## Topics in Absolute Anabelian Geometry II: Decomposition Groups and Endomorphisms

J. Math. Sci. Univ. Tokyo
Vol. 20 (2013), No. 2, Page 171–269.

Mochizuki, Shinichi
Topics in Absolute Anabelian Geometry II: Decomposition Groups and Endomorphisms
The present paper, which forms the second part of a three-part series in which we study $\textit{absolute anabelian geometry}$ from an $\textit{algorithmic}$ point of view, focuses on the study of the closely related notions of $\textit{decomposition groups}$ and $\textit{endomorphisms}$ in this anabelian context. We begin by studying an $\textit{abstract combinatorial analogue}$ of the algebro-geometric notion of a stable polycurve (i.e., a successive extension of families of stable curves'') and showing that the geometry of log divisors on stable polycurves'' may be extended, in a purely group-theoretic fashion, to this abstract combinatorial analogue; this leads to various $\textit{anabelian}$ results concerning $\textit{configuration spaces}$. We then turn to the study of the $\textit{absolute pro-$\Sigma$anabelian geometry}$ of hyperbolic curves over mixed-characteristic local fields, for $\Sigma$ a set of primes of cardinality $\ge 2$ that contains the residue characteristic of the base field. In particular, we prove a certain $\textit{pro-$p$resolution of nonsingularities''}$ type result, which implies a conditional'' anabelian result to the effect that the condition, on an isomorphism of arithmetic fundamental groups, of preservation of decomposition groups of most'' closed points implies that the isomorphism arises from an isomorphism of schemes --- i.e., in a word, $\textit{point-theoreticity implies geometricity''}$; a non-conditional'' version of this result is then obtained for $\textit{pro-curves''}$ obtained by removing from a proper curve some set of closed points which is $\textit{$p$-adically dense in a Galois-compatible fashion''}$. Finally, we study, from an algorithmic point of view, the theory of $\textit{Belyi}$ and $\textit{elliptic cuspidalizations}$, i.e., group-theoretic reconstruction algorithms for the arithmetic fundamental group of an $\textit{open subscheme}$ of a hyperbolic curve that arise from consideration of certain $\textit{endomorphisms}$ determined by $\textit{Belyi maps}$ and $\textit{endomorphisms of elliptic curves}$.
Keywords: absolute anabelian geometry, hyperbolic curves, absolute $p$-adic Grothendieck Conjecture, $p$-adic Section Conjecture, configuration spaces, hidden endomorphisms, point-theoreticity, Belyi cuspidalization, elliptic cuspidalization.