed by
Jean-Marc Fontaine and Barry Mazur, Geometric Galois representations (1995); the p = 2 two-dimensional case is the part left open after Pan and Zhang, as the manuscript explains
Year posed
1995
Years open
31y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
45 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: every continuous, irreducible, odd r:GQ→GL2(Q‾2)r:G_{\mathbb Q}\to\mathrm{GL}_2(\overline{\mathbb Q}_2) unramified outside finitely many primes and de Rham at 22 with distinct Hodge-Tate weights is a Tate twist of the representation of a classical cuspidal eigenform; no residual hypothesis. With the odd-prime results this gives the regular odd two-dimensional case over Q\mathbb Q at every prime. It does not address equal Hodge-Tate weights, even representations, higher dimension or other number fields, so the general Fontaine-Mazur conjecture stays open. Companions prove 2-adic pro-modularity without a de Rham hypothesis (October 4) and Emerton's dimension conjecture at p=2p=2 (October 5), entered separately.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The manuscript says this theorem supplies the weight-two modularity input to the release's Hilbert's tenth problem over Q manuscript. It cites several 2025-2026 works (Thorne, Zhang, Emerton-Gee-Pan-Zhu, Pan) as inputs.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the September 23 manuscript was read against the odd regular two-dimensional case of the Fontaine-Mazur conjecture as the paper formulates it; it claims modularity up to Tate twist for every such 2-adic representation, including scalar and reducible residual representations. The proof was not refereed. It depends on recent external results whose status was not checked here, notably Thorne's modularity of ordinary weight-{0,1} representations (cited as 2026), Zhang (2025), Pan (2022), Paskunas-Tung (2021) and the release's own companion results. No Lean formalization exists for this family.

Sources

Changelog1 change

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