Emerton's dimension conjecture for the full Hecke algebra at p = 2 and odd level

For a prime $p$ and level $N$ prime to $p$ , let $\mathbb T_p(N)$ be the inverse limit over $k$ of the $\mathbb Z_p$ -Hecke algebras generated by $T_\ell$ and $\ell S_\ell$ ( $\ell\nmid pN$ ) acting on all modular forms of weights $1,\dots,k$ and level $\Gamma_1(N)$ , Eisenstein forms included. Emerton, in his 2011 Bourbaki account of $p$ -adic families (Conjecture 2.9), conjectured that every irreducible component of $\mathrm{Spec}\,\mathbb T_p(N)$ has Krull dimension four, and proved the lower bound (Corollary 2.28). For $p=2$ and every odd $N$ , does every irreducible component have dimension exactly four?

Result: Proved(see note)
Status: Partial result
AI contribution: AI-discovered
Method: Argument
Field: p-adic families of modular forms; Hecke algebras
Posed by: Matthew Emerton, p-adic families of modular forms [after Hida, Coleman, and Mazur], Seminaire Bourbaki 2009/2010, Asterisque 339 (2011), Conjecture 2.9
Year posed: 2011
Years open: 15y
Solved: 2026-10-05
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 20 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: for every odd $N$ , every irreducible component of $\mathrm{Spec}\,\mathbb T_2(N)$ has Krull dimension exactly $4$ . The upper bound comes from a cuspidal point of weight at least 3 on each component, a tangent bound of 3 from Newton-Thorne Selmer vanishing, and one arithmetic dimension. Only the case $p=2$ is treated; the paper does not claim the conjecture for odd $p$ .

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.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the October 5 manuscript was read against Emerton's Conjecture 2.9 as the paper cites it; it claims dimension exactly four for every component of the full 2-adic Hecke algebra of level $\Gamma_1(N)$ , $N$ odd, residually reducible and scalar components included. The proof was not refereed. It uses Emerton's lower bound, Chenevier's determinant laws and the Newton-Thorne adjoint Selmer vanishing theorem as external inputs. No Lean formalization exists for this family.

Emerton's dimension conjecture for the full Hecke algebra at p = 2 and odd level

What was actually shown

What the AI did

Verification

Sources

Discussion