Étale covers of a punctured p-adic disc

Abstract

The aim of this paper is to report on recent work on éetale covers of the punctured disc. The paper surveys basic results on curves over p-adic fields and explains ideas of Riemann's existence problem for a p-adic field giving full details and as well new results. In [17] the problem of the extension of an étale cover φ* : X* → D* of a punctured disc D*, defined over an p-adic field K, to a (ramified) cover φ : X → D was proved in the case where the base field K has characteristic 0. The behavior of the discriminant of such an étale cover is now well understood. Moreover, new results of the second author [22] in the case of positive characteristic char(K) > 0 are presented as well. The paper ends with two interesting examples of covers in positive characteristic which are not extendable.