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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.

edited Sep 20 2010 at 12:33

asked Oct 18 2009 at 14:27

Thomas Riepe
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	The "Sketch of a program" is available at grothendieckcircle.org. On page 40 of the pdf-file there is certainly a reference to FLT. – Philipp Lampe Oct 18 2009 at 18:43
	Thanks. Has anyone an idea what he had in mind about that? – Thomas Riepe Oct 19 2009 at 9:42

JS Milne · Accepted Answer · 2009-10-21 14:11:28Z UTC

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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.

answered Oct 21 2009 at 14:11

JS Milne
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Thanks! I'd be happy about further explanations on arithmetics and anabelian geometry. – Thomas Riepe Oct 23 2009 at 9:46

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See also ucl.ac.uk/~ucahmki/cambridgews.pdf – Seamus Sep 20 2010 at 15:48

Mahesh Kakde · Answer 2 · 2009-12-02 23:15:58Z UTC

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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.

answered Dec 2 2009 at 23:15

Mahesh Kakde
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I believe that it is not yet known that the section conjecture implies Mordell's conjecture (Faltings' theorem). – Emerton Mar 10 2010 at 14:21

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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 – Felipe Voloch Mar 10 2010 at 16:41

“Fermat’s last theorem” and anabelian geometry??

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