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

Revised on December 8, 2014 19:05:45 by Urs Schreiber (87.183.144.209)