Height pairings on orthogonal Shimura varieties

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.

Final version. To appear in Compos. Math

References in corpus (3)

• Revêtements étales et groupe fondamental (SGA 1)
• 2-adic integral canonical models and the Tate conjecture in characteristic 2
• Faltings heights of big CM cycles and derivatives of L-functions

Cited by in corpus (8)

• On two arithmetic theta lifts
• Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields
• Picard rank jumps for K3 surfaces with bad reduction
• On a question of Ekedahl and Serre
• Pullback formulas for arithmetic cycles on orthogonal Shimura varieties
• On a Conjecture of Yui and Zagier
• Nonvanishing of self-dual $L$-values via spectral decomposition of shifted convolution sums
• Derivatives of L-functions