Height pairings on orthogonal Shimura varieties. (2nd March 2017)

• Record Type: Journal Article
• Authors: Andreatta, Fabrizio; Goren, Eyal Z.; Howard, Benjamin; Madapusi Pera, Keerthi
• Is Part Of: Compositio mathematica. Volume 153:Number 3(2017)
• Page Start: 474
• Page End: 534
• Publication Date: 2017-03-02
• Subjects: 11G18, -- 14G40 (primary); Shimura varieties, -- arithmetic intersection theory, -- complex multiplication, -- special divisor; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X1600779X
• Languages: English
• ISSNs: 0010-437X

Abstract

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 complex multiplication 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.