Anabelian Group-theoretic Properties of the Pro-p Absolute Galois Groups of Henselian Discrete Valuation Fields

Abstract

Let p be a prime number; K a Henselian discrete valuation field of characteristic 0 such that the residue field is an infinite field of characteristic p. Write GK for the absolute Galois group of K. In our previous papers, under the assumption that K contains a primitive p-th root of unity ζp, we proved that any almost pro-p-maximal quotient of GK satisfies certain “anabelian” group-theoretic properties called very elasticity and strong internal indecomposability. In the present paper, we generalize this result to the case where K does not necessarily contain ζp. Then, by applying this generalization, together with some facts concerning Hilber-tian fields, we prove the semiabsoluteness of isomorphisms between thepro-p etale fundamental groups of smooth varieties over certain classes offields of characteristic 0. Moreover, we observe that there are various sim-ilarities between the maximal pro-p quotient GpK of GK and non abelianfree pro-p groups. For instance, we verify that every topologically finitely generated closed subgroup of GpK is a free pro-p group. One of the key ingredients of our proofs is “Artin-Schreier theory in characteristic zero”introduced by MacKenzie and Whaples.