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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    The "Sketch of a program" is available at On page 40 of the pdf-file there is certainly a reference to FLT. – Philipp Lampe Oct 18 '09 at 18:43
  • Thanks. Has anyone an idea what he had in mind about that? – Thomas Riepe Oct 19 '09 at 9:42
up vote 12 down vote accepted

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.

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.

  • I believe that it is not yet known that the section conjecture implies Mordell's conjecture (Faltings' theorem). – Emerton Mar 10 '10 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: – Felipe Voloch Mar 10 '10 at 16:41

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