Contents

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)

References

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