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Inventiones mathematicae

, Volume 184, Issue 3, pp 629–682 | Cite as

Special cycles on unitary Shimura varieties I. Unramified local theory

  • Stephen Kudla
  • Michael Rapoport
Article

Abstract

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 \(\mathcal{N}\). In the case when p is an inert prime, we define special cycles \({\mathcal{Z}}({\bold x})\) in \(\mathcal{N}\), associated to collections \({\bold x}\) of m ‘special homomorphisms’ with fundamental matrix T∈Herm m (O k ). When m=n and T is nonsingular, we show that the cycle \({\mathcal{Z}}({\bold x})\) 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 \({\mathcal{Z}}({\bold x})\) to be irreducible. When \({\mathcal{Z}}({\bold x})\) 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.

Keywords

Modulus Space Eisenstein Series Fundamental Matrix Hermitian Form Jordan Decomposition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada
  2. 2.Mathematisches Institut der Universität BonnBonnGermany

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