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 $\mathbf{P}^1-\{0,1,\infty\}$?
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2 Answers
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The Jacobian is G_m^2 and the embedding sends t to (t,t-1).
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$\begingroup$ @JSE I would avoid the term Jacobian - these belong to abelian varieties, that are proper. Generalized Jacobian is more standard. $\endgroup$– NielsCommented May 14, 2016 at 12:37
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[Cornell-Silverman], p. 107, Corollary 3.8 (see http://jmilne.org/math/articles/1986b.pdf): Every rational map $\mathbf{P}^1 \to A$ is constant.
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$\begingroup$ But, then, the proof using Mordell--Weil does not work for $\mathbf{P}^1-\{0,1,\infty\}$? $\endgroup$– StudentCommented May 12, 2016 at 16:51