Georgia Institute of Technology College of Sciences

The structure of non-archimedean curves X and their tame covers f:Y-->X is well understoodand can be adequately described in terms of a (simultaneous) semistable model. In particular, asindicated by the lifting theorem of Amini-Baker-Brugalle-Rabinoff, it encodes all combinatorialand residual algebra-geometric information about f. My talk will be mainly concerned with the morecomplicated case of wild covers, where new discrete invariants appear, with the different function being the most basic one. I will recall its basic properties following my joint work with Cohen and Trushin,and will then pass to the latest results proved jointly with U. Brezner: the different functioncan be refined to an invariant of a residual type, which is a (sort of) meromorphic differential form on the reduction, so that a lifting theorem in the style of ABBR holds for simplest wild covers.