Geodesic
The absolute anabelian geometry of canonical curves
Documenta mathematica, Kazuya Kato's Fiftieth Birthday (2003), pp. 609-640.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

We continue our study of the issue of the extent to which a hyperbolic curve over a finite extension of the field of $p$-adic numbers is determined by the profinite group structure of its étale fundamental group. Our main results are that: (i) the theory of correspondences of the curve -- in particular, its arithmeticity -- is completely determined by its fundamental group; (ii) when the curve is a canonical lifting in the sense of "$p$-adic Teichmüller theory", its isomorphism class is functorially determined by its fundamental group. Here, (i) is a consequence of a "$p$-adic version of the Grothendieck conjecture for algebraic curves" proven by the author, while (ii) builds on a previous result to the effect that the logarithmic special fiber of the curve is functorially determined by its fundamental group.
Classification : 14G32, 14H30, 14H25, 11G20
Keywords: correspondences, Grothendieck conjecture, canonical lifting, $p$-adic Teichmüller theory 
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     title = {The absolute anabelian geometry of canonical curves},
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Mochizuki, Shinichi. The absolute anabelian geometry of canonical curves. Documenta mathematica, Kazuya Kato's Fiftieth Birthday (2003), pp. 609-640. http://geodesic.mathdoc.fr/item/DOCMA_2003__S6__a7/