Special cycles on unitary Shimura varieties I. unramified local theory

TL;DR

This paper studies special cycles on unitary Shimura varieties at inert primes, characterizing their structure, irreducibility, and local properties, and linking their local ring length to derivatives of hermitian form representation densities.

Contribution

It provides a detailed analysis of special cycles on unitary Shimura varieties, including criteria for irreducibility and explicit computation of local ring lengths in terms of hermitian form densities.

Findings

Z(x) cycles are unions of Ekedahl-Oort strata components.

Criteria for irreducibility of Z(x) based on the fundamental matrix T.

Length of local rings at zero-dimensional cycles matches derivatives of hermitian form representation densities.

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 \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.