Apparaît dans la collection : Arithmetic Geometry – A Conference in Honor of Hélène Esnault on the Occasion of Her 70th Birthday

For an étale morphism $f:Y \to X$ of schemes over a base~$S$ there are different approaches to define what it means that~$f$ is tame. Behind all of them lies the intuition that the induced morphism of compactifications $\bar{f} : \bar{Y} \to \bar{X}$ is tamely ramified along the boundary $\bar{Y} \setminus Y$ (in an appropriate sense). Many of the tameness definitions work with valuations without relying on the choice of a compactification. Kerz and Schmidt compare these different notions of tameness in their article \emph{On different notions of tameness} mainly working with compactifications. The disadvantage of this approach is that they need to assume resolution of singularities in order to obtain nice compactifications. In my talk I want to present work in progress with Michael Temkin that approaches the problem purely valuation theoretic by using nonachimedean geometry. As a consequence we can drop the assumption on resolution of singularities. The heart of the project lies in a careful study of the geometry of adic curves over an arbitrary affinoid field (of higher rank) and of the wild locus of an étale morphism of such curves.