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Reconstruction of Function Fields from Mod-ℓ Milnor K-Theory
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Abstract
We prove a mod-ℓ Bogomolov–Pop reconstruction theorem for function fields of transcendence degree at least two over arbitrary algebraically closed fields of characteristic different from ℓ. The groups $K^{\mathrm M}_1/\ell$ and $K^{\mathrm M}_2/\ell$, together with their full bilinear product, determine the perfect closure and its constant field. Every compatible isomorphism of these data is induced by a field isomorphism up to a single scalar in $\mathbb F_\ell^\times$, with only Frobenius ambiguity in the field isomorphism in positive characteristic.
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Comments and discussion
- AHGT project / Adam Topaz, On the reconstruction of function fields from mod-ℓ Milnor K-theory (Taming the Flow seminar announcement, autumn 2026; date TBA). The programme names Topaz for a forthcoming talk targeting this manuscript and situating its degree-one and degree-two invariants in birational anabelian geometry.
Related prior literature mentioned by others
- Adam Topaz, Reconstructing function fields from rational quotients of mod-ℓ Galois groups (arXiv:1408.5194, 2014; revised 2015). Listed in the AHGT talk's references, this predecessor reconstructs function fields of transcendence degree at least five with rational quotients supplied, whereas the manuscript claims recovery from the Milnor product alone in degree at least two.
- 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)
- A. M. Silberstein, Anabelian Intersection Theory I: The Conjecture of Bogomolov-Pop and Applications (arXiv:1211.4608, 2012). Earlier work on the Bogomolov–Pop conjecture for the full absolute Galois group over the algebraic closures of ℚ and 𝔽_p; it is cited in the companion Bogomolov–Pop paper but not in the historical context section of this paper.
- A. Topaz, Abelian-by-Central Galois groups of fields I: a formal description (arXiv:1310.5613, 2013). Describes the mod-n abelian-by-central Galois layer via Merkurjev–Suslin, which links the mod-ℓ Milnor datum of Theorem 1.1 to the corresponding Galois-theoretic datum.
- F. Pop, Pro-l birational anabelian geometry over algebraically closed fields I (arXiv:math/0307076, 2003). Pop's earlier preprint on pro-ℓ reconstruction over algebraic closures of finite fields, relevant to the historical attributions in Section 1.2.
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
- Emergent Mind: family 009
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026,
author = {{OpenAI}},
title = {{Reconstruction of Function Fields from Mod-$\ell$ Milnor $K$-Theory}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026/mod-ell-bogomolov-pop.pdf}{OAI:Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026}},
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 from Milnor K-theory modulo the characteristic · October 5, 2026
- The Bogomolov-Pop reconstruction theorem · September 23, 2026