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Height pairings on orthogonal Shimura varieties
- Part of
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- Journal:
- Compositio Mathematica / Volume 153 / Issue 3 / March 2017
- Published online by Cambridge University Press:
- 02 March 2017, pp. 474-534
- Print publication:
- March 2017
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- Article
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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.
