Galois covers of the open p-adic disc

TL;DR

This paper studies Galois covers of the open p-adic disc, linking their reductions to characteristic p with properties of their generic fibers, and applies these insights to criteria for good reduction and the local Oort Conjecture.

Contribution

It introduces a new approach using the field of norms to relate the special fiber of Galois covers to their generic fibers and characteristic zero specializations.

Findings

Derived a criterion for good reduction in the abelian case

Provided an arithmetic reformulation of the local Oort Conjecture

Connected the special fiber structure to properties of the generic fiber

Abstract

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.