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

Abstract — v1

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.