Reconstruction of Function Fields

Abstract.

For k an algebraic closure of the finite field $\mathbb{F}_p$, ℓ prime distinct from p and X a surface over k, we prove that the field of rational functions k(X) can be recovered from the maximal pro-ℓ-quotient ${\mathcal{G}}_{K}$ of its absolute Galois group – in fact already from the second central descending series quotient of ${\mathcal{G}}_{K}$.