Coverings in p-adic analytic geometry and log coverings I: Cospecialization of the (p')-tempered fundamental group for a family of curves

TL;DR

This paper develops cospecialization homomorphisms for the (p')-tempered fundamental groups in p-adic analytic geometry, linking fiber fundamental groups in families of curves with semistable reduction through log geometry techniques.

Contribution

It introduces a method to construct cospecialization homomorphisms for tempered fundamental groups in p-adic families using log geometric approaches.

Findings

Constructed cospecialization homomorphisms for tempered fundamental groups.

Established invariance of geometric log fundamental groups under change of log points.

Linked fundamental groups of fibers via log geometry in p-adic analytic spaces.

Abstract

The tempered fundamental group of a p-adic analytic space classifies coverings that are dominated by a topological covering (for the Berkovich topology) of a finite etale covering of the space. Here we construct cospecialization homomorphisms between (p') versions of the tempered fundamental groups of the fibers of a smooth family of curves with semistable reduction. To do so, we will translate our problem in terms of cospecialization morphisms of fundamental groups of the log fibers of the log reduction. In particular, we will have to study invariance of the geometric log fundamental group of saturated log smooth log schemes over a log point by change of log point.