nLab section conjecture

Idea

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:

$$1 \to\pi_1 (\overline{X})\to\pi_1 (X) \to \Gamma_k \to 1 \,.$$

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)

Blog discussions

• n-Category café

• Motivic stuff: Cohomology, homotopy theory, and arithmetic geometry (Andreas Holmstrom on October 4, 2009)

Related entries

• anabelian geometry

References

• 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.