Local-to-Global Extensions to Wildly Ramified Covers of Curves

TL;DR

This paper investigates when local Galois extensions of Laurent series rings can be realized as global covers of algebraic curves, extending Harbater's local-to-global principles to more general fields and non-abelian p-groups.

Contribution

It generalizes Harbater's local-to-global lifting criteria for wildly ramified covers to broader fields and non-abelian p-groups using a generalized Artin-Schreier theory.

Findings

Characterizes curves Y for which local extensions lift globally.

Establishes conditions for the uniqueness of such liftings.

Extends the theory to non-abelian p-groups over general fields.

Abstract

Given a Galois cover of curves X to Y with Galois group G which is totally ramified at a point x and unramified elsewhere, restriction to the punctured formal neighborhood of x induces a Galois extension of Laurent series rings k((u))/k((t)). If we fix a base curve Y , we can ask when a Galois extension of Laurent series rings comes from a global cover of Y in this way. Harbater proved that over a separably closed field, this local-to-global principle holds for any base curve if G is a p-group, and gave a condition for the uniqueness of such an extension. Using a generalization of Artin-Schreier theory to non-abelian p-groups, we characterize the curves Y for which this lifting property holds and when it is unique, but over a more general ground field.