Let $k$ be a field of characteristic zero with algebraic closure $\overline{k}$ and absolute Galois group $\Gamma_k := Gal (\overline{k}/k)$. Let $X$ be a geometrically connected variety over $k$. Fix a geometric point, $\overline{x}\in X (\overline{k})$ and let $\pi_1 (X ) \coloneqq \pi_1 (X, \overline{x})$ be the étale fundamental group of $X$. Set $\overline{X} = X \times_k \overline{k}$ and denote by $\pi_1 (\overline{X}) := \pi_1 (\overline{X} , \overline{x})$ the étale fundamental group of $\overline{X}$.
Grothendieck’s fundamental short exact sequence of profinite groups (SGA1), is:
By functoriality of $\pi_1$, the existence of a $k$-point on $X$ implies that the above exact sequence has a section (is a split exact sequence).
This suggests that the converse statement is true whenever $X$ 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)
Motivic stuff: Cohomology, homotopy theory, and arithmetic geometry (Andreas Holmstrom on October 4, 2009)
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)
Jakob Stix, Evidence for the section conjecture in the theory of arithmetic fundamental groups Habilitationsschrift, School of Mathematics and Computer Science at the Ruprecht-Karls-Universität Heidelberg, January 2011, x+190 pages, to appear as Rational Points and Arithmetic of Fundamental Groups, Evidence for the Section Conjecture. Springer Lecture Notes in Mathematics 2054, xx+pp.249, Springer, 2013.
Last revised on January 19, 2017 at 17:05:01. See the history of this page for a list of all contributions to it.