  This paper investigates Galois branched covers of the open $p$-adic disc and
their reductions to characteristic $p$. Using the field of norms functor of
Fontaine and Wintenberger, we show that the special fiber of a Galois cover is
determined by arithmetic and geometric properties of the generic fiber and its
characteristic zero specializations. As applications, we derive a criterion for
good reduction in the abelian case, and give an arithmetic reformulation of the
local Oort Conjecture concerning the liftability of cyclic covers of germs of
curves.
