Mochizuki, Shinichi

The absolute anabelian geometry of canonical curves

Doc. Math. (Bielefeld) Extra Vol. Kazuya Kato's Fiftieth Birthday, 609-640 (2003)


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.

Mathematics Subject Classification

14G32, 14H30, 14H25, 11G20


correspondences, Grothendieck conjecture, canonical lifting, $p$-adic Teichmüller theory