Modularity of generating series of divisors on unitary Shimura varieties II: arithmetic applications

TL;DR

This paper establishes formulas linking derivatives of L-functions to arithmetic intersections on unitary Shimura varieties and proves a case of Colmez's conjecture, advancing the understanding of arithmetic geometry and automorphic forms.

Contribution

It introduces new formulas connecting L-function derivatives with intersection pairings and confirms a special case of Colmez's conjecture, building on prior modularity results.

Findings

Formulas relating L-function derivatives to intersection pairings

Proof of a case of Colmez's conjecture on Faltings heights

Extension of modularity results for generating series of divisors

Abstract

We prove two formulas in the style of the Gross-Zagier theorem, relating derivatives of L-functions to arithmetic intersection pairings on a unitary Shimura variety. We also prove a special case of Colmez's conjecture on the Faltings heights of abelian varieties with complex multiplication. These results are derived from the authors' earlier results on the modularity of generating series of divisors on unitary Shimura varieties.