Special cycles on unitary Shimura varieties I. Unramified local theory
Special cycles on unitary Shimura varieties I. Unramified local theory
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单一志村品种的特殊周期 I 无分支的当地理论
DOI：
10.1007/s00222-010-0298-z
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发表时间：
2011
影响因子：
3.1
通讯作者：
M. Rapoport
中科院分区：
文献类型：
--
作者：
S. Kudla;M. Rapoport
The supersingular locus in the fiber atpof a Shimura variety attached to a unitary similitude group GU(1,n−1) over ℚ is uniformized by a formal scheme. In the case whenpis an inert prime, we define special cycles ${\mathcal{Z}}({\bold x})$ in, associated to collections ${\bold x}$ ofm‘special homomorphisms’ with fundamental matrixT∈Hermm(Ok). Whenm=nandTis 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 ofT, 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.
The supersingular locus in the fiber atpof a Shimura variety attached to a unitary similitude group GU(1,n−1) over ℚ is uniformized by a formal scheme. In the case whenpis an inert prime, we define special cycles ${\mathcal{Z}}({\bold x})$ in, associated to collections ${\bold x}$ ofm‘special homomorphisms’ with fundamental matrixT∈Hermm(Ok). Whenm=nandTis 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 ofT, 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.
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