The Absolute Anabelian Geometry of Canonical Curves

In this paper, 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. 2000 Mathematics Subject Classification: 14H25, 14H30