Heights of Kudla–Rapoport divisors and derivatives of L-functions

Jan Hendrik Bruinier Technische Universität Darmstadt Tonghai Yang University of Wisconsin Madison Benjamin Howard Boston University

Number Theory mathscidoc:1705.24004

Invent. Math., 201, (1), 1-95, 2015
We study special cycles on integral models of Shimura varieties associated with unitary similitude groups of signature (n.1, 1).We construct an arithmetic theta lift from harmonic Maass forms of weight 2 . n to the arithmetic Chow group of the integral model of a unitary Shimura variety, by associating to a harmonic Maass form f a linear combination of Kudla– Rapoport divisors, equipped with the Green function given by the regularized theta lift of f . Our main result is an equality of two complex numbers: (1) the height pairing of the arithmetic theta lift of f against a CM cycle, and (2) the central derivative of the convolution L-function of a weight n cusp form (depending on f ) and the theta function of a positive definite hermitian lattice of rank n − 1. When specialized to the case n = 2, this result can be viewed as a variant of the Gross–Zagier formula for Shimura curves associated to unitary groups of signature (1, 1). The proof relies on, among other things, a new method for computing improper arithmetic intersections.
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@inproceedings{jan2015heights,
  title={Heights of Kudla–Rapoport divisors and derivatives  of L-functions},
  author={Jan Hendrik Bruinier, Tonghai Yang, and Benjamin Howard},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170530153951890673768},
  booktitle={Invent. Math.},
  volume={201},
  number={1},
  pages={1-95},
  year={2015},
}
Jan Hendrik Bruinier, Tonghai Yang, and Benjamin Howard. Heights of Kudla–Rapoport divisors and derivatives of L-functions. 2015. Vol. 201. In Invent. Math.. pp.1-95. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170530153951890673768.
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