The Bogomolov-Pop conjecture: function fields over algebraically closed fields are determined by their pro-l abelian-by-central Galois groups
Let be a function field of transcendence degree at least two over an algebraically closed field of characteristic different from a prime , and let be the maximal pro- quotient of its absolute Galois group in which commutators are central, with abelianization . Bogomolov (1991) proposed that this small quotient determines the field; the precise Isom form, with its scalar and Frobenius ambiguities, is Conjecture 1 in Topaz (2016). Known cases had constants an algebraic closure of a finite field (Bogomolov-Tschinkel, Pop) or extra inertia data (Pop). Is the natural map from field isomorphisms of perfect closures, modulo Frobenius, to bracket-compatible isomorphisms modulo a bijection?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Birational anabelian geometry
- Posed by
- Fedor Bogomolov; Isom form recorded by Adam Topaz
- Year posed
- 1991
- Years open
- 35y
- 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
- 32 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1 (principal paper): for function fields , of transcendence degree at least two over arbitrary algebraically closed fields of characteristic different from , the map from isomorphisms of perfect closures modulo Frobenius to bracket-compatible isomorphisms modulo is bijective; equal characteristic and transcendence degree are forced. Companions: the same reconstruction from , and their product (removing Topaz's rational-subgroup data and transcendence degree at least five), and from with Steinberg relations in characteristic . Not shown: transcendence degree one, or fields with non-algebraically-closed constants.
What the AI did
The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's two 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 principal manuscript (September 23, 2026) proves the pro-l Galois form; two companions (October 5, 2026) prove analogues from mod-l Milnor K-theory (away from the characteristic) and from mod-p Milnor K-theory in characteristic p.
Verification
No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against the conjecture as cited; Theorem 1.1 states the bijection for arbitrary algebraically closed constants of characteristic different from , transcendence degree at least two, including surfaces and . Lean: the challenge BogomolovPopInjectivity (theorem OAI.BogomolovPop.main_graph_injective) exists at the pinned commit but is not in the formalization catalogue; its scope page says it proves only injectivity of the reconstruction map, and existence of a field isomorphism for every admissible Galois isomorphism is outside it. Because the formal statement covers one direction, not the headline, the tier is unreviewed. Not rebuilt.
Sources
- PaperCompanion: Reconstruction of Function Fields from Mod-l Milnor K-TheoryCompanion: Reconstruction from Milnor K-theory modulo the characteristic
- Lean proofLean formalization (injectivity only): BogomolovPop/Injectivity.lean
- CodeOpenAI math release: The Bogomolov-Pop reconstruction theorem
- Problem recordTopaz, Reconstructing function fields from rational quotients of mod-l Galois groups (Conjecture 1)