nLab
section conjecture

Contents

Idea

Let kk be a field of characteristic zero with algebraic closure k¯\overline{k} and absolute Galois group Γ k:=Gal(k¯/k)\Gamma_k := Gal (\overline{k}/k). Let XX be a geometrically connected variety over kk. Fix a geometric point, x¯X(k¯)\overline{x}\in X (\overline{k}) and let π 1(X)π 1(X,x¯)\pi_1 (X ) \coloneqq \pi_1 (X, \overline{x}) be the étale fundamental group of XX. Set X¯=X× kk¯\overline{X} = X \times_k \overline{k} and denote by π 1(X¯):=π 1(X¯,x¯)\pi_1 (\overline{X}) := \pi_1 (\overline{X} , \overline{x}) the étale fundamental group of X¯\overline{X}.

Grothendieck’s fundamental short exact sequence of profinite groups (SGA1), is:

1π 1(X¯)π 1(X)Γ k1. 1 \to\pi_1 (\overline{X})\to\pi_1 (X) \to \Gamma_k \to 1 \,.

By functoriality of π 1\pi_1, the existence of a kk-point on XX implies that the above exact sequence has a section (is a split exact sequence).

This suggests that the converse statement is true whenever XX 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

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.

Revised on January 19, 2017 17:05:01 by Mateo Carmona? (186.85.202.64)