On the cohomology of Rapoport-Zink spaces of EL-type (Q2881359)

Rapoport-Zink spaces are moduli spaces of \(p\)-divisible groups with additional structure. They are an important tool in the study of the reduction to positive characteristic of Shimura varieties of PEL-type. In fact, in many ways Rapoport-Zink spaces can be seen as a local counterpart (in the sense of a local-global principle) of these Shimura varieties. In particular, their \(\ell\)-adic cohomology is closely related to the local Langlands correspondence.NEWLINENEWLINEIn the paper under review, the so-called EL case is considered. Essentially this means that the underlying \(p\)-adic group is the general linear group \(\mathrm{GL}_n\). The main result is a description of the cohomology of a Rapoport-Zink space (more precisely, of a certain sum of \(\mathrm{Ext}\) groups of such cohomology groups for a certain collection of Rapoport-Zink spaces) in terms of the local Langlands correspondence and other natural representation-theoretic operations. The main result is such a description, proved using results of Mantovan and generalizing results of Harris and Taylor. We will not state it in detail, because it is a bit technical, but refer to the introduction of the paper instead.NEWLINENEWLINEThe proof uses global methods. It makes use of the interplay between Shimura varieties, Igusa varieties and Rapoport-Zink spaces. The proof of the precise relationship between the cohomology of the Shimura variety and the Igusa variety in question is based on a comparison of trace formulas.