## Journal of Mathematics of Kyoto University

### Absolute anabelian cuspidalizations of proper hyperbolic curves

Shinichi Mochizuki

#### Abstract

In this paper, we develop the theory of “cuspidalizations” of the étale fundamental group of a proper hyperbolic curve over a finite or nonarchimedean mixed-characteristic local field. The ultimate goal of this theory is the group-theoretic reconstruction of the étale fundamental group of an arbitrary open subscheme of the curve from the étale fundamental group of the full proper curve. We then apply this theory to show that a certain absolute $p$-adic version of the Grothendieck Conjecture holds for hyperbolic curves “of Belyi type”. This includes, in particular, affine hyperbolic curves over a nonarchimedean mixed-characteristic local field which are defined over a number field and isogenous to a hyperbolic curve of genus zero. Also, we apply this theory to prove the analogue for proper hyperbolic curves over finite fields of the version of the Grothendieck Conjecture that was shown in [Tama].

#### Article information

Source
J. Math. Kyoto Univ., Volume 47, Number 3 (2007), 451-539.

Dates
First available in Project Euclid: 14 August 2009

https://projecteuclid.org/euclid.kjm/1250281022

Digital Object Identifier
doi:10.1215/kjm/1250281022

Mathematical Reviews number (MathSciNet)
MR2402513

Zentralblatt MATH identifier
1143.14305

#### Citation

Mochizuki, Shinichi. Absolute anabelian cuspidalizations of proper hyperbolic curves. J. Math. Kyoto Univ. 47 (2007), no. 3, 451--539. doi:10.1215/kjm/1250281022. https://projecteuclid.org/euclid.kjm/1250281022