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Are there any good references which explain the injectivity part of Grothendieck's section conjecture in details? I heard that Grothendieck himself gave a proof in the letter to G.Faltings. But I need more kind explanation.

I already read a book `Galois groups and fundamental groups' by T.Szamuely. I do not know whether the book contains a proof of the injectivity part or not, maybe my understanding is not enough.

Any suggestion would be very helpful to me.

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You can find it in http://www.springer.com/mathematics/algebra/book/978-3-642-30673-0, Chapter 7 by Jakob Stix.

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See also the related

The use of embedding a curve into its Jacobian

If you don't have access to Timo's reference there is also the free resource

J.Stix On cuspidal sections of algebraic fundamental groups http://arxiv.org/abs/0809.0017 Appendix B

Beware that this appendix was suppressed from the published version (eponymous article on Jakob's webpage).

And yes, the argument is due to Grothendieck. You can find the original letter (translated from German into English) here :

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

p. 282/4

"The proof follows rather easily from the Mordell-Weil theorem stating that the group $A(K)$ is a finitely generated $\mathbb Z$-module, where $A$ is the “jacobienne généralisée” of $Y$ , corresponding to the “universal” embedding of $Y$ into a torsor under a quasi-abelian variety."

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