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Do I remember a remark in "Sketch of a program" or "Letter to Faltings" correctly, that acc. to Grothendieck anabelian geometry should not only enable finiteness proofs, but a proof of FLT too? If yes, how?

Edit: In this transcript, Illusie makes a remark that Grothendieck looked for a connection between "FLT" and "higher stacks". BTW, here a note on (acc. to Illusie) Grothendieck's favored landscape.

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See the papers of Minhyong Kim. For example, begin by looking at the MR review 2181717 of his paper Invent. Math. 161 (2005), no. 3, 629--656.

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  • $\begingroup$ Thanks! I'd be happy about further explanations on arithmetics and anabelian geometry. $\endgroup$
    – Thomas Riepe
    Commented Oct 23, 2009 at 9:46
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    $\begingroup$ See also ucl.ac.uk/~ucahmki/cambridgews.pdf $\endgroup$
    – Seamus
    Commented Sep 20, 2010 at 15:48
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As Minhyong Kim points out in one of his papers on unipotent fundamental group- it is not quite clear how the section conjecture would imply Faltings theorem. The nature of implication (i.e. section conjecture implies Faltings or FLT) may be known to some experts but I don't know if it is explicitly written in literature.

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  • $\begingroup$ I believe that it is not yet known that the section conjecture implies Mordell's conjecture (Faltings' theorem). $\endgroup$
    – Emerton
    Commented Mar 10, 2010 at 14:21
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    $\begingroup$ 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: arxiv.org/abs/0910.5009 $\endgroup$
    – Felipe Voloch
    Commented Mar 10, 2010 at 16:41

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