Anabelian group-theoretic properties of the absolute Galois groups of discrete valuation fields

Abstract

Let K be a field. Write GK for the absolute Galois group of K. In the present paper, we discuss the slimness [i.e., the property that every open subgroup is center-free] and the elasticity [i.e., the property that every nontrivial topologically finitely generated normal closed subgroup of an open subgroup is open] of GK. These two group-theoretic properties are closely related to [various versions of] the Grothendieck Conjecture in anabelian geometry. For instance, with regard to the slimness, Mochizuki proved that GK is slim if K is a subfield of a finitely generated extension of the field of fractions of the Witt ring W(F‾p) as a consequence of a [highly nontrivial] Grothendieck Conjecture-type result. In the present paper, we generalize this result to the case where K is a subfield of the field of fractions of an arbitrary mixed characteristic Noetherian local domain. Our proof is based on elementary field theories such as Kummer theory. On the other hand, with regard to the elasticity, Mochizuki proved that GK is elastic if K is a finite extension of the field of p-adic numbers. In the present paper, we generalize this result to the case where K is an arbitrary mixed characteristic Henselian discrete valuation field. As a corollary of this generalization, we prove the semi-absoluteness of isomorphisms between the étale fundamental groups of smooth varieties over mixed characteristic Henselian discrete valuation fields. Moreover, we also prove the weak version of the Grothendieck Conjecture for hyperbolic curves of genus 0 over subfields of finitely generated extensions of mixed characteristic higher local fields.