On \'etale fundamental groups of formal fibres of $p$-adic curves

TL;DR

This paper studies the structure of étale fundamental groups of formal fibers of p-adic curves, showing they can be compactified and describing their prime-to-p quotients, advancing understanding of p-adic geometric coverings.

Contribution

It demonstrates that certain Galois coverings of formal fibers can be extended to proper p-adic curves and characterizes the prime-to-p quotient of their fundamental groups.

Findings

Galois coverings of formal fibers can be compactified to proper p-adic curves.

The prime-to-p quotient of the fundamental group is pro-prime-to-p free.

The rank of this free group is finite and computable.

Abstract

We investigate a certain class of (geometric) finite (Galois) coverings of formal fibres of $p$-adic curves and the corresponding quotient of the (geometric) \'etale fundamental group. A key result in our investigation is that these (Galois) coverings can be compactified to finite (Galois) coverings of proper $p$-adic curves. We also prove that the maximal prime-to-$p$ quotient of the geometric \'etale fundamental group of a (geometrically connected) formal fibre of a $p$-adic curve is (pro-)prime-to-$p$ free of finite computable rank.