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As in the title, I want to know the reason for importance of the section conjecture. Of course, the statement of conjecture is important as itself, even I cannot fully grasp the soul of it. However, what I really want to know is applications of the section conjecture. For example, can we derive properties on the set of rational points through the section conjecture such as finiteness?

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up vote 14 down vote accepted

You can start here "Fermat's last theorem" and anabelian geometry??

In particular, I mention there that: At some point Deligne thought he had a proof that the section conjecture implied Mordell, but the proof doesn't work. This is all explained in an appendix by Deligne to a paper of Stix:

Finally, for an actual application of the section conjecture. Its truth implies the existence of an algorithm to decide whether a curve of genus bigger than one has a rational point.

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