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. (Lean)
Manuscripts (3)
Reconstruction of Function Fields from Mod-ℓ Milnor K-Theory
October 5, 2026 · source · cite as \cite{OAI:Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026}
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.
Reconstruction from Milnor K-theory modulo the characteristic
October 5, 2026 · source · cite as \cite{OAI:Reconstruction-from-Milnor-K-theory-modulo-the-characteristic-October-5-2026}
Let K and L be finitely generated extensions of transcendence degree at least two over algebraically closed fields of characteristic p. We prove that every isomorphism of their first Milnor K-groups modulo p preserving the degree-two Steinberg relations is a nonzero scalar multiple of the map induced by a unique field isomorphism. The proof recovers projective lines over the subfields of pth powers by an elementary calculation with derivations.
The Bogomolov-Pop reconstruction theorem
September 23, 2026 · source · cite as \cite{OAI:The-Bogomolov-Pop-reconstruction-theorem-September-23-2026}
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.
Result 009 · Number theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md