In the section conjecture for a number field $k$: the proof of the injectivity of the map

$$X(k)\to \mathrm{HomExt}_{G_k}(G_k,\pi_1(X,\overline{x}))$$

that attributes to a rational point a section of the fundamental exact sequence

$$ 1\to \pi_1(\overline{X},\overline{x})) \to \pi_1(X,\overline{x})\to G_k \to 1 $$

uses an embedding of $X$ into its jacobian to reduce to an abelian variety $A$. The map above is then interpreted as limit of coboundary maps in étale cohomology for the Kummer exact sequences for $A$. One then applies Mordell-Weil theorem ($A(k)$ is an abelian group of finite type) to conclude.

See

Jakob Stix

On cuspidal sections of algebraic fundamental groups

http://arxiv.org/abs/0809.0017

appendix B

for details. This was known to Grothendieck back in 1983, see

Grothendieck, Alexander

Brief an G. Faltings. (German) [Letter to G. Faltings]

http://www.math.jussieu.fr/~leila/grothendieckcircle/GtoF.pdf

In the section conjecture for a number field $k$: the proof of the injectivity of the map

$$X(k)\to \mathrm{HomExt}_{G_k}(G_k,\pi_1(X,\overline{x}))$$

that attributes to a rational point a section of the fundamental exact sequence

$$ 1\to \pi_1(\overline{X},\overline{x})) \to \pi_1(X,\overline{x})\to G_k \to 1 $$

uses an embedding of $X$ into its jacobian to reduce to an abelian variety $A$. The map above is then interpreted as limit of coboundary maps in étale cohomology for the Kummer exact sequences for $A$. One applies Mordell-Weil theorem ($A(k)$ is an abelian group of finite type) to conclude.

See

Jakob Stix

On cuspidal sections of algebraic fundamental groups

http://arxiv.org/abs/0809.0017

appendix B

for details. This was known to Grothendieck back in 1983, see

Grothendieck, Alexander

Brief an G. Faltings. (German) [Letter to G. Faltings]

http://www.math.jussieu.fr/~leila/grothendieckcircle/GtoF.pdf