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

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