On the Semi-absoluteness of Isomorphisms between the Pro-$p$ Arithmetic Fundamental Groups of Smooth Varieties

Abstract

Let p be a prime number. In the present paper, we consider a certain pro- p analogue of the semi-absoluteness of isomorphisms between the étale fundamental groups of smooth varieties over p -adic local fields [i.e., finite extensions of the field of p -adic numbers \mathbb{Q}_p ] obtained by Mochizuki. This research was motivated by Higashiyama’s recent work on the pro- p analogue of the semi-absolute version of the Grothendieck conjecture for configuration spaces [of dimension \geq 2 ] associated to hyperbolic curves over generalized sub- p -adic fields [i.e., subfields of finitely generated extensions of the completion of the maximal unramified extension of \mathbb{Q}_p ].