Height pairings on orthogonal Shimura varieties. (2nd March 2017) - Andreatta, Fabrizio; Goren, Eyal Z.; Howard, Benjamin; Madapusi Pera, Keerthi
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.