On Certain Isomorphisms between Absolute Galois Groups

Let k be an algebraically closed field of characteristic zero, L its finitely generated extension of transcendence degree ≥ 2, and L another finitely generated extension of k. It is a result of Bogomolov [B2] that any isomorphism between Gal(L/L) and Gal(L′/L) is induced by an isomorphism of fields L −→ L′ identifying L with L. If the transcendence degree of L over k is one, the group Gal(L/L) is free, and therefore, its structure tells nothing about the field L. Let F be an algebraically closed extension of k of transcendence degree one, and G = GF/k be the group of automorphisms over k of the field F . Let the set of subgroups UL := Aut(F/L) for all subfields L finitely generated over k be the basis of neighborhoods of the unity in G. Let λ be a continuous automorphism of G. The purpose of this note is to show that if λ induces an isomorphism Gal(F/L) ∼ −→ Gal(F/L) then the fields L and L are isomorphic (see Theorem 4.2 below for a more precise statement).