Do I remember a remark in "Sketch of a program" or "Letter to Faltings" correctly, that acc. to Grothendieck [anabelian geometry](http://www.math.okayama-u.ac.jp%2F~h-naka%2Fzoo%2Fpeacock%2FNTM.ps "anabelian survey") should not only enable finiteness proofs, but a proof of FLT too? If yes, how?


Edit: [In this](http://www.ams.org/notices/201009/rtx100901106p.pdf "AMS-Notices-pdf") transcript, Illusie makes a remark that Grothendieck looked for a connection between "FLT" and "higher stacks". BTW, [here](http://madameboisvert.blogspot.com/2009/04/discovering-paca.html "link") a note on (acc. to Illusie) Grothendieck's [favored landscape](http://www.cabris.fr/#/galerie/sample-album-1/point-de-vue-9/ "link"). 



"Fermat's last theorem" and anabelian geometry?
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.


