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 absolute anabelian geometry from the point of view of developing abstract algorithms, or "software", that may be applied to abstract profinite groups that "just happen" to arise as (quotients of) étale fundamental groups from algebraic geometry. One central theme of the present paper is the issue of understanding the gap between relative, "semi-absolute, and absolute anabelian geometry. We begin by studying various 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 absolute anabelian algorithm for constructing the quotient of an arithmetic fundamental group determined by the absolute Galois group of the base field in the case of quite general varieties of arbitrary dimension. Next, we take a more detailed look at certain 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 "semi-absolute" Hom-version --- whose absolute analogue is false! --- of the anabelian conjecture for hyperbolic curves over mixed-characteristic local fields. Finally, we generalize to the case of varieties of arbitrary dimension over arbitrary sub-p-adic fields certain techniques developed by the author in previous papers over mixed-characteristic local fields for applying relative anabelian results to obtain "semi-absolute" group-theoretic contructions of the étale fundamental group of one hyperbolic curve from the étale fundamental group of another closely related hyperbolic curve.
- Journal of mathematical sciences, the University of Tokyo
Journal of mathematical sciences, the University of Tokyo 19(2), 139-242, 2012-09-10
Graduate School of Mathematical Sciences, The University of Tokyo