Kudla--Rapoport cycles and derivatives of local densities

TL;DR

This paper proves the local Kudla--Rapoport conjecture linking intersection numbers of special cycles to derivatives of local densities, and applies this to establish the global conjecture and cases of the arithmetic Siegel--Weil formula.

Contribution

It provides a proof of the local Kudla--Rapoport conjecture and derives the global conjecture and new cases of the arithmetic Siegel--Weil formula.

Findings

Proof of the local Kudla--Rapoport conjecture

Verification of the global Kudla--Rapoport conjecture

Cases of the arithmetic Siegel--Weil formula confirmed

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.