2013/03/31 by Jan Hendrik Bruinier, Benjamin Howard, Tonghai Yang · 15 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebra over a field #Algebraic Geometry and Number Theory #Cusp form #Hermitian matrix #Lift (data mining) #Shimura variety #Signature (topology) #Theta function #Unitary state #math.NT
paper · pdf · doi:10.1007/s00222-014-0545-9
published in Inventiones mathematicae 201(1), 1-95 (Springer Science+Business Media) · Final version. To appear in Invent. Math
openalex publication_date 2014/09/23 · arxiv created 2014/10/19 · arxiv updated 2014/10/21 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.