# Special cycles on unitary Shimura varieties II: global theory

@article{Kudla2009SpecialCO, title={Special cycles on unitary Shimura varieties II: global theory}, author={Stephen S. Kudla and Michael Rapoport}, journal={arXiv: Algebraic Geometry}, year={2009} }

We introduce moduli spaces of abelian varieties which are arithmetic models of Shimura varieties attached to unitary groups of signature (n-1, 1). We define arithmetic cycles on these models and study their intersection behaviour. In particular, in the non-degenerate case, we prove a relation between their intersection numbers and Fourier coefficients of the derivative at s=0 of a certain incoherent Eisenstein series for the group U(n, n). This is done by relating the arithmetic cycles to their… Expand

#### 68 Citations

Intersections of special cycles on the Shimura variety for

- Mathematics
- 2010

We establish a close connection between intersection multiplicities of special cycles on arithmetic models of the Shimura variety for GU(1,2) and Fourier coefficients of derivatives of certain… Expand

IMPROPER INTERSECTIONS OF KUDLA–RAPOPORT DIVISORS AND EISENSTEIN SERIES

- Mathematics
- Journal of the Institute of Mathematics of Jussieu
- 2015

We consider a certain family of Kudla–Rapoport cycles on an integral model of a Shimura variety attached to a unitary group of signature (1, 1), and prove that the arithmetic degrees of these cycles… Expand

On the Global Structure of Special Cycles on Unitary Shimura Varieties

- Mathematics
- Canadian Journal of Mathematics
- 2013

Abstract In this paper, we study the reduced loci of special cycles on local models of the Shimura variety for $\text{GU}\left( 1,\,n\,-\,1 \right)$ . Those special cycles are defined by Kudla and… Expand

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

- Mathematics
- 2017

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… Expand

Special cycles on unitary Shimura varieties at ramified primes I

- Mathematics
- 2018

In this paper, we study special cycles on unitary Shimura varieties at the ramified primes of a imaginary quadratic number field ${\bf k}$ and special cycles on the corresponding Rapoport-Zink… Expand

On Shimura varieties for unitary groups

- Mathematics
- 2019

This is a largely expository article based on our previous work on arithmetic diagonal cycles on unitary Shimura varieties. We define a class of Shimura varieties closely related to unitary groups… Expand

Unitary cycles on Shimura curves and the Shimura lift II

- Mathematics
- Compositio Mathematica
- 2014

Abstract We consider two families of arithmetic divisors defined on integral models of Shimura curves. The first was studied by Kudla, Rapoport and Yang, who proved that if one assembles these… Expand

Green forms and the arithmetic Siegel–Weil formula

- Mathematics
- Inventiones mathematicae
- 2018

We construct natural Green forms for special cycles in orthogonal and unitary Shimura varieties, in all codimensions, and, for compact Shimura varieties of type $$\mathrm {O}(p,2)$$O(p,2) and… Expand

GREEN FORMS AND THE LOCAL ARCHIMEDEAN ARITHMETIC SIEGEL-WEIL FORMULA

- 2018

The goal of this paper is to construct natural Green forms for special cycles in orthogonal and unitary Shimura varieties, in any codimension, and to show that the resulting local archimedean height… Expand

Special Cycles on Shimura Curves and the Shimura Lift

- Mathematics
- 2012

Special cycles on Shimura curves and the Shimura lift Siddarth Sankaran Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2012 The main results of this thesis describe a… Expand

#### References

SHOWING 1-10 OF 66 REFERENCES

COHOMOLOGICAL ARITHMETIC CHOW RINGS

- Mathematics
- Journal of the Institute of Mathematics of Jussieu
- 2006

We develop a theory of abstract arithmetic Chow rings, where the role of the fibres at infinity is played by a complex of abelian groups that computes a suitable cohomology theory. As particular… Expand

Derivatives of Eisenstein Series and Arithmetic Geometry

- Mathematics, Materials Science
- 2002

We describe connections between the Fourier coefficients of derivatives of Eisenstein series and invariants from the arithmetic geometry of the Shimura varieties M associated to rational quadratic… Expand

Cycles on Siegel threefolds and derivatives of Eisenstein series

- Mathematics
- 2000

Abstract We consider the Siegel modular variety of genus 2 and a p-integral model of it for a good prime p>2, which parametrizes principally polarized abelian varieties of dimension two with a level… Expand

Arithmetic Hirzebruch Zagier cycles

- Mathematics
- 1999

We define special cycles on arithmetic models of twisted Hilbert-Blumenthal surfaces at primes of good reduction. These are arithmetic versions of these cycles. In particular, we characterize the… Expand

Intersection theory on Shimura surfaces

- Mathematics
- Compositio Mathematica
- 2009

Abstract Kudla has proposed a general program to relate arithmetic intersection multiplicities of special cycles on Shimura varieties to Fourier coefficients of Eisenstein series. The lowest… Expand

On Local Models with Special Parahoric Level Structure

- Mathematics
- 2009

We consider the local model of a Shimura variety of PEL type, with the unitary similitudes corresponding to a ramified quadratic extension of $\mathbb{Q}_p$ as defining group. We examine the cases… Expand

Hodge Cycles, Motives, and Shimura Varieties

- Mathematics
- 1989

General Introduction.- Notations and Conventions.- Hodge Cycles on Abelian Varieties.- Tannakian Categories.- Langlands's Construction of the Taniyama Group.- Motifs et Groupes de Taniyama.-… Expand

ARITHMETIC OF UNITARY GROUPS

- Mathematics
- 1964

The purpose of this paper is to develop the theory of elementary divisors, to prove the approximation theorem, and to determine the class number for the following two types of algebraic groups: (i)… Expand

Theta dichotomy for unitary groups

- Mathematics
- 1996

Some recent work of Gross and Prasad [14] suggests that the root numbers attached to certain symplectic representations of the Weil-Deligne group of a local field F control certain branching rules… Expand

Local models in the ramified case I. The EL-case

- Mathematics
- 2000

Local models are schemes defined in linear algebra terms that describe the 'etale local structure of integral models for Shimura varieties and other moduli spaces. We point out that the flatness… Expand