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Height pairings on orthogonal Shimura varieties

Published online by Cambridge University Press:  02 March 2017

Fabrizio Andreatta
Affiliation:
Dipartimento di Matematica ‘Federigo Enriques’, Università di Milano, via C. Saldini 50, Milano, Italia email fabrizio.andreatta@unimi.it
Eyal Z. Goren
Affiliation:
Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St. West, Montreal, QC, Canada email eyal.goren@mcgill.ca
Benjamin Howard
Affiliation:
Department of Mathematics, Boston College, 140 Commonwealth Ave, Chestnut Hill, MA, USA email howardbe@bc.edu
Keerthi Madapusi Pera
Affiliation:
Department of Mathematics, University of Chicago, 5734 S University Ave, Chicago, IL, USA email keerthi@math.uchicago.edu

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 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.

Type
Research Article
Copyright
© The Authors 2017 

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References

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