Special cycles on unitary Shimura varieties I. Unramified local theory

• 成果类型: Article
• 署名作者: Kudla, Stephen; Rapoport, Michael
• 署名单位: University of Bonn; University of Toronto
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-010-0298-z
• 发表日期: 2011
• 页码: 629-682
• 关键词: models; derivatives

摘要

The supersingular locus in the fiber at p of a Shimura variety attached to a unitary similitude group GU(1, n - 1) over Q is uniformized by a formal scheme N. In the case when p is an inert prime, we define special cycles Z(x) in N, associated to collections x of m 'special homomorphisms' with fundamental matrix T is an element of Herm(m)(O-k). When m = n and T is nonsingular, we show that the cycle Z(x) is either empty or is a union of components of the Ekedahl-Oort stratification, and we give a necessary and sufficient condition, in terms of T, for Z(x) to be irreducible. When Z(x) is zero dimensional-in which case it reduces to a single point-we determine the length of the corresponding local ring by using a variant of the theory of quasi-canonical liftings. We show that this length coincides with the derivative of a representation density for hermitian forms.