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The Bogomolov-Pop reconstruction theorem
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Abstract
We prove the Bogomolov–Pop reconstruction conjecture for function fields of transcendence degree at least two over arbitrary algebraically closed fields of characteristic different from ℓ. The pro-ℓ abelian-by-central datum determines the perfect closure and its constant field, with precisely the Frobenius and ℓ-adic unit ambiguities.
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Comments and discussion
- AHGT project, On the Bogomolov–Pop reconstruction theorem (Taming the Flow seminar announcement, autumn 2026; speaker and date TBA). The programme announces a talk targeting this manuscript and its relation to the Milnor K-theoretic reconstruction results.
Formalization
- OpenAI, Function-field reconstruction from Milnor K-theory and Galois data: Lean scope note (OpenAI math release, lean/docs/009.md, October 6, 2026). Describes a Lean formalization of injectivity of the reconstruction map in Theorem 1.1 (field isomorphisms inducing the same Galois isomorphism up to ℓ-adic units agree modulo Frobenius); it states that existence (surjectivity) is outside its scope.
- OpenAI math release, Injectivity.lean (openai/math, lean/OAI/Algebra/BogomolovPop, 2026). Lean source proving
main_canonical_injective(injectivity of the canonical map) and the corollarymain_graph_injective; it covers only the injectivity half of Theorem 1.1. [AI-generated per source]
Related prior literature mentioned by others
- Florian Pop, Reconstruction of Function Fields from their pro-ℓ abelian divisorial Inertia (arXiv:1810.04768, 2018). The AI Proved What? exposition cites this predecessor, which supplies divisorial inertia as extra input to reconstruction, as background to the family's claims of recovering the needed geometric data. [family-level]
Possibly relevant literature (found by the agent, not checked)
- F. Pop, Pro-l birational anabelian geometry over algebraically closed fields I (arXiv:math/0307076, 2003). Pop's earlier preprint on functorial reconstruction from pro-ℓ Galois theory over algebraic closures of finite fields, relevant to the history of the case the introduction attributes to Pop.
- A. Topaz, Abelian-by-Central Galois groups of fields I: a formal description (arXiv:1310.5613, 2013). Uses Merkurjev–Suslin to describe the abelian-by-central layer of the Galois group explicitly, which relates to how Section 2 passes between the Galois datum and Kummer/Milnor-theoretic data.
- A. Topaz, Abelian-by-Central Galois Groups of Fields II: Definability of Inertia/Decomposition Groups (arXiv:1503.04368, 2015). Shows that minimized inertia and decomposition groups of many valuations can be defined from commuting-liftable pairs, which bears on the local valuation theory used in Section 3.
Digests and expositions
- AI Proved What?, The whole field from a shadow (Episode 009, October 2026). This AI-assisted video and transcript explain the three reconstruction claims, outline the mod-ℓ argument, and distinguish the Galois uniqueness formalization from the full theorem. [AI-generated per source] [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 009
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-Bogomolov-Pop-reconstruction-theorem-September-23-2026,
author = {{OpenAI}},
title = {{The Bogomolov--Pop reconstruction theorem}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/paper.pdf}{OAI:The-Bogomolov-Pop-reconstruction-theorem-September-23-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/009.md. The Lean statements have not been checked by people against the paper's claims.
Family 009: Function-field reconstruction from Milnor K-theory and Galois data
Reconstructs function fields of transcendence degree at least two over algebraically closed constants from $K^{\mathrm M}_1/\ell$, $K^{\mathrm M}_2/\ell$, and their product. These data recover the perfect closure and constants when ℓ differs from the characteristic, and the original field and its named base in equal characteristic. Also proves Bogomolov–Pop reconstruction from abelian-by-central pro-ℓ Galois data away from the characteristic.
Other papers in this family:
- Reconstruction of Function Fields from Mod-ℓ Milnor K-Theory · October 5, 2026
- Reconstruction from Milnor K-theory modulo the characteristic · October 5, 2026