Non-Archimedean analytic curves in Abelian varieties

Abstract

One of the main subtleties of non-Archimedean analysis is that the natural topology that one puts on non-Archimedean analytic spaces is totally disconnected, meaning that there is a base for the topology consisting of sets which are both open and closed. This makes it difficult, for instance, to define a good notion of analytic function so that one has analytic continuation properties. Of course one can make a sensible definition for what one means by an analytic function, and one of the more successful approaches has been the notion of a analytic function. This approach was started by Tate in [Ta], and a systematic introduction to rigid analysis is given in [BGR]. However, rigid analysis is very different from classical analysis in the sense that essentially none of the topological techniques which are commonplace in classical analysis can be used in rigid analysis. Recently, Berkovich, in [Ber], has come up with a new notion of non-Archimedean analytic spaces which have nice topological properties. For instance, Berkovich's spaces are locally arc-connected, locally compact, Hausdorff spaces. Using Berkovich's theory, one is able to add such topological techniques as covering spaces and map liftings to the study of nonArchimedean analysis. In his book, Berkovich proves the following theorem with the aid of the topological techniques his new theory encourages.