Modular Heights of Unitary Shimura Varieties II: Arithmetic Generating Series of Divisors

Abstract

This is the second of a series of three papers, in which we prove a formula expressing the modular height of a unitary Shimura variety over a CM number field in terms of the logarithmic derivative of the Hecke L-function associated with the CM extension. The main idea of our proof is to compare the holomorphic projection of the derivative of a certain mixed Eisenstein-theta series and the arithmetic degree of a generating series of divisors on unitary Shimura varieties. In this paper, we define the arithmetic generating series of divisors on unitary Shimura varieties, compute the corresponding arithmetic intersection numbers, and derive the modular height formula for unitary Shimura curves as well as the height formula for a CM point on them.