Height pairings on orthogonal Shimura varieties

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 t...

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Bibliographic Details
Published in:Compositio mathematica 2017-03, Vol.153 (3), p.474-534
Main Authors: Andreatta, Fabrizio, Goren, Eyal Z., Howard, Benjamin, Madapusi Pera, Keerthi
Format: Article
Language:eng ; fre
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Summary: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.
ISSN:0010-437X
1570-5846
DOI:10.1112/S0010437X1600779X