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.

14G32, 14H30, 14H25, 11G20

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