Complex multiplication cycles and Kudla-Rapoport divisors II

TL;DR

This paper advances the understanding of Kudla-Rapoport divisors on Shimura varieties by constructing compactifications, Green functions, and computing arithmetic intersections, linking these to Fourier coefficients of Eisenstein series.

Contribution

It constructs a toroidal compactification of the integral model, develops Green functions with boundary behavior, and computes arithmetic intersections related to complex multiplication points.

Findings

Green functions have logarithmic singularities near the boundary.

Arithmetic intersection multiplicities relate to Fourier coefficients of Eisenstein series.

The work extends previous models to ramified primes and boundary components.

Abstract

This paper is about the arithmetic of Kudla-Rapoport divisors on Shimura varieties of type GU(n-1,1). In the first part of the paper we construct a toroidal compactification of N. Kramer's integral model of the Shimura variety. This extends work of K.-W. Lan, who constructed a compactification at unramified primes. In the second, and main, part of the paper we use ideas of Kudla to construct Green functions for the Kudla-Rapoport divisors on the open Shimura variety, and analyze the behavior of these functions near the boundary of the compactification. The Green functions turn out to have logarithmic singularities along certain components of the boundary, up to log-log error terms. Thus, by adding a prescribed linear combination of boundary components to a Kudla-Rapoport divisor one obtains a class in the arithmetic Chow group of Burgos-Kramer-Kuhn. In the third and final part of the paper we compute the arithmetic intersection of each of these divisors with a cycle of complex multiplication points. The computation is quickly reduced to the calculations of the author's earlier work Complex multiplication cycles and Kudla-Rapoport divisors. The arithmetic intersection multiplicities are shown to appear as Fourier coefficients of the diagonal restriction of the central derivative of a Hilbert modular Eisenstein series.