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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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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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  • $\begingroup$ Sorry if this is silly, but would you mind to explain why Deligne’s proposed argument for the implication fails? $\endgroup$ Aug 16, 2021 at 4:53
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    $\begingroup$ @Heavensfall He proved that the set $X(F)$ is compact under the weird topology coming the section conjecture. He thought he could prove that this set was also discrete under this topology (and conclude that it was finite). But proving it was discrete was not as easy as he anticipated and, in the end, he couldn't prove it. $\endgroup$ Aug 16, 2021 at 8:24

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