Weight zero Eisenstein cohomology of Shimura varieties via Berkovich spaces
The main aim of the article under review is to propose a geometric framework for the construction of weight zero Eisenstein classes in the cohomology of Shimura varieties, via Berkovich spaces. The idea begins with the abstract simplicial complex \(\Sigma\) defined by the configuration of boundary divisors of a toroidal compactification \(S'\) of a Shimura variety \(S\). In good situations, the associated Berkovich spaces \(|S|\) and \(|S'|\) (relative to a place \(v\) of the reflex field) will be contractible, and \(\Sigma\) will be homotopy equivalent to the boundary \(|S|'\smallsetminus |S|\). Using the long exact sequence on cohomology and a comparison isomorphism of Berkovich, one thus obtains an identification of the cohomology of \(\Sigma\) with the weight zero cohomology of \(S\) in degree one higher. In this article, the author carries through this idea in the simplest situation.
scientific article; zbMATH DE number 6335994