Throughout the proof of injectivity of the section conjecture, for example in Appendix B of <https://arxiv.org/abs/0809.0017>, one uses Mordell--Weil Theorem and for that embeds hyperbolic curve into an abelian variety. So, it may natural to take the embedding into the Jacobian variety. However, is it still possible to consider ''Jacobian'' of $\mathbf{P}^1-\{0,1,\infty\}$? Or, which abelian variety contains $\mathbf{P}^1-\{0,1,\infty\}$?



Embedding of punctured projective line to abelian variety
The Jacobian is G\_m^2 and the embedding sends t to (t,t-1).


