Kudla--Rapoport cycles and derivatives of local densities

Abstract

We prove the local Kudla--Rapoport conjecture, which is a precise identity between the arithmetic intersection numbers of special cycles on unitary Rapoport--Zink spaces and the derivatives of local representation densities of hermitian forms. As a first application, we prove the global Kudla--Rapoport conjecture, which relates the arithmetic intersection numbers of special cycles on unitary Shimura varieties and the central derivatives of the Fourier coefficients of incoherent Eisenstein series. Combining previous results of Liu and Garcia--Sankaran, we also prove cases of the arithmetic Siegel--Weil formula in any dimension.

new subsection 2.8 on "independence of basis"; new subsection 6.4 on "higher local modularity"; several proofs (5.4.1, 7.3.4, 10.4.3) expanded

References in corpus (2)

• Intersections of special cycles on the Shimura variety for GU(1,2)
• Chow groups and $L$-derivatives of automorphic motives for unitary groups

Cited by in corpus (11)

• The endoscopic fundamental lemma for unitary Friedberg-Jacquet periods
• Weil representation and Arithmetic Fundamental Lemma
• Anticyclotomic Euler systems for unitary groups
• Chow groups and $L$-derivatives of automorphic motives for unitary groups
• On the Northcott property for special values of L-functions
• Endoscopy for unitary symmetric spaces
• Special cycles on unitary Shimura curves at ramified primes
• Higher Siegel--Weil formula for unitary groups: the non-singular terms
• Rapoport--Zink spaces for spinor groups with special maximal parahoric level structure
• On local representation densities of hermitian forms and special cycles
• Higher theta series for unitary groups over function fields