Let be a field of characteristic zero with algebraic closure and absolute Galois group . Let be a geometrically connected variety over . Fix a geometric point, and let be the étale fundamental group of . Set and denote by the étale fundamental group of .
By functoriality of , the existence of a -point on implies that the above exact sequence has a section.
This predicts that the converse statement is true whenever is a proper hyperbolic curve over a number field.
(Needs more from Galois Theory and Diophantine geometry below)
Motivic stuff: Cohomology, homotopy theory, and arithmetic geometry (Andreas Holmstrom on October 4, 2009)