On the cohomology of certain PEL-type Shimura varieties
The article under review is dedicated to the study of the local geometry at a prime \(p\) of certain PEL-type Shimura varieties, based on the Newton polygon stratification of the fibers modulo \(p\) of these varieties. Although fascinating in their own right, these results are most favorably viewed from the perspective of the local Langlands correspondence. The prototype for this important application is the proof of the local Langlands correspondence for \(\text{GL}_n\) over \(p\)-adic fields by [\textit{M. Harris} and \textit{R. Taylor}, The geometry and cohomology of some simple Shimura varieties. With an appendix by Vladimir G. Berkovich. Annals of Mathematics Studies 151. Princeton, NJ: Princeton University Press (2001; Zbl 1036.11027)]. Very simplistically, the idea is to realize the correspondence in the cohomology of certain unitary Shimura varieties; in turn, the required results on \(\ell\)-adic cohomology follow from a careful study of the geometry of these varieties and their reductions. In a previous article [On certain unitary group Shimura varieties. Shimura varieties, Rapoport-Zink spaces and local Langlands correspondences. Paris: Société Mathématique de France. Astérisque 291, 201--331 (2004; Zbl 1062.11036)], the author generalized the approach of Harris-Taylor to a larger class of Shimura varieties coming from unitary groups. The present paper generalizes these results and techniques even further, more precisely to any PEL-type Shimura variety with a hyperspecial level subgroup -- the significance of this condition is that it guarantees good reduction at the fixed prime \(p\). The PEL-type Shimura varieties are moduli spaces of abelian varieties with additional (polarization-\-endomorphism-\-level) structure. Their reductions modulo \(p\) have a stratification by Newton polygons -- each stratum corresponds to a certain isogeny class of the \(p\)-divisible group of the abelian variety. The author undertakes the study of the geometry of these strata by making use of two main tools: Igusa varieties and Rapoport-Zink spaces. Igusa varieties, introduced by Harris and Taylor in their aforementioned book, are generalizations of the classical Igusa curves. They describe the loci of constant isomorphism class of the \(p\)-divisible group of the abelian variety, and are finite étale covers of certain subvarieties of the Newton strata. On the other hand, the Rapoport-Zink spaces (introduced in [\textit{M. Rapoport, Th. Zink}, Period spaces for \(p\)-divisible groups. Annals of Mathematics Studies. 141. Princeton, NJ: Princeton Univ. Press. (1996; Zbl 0873.14039)]) uniformize the loci corresponding to prime-to-\(p\) isogeny classes of abelian varieties. The key observation is then that products of Igusa varieties and Rapoport-Zink spaces cover entire Newton strata, and it remains to obtain precise information on the geometry and cohomology of these products. In the last sections of the paper, the author uses this information to compute and compare the sheaves of vanishing cycles on the Shimura varieties to those on the Rapoport-Zink spaces. This comparison is then applied to a comparison between the \(\ell\)-adic cohomology of a proper Shimura variety and the \(\ell\)-adic cohomology with compact supports of the Igusa varieties and the Rapoport-Zink spaces.
scientific article; zbMATH DE number 5010590
scientific article