VibeMathedMath problems solved with AI
Sign in

Chai's conjecture on stabilizer-invariant ideals of the Lubin-Tate deformation ring

Let Γn\Gamma_n be the height-nn Honda formal group over F‾p\overline{\mathbb F}_p and E0=W(Fpn)[[u1,…,un−1]]E_0=W(\mathbb F_{p^n})[[u_1,\ldots,u_{n-1}]] its Lubin-Tate deformation ring, on which the Morava stabilizer group An×A_n^\times acts by changing the marking. The height ideals Ih=(p,u1,…,uh−1)I_h=(p,u_1,\ldots,u_{h-1}), 1≤h≤n1\le h\le n, cut out the loci where the deformed formal group has height at least hh; with 00 they are invariant primes. Chai (1996), studying the group action on the closed fiber of Lubin-Tate space, proposed that invariant irreducible formal loci should be height loci; Hovey and Strickland asked the same about invariant primes (Problem 16.8), and Barthel, Heard and Naumann made it precise as Chai's Hope for finite residue fields. For an open subgroup UU of the stabilizer, are 0,I1,…,In0,I_1,\ldots,I_n the only proper UU-stable primes of E0E_0, and 0,I1,…,In,E00,I_1,\ldots,I_n,E_0 the only UU-invariant radical ideals?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Arithmetic of formal groups; Lubin-Tate space
Posed by
Ching-Li Chai (Duke Math. J. 1996); formulated as Chai's Hope 4.1 by Barthel, Heard and Naumann (2022); compare Hovey-Strickland Problem 16.8
Year posed
1996
Years open
30y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
26 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for any open subgroup U⊆An×U\subseteq A_n^\times, the proper UU-stable primes of E0E_0 are exactly 0,I1,…,In0,I_1,\ldots,I_n and the UU-invariant radical ideals are exactly 0,I1,…,In,E00,I_1,\ldots,I_n,E_0; the radical ideals invariant under the full extended group GnG_n are the same list. This is Chai's Hope in the finite-residue-field formulation, at every prime and height, extended to open subgroups. Through Barthel-Heard-Naumann it yields the Hovey-Strickland conjecture (separate entry). It is stated for the Honda formal group over finite-field coefficients and does not treat other residue fields or the full scope of Chai's closed-fiber questions.

What the AI did

The release README says all results in the release were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The invariant-ideal classification is the main step of the single manuscript in this family, which then deduces the Hovey-Strickland conjecture (a separate catalog entry).

Verification

No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against Chai's question and the Barthel-Heard-Naumann formulation (Hope 4.1 and Remark 4.2) as the manuscript cites them. No Lean formalization exists for this family. The manuscript says it proves the standard finite-residue-field formulation and extends it from the full stabilizer to every open subgroup; the core is a local orbit-density theorem at characteristic-p points of height between 1 and n-1, using perfectoid methods, Le Bras's theorem on Banach-Colmez sheaves and the Fargues-Fontaine curve. Chai's unpublished characteristic-zero result and the Gross-Hopkins period map handle primes not containing p. Not refereed here.

Sources

Changelog1 change

Discussion