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arXiv · 1504.00852

Height pairings on orthogonal Shimura varieties

Fabrizio Andreatta·Eyal Z. Goren·Benjamin Howard·Keerthi Madapusi Pera
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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.

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Fabrizio Andreatta, Eyal Z. Goren, Benjamin Howard, Keerthi Madapusi Pera. 2015-04-03. Height pairings on orthogonal Shimura varieties. https://doi.org/10.1112/s0010437x1600779x

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