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 .
Grothendieck’s fundamental short exact sequence of profinite groups (SGA1), is:
By functoriality of , the existence of a -point on implies that the above exact sequence has a section (is a split exact sequence).
This suggests that the converse statement is true whenever is a proper hyperbolic algebraic curve over a number field. This conjecture is the section conjecture (Grothendieck 97).
(Needs more from Galois Theory and Diophantine geometry below)
Alexander Grothendieck, letter to Gerd Faltings, London Math. Soc. Lecture Note Ser., 242, Geometric Galois actions, 1, 49–58, Cambridge Univ. Press, Cambridge, 1997.
Mohamed Saidi, Around the Grothendieck Anabelian Section Conjecture, ArXiv:1010.1314v2.
Minhyong Kim, Galois Theory and Diophantine geometry, 2009 (pdf)
Revised on December 8, 2014 19:05:45
by Urs Schreiber