# 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 $\pi_1 (X ) := \pi_1 (X, \overline{x})$ 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 exact sequence of profinite groups, cf. SGA1 IX Thm 6.1, (see also 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.

## Grothendieck’s section conjecture

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

(Needs more from Galois Theory and Diophantine geometry below)

## References

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