Arithmetic and geometry of the open p-adic disc
ARITHMETIC AND GEOMETRY OF THE OPEN P -ADIC DISC Scott Corry Florian Pop, Advisor Motivated by the local lifting problem for Galois covers of curves, this thesis investigates Galois branched covers of the open p-adic disc. Our main result is that the special fiber of an abelian cover is completely determined by arithmetic and geometric properties of the generic fiber and its characteristic zero specializations. This determination of the special fiber in terms of characteristic zero data is accomplished via the field of norms functor of Fontaine and Wintenberger. As a consequence of our result, we derive a characteristic zero reformulation of the abelian local lifting problem, and as an application we give a new proof of the p-cyclic case of the Oort Conjecture, which states that cyclic covers should always lift.