Abstract : Consider function fields K|k of transcendence degree > 1 and k an algebraically closed base field. In my talk I will describe a strategy for recovering K from its pro-l meta-abelian Galois group in a functorial way in the case k is an algebraic closure of a finite field. This completes -for such base fields k- a Program started by Bogomolov at the beginning of the 1990's. Using this fact, I will further give some hints for a possible strategy which might work for algebraically closed base fields k of positive characteristic in the case the transcendence degree of K|k is > 2.