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section conjecture

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 ) := \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 exact sequence of profinite groups, cf. SGA1 IX Thm 6.1, (see also 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.

Grothendieck’s section conjecture

This predicts that the converse statement is true whenever XX is a proper hyperbolic curve over a number field.

(Needs more from Galois Theory and Diophantine geometry below)

Blog discussions

References

Revised on September 25, 2012 18:01:36 by Tim Porter (95.147.237.67)