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
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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.

Mathematics Subject Classification (2010): Primary 14H30, Secondary 14H25.
Received: 2008-03-27