The Absolute Anabelian Geometry of Canonical Curves

In this paper, we continue our study of the issue of the extent to which a {\iitt hyperbolic curve over a finite extension of the field of $p$-adic numbers} is determined by the profinite group structure of its {\iitt étale fundamental group}. Our main results are that: (i) the theory of {\iitt correspondences} of the curve --- in particular, its {\iitt arithmeticity} --- is completely determined by its fundamental group; (ii) when the curve is a {\iitt canonical lifting} in the sense of {\iitt ``$p$-adic Teichmüller theory'',} its {\iitt isomorphism class} is functorially determined by its fundamental group. Here, (i) is a consequence of a {\iitt ``$p$-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 {\iitt logarithmic special fiber} of the curve is functorially determined by its fundamental group.

2000 Mathematics Subject Classification: 14H25, 14H30

Keywords and Phrases: {\iitt hyperbolic curve, étale fundamental group, anabelian, correspondences, Grothendieck Conjecture, canonical lifting, $p$-adic Teichmüller theory}

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