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 $\ge 2$] 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 ${\mathbb{Q}}_p$].