Intersections of special cycles on the Shimura variety for GU(1,2)

Abstract

We establish a close connection between intersection multiplicities of special cycles on arithmetic models of the Shimura variety for GU(1,2) and Fourier coefficients of derivatives of certain incoherent Eisenstein series, confirming a conjecture of Kudla and Rapoport. The paper is divided into two parts. The first part is the bulk of the paper and investigates intersections of local special cycles which are defined on a certain moduli space of p-divisible groups. This moduli space of p-divisible groups can be used for the uniformization of the completion of the Shimura variety along the supersingular locus in the fiber at an inert prime p. We investigate the structure of local special cycles and of their intersections. In particular, we give an explicit description of the special fiber of a local special cycle. We then prove an explicit formula for the intersection multiplicity of three local special cycles and relate it to certain hermitian representation densities, also confirming a conjecture of Kudla and Rapoport. To this end, we also give an explicit inductive formula for the relevant hermitian representation densities α_p(1_s,T), where T is of the form diag(p^{a_1},...,p^{a_n}). In the second part, we apply the main result of the first part to prove the relation of intersection multiplicities of (global) special cycles and Fourier coefficients of derivatives of Eisenstein series.