#### DOCUMENTA MATHEMATICA,
Extra Volume: Kazuya Kato's Fiftieth Birthday (2003), 609-640

** Shinichi Mochizuki **
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}

Full text: dvi.gz 53 k,
dvi 125 k,
ps.gz 651 k,
pdf 284 k.

Home Page of
DOCUMENTA MATHEMATICA