Faltings heights of abelian varieties with complex multiplication

TL;DR

This paper proves a conjecture linking intersection multiplicities on Shimura varieties to derivatives of L-functions and applies it to confirm an averaged Colmez conjecture on Faltings heights of CM abelian varieties.

Contribution

It establishes a new connection between arithmetic geometry and automorphic L-functions, confirming an averaged Colmez conjecture.

Findings

Proved a conjecture relating intersection multiplicities to L-function derivatives.

Confirmed an averaged version of Colmez's conjecture on Faltings heights.

Established new links between Shimura varieties and CM abelian varieties.

Abstract

Let M be the Shimura variety associated with the group of spinor similitudes of a rational quadratic space over of signature (n,2). We prove a conjecture of Bruinier-Kudla-Yang, relating the arithmetic intersection multiplicities of special divisors and big CM points on M to the central derivatives of certain $L$-functions. As an application of this result, we prove an averaged version of Colmez's conjecture on the Faltings heights of CM abelian varieties.