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

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 $K_1^{\mathrm{M}}/\ell$ and $K_2^{\mathrm{M}}/\ell$, 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.