Height pairings on orthogonal Shimura varieties

TL;DR

This paper proves a conjecture connecting arithmetic intersection numbers on certain orthogonal Shimura varieties to derivatives of specific L-functions, generalizing the Gross-Zagier theorem to higher dimensions.

Contribution

It establishes a new relation between intersection multiplicities on orthogonal Shimura varieties and derivatives of Rankin-Selberg L-functions, extending known results to higher-dimensional cases.

Findings

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

Extended Gross-Zagier type results to higher-dimensional Shimura varieties.

Connected special divisors and CM points through explicit formulas.

Abstract

Let $M$ be the Shimura variety associated to the group of spinor similitudes of a quadratic space over $\mathbb{Q}$ of signature $(n,2)$. We prove a conjecture of Bruinier and Yang, relating the arithmetic intersection multiplicities of special divisors and CM points on $M$ to the central derivatives of certain $L$-functions. Each such $L$-function is the Rankin-Selberg convolution associated with a cusp form of half-integral weight $n/2 +1 $, and the weight $n/2$ theta series of a positive definite quadratic space of rank $n$. When $n=1$ the Shimura variety $M$ is a classical quaternionic Shimura curve, and our result is a variant of the Gross-Zagier theorem on heights of Heegner points.