Galois covers of the open p - adic disc

Motivated by the local lifting problem for Galois covers of curves, this paper investigates Galois branched covers of the open p-adic disc. Using the field of norms functor of Fontaine and Wintenberger, we show that the special fiber of a Galois cover is completely determined by arithmetic and geometric properties of the generic fiber and its characteristic zero specializations. As applications, we derive a characteristic zero reformulation of the local lifting problem, and give a new proof of the liftability of p-cyclic covers.