Function-field reconstruction from Milnor K-theory and Galois data

What OpenAI claims

Reconstructs function fields of transcendence degree at least two over algebraically closed constants from K^M₁/ℓ, K^M₂/ℓ, 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)

Papers (3)

1. Reconstruction of Function Fields from Mod-ℓ Milnor K-Theory

   5 October 2026 · Preprint, not peer reviewed · PDF

   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^M₁/ℓ and K^M₂/ℓ, 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 𝔽_ℓ^×, with only Frobenius ambiguity in the field isomorphism in positive characteristic.

2. Reconstruction from Milnor K-theory modulo the characteristic

   5 October 2026 · Preprint, not peer reviewed · PDF

   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.

3. The Bogomolov-Pop reconstruction theorem

   23 September 2026 · Preprint, not peer reviewed · PDF

   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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.