# Modularity of generating series of divisors on unitary Shimura varieties

@article{Bruinier2020ModularityOG, title={Modularity of generating series of divisors on unitary Shimura varieties}, author={Jan H. Bruinier and Benjamin J. Howard and Stephen S. Kudla and Michael Rapoport and Tonghai Yang}, journal={Ast{\'e}risque}, year={2020} }

We form generating series of special divisors, valued in the Chow group and in the arithmetic Chow group, on the compactified integral model of a Shimura variety associated to a unitary group of signature (n-1,1), and prove their modularity. The main ingredient of the proof is the calculation of the vertical components appearing in the divisor of a Borcherds product on the integral model.

## 10 Citations

Modularity of generating series of divisors on unitary Shimura varieties II: arithmetic applications

- MathematicsAstérisque
- 2020

We form generating series of special divisors, valued in the Chow group and in the arithmetic Chow group, on the compactified integral model of a Shimura variety associated to a unitary group of…

Special cycles on unitary Shimura curves at ramified primes

- Mathematics
- 2020

In this paper, we study special cycles on the Kramer model of $\mathrm{GU}(1,1)(F)$ Rapoport-Zink spaces where $F$ is a ramified extension of $\mathbb{Q}_p$ with the assumption that the underlying…

Deformations of Theta Integrals and A Conjecture of Gross-Zagier

- Mathematics
- 2022

In this paper, we complete the proof of Gross-Zagier’s conjecture concerning algebraicity of higher Green functions at a single CM point on the product of modular curves. The new ingredient is an…

Kudla program for unitary Shimura varieties

- MathematicsSCIENTIA SINICA Mathematica
- 2021

In this paper, we first review and summarize some recent progress in Kudla program on unitary Shimura varieties. We show how the local arithmetic Siegel-Weil formula implies the global arithmetic…

On the Arithmetic Fundamental Lemma conjecture over a general $p$-adic field

- Mathematics
- 2021

We prove the arithmetic fundamental lemma conjecture over a general p-adic field with odd residue cardinality q ≥ dimV . Our strategy is similar to the one used by the second author during his proof…

Picard rank jumps for K3 surfaces with bad reduction

- Mathematics
- 2022

Let X be a K3 surface over a number field. We prove that X has infinitely many specializations where its Picard rank jumps, hence extending our previous work with Shankar–Shankar–Tang to the case…

Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields

- Mathematics
- 2019

Given a K3 surface $X$ over a number field $K$, we prove that the set of primes of $K$ where the geometric Picard rank jumps is infinite, assuming that $X$ has everywhere potentially good reduction.…

Theta series and generalized special cycles on Hermitian locally symmetric manifolds

- MathematicsMathematische Annalen
- 2022

We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G=\mathrm{U}(p,q)$, $\mathrm{Sp}(2n,\mathbb{R}) $ and $\mathrm{O}^*(2n)$.…

Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture

- MathematicsInventiones mathematicae
- 2022

Let $\mathscr{X} \rightarrow C$ be a non-isotrivial and generically ordinary family of K3 surfaces over a proper curve $C$ in characteristic $p \geq 5$. We prove that the geometric Picard rank jumps…

Special cycles on toroidal compactifications of orthogonal Shimura varieties

- MathematicsMathematische Annalen
- 2021

We determine the behavior of automorphic Green functions along the boundary components of toroidal compactifications of orthogonal Shimura varieties. We use this analysis to define boundary…