Special cycles on unitary Shimura varieties I. unramified local theory

Details Special cycles on unitary Shimura varieties I. unramified local theory Special cycles on unitary Shimura varieties I. unramified local theory

2008-04-03

arxiv.org/abs/0804.0600v2

Mathematics

Algebraic Geometry

In this new version, suggestions by B. Howard, U. Terstiege, and the referee for Inventiones are taken into account. Also a mi

Scientific paper

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 \Cal N. In the case when p is inert, we define special cycles Z(x) in \Cal N, associated to a collection x of m `special homomorphisms' with fundamental matrix T in Herm_m(OK). When m=n and T is nonsingular, we show that the cycle Z(x) is a union of components of the Ekedahl-Oort stratification, and we give a necessary and sufficient conditions, 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.

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