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?

up vote 18 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: http://arxiv.org/abs/0910.5009

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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