## Topics in Absolute Anabelian Geometry I: Generalities

J. Math. Sci. Univ. Tokyo
Vol. 19 (2012), No. 2, Page 139--242.

Mochizuki, Shinichi
Topics in Absolute Anabelian Geometry I: Generalities
This paper forms the first part of a three-part series in which we treat various topics in {\it absolute anabelian geometry} from the point of view of developing {\it abstract algorithms}, or {\it software''}, that may be applied to abstract profinite groups that just happen'' to arise as (quotients of) \'etale fundamental groups from algebraic geometry. One central theme of the present paper is the issue of understanding the gap between {\it relative}, {\it semi-absolute''}, and {\it absolute anabelian geometry}. We begin by studying various {\it abstract combinatorial properties} of profinite groups that typically arise as absolute Galois groups or arithmetic/geometric fundamental groups in the anabelian geometry of {quite general varieties in arbitrary dimension} over number fields, mixed-characteristic local fields, or finite fields. These considerations, combined with the classical theory of Albanese varieties, allow us to derive an {\it absolute anabelian algorithm} for constructing the {\it {\it quotient} of an arithmetic fundamental group} determined by the {\it absolute Galois group} of the base field in the case of {\it quite general varieties of {\it arbitrary dimension}}. Next, we take a more detailed look at certain {\it $p$-adic Hodge-theoretic} aspects of the absolute Galois groups of mixed-characteristic local fields. This allows us, for instance, to derive, from a certain result communicated orally to the author by A. Tamagawa, a {\it semi-absolute'' $\text{{\rm Hom}}$-version} --- whose {\it absolute} analogue is {\it false}! --- of the {\it anabelian conjecture for hyperbolic curves} over mixed-characteristic local fields. Finally, we generalize to the case of {\it varieties of {\it arbitrary dimension} over arbitrary sub-$p$-adic fields} certain techniques developed by the author in previous papers over mixed-characteristic local fields for applying {\it relative anabelian} results to obtain {\it semi-absolute'' group-theoretic contructions} of the \'etale fundamental group of one hyperbolic curve from the \'etale fundamental group of another closely related hyperbolic curve.