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
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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.

Mathematics Subject Classification (2010): Primary 14H30; Secondary 14H25.
Mathematical Reviews Number: MR3154380

Received: 2008-03-26