On the cohomology of certain PEL-type Shimura varieties
type: Item
instance of: scholarly article
title: On the cohomology of certain PEL-type Shimura varieties
zbMATH Open document ID: 1112.11033
DOI: 10.1215/S0012-7094-05-12935-0
author: Elena Mantovan
published in: Duke Mathematical Journal
publication date: 2006-03-09T00:00:00Z
review text: 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 strat…
reviewed by: Alexandru Ghitza
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 14L05
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 5010590
zbMATH Keywords: Shimura varieties
zbMATH Keywords: l-adic cohomology
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5010590