Pro-modularity at p = 2: odd absolutely irreducible 2-adic representations occur in the completed Hecke algebra (auxiliary tame level allowed)

Let $\mathbb T_p(N)$ be the full varying-weight $p$ -adic Hecke algebra of tame level $N$ . Emerton (2011, Conjecture 2.12) records the expectation that every continuous odd two-dimensional $p$ -adic representation of $G_{\mathbb Q}$ unramified outside finitely many primes occurs in such an algebra, with no Hodge-theoretic condition at $p$ and with tame level prescribed by the ramification. Earlier pro-modularity theorems (Skinner-Wiles, Emerton, Pan) need $p>2$ or residual restrictions. For $p=2$ , does every continuous, odd, absolutely irreducible $r:G_{\mathbb Q}\to\mathrm{GL}_2(E)$ unramified outside finitely many primes occur in $\mathbb T_2(N)$ for some odd $N$ ?

Result: Proved(see note)
Status: Variant only
AI contribution: AI-discovered
Method: Argument
Field: Completed Hecke algebras; p-adic Langlands for GL2 over Q
Posed by: Matthew Emerton, p-adic families of modular forms, Seminaire Bourbaki 2009/2010, Asterisque 339 (2011), Conjecture 2.12, as the manuscript cites
Year posed: 2011
Years open: 15y
Solved: 2026-10-04
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 22 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: for every continuous, odd, absolutely irreducible $r:G_{\mathbb Q}\to\mathrm{GL}_2(E)$ , $E/\mathbb Q_2$ finite, unramified outside finitely many primes, there are an odd $N$ divisible by the odd ramified primes and a continuous $\lambda:\mathbb T_2(N)\to\mathcal O_E$ with $\lambda(T_\ell)=\mathrm{tr}\,r(\mathrm{Frob}_\ell)$ and $\lambda(\ell S_\ell)=\det r(\mathrm{Frob}_\ell)$ for $\ell\nmid2N$ . Scalar and reducible residual representations are allowed; no de Rham hypothesis. It is an occurrence result and does not assert classical modularity; the tame level may contain auxiliary primes, unlike Emerton's formulation.

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 proof uses the release's Fontaine-Mazur-at-2 theorem and its Hecke dimension theorem as inputs.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the October 4 manuscript was read against Emerton's Conjecture 2.12 as the paper describes it. The paper itself says Emerton's formulation prescribes the allowed tame primes, while the theorem allows auxiliary primes in the level; hence Variant. The proof was not refereed and depends on the two companion manuscripts and on Newton-Thorne Selmer vanishing. No Lean formalization exists for this family.

Pro-modularity at p = 2: odd absolutely irreducible 2-adic representations occur in the completed Hecke algebra (auxiliary tame level allowed)

What was actually shown

What the AI did

Verification

Sources

Discussion