Heights of Kudla-Rapoport divisors and derivatives of L-functions

TL;DR

This paper establishes a deep connection between heights of Kudla-Rapoport divisors on unitary Shimura varieties and derivatives of L-functions, extending the Gross-Zagier formula to higher dimensions.

Contribution

It constructs an arithmetic theta lift from harmonic Maass forms to the Chow group and proves an explicit formula relating heights to L-function derivatives.

Findings

Height pairing equals the central derivative of a convolution L-function.

Introduces a new method for computing improper arithmetic intersections.

Generalizes Gross-Zagier formula to higher-dimensional unitary Shimura varieties.

Abstract

We study special cycles on integral models of Shimura varieties associated with unitary similitude groups of signature (n-1,1). We construct an arithmetic theta lift from harmonic Maass forms of weight 2-n to the arithmetic Chow group of the integral model of a unitary Shimura variety, by associating to a harmonic Maass form f a linear combination of Kudla-Rapoport divisors, equipped with the Green function given by the regularized theta lift of f. Our main result is an equality of two complex numbers: (1) the height pairing of the arithmetic theta lift of f against a CM cycle, and (2) the central derivative of the convolution L-function of a weight n cusp form (depending on f) and the theta function of a positive definite hermitian lattice of rank n-1. When specialized to the case n=2, this result can be viewed as a variant of the Gross-Zagier formula for Shimura curves associated to unitary groups of signature (1,1). The proof relies on, among other things, a new method for computing improper arithmetic intersections.