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 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.