On etale fundamental groups of formal fibres of p-adic curves

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