Arithmetic inner product formula for unitary groupsyl2269/thesis_d.pdf · 2012-09-06 · Arithmetic...

Arithmetic inner product formula

for unitary groups

Yifeng Liu

Submitted in partial fulfillment of therequirements for the degree of

Doctor of Philosophyin the Graduate School of Arts and Sciences

COLUMBIA UNIVERSITY

2012

c© 2012

Yifeng Liu

All rights reserved

Abstract


for unitary groups

Yifeng Liu

We study central derivatives of L-functions of cuspidal automorphic representations

for unitary groups of even variables defined over a totally real number field, and their

relation with the canonical height of special cycles on Shimura varieties attached to

unitary groups of the same size. We formulate a precise conjecture about an arithmetic

analogue of the classical Rallis’ inner product formula, which we call arithmetic inner

product formula, and confirm it for unitary groups of two variables. In particular,

we calculate the Neron–Tate height of special points on Shimura curves attached to

certain unitary groups of two variables.

For an irreducible cuspidal automorphic representation of a quasi-split unitary

group, we can associate it an ε-factor, which is either 1 or −1, via the dichotomy

phenomenon of local theta lifting. If such factor is −1, the central L-value of the rep-

resentation always vanishes and the Rallis’ inner product formula is not interesting.

Therefore, we are motivated to consider its central derivative, and propose the arith-

metic inner product formula. In the course of such formulation, we prove a modularity

theorem of the generating series on the level of Chow groups. We also show the coho-

mological triviality of the arithmetic theta lifting, which is a necessary step to consider

the canonical height. As evidence, we also prove an arithmetic local Siegel–Weil for-

mula at archimedean places for unitary groups of arbitrary sizes, which contributes as

a part of the local comparison of the conjectural arithmetic inner product formula.

Contents

Table of Contents i

Acknowledgments v

1 Introduction 1

2 Doubling method and analytic kernel functions 12

2.1 Siegel–Weil formula and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Degenerate principal series and Eisenstein series . . . . . . . . . . . . . . . . . 13

2.1.2 Real quasi-split unitary groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.3 Weil representations and theta functions . . . . . . . . . . . . . . . . . . . . . . 16

2.1.4 Siegel–Weil formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Doubling integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Decomposition of global period integrals . . . . . . . . . . . . . . . . . . . . . . 19

2.2.2 Local zeta integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3 Central special values of L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.1 Theta lifting and central L-values . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.2 Vanishing of central L-values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Analytic kernel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4.1 Regular test functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4.2 Test functions of higher discriminant . . . . . . . . . . . . . . . . . . . . . . . . 36

2.4.3 Density of test functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 Arithmetic theta lifting and arithmetic kernel functions 41

3.1 Modularity of generating series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.1.1 Shimura varieties of unitary groups . . . . . . . . . . . . . . . . . . . . . . . . . 42

i

3.1.2 Special cycles and generating series . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.1.3 Pullback formula and modularity theorem . . . . . . . . . . . . . . . . . . . . . 45

3.2 Smooth compactification of unitary Shimura varieties . . . . . . . . . . . . . . . . . . 50

3.2.1 Compactified unitary Shimura varieties . . . . . . . . . . . . . . . . . . . . . . 50

3.2.2 Compactified generating series . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3 Arithmetic theta lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.3.1 Cohomological triviality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.3.2 Arithmetic inner product formula: the general conjecture . . . . . . . . . . . . 59

3.4 Arithmetic kernel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.4.1 Neron–Tate height pairing on curves . . . . . . . . . . . . . . . . . . . . . . . . 60

3.4.2 Degree of generating series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.4.3 Decomposition of arithmetic kernel functions . . . . . . . . . . . . . . . . . . . 67

4 Comparison at infinite places 69

4.1 Archimedean Whittaker integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.1.1 Elementary reduction steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.1.2 Analytic continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.1.3 First-order derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2 Archimedean local height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.2.1 Green currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.2.2 Invariance under U(2): an exercise in Calculus . . . . . . . . . . . . . . . . . . 87

4.3 An archimedean local Siegel–Weil formula . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.3.1 Comparison on the hermitian domain . . . . . . . . . . . . . . . . . . . . . . . 95

4.3.2 Proof of Lemma 4.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.3.3 Comparison on Shimura varieties . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5 Comparison at finite places: good reduction 104

5.1 Integral models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.1.1 Change of Shimura data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.1.2 Moduli interpretations and integral models: minimal level . . . . . . . . . . . . 108

5.1.3 Basic abelian scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.1.4 The nearby space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.1.5 Integral special subschemes: minimal level . . . . . . . . . . . . . . . . . . . . . 116

5.1.6 Remark on the case F = Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

ii

5.2 Local intersection numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.2.1 p-adic uniformization of supersingular locus . . . . . . . . . . . . . . . . . . . . 119

5.2.2 Special formal subschemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.2.3 A formula for local intersection multiplicity . . . . . . . . . . . . . . . . . . . . 121

5.2.4 Proof of Proposition 5.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.3 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.3.1 Non-archimedean Whittaker integrals . . . . . . . . . . . . . . . . . . . . . . . 130

5.3.2 Comparison on Shimura curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6 Comparison at finite places: bad reduction 135

6.1 Split case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.1.1 Integral models and ordinary reduction . . . . . . . . . . . . . . . . . . . . . . 136

6.1.2 Coherence for intersection numbers . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.2 Quasi-split case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.2.1 Integral models and supersingular reduction . . . . . . . . . . . . . . . . . . . . 139


6.3 Nonsplit case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.3.1 Integral models and Cerednik–Drinfeld uniformization: minimal level . . . . . . 146

6.3.2 Integral models and Cerednik–Drinfeld uniformization: higher level . . . . . . . 147


7 Arithmetic inner product formula: the main theorem 154

7.1 Holomorphic projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

7.1.1 Holomorphic and quasi-holomorphic projection: generality . . . . . . . . . . . . 155

7.1.2 Siegel–Fourier expansion of Eisenstein series on U(2, 2) . . . . . . . . . . . . . . 158

7.1.3 Holomorphic projection of analytic kernel functions . . . . . . . . . . . . . . . . 161

7.1.4 Quasi-holomorphic projection of analytic kernel functions . . . . . . . . . . . . 163

7.2 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

7.2.1 Difference of kernel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

7.2.2 The final step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Appendix 169

A.1 Theta dichotomy for unitary groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

A.2 Uniqueness of local invariant functionals . . . . . . . . . . . . . . . . . . . . . . . . . . 174

iii

A.3 Theta correspondence of unramified representations . . . . . . . . . . . . . . . . . . . 177

Bibliography 184

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Index of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

iv

Acknowledgments

Without the advise and encouragement of many people, this work would never have come into ex-

istence. First of all, I am grateful to my advisor, Shou-Wu Zhang for introducing me this subject

including the original problem, and his consistent encouragement and interest in this work. I am

indebted to Xinyi Yuan, Shou-Wu Zhang and Wei Zhang who were kind enough to share with me

some of the ideas in their recent joint work, which are crucial to this one. I am also thankful to Wee

Teck Gan, Atsushi Ichino, Aise Johan de Jong, Luis Garcia Martinez, Frans Oort, Michael Rapoport,

Mingmin Shen, Ye Tian, Chenyan Wu, Shunsuke Yamana, Tonghai Yang and Weizhe Zheng for useful

conversations during the long-term preparation.

I would like to thank Stephen S. Kudla and Joachim Schwermer for inviting me to the workshop

Automorphic Forms: New Directions at Mathematisches Forschungsinstitut Oberwolfach in March

2011 and to present this work there; YoungJu Choie and Sug Woo Shin for inviting me to give a mini

course on this subject in the Theta Festival : The 4th ILJU School of Mathematics held at Pohang

University of Science and Technology in August 2011. Without their wonderful organization and

hospitality, I would have missed very good opportunities to communicate with other mathematicians

on this work and related subjects.

I also deeply appreciate the Morningside Center of Mathematics, Chinese Academy of Sciences,

in Beijing for providing me a perfect research environment with full hospitality during the summers

of last four years. Especially in July 2008, I was able to attend a workshop on Arithmetic Geometry

held at the Center, where Jianshu Li and Shou-Wu Zhang gave two talks that directly motivated me

to work on this problem in the following years.

For my Ph.D. life from 2007 to 2012, I must thank all the staff, faculty and students at Department

of Mathematics, Columbia University for making it enjoyable.

Last but not least, I thank my family and my friends for their everlasting encouragement and

support in every aspects of my life.

v

1

Chapter 1

Introduction

A central question in Number Theory is to solve Diophantine equations, that is, to study the poly-

nomial equations in the field of rational numbers, or more generally, in number fields. From the

viewpoint of algebraic geometry, the zero locus of a set of polynomial equations in an affine or pro-

jective space defines naturally some geometric object, which is called an algebraic variety. Therefore,

it is important to study geometry objects defined over number fields. Among them, there is a special

class of algebraic varieties, called Shimura varieties, for which one can systematically construct a large

supply of points (i.e., solutions), or more generally, cycles (i.e., families of solutions).

In this article, we study Shimura varieties associated to certain unitary groups and their special

cycles. Moreover, we relate the arithmetic of such cycles to L-functions of automorphic representations,

which are analytic objects. The method we use to set up such a relation is an arithmetic analogue of the

theta lifting in the classical theory of automorphic representation. This is first observed by S. Kudla

[Kud1997,Kud2002,Kud2003] and later developed by S. Kudla, M. Rapport and T. Yang [KRY2006].

We formulate a precise conjecture about an arithmetic analogue of the classical Rallis’ inner product

formula, which we call arithmetic inner product formula, and confirm it for unitary groups of two

variables. In particular, we calculate the Neron–Tate height of special points on Shimura curves

attached to certain unitary groups of two variables.

2

Rallis’ inner product formula

Let us briefly review the Rallis’ inner product formula in the classical theory of theta lifting, which

first appeared in [Ral1984]. The original formula is to calculate the Petersson inner product of two

automorphic forms on an orthogonal group that are lifted from a symplectic group through theta

lifting. It turns out, using the Siegel–Weil formula, that the inner product is related to a diagonal

integral on the doubling symplectic group of the original automorphic forms with certain Eisenstein

series. This doubling method was later generalized to other cases by S. Gelbart, I. Piatetski-Shapiro

and S. Rallis [GPSR1987]. This diagonal integral is in fact Eulerian. In other words, it decomposes

into so-called local zeta integrals. These local zeta integrals are directly related to the L-factors of the

corresponding representations. In fact, they prove in many cases that when everything is unramified,

the local zeta integral coincides with the local Langlands L-factor, modified by some Tate L-factors.

Later, J. Li [Li1992] extended such results to unitary groups.

In the introduction, we only look at a special case of Rallis’ inner product formula, which is parallel

to the arithmetic theory developed later. Let F be a totally real field, and E/F a totally imaginary

quadratic extension. For an integer n ≥ 1, let H ′ = U(n, n)F be the unique quasi-split unitary group

of a skew-hermitian space over E (with respect to the Galois involution of E/F ) of rank 2n. Let H

be the unitary group of a hermitian space V over E of rank 2n. Both H ′ and H are reductive groups

over F . We have a Weil representation 1 ω of H ′(AF )×H(AF ), realizing on S(V (AE)n): the space

of Schwartz functions on V (AE)n. For such an element φ ∈ S(V (AE)n), we have the following theta

series

θ(g, h;φ) =∑

x∈V n(E)

(ω(g, h)φ) (x),

which is a smooth, slowly increasing function on H ′(F )\H ′(AF )×H(F )\H(AF ). Let π ⊂ A0(H ′) be

an irreducible representation of H ′(AF ) contained in the space of cusp forms of H ′. For every f ∈ π

and φ ∈ S(V (AE)n), we define the theta lifting to be

θfφ(h) =

∫H′(F )\H′(AF )

θ(g, h;φ)f(g)dg.

Similarly, we have θf∨

φ∨ for f∨ ∈ π∨ =f | f ∈ π

and φ∨ ∈ S(V (AE)n), viewed as the underlying

space of the contragredient representation ω∨. Applying a regularized Siegel–Weil formula of A. Ichino

1It depends on the choice of a nontrivial additive character ψ of F\AF , and two characters χα (α = 1, 2) ofE×A×F \A

×E . For simplicity, we will assume that χα are both trivial.

3

[Ich2004, Ich2007], we obtain the following formula2

〈θfφ, θf∨

φ∨〉H =L( 1

2 , π)

2∏2ni=1 L(i, εiE/F )

∏v∈S

Z∗(0, fv, f∨v , φv ⊗ φ∨v ), (1.1)

where

• 〈−,−〉H denotes the Petersson inner product on H (for a suitable Haar measure);

• S is a finite set of primes of F containing all archimedean places;

• L(s, π) = LS(s, π)LS(s, π) is the L-function of π, whose unramified part LS(π) is defined from

the Satake parameter and the ramified part LS(π) is defined in [HKS1996] as a greatest common

divisor;

• εE/F is the quadratic character of F×\A×F associated to the quadratic extension E/F ; and

• Z∗(0, fv, f∨v , φv ⊗ φ∨v ) is a (normalized) local zeta integral for a ramified place v ∈ S.

By the theta dichotomy, for each place v of F , we can define a factor ε(πv) ∈ ±1, which is 1 if v 6∈ S,

and such that Z∗(0, fv, f∨v , φv ⊗ φ∨v ) is not always zero if and only if ε(πv) = ηEv/Fv ((−1)n detVv).

We let ε(π) =∏v ε(πv). There are two cases.

1. ε(π) = 1. Then we can choose a suitable hermitian space V such that the set of theta lifting θfφ

for φ ∈ S(V (AE)n) and f ∈ π contains a nonzero function if and only if L( 12 , π) 6= 0.

2. ε(π) = −1. Then whatever V we choose, all theta lifting θfφ is trivial. In this case, Rallis’ inner

product formula is not interesting. In fact, as we will see in Theorem 2.3.9, L( 12 , π) = 0 for such

π.

Therefore, it is natural in case (2) to ask the information about L′( 12 , π). Parallel to the classical

theory where the central value of the L-function relates to the lifting on the level of functions, we

propose an arithmetic theory where the central derivative of the L-function relates to the lifting on

the level of cycles. Such formulation will be elaborated in the next subsection.


We now consider the second case, that is, π has the factor ε(π) = −1. Therefore, L( 12 , π) = 0. We also

assume that the archimedean component π∞ of π is a discrete series representation of certain type.

2Here, we assume that V is anisotropic for simplicity to avoid the regularization process.

4

From such π, we can construct, instead of the hermitian space V over E, a hermitian space (or rather

a hermitian module) V over AE of rank 2n. Such V is incoherent in the sense that there does not

exist a hermitian space V over E making V ∼= V ⊗E AE , which is parallel to the fact that ε(π) = −1.

Moreover, by our assumption on π∞, V is totally positive definite. Let H = U(V) be the group of

isometry ofV, which is a reductive group overAF . By the theory of Shimura variety, we attach toH (a

projective system of) Shimura varieties (Sh(H)K)K for open compact subgroups K ⊆ H(AF,fin). They

are smooth quasi-projective varieties over SpecE. Let S(V(AE)n)U∞ ⊂ S(V(AE)n) be the subspace

of those Schwartz functions whose archimedean components are essentially the Gaussian. Following

S. Kudla, for φ ∈ S(V(AE)n)U∞ , we define the generating series Zφ(g), which is a “function” on

H ′(AF ) whose values are formal series in CHn(Sh(H))C, the injective limit of groups of Chow cycles

(with coefficients in C) of codimension n on Sh(H)K for all K. The series Zφ(g) should be viewed

as the arithmetic analogue of the classical theta series θ(g, •;φ), where the later is a “function” on

H ′(AF ) whose values are automorphic forms of H. Parallel to the automorphy property of θ(g, •;φ)

(as a “function” on H ′(AF )), we have the following result.

Theorem (Modularity of the generating series, Theorem 3.1.6). Let l be a linear functional on

CHn(Sh(H))C. Then

1. If l(Zφ)(g) is absolutely convergent, it is an automorphic form of H ′.

2. If n = 1, l(Zφ)(g) is absolutely convergent for every l.

In fact, the above theorem holds for all codimensions, not just n. There is also a version in the case

of symplectic-orthogonal pairs, which is proved by X. Yuan, S.-W. Zhang and W. Zhang [YZZ2009].

The proof for both cases use the induction process on the codimension, which originally comes from

the idea of W. Zhang [Zha2009]. Moreover, the proof for the case where the generating series has

codimension 1 reduces to the result in [YZZ2009].

Recall that in the case of classical theta lifting, we construct θfφ simply by taking the inner product

of f and the theta series. In view of the above theorem, we have the following parallel definition. We

define the arithmetic theta lifting to be

Θfφ =

∫H′(F )\H′(AF )

f(g)Zφ(g)dg,

for f ∈ π. Rigorously speaking, the above integration is formal and we should justify such expression

by applying a linear functional l as in the above theorem. Nevertheless, the (Betti) cohomology class

cl(Θfφ) of Θf

φ is always well-defined. To find an arithmetic analogue of the Petersson inner product, we

5

invoke the conjectural Beilinson–Bloch height paring [Beı1987, Blo1984], which is, in this particular

case, a hermitian paring

〈−,−〉BB : CHn(Sh(H))0C × CHn(Sh(H))0

C → C.

Here, CHn(Sh(H))0C ⊂ CHn(Sh(H))C is the kernel of the cycle class map cl. Such paring is conjectured

to be positive definite. Even modulo the conjectural construction, there are still two remaining issues.

First, we need Sh(H)K to be proper. Second, we would like to have Θfφ ∈ CHn(Sh(H))0

C. The first

issue will be discussed in 3.2.1, where we propose some constructions as well as some conjectures. To

simplify the discussion in the introduction, we assume that Sh(H)K is already proper, which is the

case when, for example, F 6= Q. Then we have the following result concerning the second issue.

Proposition (Proposition 3.3.4). Assume that Sh(H)K is proper for all K. Then the cohomology

class cl(Θfφ) is trivial.

In view of the above result, we formulate the following conjecture of the arithmetic inner product

formula.

Conjecture (Arithmetic inner product formula, Conjecture 3.3.6). Let π be an irreducible cuspidal

automorphic representation of H ′(AF ) as above. In particular, ε(π) = −1. Then for every f ∈ π,

f∨ ∈ π∨ and every φ, φ∨ ∈ S(V(AE)n)U∞ , we have

〈Θfφ,Θ

f∨

φ∨〉BB =L′( 1

2 , π)∏2ni=1 L(i, εiE/F )

∏v∈S

Z∗(0, fv, f∨v , φv ⊗ φ∨v ),

where S is a finite set of primes, and the local factors Z∗ are same to those in the Rallis inner product

formula (1.1).

The main result of the article is the following theorem, which justifies the above conjecture in the

case n = 1. We point out that when n = 1, Sh(H)K is a curve. Therefore, the Beilinson–Bloch height

paring is simply the well-known Neron–Tate height paring, denoted by 〈−,−〉NT.

Theorem (Arithmetic inner product formula, Theorem 7.2.1). Let n = 1. Let π be an irreducible

cuspidal automorphic representation of H ′(AF ) as in the above conjecture. Then for every f ∈ π,

f∨ ∈ π∨ and every φ, φ∨ ∈ S(V(AE)n)U∞ , we have

〈Θfφ,Θ

f∨

φ∨〉NT =L′( 1

2 , π)

LF (2)L(1, εE/F )

∏v∈S

Z∗(0, fv, f∨v , φv ⊗ φ∨v ),

6

where S is a finite set of primes, and the local factors Z∗ are same to those in the Rallis inner product

formula (1.1).

The first appearance of such arithmetic analogue of Rallis’ inner product formula is the main

result of [KRY2006]. The authors studied the case where f is a new form of PGL2(Q) of weight 2 and

square-free level. In particular, the corresponding Shimura curve where the height paring is taken on

is the one attached to a division quaternion algebra over Q. More recently, J. Bruinier and T. Yang

[BY2009] studied the case where the Shimura curve is the modular curve. They obtain a formula that

is very similar to ours here, and from which they deduce certain cases of the Gross–Zagier formula.

In fact, the study of derivative of L-functions was initiated by Gross and Zagier about thirty years

ago in the pioneer paper [GZ1986]. The recent work of X. Yuan, S.-W. Zhang and W. Zhang [YZZa]

generalizes this formula in an extremely broad and conceptual form, based on the connection with the

representation theory of the so-called restriction problem. As for this work, we generalize the formulas

of Kudla et al. in a uniform and explicit form, by exploring the theory of local theta correspondence,

as we will see in the next subsection where we outline the idea of the proof.

Outline of the proof

The proof of the main theorem consists of six parts, which occupy the following six chapters respec-

tively. Throughout the process, we make our argument as general as possible. In other words, we deal

with the problem for general n, not just 1, once we are able to do so.

In Chapter 2, we introduce analytic kernel functions that compute the derivative of L-functions.

Such kernel functions are derivatives of Siegel Eisenstein series associated to degenerate principal

series on the doubling unitary group. Precisely, given a pair of Schwartz functions φα ∈ S(V(AE)n)

(α = 1, 2), we have a kernel function E′(0, g, φ1⊗φ2). We show that one can choose φα carefully such

that E′(0, g, φ1 ⊗ φ2) can be expressed as a sum of local terms indexed by (almost all) places of F

that are nonsplit in E, for g in an open dense subset. Precisely, we have the following decomposition

(2.22)

E′(0, ι(g1, g∨2 ), φ1 ⊗ φ2) =

∑v 6∈S

Ev(0, ι(g1, g∨2 ), φ1 ⊗ φ2),

where S is a finite set of finite places of F at which ramification occurs. To compute the actual

derivative L′( 12 , π), we only need to take the inner product of E′(0, •, φ1⊗φ2) with f and f∨. Therefore,

we should study Ev(0, g, φ1 ⊗ φ2). Throughout this chapter, all discussions work for general n except

in 2.4.3.

7

We introduce the Shimura varieties attached to certain unitary groups, their special cycles and

generating series in Chapter 3. Moreover, we discuss the case of non-proper Shimura varieties, where

we need to compactify everything we introduce above. As mentioned previously, we prove the modu-

larity of the generating series, and the cohomological triviality of the arithmetic theta lifting. Finally,

in 3.4, where we restrict ourselves to the case n = 1, we introduce the arithmetic kernel functions

E(g1, g2;φ1 ⊗ φ2) that compute the Neron–Tate height paring of the arithmetic theta lifting. Then

modulo certain volume factors and terms that are perpendicular to f and f∨, we have the following

decomposition of the arithmetic kernel function (3.15)

E(g1, g2;φ1 ⊗ φ2) =∑v∈Σ

Ev(g1, g2;φ1 ⊗ φ2),

where Σ is the set of places of E, and

Ev(g1, g2;φ1 ⊗ φ2) = 〈Zφ1(g1), Zφ2

(g2)〉v

is a local height paring on a certain model of the Shimura curve. It is clear that to prove the main

theorem, we need to compare the analytic and arithmetic kernel functions place by place. There are

three cases. First, v is an archimedean place. Second, v is a finite nonsplit place that is not in S.

Third, v is either finite split or in S.

The first case is treated in Chapter 4. We reduce the comparison to a local question, which can be

formulated for general dimensions. We prove a formula which we call the archimedean local arithmetic

Siegel–Weil formula. Let m ≥ 2 be an integer. Let T be a (nondegenerate) hermitian matrix in

GLm(C) of signature (m − 1, 1). Let Φ0 be the Gaussian on V m, where V is the standard positive

definite complex hermitian space of dimension m. On the one hand, we have the Whittaker integral

WT (s, e,Φ0) that is a holomorphic function in s. It is not hard to see that WT (0, e,Φ0) = 0. On

the other hand, we define an archimedean local intersection number H(T )∞, which is the volume of

the open unit ball D in Cm−1 with respect to a star product of Green currents constructed from

Kudla–Millson forms. We prove the following result.

Theorem (Archimedean local arithmetic Siegel–Weil formula, Theorem 4.3.1). Let T be a (nonde-

generate) hermitian matrix in GLm(C) of signature (m− 1, 1). Then we have

W ′T (0, e,Φ0) = Cm exp(−2π trT )H(T )∞,

8

where Cm is some nonzero constant depending only on m, not on T .

The above theorem specified to m = 2 will pose a close relation between Ev(0, ι(g1, g∨2 ), φ1 ⊗ φ2)

and Ev(g1, g2;φ1 ⊗ φ2) for an archimedean place v dividing v. More precisely, they are related by

the process of holomorphic projection, which we discuss in 7.1.

The second case is treated in Chapter 5. We prove that for a finite place v = p outside S that is

nonsplit in E, Ep(0, ι(g1, g∨2 ), φ1⊗φ2) and Ep(g1, g2;φ1⊗φ2) are equal, where p is the unique place

of E over p. This comparison is again reduced to a local question as follows. For a 2-by-2 hermitian

matrix T with entries in OEp , the ring of integers of Ep , such that its determinant has odd valuation,

we define a number Hp(T ) that is some intersection multiplicity on the smooth integral model of the

Shimura curve at p. Let V + be the 2-dimensional hermitian space over E with a selfdual lattice

Λ+, and Φ0+ the characteristic function of Λ+. We have a p-adic Whittaker integral WT (s, e,Φ0+),

holomorphic in s. Since detT has odd valuation, WT (0, e,Φ0+) = 0. We prove the following result.

Theorem (Non-archimedean local arithmetic Siegel–Weil formula, Theorem 5.2.3 and Corollary

5.3.2). Let T be a 2-by-2 hermitian matrix with entries in OEp such that detT has odd valuation.

Then we have

W ′T (0, e,Φ0+) = Cp ·Hp(T ),

where Cp is some nonzero constant depending only on the place p, not on T .

In the entire chapter, we restrict ourselves to the case n = 1.

The third case is treated in Chapter 6. We prove that under careful choices of φα (α = 1, 2), there

is no contribution of terms Ev(g1, g2;φ1⊗φ2) for a finite place v either split in E or in S, after taking

inner product with f and f∨. This is compatible with the analytic side since the corresponding terms

are all zero. We are only able to make such argument when n = 1. Now we come to the final stage

of the proof of the main theorem, which is accomplished in Chapter 7. As we have mentioned before,

we apply the holomorphic projection to the analytic kernel function to make its archimedean part

coincide with the corresponding part of the arithmetic kernel function. To prove the arithmetic inner

product formula in the full generality, we apply the result of multiplicity one proved in A.2. The last

step is extremely crucial to make us able to avoid explicit computations at all bad places. We would

like to remark that such idea originally comes from [YZZa].

Conventions and notations

Convention 1.0.1. All rings will have a unit.

9

Notation 1.0.2. • We denote by Z the ring of (rational) integers. We let Q, R and C be the

fields of rational, real and complex numbers, respectively.

• For a Z-module M and a commutative ring R, we denote MR = M ⊗Z R the base change

R-module.

• We denote by Afin = Z⊗Z Q =(

lim←−N Z/NZ)⊗Z Q the ring of finite adeles; A=R×Afin the

ring of full adeles.

• For a number field K, we let AK = A⊗Q K, AK,fin = Afin ⊗Q K and K∞ = R⊗Q K.

• As usual, for a subset S of places, −S (resp. −S) means the S-component (resp. component

away from S) for the corresponding (decomposable) adelic object; −∞ (resp. −fin) means the

infinite/archimedean (resp. finite) part.

• For a field K, we fix a separable closure Ksep of K and denote ΓK = Gal(Ksep/K) the Galois

group of K.

• The symbol Tr and Nm stand for the trace (resp. reduced trace) and norm (resp. reduced

norm), respectively, if they are applied to fields or rings of adeles (resp. simple algebras). The

symbol tr stands the trace for matrix and linear transforms.

Notation 1.0.3. • For a ring R and integers m,n > 0, we denote by Matm,n(R) the ring of

m-by-n matrix with entries in R. We also set Matn(R) = Matn,n(R).

• We denote by 1n (resp. 0n) the identity (resp. zero) matrix of rank n.

• We denote by tg the transpose of a matrix g.

Definition 1.0.4. • Let R be a commutative ring and R′ an etale (commutative) algebra over R

of rank 2. Let τ : r 7→ rτ for r ∈ R′ be the nontrivial automorphism of R′ over R. For n ≥ 0, a

hermitian (resp. skew-hermitian) space of rank n over R′ with respect to τ is a free module V

over R′ of rank n equipped with a map

(−,−) : V × V → R′

satisfying that

1. It is R′-linear in the first variable, i.e., for r ∈ R and v, v′ ∈ V , (rv, v′) = r(v, v′).

2. It is (R′, τ)-linear in the second variable, i.e., for r ∈ R and v, v′ ∈ V , (v, rv′) = rτ (v, v′).

10

3. It is τ -symmetric (resp. τ -antisymmetric), i.e., for v, v′ ∈ V , (v, v′) = (v′, v)τ (resp. (v, v′) =

−(v′, v)τ ).

In fact, under the assumption (3), assumptions (1) and (2) imply each other.

• A hermitian or skew-hermitian space V over R′ of rank n is nondegenerate if their is a basis

v1, . . . , vn of V over R′ such that the matrix ((vi, vj))1≤i,j≤n has determinant in R′×: the

group of invertible elements in R′. It is clear that this property does not depend on the choice

of the basis.

• In practice, R will be a field and R′/R a (possibly split) extension of degree 2, or R the ring of

adeles of a number field and R′ that of a quadratic field extension. The involution τ will always

be clear in the context and hence we will not say with respect τ in general.

• In the main part of the article, all hermitian spaces will be of finite rank and nondegenerate.

Notation 1.0.5. Let r ≥ 1 be an integer, we set

wr =

1r

−1r

.

For 0 ≤ d ≤ r, we set

wr,d =

1d

1r−d

1d

−1r−d

.

For r elements a1, . . . , ar in a ring, we set

diag[a1, . . . , ar] =

a1

. . .

ar

to be the diagonal matrix.

Notation 1.0.6. Let G be a Lie group over a local field. We denote by λG : G → C× the modulus

character of G. Precisely, for g ∈ G, we have the adjoint action Adg on the Lie algebra LieG. Take a

nonzero Haar measure dx on LieG. Then λG(g) = Ad∗g dx/dx.

11

Notation 1.0.7. Let K be a field and X a scheme of finite type over SpecK,

• For an integer r ≥ 0, we denote by CHr(X) the Chow cohomology group of codimension r. In

practice, X will be smooth over K. Therefore, CHr(X) can be canonically identified with the

abelian group of Chow cycles of X of codimension r over K.

• If r = 1, we set Pic(X) = CH1(X). In other words, Pic(X) is the Picard group of X, which

should not be understood as the Picard scheme or Picard stack of X.

12

Chapter 2

Doubling method and analytic

kernel functions

The goal of this chapter is to introduce the analytic kernel functions. In 2.1, we review the Siegel–Weil

formula and some of its generalization that are related to our problem. In 2.2, we review the theory of

I. Piatetski-Shapiro and S. Rallis on the doubling method. In particular, we deduce the Rallis’ inner

product formula in certain cases from the doubling method and the Siegel–Weil formula. We also

introduce the (normalized) local zeta integrals that will serve as the local terms in both classical and

arithmetic inner product formula. In 2.3, we introduce the L-function and the formula representing

it. We show that the central L-value of an automorphic representation vanishes if its global epsilon

factor equals −1. Then we introduce the analytic kernel functions that compute the L-derivatives. In

2.4, we study these analytic kernel functions. We prove that for certain nice choices of test functions,

the analytic kernel function can be decomposed into terms indexed by archimedean and unramified

nonsplit finite places of F .

13

2.1 Siegel–Weil formula and generalizations

2.1.1 Degenerate principal series and Eisenstein series

Let F be a totally real number field and E a totally imaginary quadratic extension of F . We denote by

τ the nontrivial element in Gal(E/F ) and εE/F : A×F /F× → ±1 the quadratic character associated

by the class field theory. Let Σ (resp. Σfin, Σ∞) be the set of all places (resp. finite places, infinite

places) of F , and Σ, Σfin, Σ∞ those of E. We fix a nontrivial additive character ψ of AF /F , that is,

a continuous character ψ : AF → C×, which is trivial on F .

For a positive integer r, we denote by Wr the standard skew-hermitian space over E with respect

to the involution τ , which is equipped with a skew-hermitian form 〈−,−〉 such that there is an E-basis

e1, . . . , e2r satisfying

• 〈ei, ej〉 = 0;

• 〈er+i, er+j〉 = 0;

• 〈ei, er+j〉 = δij for 1 ≤ i, j ≤ r.

Let Hr = U(Wr) be the unitary group of Wr, which is a reductive group over F . The group Hr(F ),

in which F can be itself or its completion at some place, is generated by the parabolic subgroup

Pr(F ) = Nr(F )Mr(F ) and the element wr. Precisely,

Nr(F ) =

n(b) =

1r b

1r

| b ∈ Herr(E)

;

Mr(F ) =

m(a) =

a

taτ,−1

| a ∈ GLr(E)

;

and

wr =

1r

−1r

as in Notation 1.0.5. Here, Herr(E) =

b ∈ Matr(E) | bτ = tb

.

We fix a place v ∈ Σ and suppress it from notations. Thus F = Fv is a local field of characteristic

zero; E = Ev := E⊗F Fv is a quadratic extension of F which could be split; and Hr = Hr,v := Hr(Fv)

is a reductive Lie group over the local field. We define Kr to be a maximal compact subgroup of Hr

14

in the following way:

• If v is finite, then

Kr = Hr ∩GL(OE〈e1, . . . , e2n〉) ⊂ GL(Wr).

• If v is (real) infinite, then

Kr = Hr ∩U(2r)R ⊂ GL(2r)C ∼= GL(Wr)

where the isomorphism is determined by the basis e1, . . . , e2n. Therefore, Kr is isomorphic to

U(r)R ×U(r)R (2.1).

For s ∈ C and a character χ of E×, we denote by Ir(s, χ) = sIndHrPr (χ| • |s+r2

E ) the degenerate principal

series representation (cf. [KS1997]) of Hr, where sInd stands for the (non-normalized) smooth Kr-

finite induction. Precisely, it is realized on the space of smooth Kr-finite functions ϕs on Hr satisfying

ϕs(n(b)m(a)g) = χ(det a)|det a|s+r2

E ϕs(g)

for all g ∈ Hr, m(a) ∈Mr and n(b) ∈ Nr. A (holomorphic) section ϕs of Ir(s, χ) is called standard if

its restriction to Kr is independent of s. It is called unramified if it takes value 1 on Kr.

Now we view F and E as number fields. For a (continuous) character χ of A×E that is trivial on

E×, and s ∈ C, we have an admissible representation Ir(s, χ) =⊗′

v∈Σ Ir(s, χv) of Hr(AF ), where

the restriction in the restricted tensor product refers to the collection of unramified sections. For a

standard section ϕs = ⊗ϕs,v ∈ Ir(s, χ), we define the Eisenstein series

E(g, ϕs) =∑

γ∈Pr(F )\Hr(F )

ϕs(γg).

The series is absolutely convergent if Re s > r2 , and has a meromorphic continuation to the entire

complex plane, which is holomorphic at s = 0 (cf. [Tan1999, Proposition 4.1]).

2.1.2 Real quasi-split unitary groups

Let r ≥ 1 be an integer. Let WR,r be a skew-hermitian space over C (with respect to the quadratic

extension C/R) of rank 2r with a basis e1, . . . , er; er+1, . . . , e2r, under which the skew-hermitian

15

form is given by the matrix

wr =

1r

−1r

.

Let U(r, r)R be the subgroup of ResC/RGL(WR,r) preserving the skew-hermitian form, which is a

reductive group over R. We also let

U(r)R = g ∈ ResC/RGLr(C) | tgg = 1r1 (2.1)

be a subgroup of ResC/RGLr(C). We have the following embedding

U(r)R ×U(r)R → U(r, r)R

(k1, k2) 7→ [k1, k2] :=1

2

k1 + k2 −ik1 + ik2

ik1 − ik2 k1 + k2

.

Moreover, the above embedding identifies U(r)R × U(r)R as a maximal compact subgroup Kr of

U(r, r)R.

Notation 2.1.1. We introduce the following notation.

1. We let ιi (i = 1, . . . , d) be all embeddings of F into C, whose image is contained in R, and ιi ,

ι•i those of E above ιi. We identify E ⊗F,ιi R with C through the embedding ιi . In particular,

we have identified Hr ×F,ιi R with U(r, r)R.

2. Let ι be an archimedean place of F . Let χι be a character of E×ι , which is identified with C×

via ι, such that χι | F×ι ∼= R× equals sgnm. Here, sgn denotes the sign character and m is an

integer. In particular, we can write

χι(z) =zkχι√|zz|

kχι,

for a unique integer kχι that has the same parity as m. If χ is an automorphic character of A×E

whose restriction to A×F equals ηmE/F , we set kχ = (kχι1 , . . . , kχιd ).

Definition 2.1.2 (Weights). We define the notions of weights as follows.

1. Let π be an irreducible (Lie U(r, r)R,Kr)-module (or its Casselman–Wallach globalization). Let

a, b be integers. We say π is of weight (a, b) if the minimal Kr-type of π is the character

detadetb, i.e., its sends [k1, k2] to (det k1)a(det k2)b.

16

2. Let π be an irreducible automorphic representation of Hr(AF ). For two d-tuples of integers a =

(a1, . . . , ad),b = (b1, . . . , bd), we say π∞ =⊗d

i=1 πιi is of weight (a,b) if for every i = 1, . . . , d,

πιi is of weight (ai, bi) in the sense above.

3. An automorphic form f ofHr(AF ) is of weight (a,b) if f(g[k1,ι, k2,ι]) = (det k1,ι)aι(det k2,ι)

bιf(g)

for every ι ∈ Σ∞.

2.1.3 Weil representations and theta functions

Let us review the classification of (nondegenerate) hermitian spaces. Let m ≥ 1 be an integer and

v ∈ Σ be a place of F . For a hermitian space V over Ev of rank m, we define

ε(V ) = εEv/Fv

((−1)

m(m−1)2 detV

)∈ ±1.

There are three cases:

• If v ∈ Σfin such that E is nonsplit at v, then up to isometry, there are two different hermitian

spaces over Ev of dimension m ≥ 1: V ± determined by ε(V ±) = ±1.

• If v ∈ Σfin such that E is split at v, then up to isometry, there is only one hermitian space V +

over Ev of dimension m.

• If v ∈ Σ∞ (which is real), then up to isometry, there are m+ 1 different hermitian spaces over

Ev of dimension m: Vs with signature (s,m− s) where 0 ≤ s ≤ m.

In the global situation, up to isometry, all hermitian spaces V over E of dimension m are classified

by signatures at infinite places and detV ∈ F×/NmE×. In particular, V is determined by Vv =

V ⊗F Fv for all v ∈ Σ.

More generally, we also need to consider nondegenerate hermitian spaces over AE of rank m.

Recall in Definition 1.0.4 that in this case, a hermitian space V is nondegenerate if there is a basis

under which the matrix representing the hermitian form is invertible in GLm(AE). For a place v ∈ Σ,

we let Vv = V ⊗AF Fv, Vfin = V ⊗AF AF,fin; and define Σ(V) = v ∈ Σ | ε(Vv) = −1, which is a

finite set. Finally, we let ε(V) =∏v∈Σ ε(Vv).

Definition 2.1.3 (Coherent/incoherent hermitian spaces). We say a (nondegenerate) hermitian space

V over AE is coherent (resp. incoherent) if the cardinality of Σ(V) is even (resp. odd), i.e., ε(V) = 1

(resp. −1).

17

By the Hasse principle, there is a hermitian space V over E such that V ∼= V ⊗F AF if and only if

V is coherent. These two terminologies are inspired from the coherent/incoherent collections of local

quadratic spaces introduced by S. Kudla in the orthogonal case in [KR1994,Kud1997].

We fix a place v ∈ Σ and suppress it from notations. For a hermitian space V of dimension

m with hermitian form (−,−) and a positive integer r, we can construct a symplectic space W =

ResE/F Wr ⊗E V of dimension 4rm over F with the skew-symmetric form TrE/F 〈−,−〉 ⊗ (−,−),

where Res stands for the Weil restriction. Let Sp(W) be the symplectic group and Mp(W) be its

C×-metaplectic cover fitting into the following exact sequence:

1 // C× // Mp(W) // Sp(W) // 1.

We let H = U(V ) be the unitary group of V and S(V r) the space of Schwartz functions on V r. Given

a character χ of E× satisfying χ|F× = εmE/F , we have a splitting homomorphism

ı(χ,1) : Hr ×H → Mp(W)

lifting the natural homomorphism ı : Hr × H → Sp(W) (cf. [HKS1996, Section 1]). We thus have

a Weil representation (with respect to ψ) ωχ = ωχ,ψ of Hr × H on the space S(V r). Explicitly, for

φ ∈ S(V r) and h ∈ H,

• (ωχ(n(b))φ) (x) = ψ(tr bT (x))φ(x);

• (ωχ(m(a))φ) (x) = |det a|m2

E χ(det a)φ(xa);

• (ωχ(wr)φ) (x) = γV φ(x);

• (ωχ(h)φ) (x) = φ(h−1x),

where

T (x) =1

2((xi, xj))1≤i,j≤r

is the moment matrix of x; γV is the Weil constant associated to the underlying quadratic space of V

(and also ψ); φ is the Fourier transform

φ(x) =

∫V rφ(y)ψ

(1

2TrE/F (x, ty)

)dy

18

using the selfdual measure dy on V r with respect to ψ.

In the global situation where F is the number field, by taking the restricted tensor product over

all local Weil representations, we obtain a representation of Hr(AF )×H(AF ) on the space S(V r) :=⊗′v∈Σ S(V rv ).

Warning 2.1.4. We abuse notation by denoting S(V r) for spaces of both local and adelic Schwartz

functions. More precisely, we should use S(V (AE)r) in the adelic case. We feel that there is little

danger of confusion since in the adelic case, V is always considered over a number field.

For V over E, χ a character of A×E/E× such that χ|A×F = εmE/F and φ ∈ S(V r), we define the theta

function

θ(g, h;φ) =∑

x∈V r(E)

(ωχ(g, h)φ) (x),

which is a smooth, slowly increasing function on Hr(F )\Hr(AF ) × H(F )\H(AF ). Consider the

integral

IV (g, φ) =

∫H(F )\H(AF )

θ(g, h;φ)dh,

if it is absolutely convergent. Here we normalize the measure dh such that Vol(H(F )\H(AF )) = 1.

It is well-known that IV (g, φ) is absolutely convergent for all φ if m > 2r or V is anisotropic.

2.1.4 Siegel–Weil formulae

It is immediate to see that

ϕφ,s(g) = (ωχ(g)φ) (0)λPr (g)s−m−r

2

is a standard section in Ir(s, χ) for every φ ∈ S(V r). Recall the following formula for the modulus

character

λPr (g) = λPr (n(b)m(a)k) = |det a|AE ,

if g = n(b)m(a)k under the Iwasawa decomposition with respect to the parabolic subgroup Pr(AF ).

Therefore, we can define the Eisenstein series E(s, g, φ) = E(g, ϕφ,s). We have the following theorem.

Theorem 2.1.5 (Siegel–Weil formula). Let s0 = m−r2 . Then we have

19

1. If m > 2r, E(s0, g, φ) is absolutely convergent and

E(s0, g, φ) = IV (g, φ).

2. If r < m ≤ 2r and V is anisotropic, E(s, g, φ) is holomorphic at s0 and

E(s, g, φ)|s=s0 = IV (g, φ).

3. if m = r and V is anisotropic, E(s, g, φ) is holomorphic at s0 = 0 and

E(s, g, φ)|s=0 = 2IV (g, φ).

Proof. 1. It is the classical Siegel–Weil formula.

2. It is a generalized Siegel–Weil formula proved in [Ich2007, Theorem 1.1].

3. It is a generalized Siegel–Weil formula proved in [Ich2004, Theorem 4.2].

In what follows, we simply write E(s0, g, φ) for E(s, g, φ)|s=s0 for simplicity if the Eisenstein series

is holomorphic at s0.

Remark 2.1.6. In Theorem 2.1.5 (3), if V is isotropic, we still have a (regularized) Siegel–Weil

formula. However, since the theta integral IV (g, φ) is not necessarily convergent, a regularization

process must be applied. The inner product introduced in the next section also requires a regularization

process. Since the classical inner product formula is not the purpose of this article, we will always

assume that V is anisotropic for simplicity, or pretend that the regularization process has been applied

for general V in the following discussion.

2.2 Doubling integrals

2.2.1 Decomposition of global period integrals

Let m = 2n and r = n with n ≥ 1 and suppress n from notations except that we will use H ′

instead of Hn; P ′ instead of Pn; N ′ instead of Nn and K′ instead of Kn. Therefore, χ|A×F = 1 is the

trivial character. Let π =⊗′

v∈Σ πv be an irreducible cuspidal automorphic representation of H ′(AF )

20

contained in A0(H ′): the space of cuspidal automorphic forms on H ′(AF ). We will not distinguish π

with its underlying space. Let π∨ be the contragredient representation which is realized on the space

of complex conjugation of functions in π.

We denote by (−W ) the skew-hermitian space over E with the skew-hermitian form −〈−,−〉.

Therefore, we have a basis e−1 , . . . , e−2n satisfying

• 〈e−i , e−j 〉 = 0;

• 〈e−r+i, e−r+j〉 = 0;

• 〈e−i , e−n+j〉 = −δij for 1 ≤ i ≤ n.

Let W ′′ = W ⊕ (−W ) be the direct sum of the two skew-hermitian spaces. There is a natural

embedding

ı : H ′ ×H ′ → U(W ′′) (2.2)

which is, under the basis

• e1, . . . , e2n of W and

• e1, . . . , en; e−1 , . . . , e−n ; en+1, . . . , e2n;−e−n+1, . . . ,−e

−2n of W ′′,

given by ı(g1, g2) = ı0(g1, g∨2 ) where

g1 =

a1 b1

c1 d1

, g2 =

a2 b2

c2 d2

, g∨ =

1n

−1n

g

1n

−1n

−1

and

ı0(g1, g2) =

a1 b1

a2 b2

c1 d1

c2 d2

.

Notation 2.2.1. Let

W ′ = spanEe1, . . . , en; e−1 , . . . , e−n

W′

= spanEen+1, . . . , e2n;−e−n+1, . . . ,−e−2n,

21

be a pair of complimentary Lagrangian subspaces of W ′′. Let P be the parabolic subgroup of U(W ′′)

stabilizing W ′ and N the unipotent radical of P .

For the complete polarization W ′′ = W ′ ⊕W ′, we can realize the Weil representation of U(W ′′),

denoted by ω′′χ (with respect to ψ), on the space S(V 2n), such that ı∗ω′′χ∼= ωχ,ψ χω∨χ,ψ, where the

latter one is realized on the space S(V n) ⊗ S(V n). Here we realize the contragredient representation

ω∨χ,ψ on the space S(V n) through the bilinear pairing

〈φ, φ∨〉V =

∫V n(AE)

φ(x)φ∨(x)dx

for φ, φ∨ ∈ S(V n). Then ω∨χ,ψ is identified with ωχ−1,ψ−1 .

For φ ∈ S(V n) and f ∈ π, we define thetheta lifting

θfφ(h) =

∫H′(F )\H′(AF )

θ(g, h;φ)f(g)dg.

It is a well-defined, slowly increasing function on H(F )\H(AF ), where dg = ⊗v∈Σdgv such that K′v

gets volume 1 for every v ∈ Σ. Similarly, for φ∨ ∈ S(V n) and f∨ ∈ π∨, we have θf∨

φ∨ . The reader should

be careful that in the contragredient side, the Weil representation used to form the theta function

should also be the contragredient one, i.e., ω∨χ . We have

〈θfφ, θf∨

φ∨〉H

: =

∫H(F )\H(AF )

θfφ(h)θf∨

φ∨(h)dh

=

∫H(F )\H(AF )

∫[H′(F )\H′(AF )]2

θ(g1, h;φ)f(g1)θ(g2, h;φ∨)f∨(g2)dg1dg2dh

=

∫H(F )\H(AF )

∫[H′(F )\H′(AF )]2

θ(ı(g1, g2), h;φ⊗ φ∨)f(g1)f∨(g2)χ−1(det g2)dg1dg2dh

=

∫[H′(F )\H′(AF )]2

f(g1)f∨(g2)χ−1(det g2)

∫H(F )\H(AF )

θ(ı(g1, g2), h;φ⊗ φ∨)dhdg1dg2. (2.3)

We assume that V is anisotropic. Then the inside integral in the last step is absolutely convergent.

By Theorem 2.1.5 (3), we have

(2.3) =1

2

∫[H′(F )\H′(AF )]2

f(g1)f∨(g2)χ−1(det g2)E(0, ı(g1, g2), φ⊗ φ∨)dg1dg2.

We should mention that the Eisenstein series on U(W ′′) appearing above is formed with respect

22

to the parabolic subgroup P (see Notation 2.2.1), i.e.,

E(s, g,Φ) = E(g, ϕΦ,s) =∑

γ∈P (F )\U(W ′′)(F )

ω′′χ(γg)Φ(0)λP (γg)s

for g ∈ U(W ′′)(AF ), Φ ∈ S(V 2n) and Re s > n. The coset P (F )\U(W ′′)(F ) can be canonically

identified with the space of isotropic n-planes in W ′′. Under the right action of H ′(F ) × H ′(F )

through ı, the orbit of an n-plane Z is determined by the invariant d = dimZ ∩W = dimZ ∩ (−W ).

Let γd be a representative of the corresponding double coset where 0 ≤ d ≤ n. In particular, we take

γ0 =

1n

1n

−1n 1n

1n 1n

and γn = 14n

(cf. [KR2005]). Let Std be the stabilizer of Pγdı(H′ ×H ′) in H ′ ×H ′. In particular St0 = ∆(H ′) is

the diagonal subgroup. Therefore, for a standard section ϕs ∈ I2n(s, χ) and Re s > n,

∫[H′(F )\H′(AF )]2

f(g1)f∨(g2)χ−1(det g2)E(ı(g1, g2), ϕs)dg1dg2

=

∫[H′(F )\H′(AF )]2

(f ⊗ f∨χ−1)(g)∑

γ∈P (F )\U(W ′′)(F )

ϕs(γı(g))dg

=

n∑d=0

∫Std(F )\H′(AF )2

(f ⊗ f∨χ−1)(g)ϕs(γdı(g))dg. (2.4)

When d > 0, Std has nontrivial unipotent radical. Since f and f∨ are cuspidal, we have

(2.4) =

∫∆(H′(F ))\H′(AF )2

(f ⊗ f∨χ−1)(g)ϕs(γ0ı(g))dg

=

∫H′(F )\H′(AF )

∫H′(AF )

f(g1g2)f∨(g1)χ−1(det g1)ϕs(γ0ı(g1g2, g1))dg1dg2

=

∫H′(F )\H′(AF )

∫H′(AF )

π(g2)f(g1)f∨(g1)χ−1(det g1)ϕs(p(g1)γ0ı(g2, 1))dg1dg2 (2.5)

where p(g1)γ0 = γ0ı(g1, g1), and under the Levi decomposition p(g1) = n(b)m(a) ∈ P (AF ), we have

23

det a = det g1. Therefore,

(2.5) =

∫H′(AF )

∫H′(F )\H′(AF )

π(g2)f(g1)f∨(g1)dg1ϕs(γ0ı(g2, 1))dg2

=

∫H′(AF )

〈π(g)f, f∨〉ϕs(γ0ı(g, 1))dg

=∏v∈Σ

∫H′v

〈πv(gv)fv, f∨v 〉ϕs,v(γ0ı(gv, 1))dgv

where we assume that f , f∨ and ϕs are all decomposable. In summary, we have the following

proposition.

Proposition 2.2.2. Let f , f∨ and ϕs be as above. Then for Re s > n, the integral

∫[H′(F )\H′(AF )]2

f(g1)f∨(g2)χ−1(det g2)E(ı(g1, g2), ϕs)dg1dg2

=∏v∈Σ

∫H′v

〈πv(gv)fv, f∨v 〉ϕs,v(γ0ı(gv, 1))dgv

defines an element in the space

HomH′(AF )×H′(AF )(I2n(s, χ), π∨ χπ) =⊗v∈Σ

HomH′v×H′v (I2n(s, χv), π∨v χvπv).

2.2.2 Local zeta integrals

We study the local functionals obtained in 2.2.1.

Fix a finite place v of F and suppress it from notations. For f ∈ π, f∨ ∈ π∨ and a holomorphic

section ϕs ∈ I2n(s, χ), define the local zeta integral

Z(χ, f, f∨, ϕs) =

∫H′〈π(g)f, f∨〉ϕs(γ0ı(g, 1))dg,

which is absolutely convergent when Re s > 2n. In [HKS1996, Section 6], the family of good sections

is introduced. For every good section ϕs, the zeta integral Z(χ, f, f∨, ϕs) is a rational function in q−s,

where q is the cardinality of the residue field of F . In particular, it has a meromorphic continuation

to the entire complex plane. Consider the family of zeta integrals

Z(χ, f, f∨, ϕs) | f ∈ π, f∨ ∈ π∨, ϕs is good

and the fractional ideal I of the ring C[qs, q−s] in its fraction field generated by the above family. In

24

fact, I is generated by P (q−s)−1, for a unique polynomial P (X) ∈ C[X] such that P (0) = 1. We let

L(s+1

2, π, χ) =

1

P (q−s)

be the local doubling L-series of Piatetski-Shapiro and Rallis. Same construction can also be applied

to the archimedean case.

Now suppose E/F is unramified (including split) at v and ψ, χ, π are also unramified. Let

f0 ∈ πK′ , f∨0 ∈ (π∨)K′

such that 〈f0, f∨0 〉 = 1, and ϕ0

s be the unramified standard section. Then the

calculation in [GPSR1987] and [Li1992] (see [Li1992, Theorem 3.1]) shows that

Z(χ, f0, f∨0 , ϕ

0s) =

L(s+ 12 ,BC(π)⊗ χ)

b2n(s)

where

bm(s) =

m−1∏i=0

L(2s+m− i, εiE/F ) (2.6)

is a product of local Tate factors. For the general case,

b2n(s)Z(χ, f, f∨, ϕs)

L(s+ 12 , π, χ)

admits a meromorphic extension to the entire complex plane which is holomorphic at s = 0. Moreover,

the normalized zeta integral

Z∗(χ, f, f∨, ϕs) =b2n(s)Z(χ, f, f∨, ϕs)

L(s+ 12 , π, χ)

|s=0 (2.7)

defines a nonzero element in HomH′×H′(I2n(0, χ), π∨χπ) (cf. [HKS1996, Proof of Theorem 4.3 (1)]).

Remark 2.2.3. It is conjectured (cf. e.g., [HKS1996]) that for all irreducible admissible representa-

tions π of H ′ and characters χ of E×, we have

L(s, π, χ) = L(s,BC(π)⊗ χ).

This is known when E/F , χ and π are all unramified due to (the same method of) Kudla–Rallis

[KR2005, Section 5]. It is also known when n = 1 due to [Har1993].

For further discussion, we need to recall a result on degenerate principal series. In what follows, we

25

will use the notation H ′′ instead of U(W ′′) for short, and recall our embedding ı : H ′×H ′ → H ′′. Let

V be a hermitian space of dimension 2n over E. Then ϕφ(g) = ωχ(g)φ(0) defines an H ′′-intertwining

map S(V 2n) → I2n(0, χ) whose image R(V, χ) is isomorphic to S(V 2n)H . Recall in 2.1.3 that we

denote by V ± the two non-isometric hermitian spaces of dimension 2n when v is finite nonsplit; V +

the only hermitian space (up to isometry) of dimension 2n when v is finite split; and Vs (0 ≤ s ≤ 2n)

the 2n+ 1 non-isometric hermitian spaces of dimension 2n when v is infinite.

Proposition 2.2.4. Let notations be as above. We have

1. If v is finite nonsplit, then R(V +, χ) and R(V −, χ) are irreducible, inequivalent, and I2n(0, χ) =

R(V +, χ)⊕R(V −, χ).

2. If v is finite split, then R(V, χ) is irreducible, and I2n(0, χ) = R(V +, χ).

3. If v is infinite, then R(Vs, χ) are irreducible, inequivalent, and I2n(0, χ) =⊕2n

s=0R(Vs, χ).

Proof. 1. It is [KS1997, Theorem 1.2].

2. It is [KS1997, Theorem 1.3].

3. It is [Lee1994, Section 6, Proposition 6.11].

2.3 Central special values of L-functions

2.3.1 Theta lifting and central L-values

We study the relation between the theta lifting θfφ defined in 2.2.1 and the central special value of the

L-function of the representation π.

Recall that we have an irreducible unitary cuspidal automorphic representation π of H ′ = Hn and

a hermitian space V over E of dimension 2n. One key question in the theory of theta lifting is whether

θfφ is nonvanishing. A sufficient condition is to look at the local invariant functional as follows. We

have the following proposition, which is usually referred as theta dichotomy.

Proposition 2.3.1. For every nonsplit place v ∈ Σ, HomH′v×H′v (R(Vv, χv), π∨v χvπv) 6= 0 for exactly

one hermitian space Vv (up to isometry) over Ev of dimension 2n.

Such Vv will be denoted as V (πv, χv).

26

Proof. If v is (real) archimedean, it is due to [Pau1998, Theorem 2.9]. If v is non-archimedean, it is

due to Proposition A.1.1 or [GG2011, Theorem 2.10], and the nonvanishing of Z∗.

In Proposition 2.3.1, if we let ϕs = ϕφ⊗φ∨,s and set

Z∗(s, χv, fv, f∨v , φv ⊗ φ∨v ) = Z∗(χv, fv, f

∨v , ϕφv⊗φ∨v ,s),

then both sides have meromorphic continuation to the entire complex plane, which are holomorphic

at the point s = 0. Namely, we have

〈θfφ, θf∨

φ∨〉H =L( 1

2 , π, χ)


∏v∈S

Z∗(0, χv, fv, f∨v , φv ⊗ φ∨v ), (2.8)

in which the product of normalized zeta integrals can actually be taken over a finite set S by the

unramified calculation. In particular, for v 6∈ S, Vv ∼= V (πv, χv). Then one necessary condition for θfφ

to be nonvanishing for some f and φ is that each local (normalized) zeta integral is not identically zero.

In other words, we should have Vv ∼= V (πv, χv) for all v ∈ Σ. Let V(π, χ) be the hermitian space over

AE such that V(π, χ)v ∼= V (πv, χv), and let ε(πv, χv) = ε(V (πv, χv)), ε(π, χ) =∏v∈Σ ε(πv, χv). If

ε(π, χ) = −1. Then V(π, χ) is incoherent, and hence for every V , the theta lifting θfφ always vanishes.

If ε(π, χ) = 1, then V(π, χ) ∼= V (π, χ)⊗F AF for a unique hermitian space V (π, χ) (up to isometry)

over E. Assume that V (π, χ) is anisotropic. Then there exist some f ∈ π and φ ∈ S(V (π, χ)n) such

that θfφ 6= 0 if and only if L( 12 , π, χ) 6= 0.

We would like to give another interpretation of the formula (2.8) when ε(π, χ) = 1, which is

heuristic for the arithmetic case. For this purpose, let us assume the following conjecture proposed

by S. Kudla and S. Rallis (in the symplectic-orthogonal case).

Conjecture 2.3.2 (cf. [HKS1996]). We have

dim HomH′v×H′v (I2n(0, χv), π∨v χvπv) = 1 (2.9)

for every irreducible admissible representation πv of H ′v.

Remark 2.3.3. 1. When n = 1, the above conjecture is proved as Proposition A.2.1.

2. In general, the above conjecture follows from the multiplicity preservation (cf. [LST2011, The-

orem A]) and the Local Howe Duality Conjecture.

3. The multiplicity preservation has been proved by Waldspurger [Wal1990] when v is non-archimedean

27

and has odd residue characteristic; by Li–Sun–Tian [LST2011] for every non-archimedean place

v; and by Howe [How1989] for every archimedean place v.

4. The Local Howe Duality Conjecture has been proved by Waldspurger [Wal1990] when v is non-

archimedean and has odd residue characteristic; by Minguez [Mın2008] for type II dual pairs for

every non-archimedean place v; and by Howe [How1989] for every archimedean place v.

5. In particular, the above conjecture is known if v does not have residue characteristic 2; or v has

residue characteristic 2 but is split in E.

Let V = V (π, χ) and R(V, χ) =⊗′

R(Vv, χv). On the one hand, the functional

β(f, f∨, φ, φ∨) = 〈θfφ, θf∨

φ∨〉H

defines an element in

HomH′(AF )×H′(AF )(R(V, χ), π∨ χπ) =⊗v∈Σ

HomH′v×H′v (R(Vv, χv), π∨v χvπv).

On the other hand, the functional

α(f, f∨, φ, φ∨) =∏v∈Σ

Z∗(0, χv, fv, f∨v , φv ⊗ φ∨v ) (2.10)

(when everything is decomposable, otherwise we take the linear combination) defines also an ele-

ment in⊗

v∈Σ HomH′v×H′v (R(Vv, χv), π∨v χvπv), which is nonzero. However, by (2.9), the space

HomH′v×H′v (R(Vv, χv), π∨v χvπv) is of dimension one. Therefore, β is a constant multiple of α. This

constant, by (2.8), is equal to

β

α=

L( 12 , π, χ)


.

In other words, vanishing of L( 12 , π, χ) is the obstruction for β to be a nontrivial global invariant

functional. This kind of formulation is first observed by X. Yuan, S.-W. Zhang and W. Zhang in

[YZZa,YZZb].

2.3.2 Vanishing of central L-values

We prove that the central L-value L( 12 , π, χ) vanishes when ε(π, χ) = −1.

28

By Proposition 2.2.4, we have a decomposition of H ′′(AF )-admissible representation

I2n(0, χ) =⊕V

R(V, χ) =⊕V

⊗′

v∈ΣR(Vv, χv),

where the direct sum is taken over all (isometry classes of) hermitian spaces over AE of rank 2n.

We note that each R(V, χ) is irreducible. Recall the group H ′′ = U(W ′′) and its standard parabolic

subgroup P with is unipotent radical N as in Notation 2.2.1. First, we need some lemmas on local

representations.

Fix a place v and suppress it from notations. For T ∈ Her2n(E), let OT = x ∈ V 2n | T (x) = T.

Define a character ψT of N ∼= Her2n(E) by ψT (n(b)) = ψ(trTb).

Lemma 2.3.4. Let notations be as above. We have

1. Suppose v is finite. Let S(V 2n)N,ψT (resp. R(V, χ)N,ψT ) be the twisted Jacquet module of S(V 2n)

(resp. R(V, χ)) associated to N and the character ψT . Then

(a) The quotient map S(V 2n)→ S(V 2n)N,ψT can be realized by the restriction S(V 2n)→ S(OT ).

(b) If T is nonsingular, then

dimR(V, χ)N,ψT =

1 if OT 6= ∅;

0 otherwise.

2. Suppose v is infinite, i.e., E/F = C/R and T is nonsingular. The space of H-invariant tempered

distribution T on S(V 2n) such that

T (ωχ(X)Φ) = dψT (X)T (Φ)

for X ∈ LieN is of dimension 1 (resp. 0) if OT 6= ∅ (resp. OT = ∅).

Proof. 1. It is [Ral1987, Lemma 4.2].

2. It is [Ral1987, Lemma 4.2] and [KR1994, Proposition 2.9].

We now construct the twisted Jacquet module R(V, χ)N,ψT or the invariant distribution explicitly

29

if it is not trivial. For a standard section ϕs ∈ I2n(s, χ), define the Whittaker integral

WT (g, ϕs) =

∫N

ϕs(wng)ψT (n)−1dn,

where w = w2n and dn is selfdual with respect to ψ. The integral WT (g, ϕs) is absolutely convergent

when Re s > n. It is easy to see that WT (e, •) : I2n(s, χ) → CN,ψT is an N -intertwining map. Let

WT (s, g,Φ) = WT (g, ϕΦ,s) for Φ ∈ S(V 2n). We have the following lemmas.

Lemma 2.3.5. Assume that T is nonsingular. Then

1. WT (g, ϕs) is entire.

2. The map Φ 7→ WT (0, e,Φ) realizes the surjective N -intertwining map S(V 2n) → R(V, χ) →

R(V, χ)N,ψT or the invariant distribution in Lemma 2.3.4 (2).

Proof. 1. It is [Kar1979, Corollary 3.6.1] for v finite and [Wal1988, Theorem 8.1] for v infinite.

2. It is [KR1994, Proposition 2.7].

Lemma 2.3.6. Assume that v is a finite place; E/F , ψ and χ are all unramified; and V = V +. Let

Φ0 be the characteristic function of (Λ+)2n for a selfdual OE-lattice Λ+, and T ∈ Her2n(OF ) with

detT ∈ O×F . Then we have

WT (s, e,Φ0) = b2n(s)−1.

Proof. This is [Tan1999, Proposition 3.2].

Now suppose we are in the global situation. We denote by A(G) the space of automorphic forms

of G for a reductive group G. For T ∈ Her2n(F ), define the T -th Fourier coefficient of f(g) ∈ A(H ′′)

as

WT (g, f) =

∫N(F )\N(AF )

f(ng)ψT (n)−1dn.

For a hermitian space V over AE of rank 2n, we have a family of linear maps

Es : R(V, χ)→ A(H ′′) (2.11)

Φ 7→ E(s, g,Φ) = E(g, ϕΦ,s)

30

for s near 0. It is an H ′′(AF )-intertwining map exactly when s = 0. Then for T nonsingular (and s

near 0), we have

ET (s, g,Φ) := WT (g,Es(Φ)) =∏v∈Σ

WT (s, gv,Φv). (2.12)

Lemma 2.3.7. For every H ′′(AF )-intertwining map E : R(V, χ)→ A(H ′′), if WT (g, •) E vanishes

for all nonsingular T , then E = 0.

Proof. Fix a finite place v, by Lemma 2.3.4 (1), we can find a section Φ0 = Φv,0Φv ∈ S(V2n) with

nonzero projection in R(V, χ) such that Φv,0 ∈ S(V2nv )reg (cf. Definition 2.4.1).

For every gv ∈ evH′′(AvF ), the functional Φv 7→ WT (0, gv,ΦvΦ

v) factors through the twisted

Jacquet module S(V2nv )Nv,ψT . If T is singular, then by our choice of Φv,0 and Lemma 2.3.4 (1-a),

WT (0, gv,Φv,0Φv) = 0. Similarly, WT (0, g,Φv,0Φv) = 0 for all g ∈ PvH′′(AvF ) since the action of

Pv stabilizes the subspace S(V2nv )reg. For T nonsingular, WT ≡ 0 by the assumption. Therefore,

E (Φ0)(g) = 0 for g ∈ PvH ′′(AvF ). It follows that E (Φ0) = 0. Therefore, E = 0 by our choice of Φ0

and the irreducibility of R(V, χ).

Proposition 2.3.8. We have

1. If V is incoherent, then dim HomH′′(AF )(R(V, χ),A(H ′′)) = 0.

2. If V is coherent, then dim HomH′′(AF )(R(V, χ),A(H ′′)) = 1 and E0 (2.11) is a basis.

Proof. 1. Assume that E is a nontrivial intertwining map. Then by Lemma 2.3.7, there is a

nonsingular T ∈ Her2n(F ) such that WT (g, •) E does not vanish. By Lemma 2.3.4 (1-b) and

(2), T is representable by Vv for every v ∈ Σ, i.e., OT 6= ∅. However, V will be coherent, which

is a contradiction.

2. Assume that E and E ′ are both nontrivial intertwining maps. By Lemma 2.3.7, there is a

nonsingular T such that WT (g, •) E does not vanish. By Lemma 2.3.4 (1-b)(2), there exists

c ∈ C such that WT (g, •) E ′ = cWT (g, •) E . Furthermore, c is independent of nonsingular T

since all of those that can be represented by V are in a single M(F )-orbit under the conjugation

action on N(F ). By Lemma 2.3.7, E ′ − cE = 0, i.e., dim HomH′′(AF )(R(V, χ),A(H ′′)) ≤ 1.

For the rest, we need to prove that E0 is actually nontrivial. Since V is coherent, we can choose

a nonsingular T ∈ Her2n(F ) that is representable by V. By (2.12), Lemma 2.3.5 (2) and Lemma

2.3.6, we can find a suitable Φ such that WT (0, e,Φ) 6= 0. Therefore, E0 6= 0.

31

Now we can state the following main result.

Theorem 2.3.9. If ε(π, χ) = −1, then L( 12 , π, χ) = 0.

Proof. Let V = V(π, χ). Then it is incoherent. We can choose suitable fv, f∨v , φv and φ∨v when some

of E,ψ, χ, π is ramified at v, such that Z∗(0, χv, fv, f∨v , φv ⊗ φ∨v ) 6= 0. Let f , f∨, φ and φ∨ be adelic

vectors with these subscribed local components and unramified ones at the places where E,ψ, χ, π are

unramified. From Proposition 2.2.2 (after analytic continuation), we have

∫[H′(F )\H′(AF )]2

f(g1)f∨(g2)χ−1(det g2)E(0, ı(g1, g2), φ⊗ φ∨)dg1dg2

=L( 1

2 , π, χ)∏2ni=1 L(i, εiE/F )

∏v∈S

Z∗(0, χv, fv, f∨v , φv ⊗ φ∨v ).

However, E0 is zero on R(V, χ) by Proposition 2.3.8 (1). We have E(0, ı(g1, g2), φ⊗φ∨) ≡ 0. Therefore,

L( 12 , π, χ) = 0 by our choices and the fact that the Tate L-values appearing above are finite.

Since L( 12 , π, χ) = 0, it motivates us to consider its derivative at 1

2 . In fact, we have

∫[H′(F )\H′(AF )]2

f(g1)f∨(g2)χ−1(det g2)d

ds|s=0 E(s, ı(g1, g2), φ⊗ φ∨)dg1dg2

=d

ds|s=0

∫[H′(F )\H′(AF )]2

f(g1)f∨(g2)χ−1(det g2)E(s, ı(g1, g2), φ⊗ φ∨)dg1dg2

=d

ds|s=0

L(s+ 12 , π, χ)∏2n

i=1 L(2s+ i, εiE/F )

∏v∈S

Z∗(s, χv, fv, f∨v , φv ⊗ φ∨v )

=L′( 1

2 , π, χ)∏2ni=1 L(i, εiE/F )

∏v∈S

Z∗(0, χv, fv, f∨v , φv ⊗ φ∨v ) + L(

1

2, π, χ)

d

ds|s=0

∏v∈S Z

∗(s, χv, fv, f∨v , φv ⊗ φ∨v )∏2n

i=1 L(2s+ i, εiE/F )

=L′( 1

2 , π, χ)∏2ni=1 L(i, εiE/F )

∏v∈S

Z∗(0, χv, fv, f∨v , φv ⊗ φ∨v ). (2.13)

Definition 2.3.10 (Analytic kernel functions). We call E′(0, g,Φ) = dds |s=0 E(s, g,Φ) the analytic

kernel function associated to the test function Φ ∈ S(V2n).

Recall that for T ∈ Her2n(F ), we let

ET (s, g,Φ) = WT (g,Es(Φ))

for s near 0. If T is nonsingular, then

WT (g,Es(Φ)) =∏v∈Σ

WT (s, gv,Φv)

32

if Φ = ⊗Φv is decomposable. Therefore,

E(s, g,Φ) =∑T sing.

ET (s, g,Φ) +∑

T nonsing.

∏v∈Σ

WT (s, gv,Φv).

Taking derivative at s = 0, we have

E′(0, g,Φ) =∑T sing.

E′T (0, g,Φ) +∑

T nonsing.

∑v∈Σ

W ′T (0, gv,Φv)∏v′ 6=v

WT (0, gv′ ,Φv′)

=∑T sing.

E′T (0, g,Φ) +∑v∈Σ

∑T nonsing.

W ′T (0, gv,Φv)∏v′ 6=v

WT (0, gv′ ,Φv′).

However,∏v′ 6=vWT (0, gv,Φv) 6= 0 only if Vv′ represents T for all v′ 6= v by Lemma 2.3.4 (1-b).

Since V is incoherent, Vv can not represent T . For T nonsingular, there are only finitely many v ∈ Σ

such that T is not represented by Vv, i.e., there does not exist x1, . . . , x2n ∈ Vv whose moment matrix

is T . We denote the set of such v by Diff(T,V). Then

E′(0, g,Φ) =∑T sing.

E′T (0, g,Φ) +∑v∈Σ

Ev(0, g,Φ),

where

Ev(0, g,Φ) =∑

Diff(T,V)=v

W ′T (0, gv,Φv)∏v′ 6=v

WT (0, gv′ ,Φv′). (2.14)

In fact, the summation in (2.14) is taken only over those v that are nonsplit in E.

2.4 Analytic kernel functions

2.4.1 Regular test functions

We prove that the summation of E′T (0, g,Φ) over singular T ’s vanishes for certain choice of Φ, and g

in a suitable subset of H ′′(AF ). We follow the idea in [YZZb].

Definition 2.4.1 (Regular test functions). Let v be a place and V a hermitian space over Ev. A test

function φ ∈ S(V r) is regular, if φ(x) = 0 for x with degenerate moment matrix, i.e., detT (x) = 0.

We denote by S(V r)reg ⊂ S(V r) the subspace of regular test functions.

Fix a finite subset S ⊂ Σfin with |S| = k > 0 and let S(V2nS )reg =

⊗v∈S S(V2n

v )reg. We have the

following proposition.

33

Proposition 2.4.2. For Φ = ΦSΦS ∈ S(V2nS )reg ⊗ S(VS,2n), ords=0ET (s, g,Φ) ≥ k for T singular

and g ∈ P (AF,S)H ′′(ASF ).

We can assume that Φ = ⊗v∈ΣΦv is decomposable with Φv ∈ S(V2nv )reg for v ∈ S and rankT =

2n− r < 2n. Choose a ∈ GL2n(E) such that

aT taτ =

0r

T

(2.15)

with T ∈ Her2n−r(E). Then

ET (s, g,Φ) = EaT taτ (s,m(a)g,Φ).

Therefore, we can assume that T is of the form (2.15).

First, we need a more explicit formula for the singular coefficient ET . By definition, for Re s > n

ET (s, g,Φ) =

∫N(F )\N(AF )

∑γ∈P (F )\H′′(F )

ϕΦ,s(γng)ψT (n)−1dn

=

∫N(F )\N(AF )

∑γ∈P (F )\H′′(F )

(r(g)ϕΦ,s) (γn)ψT (n)−1dn (2.16)

where r(g) standards for the action of H ′′ on I2n(s, χ) by right translation. We need to unfold this

summation. Recall that for 0 ≤ d ≤ 2n,

w2n,d =

1d

12n−d

1d

−12n−d

(2.17)

in Notation 1.0.5. Then w2n,d | 0 ≤ d ≤ 2n is a set of representatives of the double coset

WP/N\WH′′/WP/N of Weyl groups. In particular, w2n,0 = w2n = w. We have a Bruhat decom-

position

H ′′(F ) =

2n∐d=0

P (F )w2n,dP (F ),

where F can be a global field or its local completions.

Lemma 2.4.3. For v ∈ S and gv ∈ Pv, the support of r(gv)ϕΦv,s is contained in P (Fv)wN(Fv).

34

Proof. It suffices to prove that ϕΦv,s vanishes on P (Fv)w2n,dP (Fv) for d > 0 since gv ∈ P (Fv). For

g = n(b1)m(a1)w2n,dn(b2)m(a2) ∈ P (Fv)w2n,dP (Fv), we have

ϕΦv,s(g) = (ωχv (g)Φ) (0)λ(g)s

= χv(det a1a2)|det a1a2|nEvλ(g)s∫V

2n−dv

ψb2(T (x))Φv(xa2)dx

where V2n−dv is viewed as a subset of V2n

v by the map (x1, . . . , x2n−d) 7→ (0, . . . , 0, x1, . . . , x2n−d).

Since Φv is regular and d > 0, Φv(xa2) = 0 for x ∈ V2n−dv . Therefore, the lemma follows.

By the above lemma, we have for g ∈ P (AF,S)H ′′(ASF ),

(2.16) =

∫N(F )\N(AF )

∑γ∈P (F )\P (F )wP (F )

(r(g)ϕΦ,s) (γn)ψT (n)−1dn

=

∫N(F )\N(AF )

∑γ∈wN(F )

(r(g)ϕΦ,s) (γn)ψT (n)−1dn

=

∫N(AF )

(r(g)ϕΦ,s) (wn)ψT (n)−1dn

=

∫N(AF )

ϕs(wn)ψT (n)−1dn

=∏v∈Σ

∫Nv

ϕv,s(wnv)ψT (nv)−1dnv (2.18)

where we denote by ϕs instead of r(g)ϕΦ,s for simplicity. Let S′ ⊂ Σ be a finite subset containing all

infinite places such that for all v 6∈ S′, v, χv and ψv are unramified; ϕv,s is the (unique) unramified

section in I2n(s, χv) (hence S′ ⊃ S) and det T ∈ O×Fv . Then

(2.18) =

(∏v∈S′

WT (e, ϕv,s)

)WT (e, ϕS

′

s ). (2.19)

By [KR1994, Page 36] and [Tan1999, Proposition 3.2],

WT (e, ϕS′

s ) =aS′

2n(s)

aS′

2n−r(s− r2 )bS

′2n(s)

where

am,v(s) =

m−1∏i=0

Lv(2s+ i−m+ 1, εiE/F )

35

and

bm,v(s) =

m−1∏i=0

Lv(2s+m− i, εiE/F )

as (2.6). Therefore, WT (e, ϕS′

s ) has a meromorphic continuation to the entire complex plane. For

v ∈ S′, we normalize the Whittaker integral to be

W ∗T (e, ϕv,s) =a2n−r,v(s− r

2 )b2n,v(s)

a2n,v(s)WT (e, ϕv,s).

From the argument (and also notations) in [KR1994, Page 35],

WT (e, ϕv,s) = WT (e, i∗ Ur,v(s)ϕv,s).

By [PSR1987, Section 4], the (local) intertwining operator Ur,v(s) has a meromorphic continuation to

the entire complex plane. By Lemma 2.3.5 (1), WT (e, ϕv,s) and hence W ∗T (e, ϕv,s) have meromorphic

continuation to the entire complex plane. Together with the meromorphic continuation of WT away

from S′ and W ∗T in S′, (2.19) has a meromorphic continuation which equals

a2n(s)

a2n−r(s− r2 )b2n(s)

∏v∈S′

W ∗T (e, ϕv,s).

Proof of Proposition 2.4.2. At the point s = 0, we have b2n(0) =∏2ni=1 L(i, εiE/F ) ∈ C× and

a2n(0)

a2n−r(− r2 )=

r−1∏i=0

L(−i, εi+1E/F ) ∈ C×.

Let κv = ords=0W∗T (e, •) be the order of the functional at s = 0 for v ∈ S′ and

κ′v = ords=0W∗T (s, e, •)|S(V2n

v )reg

for v ∈ S. Since ET (e, ϕΦ) = 0 if Φ = ⊗Φv for at least one Φv regular, by (2.19) and the proof of

Lemma 2.3.7, we have κ′v0 +∑v0 6=v∈S′ κv ≥ 1 for every v0 in S. Also by the definition of WT , we see

that

ϕv,0 7→ s−κvW ∗T (e, ϕv,s)|s=0

36

is a nontrivial N -intertwining map from I2n(0, χ) to CN,ψT . Now if v ∈ S, ϕv,0 = ϕΦv,0 for a

regular test function Φv ∈ S(V2nv )reg. By Lemma 2.3.4 (1-a), ϕv,0 goes to 0 under the above map,

i.e., κ′v ≥ κv + 1 for v ∈ S. Therefore,

ords=0

∏v∈S′

W ∗T (e, ϕv,s) ≥∑v∈S

κ′v +∑

v∈S′−Sκv ≥ k − 1 + κ′v0 +

∑v0 6=v∈S′

κv ≥ k.

The proposition follows.

In conclusion, if we choose S such that |S| ≥ 2, and Φ = ΦSΦS ∈ S(V2nS )reg ⊗ S(VS,2n) which is

decomposable, then for g ∈ P (AF,S)H ′′(ASF ), we have

E′(0, g,Φ) =∑v∈Σ

Ev(0, g,Φ). (2.20)

2.4.2 Test functions of higher discriminant

We show that if we have a finer choice of Φv for v ∈ S, we can even make W ′T (0, e,Φv) = 0 for all

nonsingular T that are not representable by Vv.

Since the issue is local, we fix a nonsplit place v ∈ S and suppress it from notations in this

subsection. Let V be one of V ± and V ′ the other one which is not isometric to V . Let

Her02n(E) = T ∈ Her2n(E) | detT 6= 0;

H = b ∈ Her02n(E) | b = T (x) for some x ∈ V 2n;

H′ = b′ ∈ Her02n(E) | b′ = T (x′) for some x′ ∈ V ′2n;

H′d = b′ + b′′ | b′ ∈ H′ and b′′ ∈ Her2n(p−dE ) ∩Her02n(E), d ∈ Z,

where pE is the maximal ideal of OE . Then

• Her02n(E) = H

∐H′;

• · · · H′−1 H′0 H′1 · · · ;

•⋂dH′d = H′;

•⋃dH′d = Her0

2n(E).

We say a test function Φ ∈ S(V 2n) is of discriminant d if

T (x) | x ∈ Supp(Φ) ∩H′d = ∅,

37

and denote by S(V 2n)d the space of such functions. Set S(V 2n)reg,d = S(V 2n)reg ∩ S(V 2n)d.

Lemma 2.4.4. For every d ∈ Z, S(V 2n)reg,d is not empty.

Proof. Fix an element d ∈ Z. We only need to prove that there exists T 6∈ H′d such that detT 6= 0.

Then(T + Her2n(p−dE )

)∩ H′ = ∅. Any test function with support whose elements have moment

matrices contained in(T + Her2n(p−dE )

)∩ Her0

2n(E), which is open, will be in S(V 2n)reg,d. Now we

want to find such a T . Take an element T1 ∈ H with detT1 6= 0. Since H is open, we can find a

neighborhood T1 + Her2n(pνE) ⊂ H for some ν ∈ Z. If ν ≤ −d, then we are done. Otherwise, let $

be the uniformizer of F . Then $−ν−d (T1 + Her2n(pνE)) ⊂ H. However, $−ν−d (T1 + Her2n(pνE)) =($−ν−dT1 + Her2n(p−dE )

). Therefore, T = $−ν−dT1 will serve for our purpose.

Since ψ is nontrivial, we can define its discriminant dψ to be the largest integer d such that the

character ψT is trivial on N(OF ) ∼= Her2n(OE) for all T ∈ Her2n(p−dE ). We slightly abuse notation

since dψ depends also on 2n. We also need to mention that this is not the usual discriminant or

different of a p-adic additive character. However, the difference between them depends only on n and

the ramification of E/F . We have the following proposition.

Proposition 2.4.5. Let d ≥ dψ be an integer and Φ ∈ S(V 2n)reg,d. Then WT (s, e,Φ) ≡ 0 for T ∈ H′

nonsingular.

Proof. For Re s > n,

WT (s, e,Φ) =

∫N

(ωχ(wn)Φ) (0)λ(wn)sψT (n)−1dn

is absolutely convergent. Therefore, it equals

∫Her2n(E)

(∫V 2n

ψ(tr bT (x))Φ(x)dx

)λ(wn(b))sψ(− trTb)db

=

∫V 2n

Φ(x)dx

∫Her2n(E)

λ(wn(b))sψ(tr(T (x)− T )b)db

=

∫V 2n

Φ(x)dx

∫Her2n(E)

λ(wn(b))sψT (x)−T (n(b))db. (2.21)

Since λ(wn(b)n(b1)) = λ(wn(b)) for b1 ∈ Her2n(OE),

(2.21) =

∫V 2n

Φ(x)dx

∫Her2n(E)/Her2n(OE)

λ(wn(b))sψT (x)−T (n(b))db

∫Her2n(OE)

ψT (x)−T (n(b1))db1,

38

in which the last integral is zero for all x ∈ Supp(Φ) by our assumption on Φ. Therefore, WT (s, e,Φ) ≡

0 after continuation. In particular, W ′T (0, e,Φ) = 0.

Remark 2.4.6. Obviously, it is not necessary to assume the dimension of V to be even. All definitions

and results above can be applied to arbitrary dimensions.

In conclusion, let S be a finite subset of Σfin with |S| ≥ 2, and Φ = ⊗Φv ∈ S(V2n) with Φv ∈

S(V2nv )reg for v ∈ S and Φv ∈ S(V2n

v )reg,dv for v ∈ S nonsplit with dv ≥ dψv . Then combining (2.20),

we have

E′(0, g,Φ) =∑v 6∈S

Ev(0, g,Φ) (2.22)

for g ∈ eSH ′′(ASF ).

2.4.3 Density of test functions

We have made particular choices of test functions to simplify the formula of the analytic kernel

function. However, for our proof of the main theorem, not arbitrary choices will work. We now show

that there are “sufficiently many” test functions satisfying these choices we have made, in the sense

of Proposition 2.4.10. We follow the idea in [YZZa].

We keep our notations in 2.4.1 and 2.4.2. In particular, v will be a place in S and suppressed from

notations. Recall that we have an H ′′-intertwining map

S(V 2n) S(V 2n)H ∼= R(V, χ) → I2n(0, χ)

through the Weil representation ω′′χ. Therefore, we obtain an H ′ × H ′ admissible representation on

S(V 2n) through the embedding ı (2.2).

Lemma 2.4.7. If v is nonsplit, then for every d ∈ Z we have

S(V 2n)reg = ω′′χ(m(F×12n))S(V 2n)reg,d.

Proof. Fix an element d ∈ Z. For every function Φ ∈ S(V 2n)reg, Supp(Φ) is a compact subset of H.

Since Her02n(E)\H′d is open and

⋃d

(Her0

2n(E)\H′d)

= Her02n(E)\

⋂d

H′d = Her02n(E)\H′ = H,

39

the family(Her0

2n(E)\H′d)d∈Z is an open covering of Supp(Φ), hence has a finite subcover. Therefore,

there exists d0 ∈ Z such that Supp(Φ) ∩ H′d0 = ∅. If d0 ≥ d, we are done. Otherwise, consider

Φ′ = ω′′χ(m($d0−d12n))Φ. Then Supp(Φ′) ∩H′d = ∅. The lemma follows.

In the rest of this subsection, let n = 1. Then H ′ = U(W1).

Lemma 2.4.8. Let π be an irreducible admissible representation of H ′ which is not of dimension 1 and

A : S(V )→ π a surjective H ′-intertwining map, where H ′ acts on S(V ) through a Weil representation

ω. Then for every φ with A(φ) 6= 0, there is φ′ ∈ S(V )reg such that A(φ′) 6= 0 and Supp(φ′) ⊂ Supp(φ).

Proof. Let f = A(φ), if there exists n ∈ N ′ such that π(n)f 6= f , then

A (ω(n)φ− φ) = π(n)f − f 6= 0,

However,

(ω(n)φ) (x)− φ(x) = (ψ(bT (x))− 1)φ(x)

where n = n(b). We see that φ′ = ω(n)φ − φ ∈ S(V )reg and Supp(φ′) ⊂ Supp(φ). If for every

n ∈ N ′, π(n)f = f , then f will be fixed by an open subgroup of H ′ containing N since π is smooth.

However, any such subgroup will contain SU(W1). Therefore, π factors through H ′/ SU(W1) =

U(W1)/ SU(W1) ∼= E×,1, which contradicts the assumption on π.

Lemma 2.4.9. Let π1 and π2 be two two irreducible admissible representations of H ′ which are not of

dimension 1. Then for every surjective H ′×H ′-intertwining map B : S(V )⊗S(V ) = S(V 2)→ π1π2

where H ′ × H ′ acts on S(V ) ⊗ S(V ) by a pair of Weil representations ω1 ω2, there is an element

Φ = φ1 ⊗ φ2 ∈ S(V 2)reg such that φα ∈ S(V )reg (α = 1, 2) and B(Φ) 6= 0.

Proof. Let Φ′ ∈ S(V 2) be an element such that B(Φ′) 6= 0, write Φ′ =∑φi,1 ⊗ φi,2 as an element

in S(V ) ⊗ S(V ). Therefore, we can assume that there is φ1 ⊗ φ2 such that B(φ1 ⊗ φ2) 6= 0. By

Lemma 2.4.8, we can also assume that φ1 ∈ S(V )reg. For x ∈ Supp(φ1), let Vx be the subspace of V

generated by x and V x its orthogonal complement. They are both nondegenerate hermitian spaces of

dimension 1. As H ′-representation, S(V ) = S(Vx)⊗ S(V x). Write φ2 =∑φi,x ⊗ φxi according to this

decomposition. We can assume that there is one φx ⊗ φx such that B(φ1 ⊗ (φx ⊗ φx)) 6= 0. Since as

H ′-representation, S(V x) is generated by the subspace S(V x)reg. We can then write

φx ⊗ φx =∑

ω2(gj)(ω−1

2 (gj)φx ⊗ φxj)

40

with φxj ∈ S(V x)reg. So we can further assume that B(φ1 ⊗ (φx ⊗ φx)) 6= 0 with φx ∈ S(V x)reg,

i.e., Supp(φx ⊗ φx) ∩ Vx = ∅. Applying Lemma 2.4.8 again, we can further assume that there

exists φ(x)2 ∈ S(V )reg such that Supp(φ

(x)2 ) ⊂ Supp(φx ⊗ φx) and B(φ1 ⊗ φ(x)

2 ) 6= 0. The condition

that Supp(φ2) ∩ V x = ∅ is open for x. Therefore, we can find a neighborhood Ux of x such that

(φ1 | Ux)⊗ φ(x)2 ∈ S(V 2)reg. Since Supp(φ1) is compact, we can find Φ of this kind such that B(Φ) 6=

0.

Recall the (normalized) zeta integrals (2.7), and that for Φ ∈ S(V 2n), we set Z∗(s, χ, f, f∨,Φ) =

Z∗(χ, f, f∨, ϕΦ,s). Combining Lemma 2.4.7 and Lemma 2.4.9, we have the following proposition.

Proposition 2.4.10. Let n = 1, v ∈ Σfin, π be an irreducible cuspidal automorphic representation of

H ′ and Vv = V (πv, χv). For every d ∈ Z, we can find fv ∈ πv, f∨v ∈ π∨v and φα ∈ S(V )reg (α = 1, 2)

with the property that φ1,v ⊗φ2,v ∈ S(V 2v )reg,d (resp. S(V 2

v )reg) if v is nonsplit (resp. split) in E, such

that the (normalized) zeta integral Z∗(0, χv, fv, f∨v , φ1,v ⊗ φ2,v) 6= 0.

41

Chapter 3

Arithmetic theta lifting and

arithmetic kernel functions

In this chapter, we study the geometric part of the arithmetic inner product. In 3.1, we introduce

the Shimura varieties of unitary groups which we work with, and on them the special cycles and

generating series. We prove the theorem of modularity of the generating series. For the case where

Shimura varieties are not proper, we need to consider their compactifications, which will be discussed

in 3.2. In 3.3, we define the arithmetic theta lifting and prove its cohomological triviality, which enable

us to formulate the explicit conjecture of the arithmetic inner product formula. Finally, we restrict

ourselves to the situation where n = 1 and hence the Shimura varieties are unitary Shimura curves,

in 3.4. We review the theory of Neron–Tate height paring, based on which we define the arithmetic

kernel functions and decompose them into local terms.

We fix an additive character ψ : F\AF → C× such that ψι is the standard one, i.e., t 7→ exp(2πit)

(t ∈ Fι = R) for any ι ∈ Σ∞ until the end of this article.

42

3.1 Modularity of generating series

3.1.1 Shimura varieties of unitary groups

We recall the notion attached to Shimura varieties of unitary groups. Let m ≥ 2 and 1 ≤ r < m be

integers. Let V be a totally positive definite incoherent hermitian space over AE of rank m. Let H =

ResAF /AU(V) be the unitary group which is a reductive group over A, and Hder = ResAF /A SU(V)

its derived subgroup. Let T ∼= ResAF /AA×,1E be the maximal abelian quotient of H, which is also

isomorphic to the center of H. Let T ∼= ResF/QE×,1 be the unique (up to isomorphism) Q-torus such

that T ×QA ∼= T. Then T has the property that T (Q) is discrete in T (Afin). For every open compact

subgroup K of H(Afin), there is a Shimura variety ShK(H) of dimension m− 1 defined over the reflex

field E. For every embedding ι : E → C over ι ∈ Σ∞, we have the following ι-adic uniformization

ShK(H)anι∼= H(ι)(Q)\

(D(ι) ×H(Afin)/K

).

We briefly explain the notations and meanings above:

• ShK(H)anι denotes the complex analytification of ShK(H) via ι.

• Let V (ι) be the nearby E-hermitian space of V at ι, i.e., V (ι) is the unique E-hermitian space

(up to isometry) such that V(ι)v∼= Vv for v 6= ι but V

(ι)ι is of signature (m − 1, 1). Then

H(ι) = ResF/QU(V (ι)).

• D(ι) is the symmetric hermitian domain consisting of all negative C-lines in V(ι)ι whose complex

structure is given by the action of Fι ⊗F Eι−→ R⊗R C ∼= C.

• The group H(ι)(Q) diagonally acts on D(ι) by conjugation, and on H(Afin)/K by identifying

H(ι)(Afin) and H(Afin) through the corresponding hermitian spaces.

In fact, the underlying real symmetric domain of D(ι) can be identified with the Hι(R)-conjugacy

class of the Hodge map h(ι) : S = ResC/RGm,C → H(ι)R∼= U(m− 1, 1)R ×U(m, 0)d−1

R given by

h(ι)(z) =

1m−1

z/z

,1m, . . . ,1m

.

Assumption 3.1.1. From now on, we will assume that K is sufficiently small, e.g., contained in the

principal congruence subgroup for N ≥ 3 for the natural embedding into the general linear group.

Then ShK(H) is a quasi-projective nonsingular E-scheme. It is proper if and only if

43

1. F 6= Q; or

2. F = Q, m = 2 and Σ(V) ! Σ∞.

The set of geometric connected components of ShK(H) can be identified with T (Q)\T (Afin)/detK.

For every other open compact subgroup K ′ ⊂ K, we have an etale covering map

πK′

K : ShK′(H)→ ShK(H). (3.1)

Let Sh(H) be the projective system ShK(H)K . On each ShK(H), we have a Hodge bundle LK ∈

Pic(ShK(H))Q which is ample. They are compatible under the pullback of πK′

K , hence define an

element

L ∈ Pic(Sh(H))Q := lim−→K

Pic(ShK(H))Q.

3.1.2 Special cycles and generating series

Let V1 be an E-subspace of Vfin, we say V1 is admissible if (−,−)|V1takes values in E and for every

nonzero x ∈ V1, (x, x) is totally positive. We have the following lemma.

Lemma 3.1.2. The E-subspace V1 is admissible if and only if for every ι ∈ Σ∞, there is an element

h ∈ Hder(Afin) such that hV1 ⊂ V (ι) ⊂ Vfin and is totally positive definite.

Proof. The “if” direction is obvious. For the “only if” direction, let us assume that V1 is admissible

and fix arbitrary ι. Take v1 ∈ V1 with nonzero norm. Then q(v1) = 12 (v1, v1) is locally a norm

for the hermitian form on V (ι) by the definition of admissibility and the signatures of V and V (ι).

Thus it is a norm for some v ∈ V (ι) by Hasse–Minkowski Theorem. Now we apply Witt Theorem

to find an element h1 ∈ U(Vfin) = H(Afin) such that h1v1 = v as elements in Vfin. Choose a

vector v′ ∈ 〈v〉⊥ ⊂ V (ι) with nonzero norm. Let h′ ∈ H(Afin) which stabilizes 〈v′〉⊥ and act on

the Afin,E-line spanned by v′ through the multiplication by (deth1)−1. Then h′h1v1 = h′v = v for

h = h′h1 ∈ SU(Vfin) = Hder(Afin).

Replacing V1 by hV1, we can assume that v1 = v ∈ Vfin. Since dimV1 < m, we can use induction

on r by considering the orthogonal complement of v in V1 and V (ι) to find an element h ∈ Hder(Afin)

such that hV1 ⊂ V (ι) ⊂ Vfin.

For admissible V1, let V1 be a totally positive definite (incoherent) hermitian space over AE such

that V1,fin∼= V ⊥1 ⊂ Vfin. Let H1 be the corresponding unitary group, we have a finite morphism of

44

Shimura varieties

ςV1 : ShK1(H1)→ ShK(H) (3.2)

where K1 = K ∩H1(Afin), such that the image of the map is represented, under the uniformization

at some ι, by the points (z, h1h) ∈ D(ι) ×H(Afin) where

• h is as in Lemma 3.1.2 (with respect to the ι dividing ι);

• z ⊥ hV1;

• h1 fixes all elements in hV1.

The image defines a special cycle Z(V1)K ∈ CHr(ShK(H))Q. It depends only on the class KV1.

For x ∈ Vrfin, let Vx be the E-subspace of Vfin generated by the components of x. We define

Z(x)K =

Z(Vx)Kc1(L ∨K)r−dimE Vx if Vx is admissible

0 otherwise.

To introduce the generating series, we need a restriction on the space S(Vrι ) of the Weil represen-

tation when ι is infinite.

We define a subspace S(Vrι )

Uι ⊂ S(Vrι ) consisting of functions of the form

P (T (x)) exp(−2π trT (x))

where P is a polynomial function on Herr(C). If we view S(Vrι ) as the corresponding (LieHr,ι,Kr,ι)-

module, then S(Vrι )

Uι is the submodule generated by the Gaussian

φ0∞(x) = exp(−2π trT (x)).

Let

S(Vr)U∞ =

(⊗ι∈Σ∞

S(Vrι )

Uι

)⊗ S(Vr

fin); S(Vr)U∞K =

(⊗ι∈Σ∞

S(Vrι )

Uι

)⊗ S(Vr

fin)K

for an open compact subgroup K of H(Afin).

45

For φ ∈ S(Vr)U∞K , we define the generating series to be

Zφ(g) =∑

x∈K\Vrfin

(ωχ(g)φ) (T (x), x)Z(x)K

as a formal series with values in CHr(ShK(H))C for g ∈ Hr(AF ). Here for φ = φ∞φfin, we denote

φ(T (x), x) = φ∞(y)φfin(x) for every y ∈ Vr∞ with T (y) = T (x), whose value does not depend on

the choice of y. This makes sense since Z(x)K 6= 0 only for Vx admissible and hence T (x) is totally

semi-positive definite. It is easy to see that Zφ(g) is compatible under the pullback of πK′

K , hence

defines a series with values in CHr(Sh(H))C := lim−→KCHr(ShK(H))C.

3.1.3 Pullback formula and modularity theorem

We briefly recall the notion of Shimura varieties attached to orthogonal groups. The AF -module

V is also a totally positive definite quadratic space over AF of rank 2m with the quadratic for-

m TrAE/AF (−,−). Then it is incoherent with discriminant in F×/F×2 ⊂ A×F /A×2F . Let G =

ResAF /AGSpin(V) be the special Clifford group of V with the adjoint (quotient) group Gadj =

ResAF /A SO(V) and the derived subgroup Gder = ResAF /A Spin(V). For every open compact sub-

group K ′ of G(Afin), there is a Shimura variety ShK′(G) defined over the reflex field F such that for

every embedding ι : F → C, we have the following ι-adic uniformization

ShK′(G)anι∼= G(ι)(Q)\

(D′(ι) ×G(Afin)/K ′

).

We briefly explain the notations and meanings above:

• ShK′(G)anι denotes the complex analytification of ShK′(G) via ι.

• Let V (ι) be the nearby F -quadratic space of V at ι, i.e., V (ι) is the unique F -quadratic space

(up to isometry) such that V(ι)v∼= Vv for v 6= ι but V

(ι)ι is of signature (2m − 2, 2). Then

G(ι) = ResF/QGSpin(V (ι)).

• D′(ι) is the symmetric hermitian domain consisting of all oriented negative definite 2-dimensional

subspaces of V(ι)ι .

• The group G(ι)(Q) diagonally acts on D′(ι) by conjugation, and on G(Afin)/K ′ by identifying

G(ι)(Afin) and G(Afin) through the corresponding quadratic spaces.

We have similar notations of Hodge bundles, special cycles, and generating series on ShK′(G),

which are denoted by L ′K′ , Z′(x)K′ for x ∈ Vr

fin, and Z ′φ(g′) for φ ∈ S(Vr)O∞K′ and g′ ∈ Gr(Afin),

46

respectively (cf. [YZZ2009]). Here we introduce the standard symplectic F -space W ′r (comparing to

the space Wr in 2.1.1) which has the basis e1, . . . , e2r with the symplectic form

• 〈ei, ej〉 = 0;

• 〈er+i, er+j〉 = 0;

• 〈ei, er+j〉 = δij for 1 ≤ i, j ≤ r.

The group Gr = Sp(W ′r) is an F -reductive group. Similarly, when defining the generating series, we

use the Weil representation ω (with respect to ψ) of Gr(AF )×G(A) on S(Vr).

Now we fix an embedding ι : E → C over ι and suppress them from the notations of nearby

objects like V = V (ι), H = H(ι), D = D(ι), etc. Therefore, we have the usual notion of Shimura

variety ShK(H,X) (resp. ShK′(G,X′) with a connected component X+ of X ′) which is defined over

ι(E) (resp. ι(F )). Its neutral component is the connected Shimura variety ShK(Hder, X) (resp.

ShK′(Gder, X+)) attached to the connected Shimura datum (Hder, X) (resp. (Gder, X+)), which is

defined over EK (resp. EK′): a finite abelian extension of ι(F ) in C. The canonical embedding

Hder → Gder (cf. Remark 3.1.4 (1) below) of reductive groups and the embedding D → D′ by

forgetting the E-action define an injective map of connected Shimura data (Hder, X) → (Gder, X+)

which gives an embedding

iK′ : ShK(Hder, X) → ShK′(Gder, X+)

which is defined over EK providing that K∩Hder(Afin) = K ′∩Hder(Afin) and K ′ is sufficiently small.

Let Z(x)K (resp. Z ′(x)K′ , Zφ(g), Z ′φ(g′)) be the restriction of Z(x)K (resp. Z ′(x)K′ , Zφ(g), Z ′φ(g′))

to the neutral component.

Proposition 3.1.3. Assume that K ′ is sufficiently small and K ∩ Hder(Afin) = K ′ ∩ Hder(Afin).

Then for x ∈ Vfin, the pullback of the special divisor i∗K′Z′(x)K′ is the sum of Z(x1)K indexed by the

classes x1 in K\K ′x, both considered as elements in Chow groups.

Proof. If x = 0, then the only class in K\K ′x is x1 = 0; the proposition follows from the compatibility

of Hodge bundles under pullbacks induced by maps of (connected) Shimura data.

Now we assume that 〈x, x〉 ∈ E and is totally positive. Suppose that (z, h) ∈ D × Hder(Afin)

represents a C-point in the scheme-theoretic intersection ShK(Hder, X) ∩ Z ′(x1)K′ for some x1 ∈

K ′x. Let g ∈ G(Afin) such that gx1 = x′1 ∈ V ⊂ Vfin. Then z ⊥ γx′1 for some γ ∈ G(Q) and

h ∈ γG(Afin)x′1gk′ for some k′ ∈ K ′, where G(Afin)x′1 is the subgroup of G(Afin) fixing x′1. We

now show that γG(Afin)x′1gk′ ∩ Hder(Afin) = G(Afin)γx′1γgk

′ ∩ Hder(Afin) 6= ∅, i.e., G(Afin)γx′1 ∩

47

Hder(Afin)k′−1g−1γ−1 6= ∅, which is true by Lemma 3.1.2. Therefore, (z, h) represents a C-point in

the special cycle Z(h−1E〈γgx1〉)K of ShK(Hder, X). If we write h = g1γgk′ with some g1 ∈ G(Afin)γx′1 ,

then

h−1E〈γgx1〉 = E〈h−1γgx1〉 = E〈k′−1g−1γ−1g−11 γgx1〉 = E〈k′−1x1〉.

Therefore, the scheme-theoretic intersection is indexed by the classes x1 in K\K ′x. This also holds

on the level of Chow groups since the intersection is proper.

Remark 3.1.4. We have the following two remarks.

1. The canonical embedding Hder → Gder is given by the following way. First, we have an embed-

ding Hder → H → Gadj by forgetting the E-action on V = V (ι). Since Hder is simply connected,

we have a canonical lifting Hder → G. Since Hder has no nontrivial abelian quotient, the image

is actually contained in Gder.

2. In the proof of Proposition 3.1.3, we can still use the adelic description of the C-points of

ShK(Hder, X) (resp. ShK′(Gder, X+)) which is compatible with that of ShK(H) (resp. ShK′(G))

since Hder (resp. Gder) is semisimple, of noncompact type and simply-connected.

The group Gr is canonically embedded in Hr by identifying the bases 〈e1, . . . , e2r〉 of W ′r and Wr.

Therefore, ωχ|Gr = ω. From Proposition 3.1.3, we have the following corollary.

Corollary 3.1.5. Let r = 1 and K, K ′ be as in Proposition 3.1.3. Then i∗K′Z′φ(g′) = Zφ(g′) for

g′ ∈ G1(AF ) and φ ∈ S(V)U∞K′ .

For a linear functional l ∈ CHr(Sh(H))∗C, we have a complex valued series

l(Zφ)(g) =∑

x∈K\Vrfin

(ωχ(g)φ) (T (x), x)l(Z(x)K)

for every K such that φ is invariant under K (which is of course independent of such choice). The

following is the main theorem.

Theorem 3.1.6 (Modularity of the generating series). We have

1. If l(Zφ)(g) is absolutely convergent, then it is an automorphic form of Hr. Moreover, the

archimedean component of (the representation generated by) l(Zφ) is a discrete series represen-

tation of weight (m+kχ

2 , m−kχ

2 ) (cf. Definition 2.1.2 and Notation 2.1.1).

48

2. If r = 1, then l(Zφ)(g) is absolutely convergent for every l.

Proof. 1. We proceed as in [YZZ2009, Section 4]. First, we can assume that φ = φ0∞ ⊗ φf since

other cases will follow from the (LieHr,ι,Kr,ι)-action. Assuming the absolute convergence of

l(Zφ)(g), we only need to check the automorphy, i.e., invariance under left translation of Hr(F ).

The weight part is clear.

It is easy to check the invariance under n(b) and m(a). For b ∈ Herr(E), the matrix bT (x) is F -

rational if Z(x)K 6= 0. Therefore, l(Zφ)(n(b)g) = l(Zφ)(g) for all g ∈ Hr(AF ). For a ∈ GLr(E),

we have Z(xa)K = Z(x)K , which implies that l(Zφ)(m(a)g) = l(Zφ)(g).

Since Hr(F ) is generated by n(b), m(a) and wr,r−1, we only need to check that l(Zφ)(wr,r−1g) =

l(Zφ)(g) for all g ∈ Hr(AF ). Assuming this for r = 1 (cf. Lemma 3.1.7 below), we now prove

for general r > 1, following [YZZ2009] and [Zha2009].

We suppress l from notations for simplicity. Then for K sufficiently small, we have

Zφ(wr,r−1g) =∑

x∈K\Vrfin

(ωχ(wr,r−1g)φ) (T (x), x)Z(x)K

=∑

x∈K\Vr−1fin

∑y∈Kx\Vfin

(ωχ(wr,r−1g)φ) (T (x, y), (x, y))Z((x, y))K , (3.3)

where Kx is the stabilizer of x in K. Then

(3.3) =∑

x∈K\Vr−1fin

∑y1∈Kx\Vxfin

∑y2∈Vx

(ωχ(wr,r−1g)φ) (T (x, y1 + y2), (x, y1 + y2))Z((x, y1 + y2))K ,

(3.4)

where Vxfin is the orthogonal complement of Vx = E〈x〉 in Vfin. Recalling the morphism ςVx in

(3.2), we have

(3.4) =∑

x∈K\Vr−1fin

∑y1∈Kx\Vxfin

∑y2∈Vx

(ωχ(wr,r−1)(ωχ(g)φ)) (T (x, y1 + y2), (x, y1 + y2))ςVx∗ Z(y1)Kx .

(3.5)

Applying the case r = 1 to the special cycle Z(Vx)Kx , we have

(3.5) =∑

x∈K\Vr−1fin

∑y1∈Kx\Vxfin

∑y2∈Vx

(ωχ(wr,r−1)ωχ(g)φ

y1)(T (x, y1 + y2), (x, y1 + y2))ςVx∗ Z(y1)Kx ,

(3.6)

49

where the superscript y1 means taking the partial Fourier transformation along the y1 direction.

Applying the Poisson Summation Formula (recall that φ∞ = φ0∞ is the Gaussian), we have

(3.6) =∑

x∈K\Vr−1fin

∑y1∈Kx\Vxfin

∑y2∈Vx


y1,y2)(T (x, y1 + y2), (x, y1 + y2))ςVx∗ Z(y1)Kx

=∑

x∈K\Vr−1fin

∑y1∈Kx\Vxfin

∑y2∈Vx


y)(T (x, y1 + y2), (x, y1 + y2))ςVx∗ Z(y1)Kx

=∑

x∈K\Vr−1fin

∑y1∈Kx\Vxfin

∑y2∈Vx

(ωχ(g)φ) (T (x, y1 + y2), (x, y1 + y2))ςVx∗ Z(y1)Kx

=∑

x∈K\Vrfin

(ωχ(g)φ) (T (x), x)Z(x)K

= Zφ(g).

2. It follows from the argument in Lemma 3.1.7, Corollary 3.1.5 and [YZZ2009, Theorem 1.3],

which uses the result in [KM1990].

Lemma 3.1.7. If r = 1, then l(Zφ)(w1g) = l(Zφ)(g) for all g ∈ H1(AF ).

Proof. We suppress l from notations. Further, we fix an element ι ∈ Σ∞ over ι ∈ Σ∞ and suppress

them as in the previous discussion. It is clear that we only need to prove Zφ(w1g) = Zφ(g) for

g ∈ G1(AF ) since G1(A∞,F )K1,∞ = H1(A∞,F ). As before, we assume that φ∞ is the Gaussian

and K is sufficiently small. Recall that π0(ShK(H,X)ι,C) ∼= T (Q)\T (Afin)/ det(K). We have the

following inclusion

CH1(ShK(H,X))C →⊕

t∈T (Q)\T (Afin)/ detK

CH1(ShK(H,X)t)C (3.7)

where ShK(H,X)t is the (canonical model of the) corresponding (geometric) connected component.

Let h ∈ H(Afin) such that deth ∈ t and Th be the Hecke operator. Then Th : ShKh(H,X) →

ShK(H,X) induces

T h : ShKh(Hder, X) = ShKh(H,X)1∼−→ ShK(H,X)t → ShK(H,X),

where Kh = hKh−1. We have T ,∗h Zφ(g) = Zφ(g) which is the image of (3.7) composed with the

projection to CH1(ShK(H,X)t)C. Here, Zφ(g) is the generating series on ShKh(Hder, X). Now

shrinking Kh if necessary such that we can apply Corollary 3.1.5, we have Zφ(g) = i∗K′Z′φ(g) for g ∈

50

G1(Afin). Applying [YZZ2009, Theorem 1.2 or Theorem 1.3], we conclude that Zφ(w1g) = Zφ(g).

The lemma follows by (3.7).

3.2 Smooth compactification of unitary Shimura varieties

We introduce the canonical smooth compactification of the unitary Shimura varieties if they are not

proper, and the compactified generating series on them.

3.2.1 Compactified unitary Shimura varieties

Let m ≥ 2 be an integer, E = Q(j) ⊂ C with Im j > 0 and j2 = −D for some square-free positive

integer D, OE its ring of integers and τ the nontrivial Galois involution on E. Let (V, (−,−)) be a

hermitian space of dimension m over E of signature (m − 1, 1). If m = 2, we further assume that

detV ∈ NmE×. Let H = U(V ) be the unitary group, we have the Hodge map h : S → HR ∼=

U(m− 1, 1)R given by

h(z) =

1

. . .

1

z/z

.

Then we have the notion of Shimura variety ShK(H,h) for every open compact subgroup K of H(Afin).

For K sufficiently small, it is smooth, quasi-projective but non-proper over E of dimension m − 1.

Therefore, we need to construct a smooth compactification of ShK(H,h) to do height paring. When

m = 2, it is easy since we only need to add cusps. When m = 3 and H is quasi-split, a canonical

smooth compactification (even of the integral model) has been constructed in [Lar1992]. In fact, the

same construction works in more general cases (just for compactification of the canonical model),

namely for every H as above. We should mention that, if the signature of V is (a, b) such that

a ≥ b > 1 or V is over a totally real field but not Q and indefinite at every archimedean place, then

we are not clear whether there exists a canonical smooth compactification (in a suitable sense).

Now let us assume that m > 2. In order to to use modular interpretations, we should work with

the group of unitary similitude. For every v, w ∈ V , the paring

(v, w)′ = TrE/Q(j(v, w))

51

defines a symplectic form of V satisfying (ev, w)′ = (v, eτw)′ for every e ∈ E. Let GH = GU(V ) such

that for every Q-algebra R,

GH(R) = h ∈ GLm(E ⊗Q R) | (hv, hw)′ = λ(h)(v, w)′ for some λ(h) ∈ R×

and the Hodge map Gh : S→ GHR ∼= GU(m− 1, 1)R is given by

Gh(z) =

z

. . .

z

z

.

For every sufficiently small open compact subgroup K of GH(Afin), we have the Shimura variety

ShK(GH,Gh) which is smooth, quasi-projective but non-proper over E of dimension m−1. Although

we do not have a map of Shimura data, ShK∩H(Afin)(H,h) and ShK(GH,Gh) have the same neutral

component for sufficiently small K. Therefore, it is same to give a canonical smooth compactification

of ShK(GH,Gh) instead of the original one. In fact, ShK(GH,Gh) is the moduli space of abelian

varieties of certain PEL type. We fix a lattice VZ of V such that VZ ⊂ V ⊥Z and let VZ

= VZ⊗Z Z. Then

(ShK(GH,Gh), Gh) represents the following functor in the category of locally noetherian E-schemes:

for every such scheme S, ShK(GH,Gh)(S) is the set of isomorphism classes of quadruples (A, θ, i, η),

where

• A is an abelian scheme over S of dimension m.

• θ : A→ A∨ is a polarization.

• i : OE → EndS(A) is a monomorphism of rings with units, such that tr(i(e); LieS(A)) =

(m− 1)e+ eτ and θ i(e) = i(eτ )∨ θ for all e ∈ OE . Here, we view (m− 1)e+ eτ as a constant

section of OS via the structure map E → OS .

• η is a K-level structure, that is, for chosen geometric point s on each connected component of

S, η is a π1(S, s)-invariant K-class of OE⊗Z Z-linear symplectic similitude η : VZ→ Het

1 (As, Z),

where the pairing on the latter space is the θ-Weil paring (hence the degree of θ is [V ⊥Z : VZ]).

In fact, such data is independent of the geometric point s we choose.

In the theory of toroidal compactification (cf. [AMRT2010]), we need to choose a rational polyhe-

dral cone decomposition. However, in our case, we have only one choice, namely the torus in a line.

52

We claim that there is a scheme Sh∼K(GH,Gh) such that

• Sh∼K(GH,Gh) is smooth and proper over E.

• iK : ShK(GH,Gh) → Sh∼K(GH,Gh) is an open immersion.

• For K ′ ⊂ K, there is a morphism πK′

K such that the following diagram commutes

ShK′(GH,Gh)

πK′

K

iK′ // Sh∼K′(GH,Gh)

πK′

K

ShK(GH,Gh)

iK // Sh∼K(GH,Gh).

• The boundary GYK := Sh∼K(GH,Gh) − ShK(GH,Gh) is a smooth divisor defined over E and

each geometric component is isomorphic to an extension of an abelian variety of dimension m−2

by a finite group.

The boundary part GYK parameterizes the degeneration of abelian varieties with above PEL

data. We consider a semiabelian variety G over an algebraically closed field of characteristic 0 with

i : OE → End(G) such that tr(i(e); Lie(G)) = (m − 1)e + eτ . For every e ∈ OE , we have following

commutative diagram

0 // Tt // G

α //

i(e)

A // 0

0 // Tt // G

α // A // 0.

Then the composition α i(e) t is trivial. Thus i induces actions of OE on both torus part T and

abelian variety part A. Suppose X(T ) ∼= Zr with r > 0. Then r is even since E is quadratic imaginary.

Further assuming that tr(i(e); Lie(T )) = ae+ beτ , then a+ b = r and a = b. Therefore, there is only

one possibility, namely a = b = 1 and r = 2. Then T is of rank 2 and A is an abelian variety of

dimension m− 2 such that OE acts on A and tr(i(e); Lie(A)) = (m− 2)e. Let A1 be an elliptic curve

of CM type (OE , e 7→ e). Then A is isogenous to Am−21 . Each geometric point s of GYK corresponds

to a semiabelian variety Gs = (Ts → Gs → As) as above with certain level structure which will be

defined later. For two geometric points s, s′ in the same geometric connected component, the abelian

variety part As ∼= As′ and the rank 1 OE-modules X(Ts) and X(Ts′) are isomorphic. It is easy to see

that if A and T are fixed, then the set of such G, up to isomorphism, is parameterized by X(T )⊗OE A

which is an abelian variety of dimension m− 2.

53

To include the level structure, we consider only one geometric component since it is same for

others. This means that we fix T and A with OE-actions but, of course, not G. Let us fix a maximal

isotropic subspace W of VZ. Then W is of rank 1. We have a filtration 0 ⊂ W ⊂ W⊥ ⊂ VZ.

Let BW be the subgroup of H(Afin) that preserves this filtration, NW ⊂ BW that acts trivially on

the associated graded modules, UW ⊂ NW that acts trivially on W⊥ and VW = NW /UW . We fix

also a generator w of W . On the other hand, we fix an OE ⊗Z Z generator wT of Het1 (T, Z) and a

polarization θA : A→ A∨ such that there exists a symplectic similitude of Het1 (A, Z) and VW⊗Z Z. For

a sufficiently small open compact subgroup K ⊂ H(Afin), let NW,K = NW ∩K, UW,K = UW ∩K and

VW,K = NW,K/UW,K . Then the level structure of (G,T → G→ A) with respect to K is a VW,K-class

of isomorphisms W⊥ ⊗Z Z→ Het1 (G, Z) which sends w to wT and induces a symplectic similitude of

VW ⊗Z Z and Het1 (A, Z) = Het

1 (G, Z)/OE ·wT . We conclude that every geometric component of GYK

is isomorphic to (a connected component of) an extension of X(T ) ⊗OE A by VW /VW,H for some T

and A as above. There is a universal object π : G→ Sh∼K(GH,Gh) which is a semiabelian scheme of

relative dimension m.

3.2.2 Compactified generating series

We come back to the Shimura variety ShK := ShK(H,h). The above canonical smooth compact-

ification induces a canonical smooth compactification for ShK which we will denote by Sh∼K and

YK = Sh∼K −ShK . They have the same properties as above. We apply the above notations also to

the trivial case m = 2. Let L ∼K be the line bundle on Sh∼K induced from∧m

π∗ΩG/ Sh∼K(GH,Gh) on

Sh∼K(GH,Gh) which is an extension of the Hodge bundle LK on ShK . By the canonicality of the

compactification, (L ∼K )K defines an element in Pic(Sh∼) := lim−→KPic(Sh∼K). We also need to extend

special cycles and the generating series. For 1 ≤ r < m and x ∈ V rfin := V r ⊗Q Afin, we define the

compactified special cycle to be

Z(x)∼K =

Z(Vx)∼Kc1(L ∼,∨K )r−dimE Vx if Vx is admissible

0 otherwise

54

where Z(Vx)∼K is just the Zariski closure of Z(Vx)K in Sh∼K . We define the compactified generating

series to be

Z∼φ (g) =

∑x∈K\V rfin

(ωχ(g)φ) (T (x), x)Z(x)∼K m > 2;∑x∈K\Vfin

(ωχ(g)φ) (T (x), x)Z(x)∼K +W0( 12 , g, φ)c1(L ∼,∨K ) r = 1,m = 2,

for g ∈ Hr(A) and φ ∈ S(Vr)U∞ . It is a formal series in CHr(Sh∼)C := lim−→KCHr(Sh∼K)C. Here,

W0(s, g, φ) =∏vW0(s, gv, φv) which is holomorphic at s = 1

2 . Moreover, we define the following

positive partial compactified generating series as

Z∼,+φ (g) =∑

x∈K\V rfinT (x)0r

(ωχ(g)φ) (T (x), x)Z(x)∼K ,

where the sum is taken over all x such that T (x) is totally positive definite. We would like to propose

the following conjecture on the modularity of the compactified generating series.

Conjecture 3.2.1. Let l be a linear functional on CHr(Sh∼)C such that l(Z∼φ )(g) is absolutely con-

vergent. Then

1. If 1 ≤ r ≤ m− 2, l(Z∼φ )(g) is a holomorphic automorphic form of Hr(AF ).

2. If r = 1,m = 2, l(Z∼φ )(g) is an automorphic form of H1(AF ), not necessarily holomorphic.

3. In general, if r = m− 1, l(Z∼,+φ )(g) is the sum of the positive definite Fourier coefficients of an

automorphic form of Hm−1(AF ).

The case (2) will be proved in Corollary 3.4.2 and is actually not far from Theorem 3.1.6 as we

point out there.

Fix a rational prime `, there are class maps

clK : CHr(Sh∼K)C → H2ret (Sh∼K ×EEac,Z`(r))

ΓE ⊗Z` C,

that are compatible under πK′

K . They induce

cl : CHr(Sh∼)C → H2ret (Sh∼×EEac,Z`(r))

ΓE ⊗Z` C ⊂ H2rBet(Sh∼,C),

55

where

H2•et (Sh∼×EEac,Z`(•)) = lim−→

K

H2•et (Sh∼K ×EEac,Z`(•));

H•Bet(Sh∼,C) = lim−→K

H•Bet(Sh∼K(C),C),

and the latter one is the Betti cohomology. Let

H•Y (Sh∼,C) = lim−→K

H•YK (Sh∼K(C),C)

be the inductive limit of cohomology groups with support in YK as K varies. Then since Y is a smooth

divisor, we have

H•Y (Sh∼,C) ∼= H•−2Bet (Y,C) := lim−→

K

H•−2Bet (YK(C),C).

Let us denote by Sh#K the Baily–Borel compactification of ShK . Therefore, we have the following

commutative diagram

Sh∼KjK // Sh#

K ,

ShK0 P

iK

bb

-

i#K

<<

which is compatible and more importantly, Hecke equivariant when K varies. We denote also by

IH•(Sh#,C) = lim−→KIH•(Sh#

K(C),C) the inductive limit of the intersection cohomology groups. Then

by [BBD1982, Theoreme 6.2.5], we have the following exact sequence

H•Y (Sh∼,C) // H•Bet(Sh∼,C)j∗ // IH•(Sh#,C) // 0. (3.8)

Let H•∂(Sh∼) be the image of the first map, which is isomorphic to a quotient of H•−2Bet (Y,C).

3.3 Arithmetic theta lifting

We assume Conjecture 3.2.1 and the following assumptions on A-packets, which are a certain part of

the Langlands–Arthur conjecture (see [Art1984,Art1989]).

• A-packets are defined for all unitary groups U(m) defined by a hermitian space over E of rank m.

56

We denote by AP(U(m)AF ) the set of A-packets of U(m) and AP(U(m)AF )disc ⊂ AP(U(m)AF )

the subset of discrete A-packets.

• If Π1 and Π2 are in AP(U(m)AF )disc such that for almost all v ∈ Σ, Π1,v and Π2,v contain the

same unramified representation, then Π1 = Π2.

• Let U(m)∗ be the quasi-split unitary group. Then we have the correspondence of A-packets of

inner forms: JL : AP(U(m)AF )disc → AP(U(m)∗AF )disc.

Remark 3.3.1. The similar assumptions for orthogonal groups will be proved in the upcoming book

of Arthur [Art]. The similar approach should be possible to handle the case of unitary groups as well.

3.3.1 Cohomological triviality

We will fix an incoherent hermitian space V as above and suppress H from the notations of Shimura

varieties.

Definition 3.3.2 (Arithmetic theta lifting). Let π be an irreducible cuspidal automorphic represen-

tation of Hr(AF ) realized in L2(Hr(F )\Hr(AF )). We assume that 1 ≤ r ≤ m − 2 or r = 1,m = 2.

For every φ ∈ S(Vr)U∞ and every cusp form f ∈ π, the following integral

Θfφ =

∫Hr(F )\Hr(AF )

f(g)Zφ(g)dg ∈ CHr(Sh)C Sh is proper;∫Hr(F )\Hr(AF )

f(g)Z∼φ (g)dg ∈ CHr(Sh∼)C Sh is non-proper,

is called the arithmetic theta lifting of f , which is a (formal integral of) codimension r cycle on certain

(compactified) Shimura variety of dimension m − 1. The cohomology class of Θfφ | Sh is well-defined

due to [KM1990]. The original idea of this construction comes from S. Kudla, see . He constructed

the arithmetic theta series as an Arakelov divisor on certain integral model of a Shimura curve.

Remark 3.3.3. The original idea of the arithmetic theta lifting comes from the work of Kudla

[Kud2003, Section 8] and Kudla–Rapoport–Yang [KRY2006, Section 9.1]. However, there is an es-

sential difference between our definition and theirs, in the sense that our arithmetic theta lifting is a

cocycle on the canonical model of Shimura variety, while theirs (just in the case of Shimura curves) is

an Arakelov divisor on certain integral model, which is not canonically defined. It is more natural to

define arithmetic theta lifting just as a cocycle on the generic fiber in order to consider the canonical

height.

57

In the following discussion, let m = 2n and r = n. Let π be an irreducible cuspidal automorphic

representation of Hn(AF ), χ a character of E×A×F \A×E such that π∞ is a discrete series representation

of weight (n− kχ

2 , n+ kχ

2 ) and ε(π, χ) = −1. Then the (equal-rank) theta correspondence of πι (under

ωχ) is the trivial representation of U(2n, 0)R for every archimedean place ι. Therefore, V(π, χ) is

a totally positive definite incoherent hermitian space over AE . Now we fix an incoherent hermitian

space V that is totally positive definite of rank 2n and let (ShK)K be the attached Shimura varieties.

We fix an embedding ι : E → C inducing ι : F → C if F 6= Q. Then similarly we have the

class map cl : CH•(ShK)C → H2•Bet(ShK,ι(C),C). By a theorem of S. Zucker [Zuc1982, Section 6]

concerning the L2-cohomology and the intersection cohomology, we have a (compatible system of)

Hecke equivariant isomorphisms

H•(2)(ShK) =

H•Bet(ShK,ι(C),C) ShK is proper;

IH•(Sh#K ,C) ShK is non-proper,

and let H•(2)(Sh) = lim−→KH•(2)(ShK). In the non-proper case, we compose the map j∗ in (3.8) to get a

class map still denoted by cl : CH•(Sh∼)→ H2•(2)(Sh). We have the following proposition.

Proposition 3.3.4. The class cl(Θfφ) = 0 in H2n

(2)(Sh), i.e., if Sh is proper (resp. non-proper), Θfφ is

cohomologically trivial (resp. such that cl(Θfφ) ∈ H2n

∂ (Sh∼)).

Proof. If Sh is non-proper, we can assume that n > 1. By our definition of the arithmetic theta lifting,

for φ = φ∞φf with fixed φ∞, cl(Θfφ) defines an element in

HomHn(AF,fin)

(S(Vn

fin)⊗ πfin,H2n(2)(Sh)

),

where Hn(AF,fin) acts trivially on the L2-cohomology.

Let V (ι) be the nearby hermitian space of V at ι (cf. 3.1.1) and H(ι) = U(V ι). Then since

Z∼ωχ(h)φ(g) = T ∗hZ∼φ (g) for all h ∈ H(ι)(AF,fin) where Th is the Hecke operator of h, we see that cl(Θf

φ)

in fact defines an element

HΘ,φ∞ ∈ HomHn(AF,fin)×H(ι)(AF,fin)

(S(Vn

fin)⊗ πfin,H2n(2)(Sh)

)= HomHn(AF,fin)×H(ι)(AF,fin)

(S(Vn

fin), π∨fin ⊗H2n(2)(Sh)

),

where Hn(AF,fin)×H(ι)(AF,fin) acts on S(Vnfin) through the Weil representation ωχ and H(ι)(AF,fin)

acts on H2•(2)(Sh) through Hecke operators and on πfin trivially. As an H(ι)(AF,fin)-representation, we

58

have the following well-known decomposition (cf. e.g., [BW2000, Chapter XIV])

H2n(2)(Sh) =

⊕σ

mdisc(σ)H2n(LieH(ι)∞ ,K

H(ι)∞

;σ∞)⊗ σfin,

where the direct sum is taken over all irreducible discrete automorphic representations of H(ι)(AF ).

If the invariant functional HΘ,φ∞ 6= 0, then some σf with

mdisc(σ)H2n(LieH(ι)∞ ,K

H(ι)∞

;σ∞) 6= 0

is the theta correspondence θ(π∨fin) of π∨fin.

We define a character χ of E×,1\A×,1E in the following way. For every x ∈ A×,1E , we can write

x = eeτ for some e ∈ A×E and define χ(x) = χ(e) which is well-defined since χ|A×F = 1.

For all finite place v such that v - 2 and ψv, χv, πv unramified, we have H(ι)v∼= Hn,v. Let

Σ ∈ AP(H(ι)AF

)disc be the A-packet containing σ and Π ∈ AP(Hn,AF )disc containing π. Then by

Corollary A.3.6, we have that for v as above, JL(Σ)v = JL(Σv) = Σv and Πv ⊗ χv contain the same

unramified representation. Therefore, JL(Σ) and Π⊗ χ coincide. In particular,

JL(Σ∞) = JL(Σ)∞ = Π∞ ⊗ χ∞

which implies that Σ∞ is a discrete series L-packet (cf. [Ada]). This contradicts our assumption since

for every discrete series representation σ∞, H•(LieH(ι)∞ ,K

H(ι)∞

;σ∞) 6= 0 happens only in the middle

dimension which is 2n− 1, not 2n (cf. [BW2000, Chapter II, Theorem 5.4]). Thus HΘ,φ∞ = 0 and we

prove the proposition.

The proposition says that Θfφ is automatically cohomologically trivial at least in the proper case.

We would like to propose the following conjecture.

Conjecture 3.3.5. When Sh is non-proper, cl(Θfφ) ∈ H2n

∂ (Sh∼) is 0 for every cusp form f ∈ π and

φ as above.

When n = 1, this follows from Corollary 3.4.2 by just computing the degree of the generating

series which is the linear combination of an Eisenstein series and (possibly) an automorphic character

(i.e., 1-dimensional automorphic representation) of H1(AF ). Therefore, cl(Θfφ) is zero since f is

cuspidal. For the general case, we believe that the same phenomenon will happen.

59

3.3.2 Arithmetic inner product formula: the general conjecture

Let us further assume Conjecture 3.3.5 and the existence of Beilinson–Bloch height paring [Beı1987,

Blo1984] on smooth proper schemes over number fields. Then Θfφ is cohomologically trivial and we

let

〈Θfφ,Θ

f∨

φ∨〉KBB

be the Beilinson–Bloch height paring on ShK (resp. Sh∼K) if it is proper (resp. non-proper) for

sufficiently small K. Let Vol(K) be the volume defined in Definition 4.3.3. Then

〈Θfφ,Θ

f∨

φ∨〉BB := Vol(K)〈Θfφ,Θ

f∨

φ∨〉KBB

is a well-defined number that is independent of K.

If V V(π, χ), then 〈Θfφ,Θ

f∨

φ∨〉BB = 0 for every f , f∨ and φ, φ∨ since otherwise, it defines a

nonzero functional

γ(f, f∨, φ, φ∨) ∈ HomHn(AF,fin)×Hn(AF,fin) (R(Vfin, χfin), π∨fin χfinπfin) ,

which contradicts the fact that the latter space is zero. This will imply that, assuming the conjecture

that the Beilinson–Bloch height pairing is non-degenerate, every arithmetic theta lifting Θfφ = 0.

If V ∼= V(π), then we have the following main conjecture.

Conjecture 3.3.6 (Arithmetic inner product formula). Let π be an irreducible cuspidal automorphic

representation of Hn(AF ), χ a character of E×A×F \A×E, such that π∞ is a discrete series represen-

tation of weight (n− kχ

2 , n+ kχ

2 ), ε(π, χ) = −1 and V ∼= V(π, χ). Then for every f ∈ π, f∨ ∈ π∨ and

every φ, φ∨ ∈ S(Vn)U∞ that are decomposable, we have

〈Θfφ,Θ

f∨

φ∨〉BB =L′( 1

2 , π, χ)∏2ni=1 L(i, εiE/F )

∏v

Z∗(0, χv, fv, f∨v , φv ⊗ φ∨v ),

where in the last product, almost all factors are 1.

We would like to remark that when n = 1, the height paring 〈Θfφ,Θ

f∨

φ∨〉BB is just the classical

Neron–Tate height paring which will be recalled in 3.4.1, hence is defined unconditionally.

60

3.4 Arithmetic kernel functions

3.4.1 Neron–Tate height pairing on curves

we will review the general theory of the Neron–Tate height paring on curves over number fields and

some related facts.

Let E be a number field, not necessarily CM; M a connected smooth projective curve over E,

not necessarily geometrically connected. Let CH1(M)0C be the group of cohomologically trivial cycles

which is the kernel of the following map

deg : CH1(M)C → H2et(MEac ,Z`(1))ΓE ⊗Z` C ∼= C

for a fixed rational prime number `. Let M be a regular model of M , i.e., a regular scheme, flat and

projective over SpecOE with the generic fibre ME∼= M .

An arithmetic divisor is a datum (Z, gι), where Z ∈ Z1(M)C is a usual divisor and gι is a Green

function, i.e., Green (0,0)-form of logarithmic type [Sou1992, II.2], for the divisor Zι(C) on Mι(C)

for each embedding ι : E → C. We denote by Z1

C(M) the group of arithmetic C-divisors. For a

nonzero rational function f on M, we define the associated principal arithmetic divisor to be

div(f) =(div(f),− log |fι,C|2

).

The quotient of Z1

C(M) divided by the C-subspace generated by principal arithmetic divisors is the

arithmetic Chow group, denoted by CH1

C(M). Inside CH1

C(M), there is a subspace CH1fin(M)C, which

is C-generated by (Z, 0) with Z supported on special fibres. Let CH1fin(M)⊥C ⊂ CH

1

C(M) be the

orthogonal complement under the C-bilinear pairing

〈−,−〉GS : CH1

C(M)× CH1

C(M)→ C.

An arithmetic divisor (Z, gι) is flat if we have the following equality in the space D1,1(Mι(C)) of

(1, 1)-currents:

ddc[gι ] + δZι (C) = 0

for every ι, where dc = (4πi)−1(∂ − ∂); [−] is the associated current; and δ is the Dirac current.

Flatness is well-defined in CH1

C(M). Now we introduce the subgroup CH1

C(M)0 of CH1

C(M) consisting

61

of elements (represented by) (Z, gι) satisfying:

1. (Z, gι) is inside CH1fin(M)⊥C.

2. ZE is inside CH1(M)0C.

3. (Z, gι) is flat.

Therefore, we have a natural map

pM : CH1

C(M)0 → CH1(M)0C (3.9)

(Z, gι) 7→ ZE

which is surjective. Now we can define the Neron–Tate height paring as

〈−,−〉NT : CH1(M)0C × CH1(M)0

C → C (3.10)

(Z1, Z2) 7→ 〈(Z1, g1,ι), (Z2, g2,ι)〉GS

where (Zi, gi,ι) (i = 1, 2) is any preimage of Zi under pM. It is easy to see that this is independent

of the choices of preimages and also the regular model M.

In practice, the cycles we are interested in are not automatically cohomologically trivial. We need

to make some modifications with respect to some auxiliary data. This is quite easy if we are working

over a curve. Let Pic(M) be the abelian group of isomorphism classes of hermitian line bundles on

M . Recall that a hermitian line bundle is the data L = (L , ‖•‖ι), where L ∈ Pic(M) and ‖•‖ι is

a (smooth) hermitian metric on the holomorphic line bundle Lι,C. We assume that deg c1(L ) 6= 0.

For every Z ∈ CH1(M), the divisor

Z0L = Z − degZ

deg c1(L )c1(L )

is in CH1(M)0C.

Now we define the modified height pairing with respect to L to be

〈Z1, Z2〉L := 〈Z01,L , Z

02,L 〉NT

for every Zi ∈ CH1(M)C (i = 1, 2). In particular, we need to choose suitable Green function on Zi

when computing via (3.10). We say that the Green function gι of Z is L -admissible if the following

62

equalities of (1, 1)-currents hold

ddc[gι ] + δZι (C) =degZ

deg c1(L )[c1(Lι,C, ‖ • ‖ι)];∫

Mι (C)

gι · c1(Lι,C, ‖ • ‖ι) = 0,

where c1(Lι,C, ‖ • ‖ι) ∈ A1,1(Mι(C)) is the Chern form associated to the hermitian holomorphic

line bundle (Lι,C, ‖ • ‖ι) which is a (1, 1)-form.

3.4.2 Degree of generating series

We adopt the construction in 3.1.1 in the case where m = 2 and r = 1. In particular, V is a totally

positive definite hermitian space over AE of rank 2 and we write H ′ = H1. Let H = ResAF /AU(V) be

the associated unitary group over A. For every (sufficiently small; Assumption 3.1.1) open compact

subgroup K of H(Afin), there is a Shimura curve ShK(H), smooth over the reflex field E. For every

embedding ι : E → C over ι ∈ Σ∞, we have the following ι-adic uniformization

ShK(H)anι∼= H(ι)(Q)\

(D(ι) ×H(Afin)/K

).

The underlying real symmetric domain D(ι) is identified with the Hι(R)-conjugacy class of the Hodge

map h(ι) : S→ H(ι)R∼= U(1, 1)R ×U(2, 0)d−1

R given by

h(ι)(z) =

1

z/z

,12, . . . ,12

.

The Shimura curves ShK(H) are non-proper if and only if F = Q and Σ(V) = ∞. In this case, we

can compactify them by add cusps. We denote by MK the compactified (resp. original) Shimura curve

if ShK(H) is non-proper (resp. proper), and M the projective system (MK)K with respect to the

projections πK′

K : MK′ →MK (3.1). On each MK , we have the Hodge bundle LK ∈ Pic(MK)Q which

is ample. They are compatible under pullbacks of πK′

K , hence define an element L ∈ Pic(M)Q :=

lim−→KPic(MK)Q.

63

Adapting the definition of (compactified) generating series in 3.2.2, we have

Zφ(g) =∑

x∈K\Vfin

ωχ(g)φ(T (x), x)Z(x)K ,

Z∼φ (g) =

Zφ(g) if Sh(H)K is proper;

Zφ(g) +W0( 12 , g, φ)c1(L ∨K) if not,

respectively, as vector-valued series in CH1(MK)C for φ ∈ S(V)U∞K and g ∈ H ′(AF ), where

W0(s, g, φ) =∏vW0(s, gv, φv) that is holomorphic at s = 1

2 . It is easy to see that Zφ(g) and Z∼φ (g)

are compatible under pullbacks of πK′

K , hence define series in CH1(M)C := lim−→KCH1(MK)C. Readers

may view the modification in the non-proper case as an analogy of the classical Eisenstein series G2(τ)

(which is not a modular form!). It becomes modular if we add a term −π/ Im τ at the price of being

non-holomorphic (cf. e.g., [DS2005, Page 18]).

We apply the construction in 3.4.1 to the curve MK . The cycles whose heights we want to compute

are the generating series Z∼φ (g) which are not necessarily cohomologically trivial. We use the dual

Hodge bundle L ∨ = (L ∨K)K ∈ Pic(M) to modify as in 3.4.1. The metric on Lι,C for some ι ∈ Σ∞

over ι ∈ Σ∞ is the one descended from the H ′ι-invariant metric

‖v‖ι =1

2(v, v)ι

for v ∈ V(ι)ι and the hermitian form (−,−)ι of V (ι) at ι. We denote by L = (LK)K ∈ Pic(M)

the corresponding metrized line bundle. Since L is ample, deg c1(LK) 6= 0. For φ ∈ S(V)U∞K and

g ∈ H ′(AF ), we define the arithmetic theta series to be

Θφ(g) = Z∼φ (g)−degZ∼φ (g)

deg c1(L ∨K′)c1(L ∨K′)

on every curve MK′ with K ′ ⊂ K. The ratio

D(g, φ) :=degZ∼φ (g)

deg c1(L ∨K′)

is independent of the choice of K ′.

We compute the degree function D(g, φ). From φ ∈ S(V)U∞K that is decomposable, we can form

64

an Eisenstein series

E(s, g, φ) =∑

γ∈P ′(F )\H′(F )

(ωχ(γg)φ) (0)λP ′(γg)s−12

on H ′(AF ), which is absolutely convergent if <(s) > 12 and has a meromorphic continuation to

the entire complex plane. We take Tamagawa measures (with respect to ψ) dh on H(A), dh on

A×,1E = H/Hder(A) and dhx on H(A)x, which is the stabilizer of x ∈ V in H(A).

For every v ∈ Σ, let b ∈ F×v such that Ωb := x ∈ Vv | T (x) = b 6= ∅. Then the local Whittaker

integral Wb(s, e, φv) has a holomorphic continuation to the entire complex plane and Wb(12 , e, φv) is

not identically zero. Therefore, on the one hand, we have an Nv-intertwining map

S(Vv)→ CNv,φb

φv 7→Wb(1

2, e, φv).

On the other hand, by [Ral1987, Lemma 4.2] for v finite, and [Ral1987, Lemma 4.2] and [KR1994,

Proposition 2.10] for v infinite (see also [Ich2004, Proposition 6.2]), we have

Wb(1

2, e, φv) = γVv

∫Ωb

φv(x)dµv,b(x) (3.11)

for the quotient measure dµv,b = dhv/dhv,x on Ωb for every x ∈ Ωb.

Proposition 3.4.1. The Eisenstein series E(s, g, φ) is holomorphic at s = 12 and

D(g, φ) = E(s, g, φ) |s= 12.

Proof. We can assume that φ is decomposable. For b ∈ F×, let

Db(g, φ) =1

deg c1(L ∨K′)

∑x∈K′\Vfin

T (x)=b

(ωχ(g)φ) (b, x) degZ(x)K′

be the b-th Fourier coefficient of D(g, φ). We first compute the degree of Z(x)K′ when T (x) = b

is totally positive. Without lost of generality, let us assume that x is contained in the image of

some (rational) nearby hermitian space V (ι) → Vfin and K ′ is sufficiently small. The isomorphism

65

det : H(ι)x → E×,1 induces a surjective map

H(ι)x \H(Afin)x/(K

′ ∩H(Afin)x)→ E×,1\A×,1fin,E/ detK ′.

Therefore,

degZ(x)K′ =

∣∣∣∣ detK ′

K ′ ∩H(Afin)x

∣∣∣∣ =Vol(detK ′,dhfin)

Vol(K ′ ∩H(Afin)x,dhf,x).

When b 6= 0 that is not totally positive, degZ(x)K′ = 0 by definition. Therefore, on the one hand,

Db(g, φ) =1

deg c1(L ∨K′)

∑x∈K′\Vfin

T (x)=b

(ωχ(g)φ) (b, x)Vol(detK ′)

Vol(K ′ ∩H(Afin)x)

=ωχ(g∞)φ∞(b) Vol(detK ′)

deg c1(L ∨K′) Vol(K ′)

∫x∈Vfin

T (x)=b

(ωχ(g)φ) (x)dµb(x)

=ωχ(g∞)φ∞(b) Vol(detK ′)

deg c1(L ∨K′) Vol(K ′)

∏v∈Σfin

∫Ωb

(ωχ(gv)φv) (x)dµv,b(x)

for b totally positive, and Db(g, φ) = 0 otherwise.

On the other hand, Eb(s, g, φ) is holomorphic at s = 12 for b 6= 0. For b not totally positive,

Eb(s, g, φ)|s= 12

= 0; otherwise,

Eb(s, g, φ)|s= 12

= Wb(1

2, g, φ) =

∏v∈Σ

Wb(1

2, gv, φv)

(3.11)=

∏v∈Σ

γVv

∫Ωb

ωχ(gv)φv(x)dµv,b(x)

= −Vol(Ω∞) (ωχ(g∞)φ∞) (b)∏

v∈Σfin

∫Ωb

(ωχ(gv)φv) (x)dµv,b(x),

where Vol(Ω∞) = Vol(Ω∞,b) for every b that is totally positive. Let

D =Vol(detK ′)

Vol(Ω∞) deg c1(LK′) Vol(K ′).

Now we compute the constant term

D0(g, φ) = ωχ(g)φ(0) +W0(1

2, g, φ),

66

on the one hand. On the other hand, the constant term of E( 12 , g, φ) is

E0(1

2, g, φ) = ωχ(g)φ(0) +W0(

1

2, g, φ).

Here the intertwining term W0( 12 , g, φ) is nonzero only if ShK(H) is not proper, i.e., |Σ(V)| = 1.

There are two cases:

• If ShK(H) is proper, then we can apply Theorem 3.1.6 to see that D(g, φ) is already an auto-

morphic form. Comparing the ratio of the constant term and non-constant terms, we find that

D = 1.

• If ShK(H) is not proper, we calculate the degree of the Hodge bundle as the classical way on

modular curves and find that D = 1.

Therefore, the proposition follows.

We let

E(g, φ) = E(s, g, φ)|s= 12−W0(

1

2, g, φ).

Then

Θφ(g) = Zφ(g)− E(g, φ)c1(L ∨K).

If |Σ(V)| > 1, W0( 12 , g, φ) = 0; otherwise, it equals C · χ det where C is a constant and χ is the

descent of χ to A×,1E , as in the proof of Proposition 3.3.4. In all cases, E(g, φ) is a linear combination

of an Eisenstein series and an automorphic character.

Proposition 3.4.1 implies the following corollary on the modularity of the generating series in the

compactified case.

Corollary 3.4.2. Let ` be a linear functional of CH1(M)C. Then `(Z∼φ )(g) and hence `(Θφ)(g) are

absolutely convergent and are automorphic forms of H ′.

Proof. Assume that φ is invariant under K ⊆ H(Afin). We only need to prove that the identity

Z∼φ (γg) = Z∼φ (g) in Pic(MK)C for every γ ∈ H ′(Q). By Theorem 3.1.6 and the fact that the Hodge

bundle is supported on the cusps, Z∼φ (γg) = Z∼φ (g) in CH1(ShK(H))C. So their difference must be

supported on the set of cusps. By a theorem of Manin–Drinfeld (cf. [Man1972,Dri1973]) which posits

that every two cusps are same in CH1(MK)C, we have an exact sequence

C→ CH1(MK)C → CH1(ShK(H))C → 0.

67

Therefore, we only need to prove that degZ∼φ (γg) = degZ∼φ (g), which is true by the Proposition

3.4.1.

In particular, Definition 3.3.2 specializes to the following unconditional definition.

Definition 3.4.3. Let π be an irreducible cuspidal automorphic representation of H ′(AF ) = H1(AF ).

For every cusp form f ∈ π and φ ∈ S(V)U∞ , we define the arithmetic theta lifting of f to be the

following integral

Θfφ =

∫H′(F )\H′(AF )

f(g)Θφ(g)dg ∈ CH1(M)0C,

which is a divisor on the projective system of (compactified) Shimura curves.

3.4.3 Decomposition of arithmetic kernel functions

For Φ =∑φi,1 ⊗ φi,2 with φi,α ∈ S(V)U∞K for α = 1, 2, we define the geometric kernel function

associated to the test function Φ to be

E(g1, g2; Φ) := Vol(K ′)∑〈Θφi,1(g1),Θφi,2(g2)〉K

′

NT,

where the superscript K ′ means that we are taking the Neron–Tate height paring on the curve MK′ for

some K ′ ⊂ K, of which the definition is independent. By Corollary 3.4.2, E(g1, g2; Φ) is in A(H ′×H ′).

Now let us work over MK and choose a regular model MK of it. We choose an arithmetic line bundle

ωK extending L∨K . In particular, the metrics on ωK at archimedean places are same as those on L

∨K .

Since the map pMK(3.9) is surjective, we may fix an inverse linear map p−1

MKand write

Θφ(g) := p−1MK

(Θφ(g)) = ([Zφ(g)]Zar, gι) + (Vφ(g), 0)− E(g, φ)ωK ,

where gι is an LK-admissible Green function of Zφ(g), and Vφ(g) is the sum of (finitely many)

vertical components supported on special fibres. We also simply write

Zφ(g) = ([Zφ(g)]Zar, gι) + (Vφ(g), 0).

68

Then we have for φα ∈ S(V)U∞K (α = 1, 2),

E(g1, g2;φ1 ⊗ φ2)

= Vol(K)〈Θφ1(g1),Θφ2

(g2)〉KNT

= −Vol(K)〈Θφ1(g1), Θφ2

(g2)〉GS

= −Vol(K)〈Zφ1(g1)− E(g1, φ1)ωK , Zφ2

(g2)− E(g2, φ2)ωK〉GS

= −Vol(K)〈Zφ1(g1), Zφ2

(g2)〉GS + E(g1, φ1) Vol(K)〈ωK , Θφ2(g2)〉GS

+ E(g2, φ2) Vol(K)〈Θφ1(g1), ωK〉GS + E(g1, φ1)E(g2, φ2) Vol(K)〈ωK , ωK〉GS, (3.12)

where the Gillet–Soule parings are taken on the model MK . By Corollary 3.4.2,

A(g, φ) := Vol(K)〈ωK , Θφ(g)〉GS

is an automorphic form of H ′, which may depend on K and also the model MK , since we do not

require any canonicality of p−1MK

. Let C = Vol(K)〈ωK , ωK〉GS. Then

(3.12) = −Vol(K)〈Zφ1(g1), Zφ2(g2)〉GS

+ E(g1, φ1)A(g2, φ2) +A(g1, φ1)E(g2, φ2) + CE(g1, φ1)E(g2, φ2). (3.13)

We assume that φ1 and φ2 are decomposable, and φ1,v ⊗ φ2,v ∈ S(V2v)reg for some v ∈ Σfin. Then

Zφ1(g1) and Zφ2

(g2) will not intersect on the generic fiber if gα ∈ P ′vH ′(AvF ) (α = 1, 2). Then

〈Zφ1(g1), Zφ2

(g2)〉GS =∑v∈Σ

〈Zφ1(g1), Zφ2

(g2)〉v , (3.14)

where the intersection 〈−,−〉v is taken on the local model MK;p := MK ×OE OEp (resp. MK,ι(C))

if v = p is finite (resp. if v = ι is infinite). Combining (3.13) and (3.14), we have for such φα and

gα (α = 1, 2),

E(g1, g2;φ1 ⊗ φ2) = −Vol(K)∑v∈Σ

〈Zφ1(g1), Zφ2(g2)〉v

+ E(g1, φ1)A(g2, φ2) +A(g1, φ1)E(g2, φ2) + CE(g1, φ1)E(g2, φ2). (3.15)

69

Chapter 4

Comparison at infinite places

We compare local terms of analytic and arithmetic kernel functions at an archimedean place. Section

4.1 is dedicated to the computation on the analytic side. We calculate certain Whittaker integral and

its derivative, following the method of G. Shimura. In 4.2, we introduce a local height function on the

hermitian domain in terms of the Kudla–Millson form, and prove an important invariance property

of such height. The actual comparison of the derivative of Whittaker integrals and the local height

function is accomplished in 4.3.

70

4.1 Archimedean Whittaker integrals

In this section, we calculate the Whittaker integral WT (s, g,Φ) and its derivative (at s = 0) at an

archimedean place. In particular, we fix archimedean places ι : F → C and ι′ : E → C over ι that

will be suppressed from notations. As in 2.1.2, we identify H ′ = H ′ι (resp. H ′′ = H ′′ι ) with U(n, n)R

(resp. U(2n, 2n)R). We have the parabolic subgroup P = Pι of H ′′ = U(2n, 2n)R as in Notation 2.2.1.

Moreover, we have the hermitian space V = Vι of rank 2n over C, which is the standard positive

definite 2n-dimensional complex hermitian space, and Φ0 ∈ S(V 2n) the Gaussian. We have also a

character χ = χι : C× → C× that is trivial on R×. Therefore χ(z) = z2`/(zz)` for some integer `.

4.1.1 Elementary reduction steps

We study the integral

WT (s, g,Φ0) =

∫Her2n(C)

ϕΦ0,s(wn(u)g)ψT (n(u))−1du (4.1)

for T ∈ Her2n(C) and Re s > n, where w = wn and du is the selfdual measure with respect to the

additive character ψ(t) = exp(2πit). We have the formula

ωχ([k1, k2])Φ0 = (det k1)n+`(det k2)−n+`Φ0

for [k1, k2] ∈ K (cf. 2.1.2). Write g = n(b)m(a)[k1, k2] under the Iwasawa decomposition. Then

(4.1) =

∫Her2n(C)

(ωχ (wn(u)n(b)m(a)[k1, k2]) Φ0

)(0)λP (wn(u)n(b)m(a)[k1, k2])sψ(− trTu)du

= ψ(trTb)(det k1)n+`(det k2)−n+`∫Her2n(C)

(ωχ(wn(u)m(a))Φ0

)(0)λP (wn(u)m(a))sψ(− trTu)du. (4.2)

Since

wn(u)m(a) = wm(a)n(a−1u ta−1) = m(ta−1)wn(a−1u ta−1),

71

changing variable du = |det a|2nC d(a−1u ta−1), we have

(4.2) = ψ(trTb)|det a|n−sC χ(det a)(det k1)n+`(det k2)−n+`∫Her2n(C)

(ωχ(wn(u))Φ0

)(0)λP (wn(u))sψ(− tr taTau)du

= ψ(trTb)|det a|n−sC χ(det a)(det k1)n+`(det k2)−n+`WtaTa(s, e,Φ0). (4.3)

Therefore, we need to only study WT (s, e,Φ0). In what follows, we will not restrict ourselves to the

case of even dimension. In other words, V will be the standard positive definite complex hermitian

space of dimension m > 0. For T ∈ Herm(C), the Whittaker integral WT (s, e,Φ0) is absolutely

convergent for Re s > m2 .

Lemma 4.1.1. For u ∈ Herm(C),

(ωχ(wn(u))Φ0

)(0) = γV det(1m − iu)m,

where γV is the Weil constant.

Proof. By definition,

γ−1V

(ωχ(wn(u))Φ0

)(0) =

∫Vm

(ωχ(n(u))Φ0

)(x)dx =

∫Vm

ψ(truT (x))Φ0(x)dx. (4.4)

Write u = k diag[u1, . . . , um]tk with uj ∈ R (j = 1, . . . ,m) and k ∈ U(m)R. Then

(4.4) =

∫Vm

ψ(tr k diag[u1, . . . , um]tkT (x)kk−1)Φ0(x)dx

=

∫Vm

ψ(tr diag[u1, . . . , um]T (xk)) exp(−2π trT (x))dx. (4.5)

Changing variable x 7→ xk and since trT (x) = trT (xk), we have

(4.5) =

∫Vm

exp(2πi tr diag[u1, . . . , um]T (x)− 2π trT (x))dx

=

m∏j=1

∫V

exp(2πiujT (xj)− 2πT (xj))dxj . (4.6)

Identifying V with Cm such that (−,−) coincides with the standard hermitian form on Cm, then the

72

selfdual measure dxj on V is simply the usual Lebesgue measure dx on Cm ∼= R2m. Therefore,

(4.6) =

m∏j=1

∫R2m

exp(−π(1− iuj)‖x‖2)dx

=

m∏j=1

(∫ ∞−∞

exp(−π(1− iuj)t2)dt

)2m

=

m∏j=1

(1− iuj)−m = det(1m − iu)−m.

Therefore, the lemma follows.

Lemma 4.1.2. For u ∈ Herm(C), λP (wn(u)) = det(1m + u2)−1.

Proof. We have the following identities

wn(u)

i1m

1m

=

1m

−1m

1m u

1m

i1m

1m

=

1m

−i1m − u

.

Then,

1m(−i1m − u)−1 = −u(1m + u2)−1 + i(1m + u2)−1.

Therefore, λP (wn(u)) = det(1m + u2)−1 that is a positive real number.

Combining Lemmas 4.1.1 and 4.1.2, we have that for Re s > m2 ,

γ−1V WT (s, e,Φ0) =

∫Herm(C)

ψ(− trTu) det(1m + iu)−s det(1m − iu)−s−mdu.

We proceed as in [Shi1982, Case II]. We introduce some new notations that may be different from

those in [Shi1982]. Set

Her+m(C) = x ∈ Herm(C) | x > 0;

hm = x+ iy | x ∈ Herm(C), y ∈ Her+m(C);

h′m = x+ iy | x ∈ Her+m(C), y ∈ Herm(C).

The following lemma is proved in [Shi1982, Section 1].

Lemma 4.1.3 (Siegel). We have

73

1. For z ∈ h′m and Re s > m− 1, we have

∫Her+m(C)

exp(− tr zx)(detx)s−mdx = Γm(s)(det z)−s,

where dx is induced from the selfdual measure on Herm(C), and

Γm(s) = (2π)m(m−1)

2

m−1∏j=0

Γ(s− j).

2. For x ∈ Herm(C), b ∈ Her+m(C) and Re s > 2m− 1, we have

Γm(s)

∫Herm(C)

exp(2πi trux) det(b+ 2πiu)−sdu =

exp(− trxb)(detx)s−m x ∈ Her+

m(C);

0 if not.

By Lemma 4.1.3 (1), for Re s > m− 1,

γ−1V WT (s, e,Φ0)

=

∫Herm(C)

ψ(− trTu)1

Γm(s)

∫Her+m(C)

exp(− tr(1m + iu)x)(detx)s−m det(1m − iu)−s−mdxdu

=1

Γm(s)

∫Her+m(C)

exp(− trx)(detx)s−m∫

Herm(C)

exp(−i tr(x+ 2πT )u) det(1m − iu)−s−mdudx.

(4.7)

Applying Lemma 4.1.3 (2) to (1m, x+ 2π, s+m), and changing variable u 7→ − u2π , we have

(4.7) =1

Γm(s)

∫x>0,x+2πT>0

exp(− trx)(detx)s−m(2π)m

2

Γm(s+m)exp(− tr(x+ 2πT )) det(x+ 2πT )sdx.

(4.8)

In [Shi1982, (1.26)], the author introduced the function

η(g, h;α, β) =

∫x>−h, x>h

exp(− tr gx) det(x+ h)α−m det(x− h)β−mdx

74

for g ∈ Her+m(C), h ∈ Herm(C), and Reα 0, Reβ 0. Changing variable x 7→ x

π + T , we have

(4.8) =(2π)m

2

π2ms

Γm(s)Γm(s+m)

∫x>−T,x>T

exp(− tr 2πx) det(x+ T )s det(x− T )s−mdx

=(2π)m

2

π2ms

Γm(s)Γm(s+m)η(2π1m, T ; s+m, s). (4.9)

In what follows, we assume that T is nonsingular with signT = (p, q) for p+ q = m. Write

T = k diag[t1, . . . , tp,−tp+1, . . . ,−tm]tk

with k ∈ U(m)R and tj ∈ R>0. Let a = k diag[√t1, . . . ,

√tm]. Then T = aεp,q

ta, where

εp,q =

1p

−1q

.

It is easy to see that

η(g, T ;α, β) = |detT |α+β−mη(a∗ga, εp,q;α, β); (4.10)

η(g, εp,q;α, β) = 2m(α+β−m) exp(− tr g)ζp,q(2g;α, β). (4.11)

We recall the definition of ζp,q(g;α, β) introduced in [Shi1982, (4.16)]. Let

εp =

1p

0q

; ε′q =

0p

1q

.

Then for g ∈ Her+m(C), and Reα 0, Reβ 0,

ζp,q(g;α, β) =

∫Xp,q

exp(− tr gx) det(x+ εp)α−m det(x+ ε′q)

β−mdx,

where

Xp,q := x ∈ Herm(C) | x+ εp > 0, x+ ε′q > 0

with the measure induced from the selfdual one on Herm(C). In particular, Xm,0 = Her+m(C).

75

4.1.2 Analytic continuation

Following [Shi1982, (4.17)], we set

ωp,q(g;α, β) = Γq(α− p)−1Γp(β − q)−1(det+εp,qg)β−q/2(det−εp,qg)α−p/2ζp,q(g;α, β), (4.12)

where for a nonsingular element h ∈ Herm(C), det+ h (resp. det− h) is the absolute value of the

product of all positive (resp. negative) (real) eigenvalues of h if they exist; 1 otherwise. It is proved in

[Shi1982, Section 4] that ωp,q(g;α, β) has a holomorphic continuation in (α, β) to the whole C2, and

satisfies the following functional equation

ωp,q(g;m− β,m− α) = ωp,q(g;α, β).

Lemma 4.1.4. If p = m and q = 0, then ωm,0(g;m,β) = ωm,0(g;α, 0) = 1.

Proof. The integral

ζm,0(g;m,β) =

∫Her+m(C)

exp(− tr gx)(detx)β−mdx

is absolutely convergent for Reβ > m−1, and equal to Γm(β)(det g)−β by Lemma 4.1.3 (1). Therefore,

ωm,0(g;m,β) = 1, which confirms the lemma by the functional equation.

Proposition 4.1.5. Suppose T ∈ Herm(C) is nonsingular with signT = (p, q). Then

1. ords=0WT (s, e,Φ0) ≥ q; and

2. If T is positive definite, i.e., p = m and q = 0, then

WT (0, e,Φ0) = γV(2π)m

2

Γm(m)exp(−2π trT ).

Proof. 1. Combining (4.9), (4.10), (4.11) and (4.12), we have

γ−1V WT (s, e,Φ0) =

Γq(m+ s− p)Γp(s− q)Γm(s)Γm(s+m)

(2π)m2+2ms|detT |2s exp(−2π tr taa)

(det+4πT )q/2−s(det−4πT )p/2−m−sωp,q(4πtaa;m+ s, s). (4.13)

All terms except the Gamma factors, are holomorphic for all s ∈ C. Since

Γq(m+ s− p)Γp(s− q)Γm(s)Γm(s+m)

=(2π)−pq−

m(m−1)2

Γ(s) · · ·Γ(s− q + 1)× Γ(s+m) · · ·Γ(s+m− p+ 1),

76

we have

ords=0WT (s, e,Φ0) ≥ −ords=0Γ(s) · · ·Γ(s− q + 1) = q.

2. If T is positive definite, then tr taa = trT . By (4.13) and Lemma 4.1.4, we have

γ−1V WT (0, e,Φ0) =

(2π)m2

Γm(m)exp(−2π trT )ωm,0(4π taa;m, 0) =

(2π)m2

Γm(m)exp(−2π trT ).

4.1.3 First-order derivatives

By Proposition 4.1.5 (1), the T -th coefficient will not contribute to the analytic kernel function

E′(0, g,Φ) if signT = (p, q) with q ≥ 2. Therefore, we focus on the case where q = 1, and study

the functions ζm−1,1(g;α, β) and ωm−1,1(g;α, β)1.

We assume that

g =

a

b

, a ∈ Her+m−1(C), b ∈ R>0.

We write elements in Xm−1,1 in the following form

x z

tz y

, x ∈ Herm−1(C), y ∈ R, z ∈ Matm−1,1(C).

Then by [Shi1982, Page 288],

Xm−1,1 = (x, y, z) | x > 0, y > 0, x+ 1m−1 > zy−1 tz, y + 1 > tzx−1z

= (x, y, z) | x+ 1m−1 > 0, y + 1 > 0, x > z(y + 1)−1 tz, y > tz(x+ 1m−1)−1z.

We have

ζm−1,1(g;α, β) =

∫Xm−1,1

exp(− tr ax− by) det

x+ 1m−1 z

tz z

α−m

det

x z

tz y + 1

β−m

dxdydz,

(4.14)

where we apply the selfdual measure dx on Herm−1(C), the Lebesgue measure dy on R, and the

measure dz that is 2m−1 times the Lebesgue measure on Matm−1,1(C). We make change of variables

1Two letters g appearing here stand for different meanings.

77

as in [Shi1982, Page 289] as follows. Put

f = (x+ 1m−1)−1/2z(y + 1)−1/2.

Then 1m−1 −tf > 0. Put

r = (1− tff)1/2; s = (1m−1 − f

tf)1/2; w = s−1f = fr−1; u = x− w tw; v = y − tww.

Then the map (x, y, z) 7→ (u, v, w) mapsXm−1,1 bijectively onto Y = Her+m−1(C)×R>0×Matm−1,1(C),

and the Jacobian

∂(x, y, z)

∂(u, v, w)= det(1m−1 + x)(1 + y)m−1(1 + tww)−m

for the measure ∂(u, v, w) on Y induced from that on Herm−1(C) × R ×Matm−1,1(C) as an open

subset. Since

det

x+ 1m−1 z

tz y

= det(u+ 1m−1 + w tw)v det(1m−1wtw)−1;

det

x z

tz y + 1

= (v + 1 + tww)(detu) det(1m−1wtw)−1,

we obtain, on the one hand, that

(4.14) =

∫Y

exp(− tr(au+ aw tw)− (bv + b tww)

)det(1m−1 + w tw)m−α−β

det(u+ 1m−1 + w tw)α−m+1(detu)β−m(v + 1 + tww)β−1vα−mdudvdw

=

∫Matm−1,1(C)

exp(− tr aw tw − b tww)ζ1,0(b(1 + tww);β, α−m+ 1)∫Her+m−1(C)

exp(− tr au) det(u+ 1m−1 + w tw)α−m−1(detu)β−mdudw. (4.15)

By (4.10), (4.11) and (4.12), we have, on the other hand, that

γ−1V WT (s, e,Φ0) =

(2π)m2+2ms|detT |2s

Γm(s)Γm(s+m)exp(−2π tr taa)ζm−1,1(4π taa;m+ s, s). (4.16)

Assume that T = k diag[a1, . . . , am−1,−b]tk with a1, . . . , am−1, b ∈ R>0 and k ∈ U(m)R. Then

78

taa = diag[a1, . . . , am−1, b]. By (4.16) and (4.13),

γ−1V W ′T (0, e,Φ0)

= lims→0

(2π)m2

sΓm(s)Γm(m)exp (−2π(a1 + · · ·+ am−1 + b)) ζm−1,1(4π diag[a1, . . . , am−1, b];m, s). (4.17)

Plugging (4.15) with (α, β) = (m, s),

(4.17) = lims→0

2m−1(2π)m2

sΓm(s)Γm(m)exp (−2π(a1 + · · ·+ am−1 + b))∫

Cm−1

exp (−4π[(a1 + am)w1w1 + · · ·+ (am−1 + am)wm−1wm−1]) ζ1,0(4πb(1 + tww); 0, 1)∫Her+m−1(C)

exp (−4π tr diag[a1, . . . , am−1]u) det(u+ 1m−1 + w tw)(detu)s−mdudw1 · · · dwm−1.

(4.18)

It is easy to see that

ζ1,0(4πb(1 + tww); 0, 1) = − exp(4πb(1 + tww)

)Ei(−4πb(1 + tww)

), (4.19)

where Ei is the exponential integral

Ei(z) = −∫ ∞

1

exp(zt)

tdt.

We evaluate the inside integral, i.e., the one over Her+m−1(C). Temporarily let g0 = 4π diag[a1, . . . , am−1],

and consider the integral

∫Her+m−1(C)

exp(− trug0) det(u+ 1m−1 + w tw)(detu)s−mdu. (4.20)

Define a differential operator

∆ = det

(∂

∂gjk

)m−1

j,k=1

.

Then

∆ exp(− trug) = (−1)m−1(detu) exp(− trug).

79

Therefore,

(4.20) = exp(tr(1m−1 + w tw)g0

)∫Her+m−1(C)

exp(− tr(u+ 1m−1 + w tw)g0

)det(u+ 1m−1 + w tw)(detx)s−mdu

= (−1)m−1 exp(tr(1m−1 + w tw)g0

)∫Her+m−1(C)

∆ |g=g0 exp(− tr(u+ 1m−1 + w tw)g

)(detx)s−mdu. (4.21)

We exchange ∆ and the integration by analytic continuation. Then

(4.21) = (−1)m−1 exp(tr(1m−1 + w tw)g0

)∆ |g=g0

∫Her+m−1(C)

exp(− tr(u+ 1m−1 + w tw)g

)(detx)s−mdu

= (−1)m−1 exp(tr(1m−1 + w tw)g0

)∆ |g=g0

(exp

(− tr(1m−1 + w tw)g

)ζm−1(g;m− 1, s− 1)

)= (−1)m−1 exp

(tr(1m−1 + w tw)g0

)∆ |g=g0

(exp

(− tr(1m−1 + w tw)g

)(det g)1−sΓm−1(s− 1)

)= (−1)m−1Γm−1(s− 1) exp

(tr(1m−1 + w tw)g0

)∆ |g=g0

(exp

(− tr(1m−1 + w tw)g

)(det g)1−s) .

(4.22)

Plugging (4.19) and (4.22) into (4.18), we obtain

(4.18) = lims→0

Γm−1(s− 1)(−2)m−1(2π)m2

sΓm(s)Γm(m)exp(−2π trT )∫

Cm−1

exp (−4π(a1w1w1 + · · ·+ am−1wm−1wm−1))

exp(tr(1m−1 + w tw)g0

)∆ |g=g0

(exp

(− tr(1m−1 + w tw)g

)(det g)1−s)

(−Ei)(−4πb(1 + tww)

)dw1 · · · dwm−1

=(2π)m

2

(−2)m−1

Γm(m)(2π)m−1exp(−2π trT )

∫Cm−1

exp (−4π(a1w1w1 + · · ·+ am−1wm−1wm−1))

exp(tr(1m−1 + w tw)g0

)∆ |g=g0

(exp

(− tr(1m−1 + w tw)g

)(det g)

)(−Ei)

(−4πb(1 + tww)

)dw1 · · · dwm−1. (4.23)

To compare with the local height later, we make a change of variables. Let

Dm−1 = z = (z1, . . . , zm−1) ∈ Cm−1 | zz := z1z1 + · · ·+ zm−1zm−1 < 1

80

be the open unit disc in Cm−1. Then the map

wj =zj

(1− zz)1/2, j = 1, . . . ,m− 1 (4.24)

is a homeomorphism from Cm−1 to Dm−1 as real manifolds. To calculate the Jacobian, let wj =

uj + vji and zj = xj + yji be the corresponding real and imaginary parts. Then

∂uj∂xk

=xjxk

(1− zz)3/2, k 6= j;

∂uj∂xj

=x2j

(1− zz)3/2+

1

(1− zz)1/2;

∂uj∂yk

=xjyk

(1− zz)3/2;

∂vj∂yk

=yjyk

(1− zz)3/2, k 6= j;

∂vj∂yj

=y2j

(1− zz)3/2+

1

(1− zz)1/2;

∂vj∂xk

=yjxk

(1− zz)3/2.

In Lemma 4.1.6 below, we let n = 2m + 2, ε = 1 − zz and c =t(c1, . . . , c2m−2) with cj = xj ,

cm+1−j = yj for j = 1, . . . ,m− 1. Then

∂(u1, v1, . . . , um−1, vm−1)

∂(x1, y1, . . . , xm−1, ym−1)

=∂(u1, . . . , um−1; v1, . . . , vm−1)

∂(x1, . . . , xm−1; y1, . . . , ym−1)

= (1− zz)−3(m−1)det((1− zz) 12m−2 + c tc

)= (1− zz)−3(m−1)

(1− zz)2m−3 (1− zz + x2

1 + · · ·+ x2m−1 + y2

1 + · · ·+ y2m−1

)= (1− zz)−m . (4.25)

Lemma 4.1.6. Let c ∈ Matn×1(C). Then

1. det(1n + c tc

)= 1 + tcc;

2. For ε > 0, det(ε1n + c tc

)= εn−1

(ε+ tcc

).

Proof. 1. It is [Shi1982, Lemma 2.2]. Since it is not difficult, we will give a proof here for com-

pleteness, following Shimura. We claim that det(1n + sc tc

)= 1+s tcc for all c ∈ R. Since they

81

are both polynomials in s, we need only to prove for s < 0. We have

1n −√−sc

1

1n

√−sc

√−s tc 1

1n

−√−s tc 1

=

1n + sc tc

1

,

and 1n

−√−s tc 1

1n

√−sc

√−s tc 1

1n −

√−sc

1

=

1n

1 + s tcc

.

Therefore, det(1n + sc tc

)= 1 + s tcc.

2. It follows from (1) immediately.

Now we write the Lebesgue measure dz1 · · · dzm−1 in the differential form of degree (m− 1,m− 1)

on Dm−1 that is

dz1 · · · dzm−1 =1

(−2i)m−1Ω,

where

Ω =

m−1∧j=1

(dzj ∧ dzj) . (4.26)

Here, we view dzj as a (1, 0)-form, not the Lebesgue measure. By (4.25), we have

(4.23) =(2π)m

2

Γm(m)(2πi)m−1exp (−2π trT )

∫Dm−1

exp (−4π(a1w1w1 + · · ·+ am−1wm−1wm−1)) (1− zz)−m

exp(tr(1m−1 + w tw

)g0

)∆ |g=g0

(exp

(− tr

(1m−1 + w tw

)g)

(det g))

(−Ei)(−4πb

(1 + tww

))Ω, (4.27)

where wj are as in (4.24). The final step is accomplished by the following lemma.

Lemma 4.1.7. For g0 = 4π diag[a1, . . . , am−1],

∆ |g=g0(exp

(− tr

(1m−1 + w tw

)g)

(det g))

= exp(− tr

(1m−1 + w tw

)g)

∑1≤s1<···<st≤m−1

(−4π)t(m− 1− t)!(as1 · · · ast) (1 + ws1ws1 + · · ·+ wstwst) ,

82

where the sum is taken over all subsets of 1, . . . ,m− 1.

Proof. Let

ujk = − (1 + wjwk) ; g = (gjk)m−1j,k=1

be the variables in matrices. For short, we use |g| to indicate the determinant of a square matrix g.

For subsets I, J ⊂ 1, . . . ,m− 1 of the same cardinality, we denote by gJ,K (resp. gJ,K) the (square)

matrix obtained by keeping (resp. discarding) the rows indexed in J and the columns indexed in

K. Therefore, gJ,K = gJ,K , where J (resp. K) is the complement set 1, . . . ,m − 1 − J (resp.

1, . . . ,m − 1 −K). Let Sm−1 be the group of (m − 1)-permutations. For σ ∈ Sm−1 and a subset

J = j1 < · · · < jt ⊂ 1, . . . ,m−1, let εJ(σ) ∈ ±1 be a factor that depends only on J and σ. This

factor comes from the combinatorics in taking successive partial derivatives. In later calculation, we

only need to know its value in the case where σ maps J to itself. Then, if we let σJ be the restriction

of σ to J , we have εJ(σ) = (−1)|σJ |.

We compute that

∂

∂g1,σ(1)(exp (trug) |g|) = uσ(1),1 exp (trug) |g|+ ε1(σ) exp(trug)

∣∣∣g1,σ(1)∣∣∣ ;

∂

∂g2,σ(2)

∂

∂g1,σ(1)(exp(trug)|g|) = uσ(2),2uσ(1),1 exp(trug)|g|+ ε2(σ)uσ(1),1 exp(trug)

∣∣∣g2,σ(2)∣∣∣

+ ε1(σ)uσ(2),2 exp(trug)∣∣∣g1,σ(1)

∣∣∣+ ε1,2(σ) exp(trug)

∣∣∣g1,2,σ(1),σ(2)∣∣∣ .

By induction, we have

∂

∂gm−1,σ(m−1)· · · ∂

∂g1,σ(1)(exp(trug)|g|)

=∑

1≤j1<···<jt≤m−1

εj1,...,jt(σ)uσ(sm−1−t),sm−1−t · · ·uσ(s1),s1 exp(trug)∣∣∣gj1,...,jt,σ(j1),...,σ(jt)

∣∣∣ ,where s1 < · · · < sm−1−t is the complement of j1, . . . , jt. Summing over σ, we have

∆ |g=g0 (exp(trug)|g|) = exp(trug0)∑

σ∈Sm−1

(−1)|σ|∑

1≤j1<···<jt≤m−1

εj1,...,jt(σ)uσ(sm−1−t),sm−1−t · · ·uσ(s1),s1

∣∣∣gj1,...,jt,σ(j1),...,σ(jt)0

∣∣∣ .

83

Changing the order of summation, since g0 is diagonal, we have

∆ |g=g0 (exp(trug)|g|)

= exp(trug0)∑

J=j1<···<jt

∑σ(J)=J

(−1)|σ|(−1)|σJ |uσ(sm−1−t),sm−1−t · · ·uσ(s1),s1

∣∣∣gJ,J0

∣∣∣= exp(trug0)

∑J=j1<···<jt

t!∣∣∣gJ,J0

∣∣∣ ∑σ′:J→J

(−1)|σ′|uσ(sm−1−t),sm−1−t · · ·uσ(s1),s1

= exp(trug0)∑

J=j1<···<jt

t!∣∣∣gJ,J0

∣∣∣ ∣∣uJ,J ∣∣= exp(trug0)

∑J′=s1<···<st

(m− 1− t)! |(g0)J′,J′ | |uJ′,J′ | .

The lemma follows by Lemma 4.1.6 (1).

In conclusion, combining (4.27), we obtain the following proposition.

Proposition 4.1.8. For T = k diag[a1, . . . , am−1,−b]tk of signature (m− 1, 1) as above, we have

W ′T (0, e,Φ0)

= γV(2π)m

2

Γm(m)(2πi)m−1exp(−2π trT )

∫Dm−1

exp (−4π(a1w1w1 + · · ·+ am−1wm−1wm−1))

∑1≤s1<···<st≤m−1

(−4π)t(m− 1− t)!(as1 · · · ast) (1 + ws1ws1 + · · ·+ wstwst)

(−Ei) (−4πb(1 + w∗w)) (1− zz)−m Ω,

where wj are functions in z as in (4.24), and Ω (4.26) is the volume form in z.

4.2 Archimedean local height

In this section, we introduce a notion of height on the symmetric domain that will eventually contribute

to the local height pairing at an archimedean place. We also prove some properties of such height. A

basic reference for archimedean Green currents and height paring is [Sou1992, Chapter II]. We keep

the notations in 4.1. We fix an integer m ≥ 2.

84

4.2.1 Green currents

Let V ′ ' Cm be the complex hermitian space with the form

(z′, z) = z′1z1 + · · ·+ z′m−1zm−1 − z′mzm; z = (z1, . . . , zm), z′ = (z′1, . . . , z′m) ∈ Cm.

In particular, the signature of V ′ is (m−1, 1). The symmetric hermitian domain D of U(V ′), introduced

in 3.1.1, can be identified with the (m− 1)-dimensional open complex unit disc Dm−1 via the map

z = [z1 : · · · : zm] ∈ D 7→(z1

zm, . . . ,

zm−1

zm

)∈ Dm−1.

In what follows, we will not distinguished between D and Dm−1.

Given any x ∈ V ′r (1 ≤ r ≤ m−1) with nonsingular moment matrix T (x), let Dx be the subspace

of Dm−1 consisting of lines perpendicular to all components in x. Then Dx is nonempty if and

only if T (x) is positive definite. Suppose r = 1, for z ∈ Dm−1, let x = xz + xz be the orthogonal

decomposition with respect to the line z, i.e., xz ∈ z and xz ⊥ z. Let R(x, z) = −(xz, xz) that is

nonnegative since z is negative definite, and R(x, z) = 0 if and only if x = 0 or z ∈ Dx. Explicitly, let

x = (x1, . . . , xm) ∈ V ′, z = (z1, . . . , zm−1) ∈ Dm−1. Then

R(x, z) =(x1z1 + · · ·+ xm−1zm−1 − xm) (x1z1 + · · ·+ xm−1zm−1 − xm)

1− zz,

where we recall that zz = z1z1 + · · ·+ zm−1zm−1. We define

ξ(x, z) = −Ei(−2πR(x, z)).

For each nonzero element x ∈ V ′, ξ(x, •) is a smooth function on Dm−1 − Dx, and has logarithmic

growth along Dx if not empty. Therefore, we can view it as a current [ξ(x)] = [ξ(x, •)] on Dm−1.

We recall the Kudla–Millson form ϕ ∈ (S(V ′r)⊗Ar,r(Dm−1))U(V ′)

(1 ≤ r ≤ m − 1) constructed in

[KM1986], and let

ω(x) = ω(x, •) = exp(2π trT (x))ϕ(x, •).

We have the following proposition.

Proposition 4.2.1. Let x ∈ V ′ be a nonzero elements. Then we have

ddc[ξ(x)] + δDx = [ω(x)]

85

as currents on Dm−1.

We will only give a proof for m = 2, and the proof for general m is similar but involves tedious

computations.

Proof. We start from showing that ddcξ(x) = ω(x) holds away from Dx. Let x = (x1, x2) and

z ∈ D1 −Dx. Sometimes we simply write R instead of R(x) for short. Then we have the formula

ddcξ(x) =1

2πi

(exp(−2πR)

R2

(R∂∂R− ∂R ∧ ∂R

)− 2π

exp(−2πR)

R∂R ∧ ∂R

). (4.28)

Computing each term, we have

R(x, z) =(x1z − x2) (x1z − x2)

1− zz;

∂R =x1 (x1z − x2) (1− zz) + (x1z − x2) (x1z − x2) z

(1− zz)2 dz;

∂R =x1 (x1z − x2) (1− zz) + (x1z − x2) (x1z − x2) z

(1− zz)2 dz;

∂∂R =

(x1x1

1− zz+x1z (x1z − x2) + (x1z − x2) (2x1z − x2) + 2Rzz

(1− zz)2

)dz ∧ dz;

∂R ∧ ∂R =

(x1x1R

1− zz+x1z (x1z − x2)R+ x1z (x1z − x2)R+R2zz

(1− zz)2

)dz ∧ dz.

Therefore,

R∂∂R− ∂R ∧ ∂R =

((x1z − x2) (x1z − x2)R

(1− zz)2 − R2zz

(1− zz)2

)dz ∧ dz = R2 dz ∧ dz

(1− zz)2 ; (4.29)

and

∂R ∧ ∂R = (x1x1 (1− zz) + x1z (x1z − x2) + x1z (x1z − x2) +Rzz)Rdz ∧ dz

(1− zz)2

= (x1x1 + x1zx2 + x1zx2 + x1x1zz)Rdz ∧ dz

(1− zz)2

= ((x, x) + (x1z − x2) (x1z − x2) +Rzz)Rdz ∧ dz

(1− zz)2

= (R(x, z) + (x, x))Rdz ∧ dz

(1− zz)2 . (4.30)

Plugging (4.29) and (4.30), we have that

(4.28) = (1− 2π(R(x, z) + (x, x))) exp(−2πR(x, z))dz ∧ dz

2πi (1− zz)2 = ω(x, z).

86

The remaining discussion is same as in the proof of [Kud1997, Proposition 11.1], from Lemma 11.2

on Page 606. We will not repeat the detail.

The above proposition says that ξ(x) is a Green function of logarithmic type for Dx. Now we

consider x = (x1, . . . , xr) ∈ V ′r with nonsingular moment matrix T (x). Then using the star product

of Green currents, we have a Green current

Ξx = [ξ(x1)] ∗ · · · ∗ [ξ(xr)]

for Dx. As currents of degree (r, r), we have

ddc ([ξ(x1)] ∗ · · · ∗ [ξ(xr)]) + δDx = [ω(x1) ∧ · · · ∧ ω(xr)] = [ω(x)].

Definition 4.2.2 (Height functions (on D)). For x = (x1, . . . , xm) ∈ V ′m with nonsingular moment

matrix T (x), we define the height function (on D) to be

H(x)∞ = 〈1,Ξx〉 = 〈1, [ξ(x1)] ∗ · · · ∗ [ξ(xm)]〉.

Since ξ(hx, hz) = ξ(x, z) for h ∈ U(V ′), the height function satisfies H(hx)∞ = H(x)∞, and thus

depends only on the (nonsingular) moment matrix T (x). Sometimes we simply write H(T )∞ for this

function.

The following proposition claims that H(T )∞ is in fact invariant under the conjugation action of

U(m)R.

Proposition 4.2.3. The height function H(T )∞ depends only on the eigenvalues of T . In other

words, for every k ∈ U(m)R, H(kTtk)∞ = H(T )∞.

Proof. We prove by induction onm. The casem = 2 in left to the next subsection. Suppose thatm ≥ 3

and the proposition holds for m − 1. Since U(m)R is generated by diagonal matrices, permutation

matrices, and the matrices of form

k′

1

, k′ ∈ U(m− 1)R,

We only need to prove that H((x′k′, xm))∞ = H((x′, xm))∞, where x = (x′, xm) ∈ V ′m−1⊕V ′ = V ′m

87

with T (x) = T . By definition,

H((x′, xm))∞ = 〈1, [ξ(x1)] ∗ · · · ∗ [ξ(xm−1)]|Dxm 〉+

∫Dm−1

ω(x1) ∧ · · · ∧ ω(xm−1) ∧ ξ(xm)

= H(x′)∞ +

∫Dm−1

ω(x′) ∧ ξ(xm).

By induction, H(x′k′)∞ = H(x′)∞ and by [KM1986, Theorem 3.2 (ii)], ω(x′) = ω(x′k′). Therefore,

H((x′k′, xm))∞ = H((x′, xm))∞.

4.2.2 Invariance under U(2): an exercise in Calculus

We consider the case m = 2. Suppose

T =

d1 m

m d2

with d1, d2 ∈ R and m ∈ C. Choose a complex number ε with norm 1 such that ε2m ∈ R. Then

ε

ε−1

d1 m

m d2

t ε

ε−1

=

d1 ε2m

ε2m d2

∈ Sym2(R).

We write elements of SO(2) in the form

kθ :=

cos θ sin θ

− sin θ cos θ

, θ ∈ R,

and write T [θ] = kθTtkθ = kθT

tkθ. Since ξ(εx) = ξ(x) for every x ∈ V ′, to finish the proof of

Proposition 4.2.3, we need only to prove the following proposition.

Proposition 4.2.4. For every T ∈ Sym2(R) with signT = (1, 1), we have H(T [θ])∞ = H(T )∞.

Suppose T = diag[a,−b] with a, b > 0,

T [θ] =

d1,θ mθ

mθ d2,θ

∈ Sym2(R).

88

Let x0 = (√

2a, 0) ∈ V ′, y0 = (0,√

2b) ∈ V ′. For θ ∈ R, let

xθ = x0kθ = (x1,θ, x2,θ) = cos θ · x0 − sin θ · y0;

yθ = y0kθ = (y1,θ, y2,θ) = sin θ · x0 + cos θ · y0.

We have

dxθdθ

= −yθ;dyθdθ

= xθ; H(T [θ])∞ = H((xθ, yθ))∞.

Let

zx,θ =x2,θ

x1,θ; zy,θ =

y2,θ

y1,θ.

Then

Dxθ = [zx,θ, 1]; Dyθ = [zy,θ, 1]

if not empty. In what follows, we adopt the convention that if |z| ≥ 1, f(z) = 0 for any function f .

We record the following lemma that is [Kud1997, Lemma 11.4].

Lemma 4.2.5. We have that

H((xθ, yθ))∞ = ξ(xθ, zy,θ) +

∫D1

ξ(yθ)ω(xθ)

= ξ(yθ, zx,θ) +

∫D1

ξ(xθ)ω(yθ)

= ξ(xθ, zy,θ) + ξ(yθ, zx,θ)−∫D1

dξ(xθ) ∧ dcξ(yθ).

We consider the last integral above in general. Write x = (x1, x2) ∈ V ′, y = (y1, y2) ∈ V ′,

R1 = R(x), and R2 = R(y). Define

I(T ) = I((x, y)) = −∫D1

dξ(x) ∧ dcξ(y)

= − 1

4πi

∫D1

(∂ + ∂

)ξ(x) ∧

(∂ − ∂

)ξ(y)

= − i

4π

∫D1

(∂ξ(x) ∧ ∂ξ(y) + ∂ξ(y) ∧ ∂ξ(x)

)= − i

4π

∫D1

exp (−2π(R1 +R2))

R1R2

(∂R1 ∧ ∂R2 + ∂R2 ∧ ∂R1

).

For z ∈ D1, let

x(z) = (1− zz)−1/2(z, 1) ∈ V ′; M = (x, x(z))(y, x(z)).

89

We have the following lemma.

Lemma 4.2.6. Let 2m = (x, y). Then

∂R1 ∧ ∂R2 + ∂R2 ∧ ∂R1 = 2(R1R2 +mM +mM

) dz ∧ dz

(1− zz)2 ;

∂R1 ∧ ∂R2 − ∂R2 ∧ ∂R1 = 2(mM −mM

) dz ∧ dz

(1− zz)2 .

Proof. By definition,

R1 =(x1z − x2) (x1z − x2)

1− zz.

Therefore,

∂R1 =(x1z − x2)x1 + zR1

1− zzdz,

and similarly for ∂R2, ∂R1, and ∂R2. We compute that

∂R1 ∧ ∂R2

= ((x1z − x2) (y1z − y2)x1y1 + (y1z − y2) y1zR1 + (x1z − x2)x1zR2 + zzR1R2)dz ∧ dz

(1− zz)2

= ((x1z − x2) (y1z − y2)x1y1 + (y1z − y2) y2R1 + (x1z − x2)x1zR2 +R1R2)dz ∧ dz

(1− zz)2

=

((x1z − x2) (y1z − y2)

(x1y1 +

y2 (x1z − x2)

1− zz

)+ (x1z − x2)x1zR2 +R1R2

)dz ∧ dz

(1− zz)2

=

((x1z − x2) (y1z − y2)

x1y1 − x2y2 − x1z (y1z − y2)

1− zz+ (x1z − x2)x1zR2 +R1R2

)dz ∧ dz

(1− zz)2

= (2mM +R1R2)dz ∧ dz

(1− zz)2 .

The lemma follows from a similar calculation for ∂R2 ∧ ∂R1.

We define a morphism

α : R×D1 → Her2(C)det=0 = h ∈ Her2(C) | deth = 0

of 3-dimensional real analytic spaces by the formula

α(θ, z) =

R1 M

M R2

=

(xθ, x(z))(xθ, x(z)) (xθ, x(z))(yθ, x(z))

(xθ, x(z))(yθ, x(z)) (yθ, x(z))(yθ, x(z))

,

90

and set αθ = α(θ, •). By an easy computation, we see that

dR1

dθ= −

(M +M

);

dR2

dθ= M +M ;

dM

dθ= R1 +R2. (4.31)

Therefore, R1 +R2 and M −M are independent of θ, which are the values at θ = 0, respectively. In

other words,

R1 +R2 =2azz + 2b

1− zz= −2a+

2(a+ b)

1− zz; (4.32)

M −M =2√ab (z − z)1− zz

. (4.33)

By Lemma 4.2.6, and the fact that 2mθ = (xθ, yθ) ∈ R, we have

I(T [θ]) = − i

2π

∫D1

exp(−2π(R1 +R2))

R1R2

(R1R2 +m

(M +M

)) dz ∧ dz

(1− zz)2

= I ′(T [θ]) + I ′′(T [θ]), (4.34)

where

I ′(T [θ]) = − i

2π

∫D1

exp(−2π(R1 +R2))dz ∧ dz

(1− zz)2 ;

I ′′(T [θ]) = − i

2π

∫D1

exp(−2π(R1 +R2))

R1R2m(M +M

) dz ∧ dz

(1− zz)2 .

By (4.32) and (4.33), the integral I ′(T [θ]) is independent of θ. There we only need to consider the

second term I ′′(T [θ]). Define a differential form of degree two on (the smooth locus of) Her2(C)det=0

as follows.

Ξ = − i

4π

exp(−2π(R1 +R2))

R1R2

M +M

M −MdR1 ∧ dR2,

which has singularities along the locus R1R2

(M −M

)= 0. We have the following lemma.

Lemma 4.2.7. We have

1. For a fixed θ ∈ R,

α∗θ(Ξ) = − i

2π

exp(−2π(R1 +R2))

R1R2m(M +M

) dz ∧ dz

(1− zz)2 ;

91

2. On Her2(C)det=0, we have

dΞ =i

π

exp(−2π(R1 +R2))(M −M

)2 (M +M

)d(M −M

)∧ dR1 ∧ dR2.

Proof. 1. It follows from Lemma 4.2.6.

2. By the equality (M +M

)2 − (M −M)2 = 4R1R2,

we have

d

d(M −M

)M +M

M −M= − 4R1R2(

M −M)2 (

M +M) .

Then it follows.

Let D+1 = z ∈ D1 | Im z ≥ 0. Since α∗θ(Ξ)/dz ∧ dz is invariant under z 7→ z, by Lemmas 4.2.5,

4.2.7 (1), and (4.34), we have

H(T [θ1])∞ −H(T [θ0])∞

= ξ(xθ1 , zy,θ1) + ξ(yθ1 , zx,θ1)− ξ(xθ0 , zy,θ0)− ξ(yθ0 , zx,θ0) + I(T [θ])− I(T [θ0])

= ξ(xθ1 , zy,θ1) + ξ(yθ1 , zx,θ1)− ξ(xθ0 , zy,θ0)− ξ(yθ0 , zx,θ0) + I ′′(T [θ])− I ′′(T [θ0])

= ξ(xθ1 , zy,θ1) + ξ(yθ1 , zx,θ1)− ξ(xθ0 , zy,θ0)− ξ(yθ0 , zx,θ0) +

∫D1

α∗θ1(Ξ)−∫D1

α∗θ0(Ξ)

= ξ(xθ1 , zy,θ1) + ξ(yθ1 , zx,θ1)− ξ(xθ0 , zy,θ0)− ξ(yθ0 , zx,θ0) + 2

∫D+

1

α∗θ1(Ξ)− 2

∫D+

1

α∗θ0(Ξ). (4.35)

We see that the form α∗θ(Ξ) has (possible) singularities when R1R2 = 0, i.e., at the (possible) points

zx,θ, zy,θ. An easy calculation shows that

zx,θ =x2,θ

x1,θ= − tan θ ·

√b

a∈ R; zy,θ =

y2,θ

y1,θ= cot θ ·

√b

a∈ R.

We now assume that 0 < θ0 ≤ θ1 < π/2. Then 0 ∈ D1 will not be a singular point for θ ∈ [θ0, θ1].

We need to evaluate ∫D+

1

α∗θ0(Ξ)−∫D+

1

α∗θ1(Ξ).

For any ε > 0 small enough, let

• B1,ε be the (oriented) path z = r exp(iε) | r ∈ [0, 1) from r = 0 to r = 1;

92

• B2,ε the path z = r exp(i(π − ε)) | r ∈ [0, 1) from r = 1 to r = 0; and

• Dε ⊂ D+1 the area containing points on or above the lines B1,ε and B2,ε.

By our assumption, α∗θ(Ξ) is nonsingular on Dε for every θ ∈ [θ0, θ1]. By Stokes’ Theorem and the

fact that exp(−2π(R1 +R2)) decays rapidly as |z| goes to 1, we have

∫[θ0,θ1]×Dε

α∗(dΞ) =

∫Dε

α∗θ1(Ξ)−∫Dε

α∗θ0(Ξ) +

∫[θ0,θ1]×(B1,ε∪B2,ε)

α∗(Ξ). (4.36)

Lemma 4.2.8. We have ∫[θ0,θ1]×Dε

α∗(dΞ) = 0.

Proof. By (4.31), (4.32) and (4.33), we have

dR1 = ∂R1 + ∂R1 −(M +M

)dθ;

dR2 = ∂R2 + ∂R2 +(M +M

)dθ;

d(M −M

)= 2√ab

(∂z − z1− zz

+ ∂z − z1− zz

).

Therefore,

α∗(d(M −M

)∧ dR1 ∧ dR2

)= 2√ab(M +M

)(∂z − z1− zz

∧ ∂(R1 +R2)− ∂(R1 +R2) ∧ ∂ z − z1− zz

)= 4√ab(a+ b)

(M +M

)(∂z − z1− zz

∧ ∂ 1

1− zz− ∂ 1

1− zz∧ ∂ z − z

1− zz

).

Then by Lemma 4.2.7 (2), we have

α∗ (dΞ) =4i√ab(a+ b)

π

exp(−2π(R1 +R2))(M −M

)2 (∂z − z1− zz

∧ ∂ 1

1− zz− ∂ 1

1− zz∧ ∂ z − z

1− zz

)

=4i√ab(a+ b)

π

exp(−2π(R1 +R2))(M −M

)2 z + z

(1− zz)3dz ∧ dz.

Therefore, z 7→ −z stabilizes the domain [θ0, θ1]×Dε, and maps α∗ (dΞ) /dz ∧ dz to its negative, the

integral is zero.

93

By the above lemma and (4.36), we have

∫D+

1

α∗θ0(Ξ)−∫D+

1

α∗θ1(Ξ) = limε→0

∫Dε

α∗θ0(Ξ)− limε→0

∫Dε

α∗θ1(Ξ) = limε→0

∫[θ0,θ1]×(B1,ε∪B2,ε)

α∗(Ξ). (4.37)

A simple computation shows that on [θ0, θ1]× (B2,ε ∪B1,ε), we have

α∗(Ξ) =−i(a+ b)

π

exp(−2π(R1 +R2))

R1R2

(M +M

)2M −M

r

(1− r2)2dr ∧ dθ

=−i(a+ b)

π

exp(−2π(R1 +R2))(M −M

)R1R2

r

(1− r2)2dr ∧ dθ

+−4i(a+ b)

π

exp(−2π(R1 +R2))

M −Mr

(1− r2)2dr ∧ dθ.

Since the integrations of the second term on two paths cancel each other, we have

(4.37) =

∫ θ1

θ0

dθ−i(a+ b)

πlimε→0

∫B1,ε∪B2,ε

exp(−2π(R1 +R2))(M −M

)R1R2

r

(1− r2)2dr

=

∫ θ1

θ0

dθ4√ab(a+ b)

πlimε→0

sin ε

∫B1,ε∪B2,ε

exp(−2π(R1 +R2))

R1R2

r2

(1− r2)3dr. (4.38)

To proceed, we need the following lemma.

Lemma 4.2.9. Let f(r) be a smooth function on [0, 1) that is rapidly decreasing as r → 1. Then for

any c1, c2, d1, d2 > 0,

limε→0+

∫ 1

0

sin ε

(c21r2 + c22 − 2c1c2r cos ε) (d2

1r2 + d2

2 + 2d1d2r cos ε)f(r)dr

=

πc1

c2(c1d2+c2d1)2f(c2c1

)c1 > c2;

0 c1 ≤ c2.

Proof. The case c1 ≤ c2 follows from the assumption that f is rapidly decreasing. For the first case,

we only need to prove that

limε→0+

∫ 1

0

sin ε

c21r2 + c22 − 2c1c2r cos ε

dr =π

c1c2. (4.39)

94

The integral of the left-hand side of (4.39) (for small ε > 0) equals

sin ε

∫ 1

0

1

(c1r − c2 cos ε)2

+ c22 (1− cos ε)

=sin ε

c1c2√

1− cos ε

∫ c1−c2 cos ε

c2√

1−cos ε

− cos ε√1−cos ε

1(c1r−c2 cos εc2√

1−cos ε

)2

+ 1d

(c1r − c2 cos ε

c2√

1− cos ε

)

=sin ε

c1c2√

1− cos ε

(arctan

c1r − c2 cos ε

c2√

1− cos ε+ arctan

cos ε√1− cos ε

).

Let ε→ 0+, the limit is π/c1c2.

Applying the above lemma, we have, on the one hand, that

(4.38) =

∫ θ1

θ0

√ab(a+ b)

(exp (−2πR1(zy,θ))

R1(zy,θ)

y1,θy2,θ

d22,θ

+exp (−2πR2(zx,θ))

R2(zx,θ)

x1,θx2,θ

d21,θ

)dθ. (4.40)

On the other hand, we have

dR1(zy,θ)

dθ=

d

dθ(R1 (zy,θ) +R2 (zy,θ)) =

4(a+ b)r

(1− r2)2|r=

y2,θy1,θ

d

dθ

(y2,θ

y1,θ

)= 2√ab(a+ b)

y1,θy2,θ

d22,θ

;

dR2(zx,θ)

dθ= 2√ab(a+ b)

x1,θx2,θ

d21,θ

.

Therefore,

(4.40) =1

2

(∫ R1(zy,θ1 )

R1(zy,θ0 )

exp (−2πR1(zy,θ))

R1(zy,θ)dR1(zy,θ) +

∫ R2(zx,θ1 )

R2(zx,θ0 )

exp (−2πR2(zx,θ))

R2(zx,θ)dR2(zx,θ)

)

=1

2(ξ(xθ1 , zy,θ1) + ξ(yθ1 , zx,θ1)− ξ(xθ0 , zy,θ0)− ξ(yθ0 , zx,θ0)) ,

which, by (4.35), implies that

H(T [θ1])∞ −H(T [θ0])∞ = 0 (4.41)

for 0 < θ0 ≤ θ1 < π/2. Same argument works for intervals (π/2, π), (π, 3π/2) and (3π/2, 2π), other

than (0, π/2). The constancy of H(T [θ])∞ for all θ ∈ R then follows from (4.41) and the continuity.

This finishes the proof of Proposition 4.2.4.

95

4.3 An archimedean local Siegel–Weil formula

In this section, we set up a relation between derivatives of Whittaker integrals and the height functions

defined in the previous section. Furthermore, we prove a local arithmetic analogue of the Siegel–Weil

formula at an archimedean place for arbitrary dimensions.

4.3.1 Comparison on the hermitian domain

We propose and prove the following theorem, which we call the archimedean local arithmetic Siegel–

Weil formula.

Theorem 4.3.1. Let T ∈ Herm(C) such that signT = (m− 1, 1). Then we have

W ′T (0, e,Φ0) = γV(2π)m

2

Γm(m)exp(−2π trT )H(T )∞,

where H(T )∞ is defined in Definition 4.2.2.

By Proposition 4.2.3, we only need to prove for T = diag[a1, . . . , am−1,−b] with a1, . . . , am−1, b ∈

R>0. We let xj = (. . . ,√

2aj , . . . ) ∈ Cm ∼= V ′ with the j-th entry√

2aj and all others zero for

j = 1, . . . ,m− 1, and xm = (0, . . . , 0,√

2b). Then H(T )∞ = H((x1, . . . , xm))∞. Since (xm, xm) < 0,

we have Dxm = ∅, and

H(T )∞ =

∫Dm−1

ω(x1) ∧ · · · ∧ ω(xm−1) ∧ ξ(xm).

Proof of Theorem 4.3.1. By Proposition 4.1.8, we need to prove that

(2πi)m−1

∫Dm−1

ω(x1) ∧ · · · ∧ ω(xm−1) ∧ ξ(xm)

=

∫Dm−1

exp (−4π (a1w1w1 + · · ·+ am−1wm−1wm−1))

∑1≤s1<···<st≤m−1

(−4π)t(m− 1− t)!(as1 · · · ast) (1 + ws1ws1 + · · ·+ wstwst)

(−Ei)(−4πb

(1 + tww

))(1− zz)−m Ω. (4.42)

96

By definition and (4.24), we have

Rj(z) : = R(xj , z) =2ajzjzj1− zz

= 2ajwjwj , j = 1, . . . ,m− 1;

Rm(z) : = R(xm, z) =−2b

1− zz= −2b

(1 + tww

).

Therefore, ξ(xm) = −Ei(−4πb

(1 + tww

)). The next step is to find an explicit formula for ω(xj). By

(4.28), we need to calculate ∂Rj , ∂Rj , and ∂∂Rj for j = 1, . . . ,m− 1. Since (1− zz)Rj = 2ajzjzj ,

∂ (1− zz)Rj + (1− zz) ∂Rj = 2ajzjdzj ; (4.43)

∂Rj =2ajzjdzj +Rj∂ (zz)

1− zz. (4.44)

Similarly,

∂Rj =2ajzjdzj +Rj∂ (zz)

1− zz. (4.45)

Differentiating (4.43) and plugging (4.44) and (4.45), we have

∂∂ (1− zz)Rj + ∂Rj∂ (1− zz) + ∂ (1− zz) ∂Rj + (1− zz) ∂∂Rj = 2ajdzjdzj ,

which implies that

Rj =2aj (1− zz) dzjdzj + 2ajzjdzj∂ (zz) + 2ajzj∂ (zz) dzj + 2Rj∂ (zz) ∂ (zz) +Rj (1− zz) ∂∂ (zz)

(1− zz)2 .

(4.46)

Taking wedge of (4.44) and (4.45), we have

∂Rj ∧ ∂Rj =4a2jzjzjdzjdzj + 2ajRjzjdzj∂ (zz) + 2ajRjzj∂ (zz) dzj +R2

j∂ (zz) ∂ (zz)

(1− zz)2 . (4.47)

Combining (4.46) and (4.47), we have

1

R2j

(Rj∂∂Rj − ∂Rj ∧ ∂Rj

)=∂ (zz) ∂ (zz)

(1− zz)2 +∂∂ (zz)

1− zz. (4.48)

97

For simplicity, we make some substitutions. Let

ω = ∂ (zz) ∂ (zz) + (1− zz) ∂∂ (zz) ;

ωj = (1− zz) zjdzjdzj + zjdzj∂ (zz) + zj∂ (zz) dzj + wjwj∂ (zz) ∂ (zz) , j = 1, · · · ,m− 1.

Then we have

2πiω(xj) = −∂∂ξ(xj) = exp (−4πajwjwj) (ω − 4πajωj) (1− zz)2.

Therefore, to prove (4.42), we only need to prove the following equality of (m − 1,m − 1)-forms on

Dm−1,

m−1∧j=1

(ω − 4πajωj) =∑

s1<···<st

(−4π)t(m−1− t)!(as1 · · · ast) (1 + ws1ws1 + · · ·+ wstwst) (1− zz)m−2Ω,

which follows from the claim that for every subset s1 < · · · < st ⊂ 1, . . . ,m− 1, we have

ωs1 ∧ · · · ∧ ωst ∧ ωm−1−t = (m− 1− t)! (1 + ws1ws1 + · · ·+ wstwst)(1− zz)m−2

Ω.

This will be proved in the next lemma where, without lost of generality, we assume that sj = j. The

theorem then follows.

Lemma 4.3.2. Let wj, Ω, ω and ωj be as above. For any integer 0 ≤ t ≤ m−1, we have the following

equality of (m− 1,m− 1)-forms on Dm−1

t∧j=1

ωj

∧ωm−1−t = (m− 1− t)!

1 +

t∑j=1

wjwj

(1− zz)m−2Ω.

The proof will occupy the next subsection.

4.3.2 Proof of Lemma 4.3.2

For j = 1, . . . ,m− 1, we let

σj = zjdzj∂ (zz) ; σ′j = zj∂ (zz) dzj ; δj = (1− zz) dzjdzj .

98

Thenm−1∑k=1

σk =

m−1∑k=1

σ′k; ω =

m−1∑k=1

(σk + δk) ; ωj = δj + σj + σ′j + wjwj

m−1∑k=1

σk;

and

σj ∧ σj = 0; σ′j ∧ σ′j = 0; δj ∧ δj = 0.

Introduce the following (m− 1)× (m− 1) matrix

Z =

z1z1 z2z1 · · · zm−1z1

z1z2 z2z2 · · · zm−1z2

......

. . ....

z1zm−1 z2zm−1 · · · zm−1zm−1

.

Recall the notation ZJ,K as in the proof of Lemma 4.1.7 for subsets J,K ⊂ 1, . . . ,m − 1 with

|J | = |K|. It is easy to see that |ZJ,K | 6= 0 only if |J | ≤ 1, where in the later case,∣∣Zj,k∣∣ = zjzk

and∣∣Z∅,∅∣∣ = 1.

Consider three subsets I, J,K ⊂ 1, . . . ,m− 1 with |I|+ |J |+ |K| = m− 1. Write

σI =∧i∈I

σi; σ′J =∧j∈J

σ′j ; δK =∧k∈K

δk.

We have the following equality

σIσ′JδK := σI ∧ σ′J ∧ δK =

εI,J,K

∣∣∣ZI,J∪K∣∣∣ ∣∣∣ZI∪K,J ∣∣∣ (1− zz)|K|Ω (I ∪ J) ∩K = ∅;

0 (I ∪ J) ∩K 6= ∅.

Here, εI,J,K ∈ 1, 0, 1 is a factor depending only on I, J,K. It is nonzero only if |I| ≤ 1 and |J | ≤ 1.

Explicitly,

σIσ′JδK =

zizizjzj (1− zz)m−3Ω i 6= j, I = i, J = j,K = I ∪ J ;

−zizjzjzi (1− zz)m−3Ω i 6= j, I = J = i,K = I ∪ j;

zizi (1− zz)m−2Ω I ∪ J = i,K = i;

(1− zz)m−1Ω I = J = ∅,K = 1, . . . ,m− 1.

99

For a subset P of 1, . . . ,m, we set wP =∏p∈P wp and wP =

∏p∈P wp. Then

t∧j=1

ωj

∧ωm−1−t =

t∧j=1

(δj + σj + σ′j + wjwj

m−1∑k=1

σk

)∧(m−1∑k=1

σk +

m−1∑k=1

δk

)m−1−t

=

∑L∐M

∐N

∐P=1,...,t

δLσMσ′NwPwP

(m−1∑k=1

σk

)|P | ∑Q⊂1,...,m−1|Q|≤m−1−t

(m− 1− t)!(m− 1− t− |Q|)!

(m−1∑k=1

σk

)m−1−t−|Q|

δQ

=

∑L∐M

∐N

∐P=1,...,t

∑Q⊂1,...,m−1|Q|≤m−1−t

TL,M,N,P,Q, (4.49)

where

TL,M,N,P,Q =(m− 1− t)!

(m− 1− t− |Q|)!δL∪QσMσ

′NwPwP

(m−1∑k=1

σk

)|P |+m−1−t−|Q|

.

It is easy to see that if TL,M,N,P,Q 6= 0, then |Q| ≥ m−2−t. We enumerate all cases where TL,M,N,P,Q

may be nonzero.

Case I: |Q| = m− 1− t. Then |P | ≤ 1:

Case I-1: |P | = 0. Then Q = t+ 1, . . . ,m− 1 and |M | ≤ 1, |N | ≤ 1:

Case I-1a: M = m and N = n for m 6= n ∈ 1, · · · , t. Then the sum of correspond-

ing terms is

∑TL,M,N,P,Q = (m− 1− t)!

t∑m,n=1m 6=n

zmzmznzn (1− zz)m−3Ω. (4.50)

Case I-1b: M ∪N = m for 1 ≤ m ≤ t. Then the sum of corresponding terms is

∑TL,M,N,P,Q = 2(m− 1− t)!

t∑m=1

zmzm (1− zz)m−2Ω. (4.51)

Case I-1c: M = N = ∅. Then the corresponding term is

TL,M,N,P,Q = T1,...,t,∅,∅,∅,t+1,...,m−1 = (m− 1− t)! (1− zz)m−1Ω. (4.52)

Case I-2: |P | = 1. Then M = N = ∅. Suppose P = p for 1 ≤ p ≤ t. Then Q =

p, t+1, . . . ,m−1−q for some q inside p, t+1, . . . ,m−1. The sum of the corresponding

100

terms is

∑TL,M,N,P,Q = (m− 1− t)!

t∑p=1

wpwp

(zpzp +

m−1∑q=t+1

zqzq

)(1− zz)m−2

Ω. (4.53)

Case II: |Q| = m − 2 − t. Then M = N = P = ∅ and |Q| = t + 1, . . . ,m − 1 − q for some q

inside t+ 1, . . . ,m− 1. The sum of the corresponding terms is

∑TL,M,N,P,Q = (m− 1− t)!

m−1∑q=t+1

zqzq (1− zz)m−2Ω. (4.54)

Taking the sum from (4.50) to (4.54), we have

(4.49) = (m− 1− t)! (1− zz)m−2Ω t∑

p=1

wpwp

(zpzp +

m−1∑q=t+1

zqzq

)+

∑tm,n=1m 6=n

zmzmznzn

1− zz+ 2

t∑m=1

zmzm +

m−1∑q=t+1

zqzq + (1− zz)

= (m− 1− t)! (1− zz)m−2

Ω1 +

t∑m=1

zmzm +

∑tm,n=1m6=n

zmzmznzn +∑tp=1 zpzp

(zpzp +

∑m−1q=t+1 zqzq

)1− zz

= (m− 1− t)! (1− zz)m−2

Ω

∑tm=1 zmzm1− zz

= (m− 1− t)!

1 +

t∑j=1

wjwj

(1− zz)m−2Ω.

Lemma 4.3.2 is proved.

4.3.3 Comparison on Shimura varieties

We use previous results to compute the archimedean local height paring on the unitary Shimura

varieties with respect to suitable Green currents. We recall some notations from 3.1.1 and 3.1.2. Let

n ≥ 1 be an integer. We have a totally positive definition incoherent hermitian space V over AE

of rank 2n, and H = ResAF /AU(V). For (sufficiently small) open compact subgroup K of H(Afin),

we have the Shimura variety ShK := ShK(H). Suppose that K =∏

p∈ΣfinKp is decomposable. Let

φα = φ0∞⊗φα,fin (α = 1, 2) be decomposable Schwartz functions with φα,fin ∈ S(Vn

fin)K . Suppose that

φ1,p ⊗ φ2,p ∈ S(V2np )reg for some finite place p of F . Then the generating series Zφ1(g1) and Zφ2(g2)

do not meet on ShK providing gα ∈ P ′pH ′(ApF ).

101

Definition 4.3.3 (Volume of open compact). For every finite place p of F , we define a measure dh

on H(Afin), which depends only on the additive character ψfin, as follows. By [Ral1987, Lemma 4.2],

there is a unique Haar measure d′hp on U(Vp) such that for every nonsingular matrix T ∈ Her2n(Ep)

such that OT is nonempty, and Φ ∈ S(V2np ),

WT (0, e,Φ) = γVpb2n,p(0)−1

∫U(Vp)

Φ(h−1v xT )d′hv,

where xT is any element in OT , and b2n,p is defined in (2.6). By Lemma 2.3.6, for almost all p, the

volume of Kv with respect to d′hp is 1. We define

dh =1

2b2n(0)

∏p∈Σfin

d′hp,

where b2n =∏v∈Σ b2n,v is (a product of) global Tate L-factors. Let Vol(K) be the volume of K with

respect to the measure dh.

In what follows, we fix an archimedean place ι of F , and ι′ ∈ ι, ι•. Let V = V (ι), H =

ResF/QU(V ), and D = D(ι′). Assume that there exists a finite place p of F such that φp(0) = 0.

Let Her+n (E) be the subset of Hern(E) consisting of totally positive definite matrices. For every

T ∈ Her+n (E), we choose an element xT ∈ V n such that T (x) = T . Then for g ∈ P ′pH ′(A

pF ), we have

the generating series

Zφ(g) =∑

T∈Her+n (E)

∑h∈HxT (Afin)\H(Afin)/K

(ωχ(g)φ) (T, h−1xT )Z(h−1xT )K ,

where HxT ⊂ H is the stabilizer of xT . Recall that under the uniformization

(ShK)anι′∼= H(Q)\D×H(Afin)/K,

the special cycle Z(h−1xT )K is represented by the points (z, h′h), where z ⊥ VxT and h′ fixes all

elements in VxT . In other words, if we identify D with D2n−1, then z is in DxT . Choose a set of

representatives h1, . . . , hl of the double coset H(Q)\H(Afin)/K such that h1 is the identity element.

Let [Zφ(g)ι′ ]anh1

be the restriction of (the ι′-analytification of) Zφ(g) to the neutral component. Then

it is the image of ∑x∈V n,T (x)∈Her+n (E)

(ωχ(g)φ) (T (x), x)Dx

under the projection map D → (H(Q) ∩ K)\D. Write gι = n(b)m(a)[k1, k2] under the Iwasawa

102

decomposition as in 4.1.1. Then

Ξφ(g)ι′,h1 =∑

x∈V n,T (x)∈Her+n (E)

(ωχ(g)φ) (T (x), x)Ξxa (4.55)

descends to a current on (H(Q)∩K)\D, which is a Green current for [Zφ(g)ι′ ]anh1

. By Hecke translation

under hi (i = 2, . . . , l), we obtain the a Green current Ξφ(g)ι′ for Zφ(g)ι′ . The following is our main

theorem of this chapter.

Theorem 4.3.4. Let ι, ι′ be as above. Let φα = φ0∞ ⊗ φα,fin (α = 1, 2) be decomposable Schwartz

functions with φα,fin ∈ S(Vnfin)K . Suppose that φ1,p ⊗ φ2,p ∈ S(V2n

p )reg for some finite place p of F .

Then for gα ∈ P ′pH ′(ApF ),

Eι(0, ı(g1, g∨2 ), φ1 ⊗ φ2) = −2 Vol(K)〈(Zφ1

(g1),Ξφ1(g1)ι′) , (Zφ2

(g2),Ξφ2(g2)ι′)〉ShK ,

where Eι is defined as (2.14); and the right-hand side is the local height pairing on ShK at the place

ι′.

Proof. It is clear that we can assume that gα,ι = m(aα) with aα ∈ GLn(Eι′) for α = 1, 2. Then we

have

〈(Zφ1(g1),Ξφ1

(g1)ι′) , (Zφ2(g2),Ξφ2

(g2)ι′)〉ShK

=

l∑i=1

∫(H(Q)∩K)\D

(Ξωχ(hi)φ1

(g1)ι′,h1

)∗(Ξωχ(hi)φ2

(g2)ι′,h1

)=

l∑i=1

∫(H(Q)∩K)\D

∑x1∈V n,T (x1)∈Her+n (E)

(ωχ(g1)φ1) (T (x1), h−1i x1)Ξx1a1

∗


(ωχ(g2)φ2) (T (x2), h−1i x2)Ξx2a2

=

l∑i=1



(ω′′χ (ı(g1, g

∨2 )) (φ1 ⊗ φ2)

)(T ((x1, x2)), h−1

i (x1, x2))

∫D

Ξx1a1 ∗ Ξx2a2 (4.56)

Let

a =

a1

a2

.

103

Then

(4.56) =

l∑i=1

∑x∈V 2n,T (x)∈Her+2n(E)

(ω′′χ (ı(g1, g

∨2 )) (φ1 ⊗ φ2)

)(T (x), h−1

i x)H(taT (x)a)∞

=∑T

H(taTa)∞∏v∈Σ∞

(ω′′χv

(ı(g1,v, g

∨2,v))

Φ0v

)(T )

∏p∈Σfin

∑hp∈U(Vp)/Kp

(ω′′χp

(ı(g1,p, g

∨2,p))

(φ1,p ⊗ φ2,p))

(h−1p xT ), (4.57)

where the sum is taken over all T ∈ Her+2n(E) that is the moment matrix of some xT ∈ V 2n. There

are three cases.

Case I: v = ι. By (4.3) and Theorem 4.3.1 for taTa, we have

H(taTa)∞(ω′′χι

(ı(g1,ι, g

∨2,ι))

Φ0ι

)(T ) = γ−1

Vι

Γ2n(2n)

(2π)4n2 W′T (0, ı(g1,ι, g

∨2,ι),Φ

0ι ).

By (2.6), we have

H(taTa)∞(ω′′χι

(ı(g1,ι, g

∨2,ι))

Φ0ι

)(T ) = γ−1

Vιb2n,ι(0)W ′T (0, ı(g1,ι, g

∨2,ι),Φ

0ι ). (4.58)

Case II: v ∈ Σ∞ and v 6= ι. By (4.3) and Proposition 4.1.5 (2), we have

(ω′′χv

(ı(g1,v, g

∨2,v))

Φ0v

)(T ) = γ−1

Vvb2n,v(0)WT (0, ı(g1,v, g

∨2,v),Φ

0v). (4.59)

Case III: v ∈ Σfin. By Definition 4.3.3,

∑hp∈U(Vp)/Kp

(ω′′χp

(ı(g1,p, g

∨2,p))

(φ1,p ⊗ φ2,p))

(h−1p xT )

= γ−1Vpb2n,p(0)

(∫Kp

d′hp

)−1

WT (0, ı(g1,p, g∨2,p), φ1,p ⊗ φ2,p) (4.60)

After plugging (4.58), (4.59) and (4.60) in (4.57), the theorem follows.

104

Chapter 5

Comparison at finite places: good

reduction

We compare local terms of analytic and arithmetic kernel functions at an unramified finite place. In

5.1, we introduce the integral models for the Shimura curves, and extend the special cycles and the

generating series to the models. At a good place, we need to calculate certain intersection multiplicity

on the smooth model. This is done in 5.2. The computation of the derivative of Whittaker integrals

at an unramified place and the comparison of the analytic and arithmetic side are the content of 5.3.

We fix some notations for this chapter, which may differ from the previous ones. Let F/Qp be a

finite extension and E/F a quadratic extension of fields with Gal(E/F ) = 1, τ. We fix a uniformizer

$ of F and let q be the cardinality of the residue field of F . Let V ± be the 2-dimensional E-hermitian

space with ε(V ±) = ±1, which is unique up to isometry and H± = U(V ±). Let Λ± be a maximal

OE-lattice in V ±, on which the hermitian form takes values in OE . Let K±0 be the stabilizer of Λ±

in H±, which is a maximal compact subgroup. Recall that we have local groups H ′ = H1, H ′′ ∼= H2,

P , etc.

Recall that we let ιi (i = 1, . . . , d) be all embeddings of F into C and ιi , ι•i those of E above ιi as

in Notation 2.1.1. Let us fix more notations for Chapter 5 and 6.

• For any rational prime p, we fix an isomorphism ι(p) : C∼−→ Cp once and for all.

• For p a finite place of F , let p (resp. p, p•) be that (resp. those) of E over p if p is nonsplit

(resp. split) in E. We fix a uniformizer $ of Fp.

• For a number field F , TF = ResF/QGm,F and T 1F ′ = ResF/Q F

′×,1 for any quadratic extension

105

F ′/F , where F ′×,1 = ker (Nm : F ′× → F×). If F is totally real, let F+ be the set of all totally

positive elements.

• For every finite extension L/Qp of local fields with ring of integers OL and maximal ideal q ⊂ OL,

let UsL be the subgroup of O×L congruent to 1 modulo qs. Denote by L0 the maximal unramified

extension of L. For s ≥ 0, let Ls be the finite extension of L0 corresponding to UsL through local

class field theory. Finally, let Ls be the completion of Ls, whose ring of integers is denoted by

OLs

.

• We fix an algebraic closure of Fp to be F.

106

5.1 Integral models

In the next four subsections, we will assume that F 6= Q. The rest case is slightly different and in

fact simpler, which will be summarized in the last subsection.

5.1.1 Change of Shimura data

Let p = p1, p2, . . . , pr (1 ≤ r ≤ d) be all places of F dividing p and p one of E above p. We assume

that the embedding ι(p) ι1 : E → Cp induces the place p. As before, we suppress ι1 and ι1 for the

nearby objects. We have the hermitian space V over E of dimension 2 whose signature is (1, 1) at

ι1 and (2, 0) elsewhere, the unitary group H over Q, and the Shimura curve MK = ShK(H,X) for a

sufficiently small open compact subgroup K ⊂ H(Afin), which is a smooth projective curve defined

over ι1(E). Recall that X is the conjugacy class of the Hodge map h : S→ HR defined by

z = x+ iy 7→

x y

−y x

−1

× z, 1, . . . , 1

∈ H(R) ⊂(GL2(R)×R× C×

)×(H× ×R× C×

)d−1,

where we identify TE(R) with (C×)d through (ι1, . . . , ιd). We denote by ν : H → T 1

E the determinant

map. Then we have the 0-dimensional Shimura variety LK = Shν(K)(T1E , ν(X)), and a smooth

morphism (also denoted by) ν : MK → LK of ι1(E)-schemes such that the fiber of each geometric

point is connected.

Let us define a subgroup Kp,n of U(Vp) for every integer n ≥ 0. Since Vp is (isometric to) either

V + or V −, we have the lattice Λ± (if p is split, we only have the positive one). We define Kp,n to be

the subgroup of K±0 consisting of elements that have trivial action on Λ±/$nΛ±. Then Kp,0 = K±0

is a maximal compact subgroup. For K = Kp,n ×Kp, we write Mn,Kp (resp. Ln,Kp) for MK (resp.

LK).

Notation 5.1.1. For simplicity, we introduce the following notation

Hpfin = U(V ⊗F Ap

F,fin).

Then Kp is an open compact subgroup of Hpfin.

Since F 6= Q, the Shimura datum (H,X) is not of PEL type. We need to change Shimura data

in order to obtain the moduli interpretations and integral models. This is analogous to the case

considered in [Car1986, YZZa, Zha2001a, Zha2001b], and we refer the detailed proof of various facts

107

to [Car1986]. We choose a negative number λ ∈ Q such that the extension Q(√λ) is split at p, and

the CM extension F † = F (√λ)/F with Gal(F †/F ) = 1, τ † is not isomorphic to E/F . We fix also a

square root λ′ of λ in C with positive imaginary part, and a square root λp of λ in Qp. Let ι1i (resp.

ι2i ) be the embeddings of F † into C above ιi (i = 1, . . . , d) which sends√λ to λ′ (resp. −λ′). Since p

is split in Q(√λ), pi for i = 1, . . . , r are all split in F †. We denote by p1

i (resp. p2i ) the place above pi

which sends√λ to λp (resp. −λp), and assume that ι(p) ι11 induces p1

1.

By the Hasse principle, there is a unique up to isometry quaternion algebra B over F , such that

B, as an F -quadratic space (of dimension 4), is isometric to V as an F -quadratic space with the

quadratic form TrE/F (−,−), where (−,−) is the hermitian form on V . More precisely, when v is

finite, Bv = B ⊗F Fv is division if and only if v is nonsplit and Vv ∼= V −; and Bι1(R) ∼= Mat2(R),

Bιi(R) ∼= H for i > 1. We identify two quadratic spaces B and V through a fixed isometry and hence

V has both left and right multiplication by B. We fix an embedding E → B, through which the

action of E induced from the left multiplication of B coincides with the E-vector space structure of

V . Let G = ResF/QB× with center T ∼= TF and

G† = G×T TF †ν†−→ T × T 1

F † ,

where

ν†(g × z) =(

Nm g · zzτ†,z

zτ†

).

Consider the subtorus T † = Gm,Q × T 1F † , and let H† be the preimage of T † under ν†. Define the

Hodge map h† : S→ H†R ⊂ GR ×TR TF †,R by

z = x+ iy 7→

x y

−y x

−1

× 1,12 × z−1, . . . ,12 × z−1

, (5.1)

and let X† be the H†(R)-conjugacy class of h†, where we identify TF †(R) with (C×)d through

(ι11, . . . , ι1d). We then have the Shimura curve M†

K†= ShK†(H

†, X†) that is defined over ι11(F †) for an

open compact subgroup K† of H†(Afin). Similarly, we have the smooth morphism

ν† : M†K†→ L†

K†.

Moreover, h†(i) defines a complex structure on Vι1 , and hence Vι1 becomes a complex hermitian space

of dimension 2 that is isometric to its original complex hermitian space structure inherited from the

108

E-hermitian space V . In such a way, X† can be identified with the set of negative definite complex

lines in Vι1 . Therefore, X† is isomorphic to X as hermitian symmetric domains.

As in [Car1986, Section 2.2], H† is a group of symplectic similitude. In fact, let B† = B ⊗F F †,

and b 7→ b be the involution of the second kind on B†, which is the tensor product of the canonical

involution on B and the conjugation on F †. Consider the underlying Q-vector space V † of B†. Define

a symplectic form by

ψ†(v, w) = TrF †/Q

(√λTrB†/F †(vw)

)for v, w ∈ B†. Then H† can be identified with the group of B†-linear symplectic similitude of

(V †, ψ†) through the left action hv = v · h−1. In particular, H†(Qp) can be identified with the group

Q×p ×∏ri=1B

×pi . For every open compact subgroups K†,pp of

∏ri=2B

×pi , and K†,p of H†(Apfin), we simply

write M†0,K†,pp ,K†,p

for M†K†

, where K† = Z×p × O×Bp×K†,pp ×K†,p, and similarly for L†

0,K†,pp ,K†,p.

We let

MK;p = MK ×E Ep ; M†K†;p

= M†K†×F F †p1

1

LK;p = LK ×E Ep ; L†K†;p

= L†K†×F F †p1

1,

where F †p11

is naturally a subfield of Ep , which is identified with Fp. Since H and H† have the same

derived subgroup, which is also the derived subgroup of G, we have the follow result of Carayol.

Proposition 5.1.2 (Section 4 of [Car1986]). Let Kp ⊂ Hpfin (Notation 5.1.1) be an open compact

subgroup that is decomposable and sufficiently small. Then there is an open compact subgroup K†,pp ×

K†,p ⊂∏ri=2B

×pi ×H†(A

pfin), such that the geometric neutral components M0,Kp;p and M†,

0,K†,pp ,K†,p;p

are defined and isomorphic over E0p .

5.1.2 Moduli interpretations and integral models: minimal level

From the Hodge map h† (5.1), we have a Hodge filtration

0 ⊂ Fil0(V †C) = (V †C)0,−1 ⊂ V †C.

We define

t†(b) = tr(b;V †C/Fil0(V †C)) ∈ ι11(F †)

for b ∈ B†. For sufficiently small open compact subgroup K†, the curve M†K†

represents the following

functor in the category of locally noetherian ι11(F †)-schemes (cf. [Kot1992]): for every such scheme

109

S, M†K†

(S) is the set of equivalence classes of quadruples (A, θ, i, η), where

• A is an abelian scheme over S of dimension 4d;

• θ : A→ A∨ is a polarization;

• i : B† → End0S(A) is a monomorphism of Q-algebras, such that tr(i(b); LieS(A)) = t†(b) and

θ i(b) = i(b)∨ θ for all b ∈ B†;

• η is a K-level structure, that is, for chosen geometric point s on each connected component of

S, η is a π1(S, s)-invariant K†-orbit of B† ⊗Afin-linear symplectic similitude η : V † ⊗Afin →

Het1 (As,Afin), where the pairing on the latter space is the θ-Weil pairing.

Two quadruples (A, θ, i, η) and (A′, θ′, i′, η′) are equivalent if there is an isogeny A→ A′ that sends θ

to a Q×-multiple of θ′, i to i′, and η to η′.

Taking the base change by ι(p), we obtain the functor M†K†;p

over the completion [ι11(F †)]∧ι(p)∼= Fp.

For every element (A, θ, i, η) in M†K†;p

(S), LieS(A) is a B†p = B† ⊗Q Qp-module. Since the algebra

B†p = B ⊗F (F † ⊗Qp) decomposes as

B†p = B11 ⊕B1

2 ⊕ · · · ⊕B1r ⊕B2

1 ⊕B22 ⊕ · · · ⊕B2

r , (5.2)

where Bji = B†⊗F F †pjiis isomorphic to Bpi as an Fpi -algebra, the B†p-module LieS(A) decomposes as

LieS(A) =

(r⊕i=1

LieS(A)1i

)⊕(r⊕i=1

LieS(A)2i

),

and

Ap∞ =

(r⊕i=1

(Ap∞)1i

)⊕(r⊕i=1

(Ap∞)2i

),

for the p-divisible group Ap∞ of A. Since the involution b 7→ b on B†p interchanges the factors B1i and

B2i , by computing the trace, we see that the condition tr(i(b); LieS(A)) = t†(b) is equivalent to the

following:

tr(b ∈ B21 ; LieS(A)2

1) = TrB21/Fp

(b); LieS(A)2i = 0, i ≥ 2. (5.3)

Fix a maximal order Λ2i = OBpi

of B2i for each i = 1, . . . , r, and let Λ1

i be the dual of Λ2i . Then

Λp =

(r⊕i=1

Λ1i

)⊕(r⊕i=1

Λ2i

)⊂

(r⊕i=1

(V †p )1i

)⊕(r⊕i=1

(V †p )2i

)= V †p := V † ⊗Qp

110

is a Zp-lattice in V †p that is selfdual under ψ†. There is a unique maximal Z(p)-order O† ⊂ B† such that

O† = O†, and O†p2i

= OBpithat acts on Λ2

i , where O†p2i

is the B2i -component of O†⊗Z(p)

Zp ⊂ B†⊗QQp =

B†p under the decomposition (5.2). Then the functor M†0,K†,pp ,K†,p;p

is isomorphic to the following one

in the category of locally noetherian Fp-schemes: for every such scheme S, M†0,K†,pp ,K†,p;p

(S) is the

set of equivalence classes of quintuples (A, θ, i, ηp, ηpp) where


• θ : A→ A∨ is a prime-to-p polarization;

• i : O† → EndS(A)⊗ Z(p) such that (5.3) is satisfied, and θ i(b) = i(b)∨ θ for all b ∈ O†;

• ηp is a K†,p-level structure, that is, a π1(S, s)-invariant K†,p-orbit of B†⊗Apfin-linear symplectic

similitude ηp : V † ⊗Apfin → Het1 (As,A

pfin);

• ηpp is a K†,pp -level structure, that is, a π1(S, s)-invariant K†,pp -orbit of isomorphisms of O†-modules

ηpp :⊕r

i=2 Λ2i →

⊕ri=2 Het

1 (As,Zp)2i .

Two quintuples (A, θ, i, ηp, ηpp) and (A′, θ′, i′, (ηp)′, (ηpp)′) are equivalent if there is a prime-to-p isogeny

A→ A′ that carries θ to a Z×(p)-multiple of θ′, i to i′, ηp to (ηp)′, and ηpp to (ηpp)′.

We will extend the previous moduli functor to the category of locally noetherian schemes over

SpecOFpto construct an integral model of M†

0,K†,pp ,K†,p;p. Let us consider an abelian scheme (A, θ, i)

that is a part of the datum defined above, but with the scheme S over OFp. Through θ, we see that

(Ap∞)1i and (Ap∞)2

i are Cartier dual to each other. We replace (5.3) by the following condition:

tr(b ∈ OBp⊂ B2

1 ; LieS(A)21) = TrB2

1/Fp(b) ∈ OFp

; LieS(A)2i = 0, i ≥ 2. (5.4)

This means that the p-divisible group (Ap∞)2i is ind-etale for i ≥ 2. We let TpA = lim←−nA[pn] that is a

pro-scheme over S. It has an action by O†⊗Z(p)Zp. Then TpA(S)2

i is isomorphic to Λ2i as O†-modules

if S is connected and simply-connected.

We define a functor M†0,K†,pp ,K†,p

in the category of locally noetherian schemes over OFp: for every

such scheme S, M†0,K†,pp ,K†,p

(S) is the set of equivalence classes of quintuple (A, θ, i, ηp, ηpp) where

• (A, θ, i) is as in the last moduli problem but satisfies (5.4);

• ηp K†,p-level structure;

• ηpp is a π1(S, s)-invariantK†,pp -orbit of isomorphisms of O†-modules ηpp :⊕r

i=2 Λ2i →

⊕ri=2 TpA(s)2

i .

111

Two quintuples (A, θ, i, ηp, ηpp) and (A′, θ′, i′, (ηp)′, (ηpp)′) are equivalent if there exists a prime-to-p

isogeny A → A′ satisfying the same requirements in the last moduli problem. For sufficiently small

K†,pp ×K†,p, this moduli functor is represented by a regular scheme denoted by M†0,K†,pp ,K†,p

, which

is flat and projective over SpecOFp. By Proposition 5.1.2, we obtain a regular scheme M0,Kp that

is flat and projective over SpecOEp whose generic fiber is (isomorphic to) M0,Kp;p . Here, we also

need to use the fact that M†0,K†,pp ,K†,p

is stable for K†,p small, and the results in [DM1969, Section 1]

to make the descent argument. By construction, the neutral components of M†0,K†,pp ,K†,p

×OFpOE0

p

and M0,Kp ×OEpOE0

pare isomorphic.

We denote by (A, θ, i) (that is a part of the datum of) the universal object over M†0,K†,pp ,K†,p

. We

also denote by X† = (Ap∞)21 → M

†0,K†,pp ,K†,p

the universal p-divisible group with the action by OBp

and another action by∏ri=2B

×pi ×H†(A

pfin) that is compatible with the one on the underlying scheme

M†0,K†,pp ,K†,p

. We have also a p-divisible group X→M0,Kp with an action by Hpfin that is compatible

with the one on M0,Kp .

Remark 5.1.3. In fact, when p | 2 and Bp is division, the condition (5.4) is not enough. One needs

to impose that (Ap∞)21 is special (cf. [BC1991, Section II.2]) for geometric points of characteristic p.

Let us consider the case where ε(Vp) = 1, that is, Vp is isometric to V +; Bp is isomorphic to

Mat2(Fp); or U(Vp) is quasi-split. Before we proceed, we introduce some notations. Let R be a

(commutative) ring (with a unit) and M a (left) R-module (or p-divisible group according to the

context). Let m > 0 be an integer. We denote by M ] = Mm (arranged in a column) as a left

Matm(R)-module in the natural way. Conversely, for any left Matm(R)-module N , we denote by

N [ = eN the (left) R-module, where e = diag[1, 0, . . . , 0] ∈ Matm(R), and the action is given by

r.(en) = (e × diag[r, . . . , r]).n for r ∈ R and n ∈ N . It is easy to see that the pair of adjoint

functors (−],−[) induce an equivalence between the category of left R-modules and the category of

left Matm(R)-modules.

We identify Λ21 = OBp

with Mat2(OFp), and hence O×Bp

with GL2(OFp), respectively. By the above

discussion, we can replace the first part of (5.4) by the following one:

tr(b ∈ OFp; LieS(A)2,[

1 ) = b, (5.5)

in the moduli problem M†0,K†,pp ,K†,p

.

Consider a geometric point s : SpecF → M†0,K†,pp ,K†,p

of characteristic p and let O(s) be the

completion of the henselization of the local ring at s. By the theorem of Serre–Tate, it is the universal

112

deformation ring of (As, θs, is) that is the isomorphic to the one of (As,p∞ , θs, is). This is in turn

isomorphic to the deformation ring of the p-divisible group X†,[s = (As,p∞)2,[1 , which is an OFp

-module

of dimension 1 and height 2. Therefore, O(s) is isomorphic to OF 0

p[[t]]. We have the following result

of Carayol [Car1986, Section 6].

Proposition 5.1.4. The scheme M†0,K†,pp ,K†,p

(resp. M0,Kp) is smooth and projective over OFp(resp.

OEp ).

For a geometric point s of characteristic p of M†0,K†,pp ,K†,p

(resp. M0,Kp), there are two cases.

We say s is ordinary if the formal part of X†s (resp. Xs) is of height 1; supersingular if X†s (resp.

Xs) is formal. We denote by [M†0,K†,pp ,K†,p

]ss (resp. [M0,Kp ]ss) the supersingular locus of the scheme


(resp. M0,Kp).

5.1.3 Basic abelian scheme

In order to obtain the moduli interpretation of special cycles, we will construct construct a special

abelian scheme, which we name the basic abelian scheme. We fix an imaginary element µ in E, that

is, an element µ 6= 0 such that µτ = −µ. Since we are interested only in the place p, we identify the

following commutative diagram

ι1(E) // ι(p)(ι1(E))

ι11(F †) // ι(p)(ι11(F †))

3 S

ff

ι1(F )?

OO

-

;;

// ι(p)(ι1(F ))

=

88

with

E // Ep

F † // Fp

0 P

``

F?

OO

/

??

// Fp,

=

>>

where the closure is taken inside Cp.

Let E† = E ⊗F F † be a CM field of degree 4d that is a subalgebra of B† extending the fixed

embedding E → B. The involution on B† induces e 7→ e that fixes the maximal totally real subfield

113

contained in E†. The maps

ιi ⊗ ιji : E ⊗F F † → C⊗R C→ C; ι•i ⊗ ι

ji : E ⊗F F † → C⊗R C→ C

for i = 1, . . . , d and j = 1, 2 provide 4d different embeddings of E† into C, where C⊗RC→ C is the

usual multiplication. We choose a CM type

Φ = ι1 ⊗ ι11, ι1 ⊗ ι21; ιi ⊗ ι1i , ι•i ⊗ ι1i | i = 2, . . . , d

of E†. Then Φ determines a Hodge map h‡ : S → T ‡R, where T ‡ is the subtorus of ResE†/QGm,E†

consisting of elements e such that ee ∈ Gm,Q. We have the Shimura varieties M‡K‡

= ShK‡(T‡, h‡)

that parameterize abelian varieties over E†,Φ with Complex Multiplication by E† of type Φ. It is

finite and projective over SpecE†,Φ, where E†,Φ is the reflex field of (E†,Φ).

To be more precise, let V ‡ be the Q-vector space underlying E†. Define a symplectic form

ψ‡(v, w) := TrF †/Q

(√λTrE†/F †(vw)

)

for v, w ∈ E†. Then T ‡ can be identified with the group of E†-linear symplectic similitude of (V ‡, ψ‡),

and T ‡(Qp) can be identified with Q×p ×∏ri=1E

×pi . The Hodge map h‡ induces a filtration 0 ⊂

Fil0(V ‡C) ⊂ V ‡C such that

t‡(e) = tr(e;V ‡C/Fil0(V ‡C)) =∑ι∈Φ

ι(e)

for e ∈ E†. Since we have identified E (resp. F †) with its embedding through ι1 (resp. ι11), we can

identify E† with its embedding through ι1 ⊗ ι11, that is, with ι1(E) · ι11(F †) ⊂ C.

Lemma 5.1.5. The reflex field E†,Φ is E†.

Proof. By definition, E†,Φ is the field generated by the elements t‡(e) for all e ∈ E†. Let e =

(x+ yµ)× (x′ + y′λ′) be an element in E† with x, y, x′, y′ ∈ F . Then

t‡(e) = (x+ yµ)(2x′) +

d∑i=2

2ιi(x)(ιi(x′) + ιi(y

′)λ′)

= 2 TrF/Q(xx′) + 2 TrF/Q(xy′)λ′ + 2yx′µ− 2xy′λ′.

Therefore, E†,Φ = E†.

114

As before, the algebra E†p = E† ⊗Q Qp decomposes as

E†p =

(r⊕i=1

E1i

)⊕(r⊕i=1

E2i

)∼=

(r⊕i=1

Epi

)⊕(r⊕i=1

Epi

),

and also for its modules. Let π11 be the projection of E†p to the first factor E1

1 . The additive map

t‡ extends to a map t‡p : E†p → E†p. From the calculation in the above lemma, we find that for

(eji ) = (e11, . . . , e

1r; e

21, . . . , e

2r) ∈ E†p,

π11 t‡p((e

ji )) =

r∑i=1

TrEpi/Qp(e1

i ) + e11 + e2

1 − TrEp1/Fp1(e1

1). (5.6)

Let O‡ = E† ∩ O† be the unique maximal Z(p)-order in E† such that O‡ = O‡ and O‡p2i

= OEpiis

the ring of integers, where O‡p2i

is the projection of O‡⊗Z(p)Zp to the E2

i component. For any abelian

variety A over an Ep -scheme S, which is equipped with an action by O‡, LieS(A) is an E†p-module,

and hence decomposes as the direct sum of LieS(A)ji (i = 1, . . . , r, j = 1, 2). In view of (5.4) and

(5.6), we introduce the following trace condition

tr(e ∈ OEp⊂ E2

1 ; LieS(A)21) = ep ∈ Ep ; LieS(A)2

i = 0, i ≥ 2. (5.7)

Let

K‡ = Z×p ×r∏i=1

O×Epi×K‡,p

be an open compact subgroup of T ‡(Afin), and we denote M‡00,K‡,p

= M‡K‡

. Let M‡00,K‡,p;p

be its

base change under ι(p) (ι1 ⊗ ι11) : E† → Ep . We fix a sufficiently small compact subgroup K‡,p of

T ‡(Apfin), M‡00,K‡,p;p

represents the following functor in the category of locally noetherian schemes

over Ep : for every such scheme S, M‡00,K‡,p;p

(S) is the set of equivalence classes of quadruples

(A, ϑ, j, ηp), where


• ϑ : A→ A∨ is a prime-to-p polarization;

• j : O‡ → EndS(A)⊗ Z(p) such that (5.7) is satisfied, and ϑ j(e) = j(e)∨ ϑ for all e ∈ O‡;

• ηp is a K‡,p-level structure, that is, a π1(S, s)-invariant K‡,p-orbit of E†⊗Apfin-linear symplectic

similitude ηp : V ‡ ⊗Apfin → Het1 (As,A

pfin).

The notion of being equivalent is similarly defined as before. Moreover, we can extend this modular

115

functor to the category of locally noetherian schemes over OEp . We omit the detailed definition.

One can similarly prove that the extended moduli functor, denoted by M‡00,K‡,p

, is connected, and

finite, projective, smooth over SpecOEp . Therefore, it is isomorphic to SpecOE\ for some finite

unramified extension of local fields E\/Ep . We fix an embedding ι\ : E\ → E0p . Let (E, ϑ, j) be the

universal object over M‡00,K‡,p

×OE\,ι\ OE0

p∼= SpecOE0

p, and denote Y = (Ep∞)2

1. Fix a geometric

point s : OE0p→ C of characteristic 0 and an O‡-generator x of HBet

1 (Es,Z(p)), where HBet• is the

Betti homology. We call the quadruple (E, ϑ, j; x) a basic unitary datum.

In what follows, we fix a basic unitary datum (E, ϑ, j; x) once and for all. Since SpecOE0p

is simply

connected, x extends to a unique section xp of the lisse Apfin-sheaf Het1 (E,Apfin) over SpecOE0

p, and

determines canonically an element xpp of

⊕ri=2(TpE)2

i . For any scheme S over SpecOE0p

, we denote

by xpS and xpp,S for the corresponding base change respectively.

Let (E, ϑE, jE) be the special fiber of (E, ϑ, j), where E is an abelian variety over SpecF. Let

Y = (Ep∞)21 and iY : OEp → End(Y) be the induced OEp -action. Therefore, Y is the special fiber

of Y, which is an OFp-module over SpecF.

5.1.4 The nearby space

In this and the next subsections, we assume that p is nonsplit in E, and fix an F-point s in the

supersingular locus of the common neutral component of M†0,K†,pp ,K†,p

and M0,Kp , which corresponds

to a quintuple (A, θA, iA, ηp, ηpp). Let

V † = Mor ((E, jE), (A, iA))⊗Q, 1

which is an E†-vector space of dimension 2. The map

Mor ((E, jE), (A, iA))×Mor ((E, jE), (A, iA))→ O‡

sending (x, y) to

(x, y)′ := j−1E ϑ−1

E y∨ θA x

induces a E†-hermitian form on V †. If we let (A0, θA0 , iA0) be the isogeny class of (A, θA, iA), then

B† = End(A0, iA0) is a quaternion algebra over F †. Moreover, the underlying F †-quadratic spaces

of V † and B† are isometric. The hermitian form (−,−)′ induces a Q-symplectic form on V †. If we

let H† be the corresponding group of symplectic similitude, then Aut(A0, θA0 , iA0) can be identified

1Here (A, iA) really means (A, iA | O‡), and we apply the similar convention in what follows.

116

with H†(Q). We fix an E-subspace V of V † of dimension 2 that is stable under the action of H†der(Q)

and such that the restricted hermitian form (−,−)′|V takes value in E. Then V † = V ⊗E E†, and Vv

is isometric to Vv exactly for v away from ι1, p. Let H = ResF/QU(V ). We fix an isometry

γp = (γpp , γp) : V ⊗F Ap

F,fin → V ⊗F ApF,fin

such that

%∗(xpp) ∈ ηpp(γpp(%)); %∗(x

p) ∈ ηp(γp(%))

for every element % ∈ V . In particular, γp identifies Hpfin with Hp

fin.

5.1.5 Integral special subschemes: minimal level

For every admissible x ∈ V ⊗ Afin,E , we have the subscheme Z(x)K on MK that is a special cycle

introduced in 3.1.2. For K = Kp,0Kp, let us consider the curve M0,Kp;p , which is the base change

MK ×E Ep , and its subscheme Z(x)0,Kp;p , which is the corresponding base change of Z(x)K .

Consider x† ∈ V † and h† ∈ H†(Afin), such that

• the (V †p )21-component of h†,−1x† is inside Λ2

1; and

• K†h†,−1x† ∩ V ⊗Afin,E (inside V † ⊗Afin) contains a K-orbit that has totally positive definite

norm in E. Here, K† = Z×p × O×Bp× K†,pp × K†,p and K = Kp,0K

p for Kp as in Proposition

5.1.2.

We define a functor Z†(x†, h†)00,K†,pp ,K†,p

in the category of locally noetherian schemes over OE0p

as

follows: for every such scheme S, Z†(x†, h†)00,K†,pp ,K†,p

(S) is the set of equivalence classes of sextuples

(A, θ, i, ηp, ηpp , %A) where

• (A, θ, i, ηp, ηpp) is an element of M†0,K†,pp ,K†,p

(S);

• %A : E×SpecOE0

pS → A is a quasi-homomorphism that satisfies the following conditions:

1. For any e ∈ O‡, the following diagram commutes:

E×SpecOE0

pS

%A //

j(e)

A

iA(e)

E×SpecO

E0pS

%A // A;

117

2. %A induces a homomorphism from Y×SpecOE0

pS to (Ap∞)2

1;

3. For the geometric point s defining the K†,p-level structure, the map ρAs,∗ : Het1 (Es,A

pfin)→

Het1 (As,A

pfin) sends xps into ηp(h†,−1x†);

4. The map ρAs,∗ :⊕r

i=2(TpEs)2i →

⊕ri=2(TpAs)

2i ⊗OFpi

Fpi sends xpp,s into ηpp(h†,−1x†).

The equivalence relations are defined in the similar way. The evident morphism

Z†(x†, h†)00,K†,pp ,K†,p

→M†0,K†,pp ,K†,p

×OEpOE0

p

is finite and its image is a 1-dimensional closed subscheme that is stable under the action of the

Galois group Gal(OE0p/OEp ). Therefore, Z†(x†, h†)0

0,K†,pp ,K†,pdefines a 1-dimensional closed sub-

scheme Z†(x†, h†)0,K†,pp ,K†,p of M†0,K†,pp ,K†,p

, which only depends on the orbit K†h†,−1x†. Moreover,

by definition, the intersection Z†(x†, h†)0,K†,pp ,K†,p ∩M0,Kp;p inside M

†,0,K†,pp ,K†,p

coincides with the

special cycle Z(x)0,Kp;p if x ∈ K†h†,−1x† has totally positive definite norm in E. Conversely, for

every admissible x ∈ V ⊗AF,fin with xp ∈ Λp, we have a 1-dimensional closed subscheme Z(x)0,Kp of

M0,Kp obtained by the Hecke translation of Z†(x†, h†)0,K†,pp ,K†,p ∩M†,0,K†,pp ,K†,p

, whose generic fiber is

Z(x)0,Kp;p . It only depends on the orbits Kp,0Kpx.

From now on, we assume that (p is nonsplit in E) and ε(Vp) = 1. Following [Car1986, 11], we have

the following isomorphisms of sets:

[M†0,K†,pp ,K†,p

]ss(F) ∼= H†(Q)\

(Z×

r∏i=2

B×pi/K†,pp × H†(A

pfin)/K†,p

);

[M0,Kp ]ss(F) ∼= H(Q)\Hpfin/K

p,

such that in both cases, the neutral double coset corresponds to the point s. Moreover, we have the

following lemma.

Lemma 5.1.6. The special fiber [Z†(x†, h†)0,K†,pp ,K†,p ]sp (resp. [Z(x)0,Kp ]sp) locates in the supersin-

gular locus [M†0,K†,pp ,K†,p

]ss (resp. [M0,Kp ]ss).

Proof. We only need to prove for [Z†(x†, h†)0,K†,pp ,K†,p ]sp. Let s = (A, θ, i, ηp, ηpp , %A) be an F-point

of Z†(x†, h†)0,K†,pp ,K†,p . We have a nontrivial homomorphism between OFp-modules %A,∗ : Y →

(Ap∞)21 = X†s

∼= (X†,[s )⊕2. Therefore, there is at least one projection (X†,[s )⊕2 → X†,[s whose composi-

tion with %A,∗ is nontrivial. Since Y is formal, X† is formal. The lemma follows.

118

5.1.6 Remark on the case F = Q

We briefly explain the constructions in the above four subsections in the case F = Q. Therefore,

E/Q is a imaginary quadratic extension. Let ι, ι• be two different embeddings of E into C such that

ι(p) ι1 : E → Cp induces the place p. We identify E with a subfield of C via the embedding ι. If

p is split in E, we denote p• the other place of E above p. We have the hermitian space V over E of

dimension 2 and signature (1, 1), the unitary group H over Q. For a sufficiently small open compact

subgroup K ⊂ H(Afin), the Shimura curve ShK(H,X) is a smooth and quasi-projective curve defined

over ι(E), and is proper if and only if V is anisotropic.

By the Hasse principle, there is a unique up to isometry quaternion algebra B over Q, such that

B, as an Q-quadratic space (of dimension 4), is isometric to V as an F -quadratic space with the

quadratic form TrE/Q(−,−), where (−,−) is the hermitian form on V . We identify two quadratic

spaces B and V through a fixed isometry and hence V has both left and right multiplication by B.

We fix an embedding E → B, through which the action of E induced from the left multiplication of

B coincides with the E-vector space structure of V . We let H† = B×, which is different from the

case F 6= Q. Then similarly, we have the Shimura curve ShK†(H†, X†) defined over Q. We can view

V as a symplectic space over Q and H† the group of E-linear symplectic similitude.

Let

Sh(H)n,Kp;p = ShKp,nKp(H,X)×E Ep ; Sh(H†)n,K†,p;p = ShK†p,nK†,p(H†, X†)×Q Qp.

We have the following proposition that is parallel to Proposition 5.1.2.

Proposition 5.1.7. Let Kp ⊂ Hpfin be an open compact subgroup that is decomposable and sufficiently

small. Then there is an open compact subgroup K†,p ⊂ H†(Apfin), such that the geometric neutral

components Sh(H)0,Kp;p and Sh(H†)0,K†,p;p are defined and isomorphic over E0p .

We are going to define a functor M†0,K†,p

in the category of locally noetherian schemes over Zp.

There are two case: the anisotropic case, i.e., B is division, and the isotropic case, i.e., B is isomorphic

to the matrix algebra. In the anisotropic case, for every Zp-scheme S, define M†0,K†,p

(S) to be the set

of equivalence classes of quadruples (A, θ, i, ηp) where

• A is an abelian surface over S;

• θ : A→ A∨ is a prime-to-p polarization;

• ι : OB → EndS(A) of a monomorphism of rings such that det(ι(b); LieS(A)) = Nm b;

119

• ηp is a K†,p-level structure, that is, a π1(S, s)-invariant K†,p-orbit of B⊗Apfin-linear symplectic

similitude ηp : V ⊗Apfin → Het1 (As,A

pfin).

In the isotropic case, for every Zp-scheme S, define M†0,K†,p

(S) to be the set of equivalence classes of

pairs (A[, η[,p) where A[ is a generalized elliptic curve over S, and η[,p is a K†,p-level structure (cf. [K-

M1985]). In both cases, the equivalence relation is described by prime-to-p isogenies, and M†0,K†,p

is

represented by a smooth and projective scheme M†0,K†,p

over Zp. Thus we obtain a scheme M0,Kp that

is smooth and projective over SpecOEp whose generic fiber is (the Baily–Borel compactification of)

Sh(H)0,Kp;p . By construction, the neutral components of M†0,K†,p

×Zp OE0p

and M0,Kp ×OEpOE0

p

are isomorphic. In the isotropic case, we will denote A =(A[)]

, ηp =(η[,p)]

when A[ is an elliptic

curve. Moreover, we have the canonical polarization θ : A → A∨ and the OB = Mat2(Z)-action

i : OB → EndS(A).

When F = Q, in the basic unitary datum (E, ϑ, j; x), E is an elliptic curve over SpecOE0p

with the

principal polarization ϑ and the OE-action j : OE → EndOE0

p(E). Let Y = (Ep∞). Let (E, ϑE, jE) be

the special fiber of (E, ϑ, j), where E is an elliptic curve over SpecF. Let Y = Ep∞ and iY : OEp →

End(Y) be the induced OEp -action. Therefore, Y is the special fiber of Y.

We now assume that p is nonsplit in E. For every admissible x ∈ V ⊗ Afin with xp ∈ Λp,

we can similarly define a 1-dimensional closed subscheme Z(x)0,Kp of M0,Kp , whose generic fiber is

Z(x)0,Kp;p . It only depends on the orbits Kp,0Kpx. In the isotropic case, Z(x)0,Kp is disjoint from

the set of cusps. If we further assume that ε(Vp) = 1, then we have the following isomorphism of sets:

[M0,Kp ]ss(F) ∼= H(Q)\Hpfin/K

p,

such that the neutral double coset corresponds to the point s. Moreover, we have that the special

fiber [Z(x)0,Kp ]sp locates in the supersingular locus [M0,Kp ]ss.

5.2 Local intersection numbers

In this section, we study the formal scheme N and its special formal subschemes Z (x). Therefore, p

will be a finite place of F that is nonsplit in E and such that ε(Vp) = 1.

5.2.1 p-adic uniformization of supersingular locus

Recall that we have fixed an F-point s on the common neutral component of M†0,K†,pp ,K†,p

and M0,Kp ,

which corresponds to a quintuple (A, θA, iA, ηp, ηpp). Let X = (Ap∞)2

1 and iX : Mat2(OFp)→ End(X)

120

be the induced Mat2(OFp)-action. Then X[ is a formal OFp

-module of dimension 1 and height 2.

We define a functor N in the category of schemes over SpecOF 0p

where $ is locally nilpotent: for

every such scheme S, N (S) is the set of equivalence classes of pairs (G, ρG) where

• G is an OFp-module over S of dimension 1 and height 2;

• ρG : X[×SpecF Ssp → G×S Ssp is a quasi-isogeny of height 0 (which is in fact an isomorphism).

Here, Ssp = S ×SpecOF0p

SpecF.

Two pairs (G, ρG) and (G′, ρG′) are equivalent if there is an isomorphism G′ → G sending ρG to

ρG′ . Then N is represented by the formal scheme N , which is isomorphic to Spf RFp,2, where

RFp,2 = OF 0

p[[t]]. Let N ′ = N ×O

F0p

OE0

p.

By the theorem of Serre–Tate, the formal completion of M†0,K†,pp ,K†,p

(resp. M0,Kp) at s is canon-

ically isomorphic to N (resp. N ′). Therefore, if we denote by [M†0,K†,pp ,K†,p

]∧ss (resp. [M0,Kp ]∧ss) the

formal completion along the supersingular locus, then we have the following p-adic uniformization:

[M†0,K†,pp ,K†,p

]∧ss ×OFpOF 0

p

∼= H†(Q)\

(N × Z×

r∏i=2


pfin)/K†,p

);

[M0,Kp ]∧ss ×OEpOE0

p∼= H(Q)\N ′ × Hp

fin/Kp.

Such uniformization is a special case of those considered in [RZ1996].

5.2.2 Special formal subschemes

Recall that we have the E†-hermitian space V † and a E-subvector space V . The obvious morphism

Mor ((E, jE), (A, iA))→ Mor ((Y, jY), (X, iX))

in fact identifies the later OEp -module as the maximal lattice Λ− in Vp ∼= V −. For every

x ∈ Mor ((Y, jY), (X, iX))reg := Mor ((Y, jY), (X, iX))− 0,

we define a subfunctor Z (x) of N as follows: for every scheme S in the previously mentioned

category, Z (x)(S) is the set of equivalence classes of (G, ρG) ∈ N (S) such that the following composed

homomorphism

(Y×SpecO

F0p

SpecF

)×SpecF Ssp = Y ×SpecF Ssp

x−→ X×SpecF Sspρ]G−−→ G] ×S Ssp

121

extends to a homomorphism Y ×SpecOF0p

S → G]. Then Z (x) is represented by a closed formal

subscheme Z (x) of N . In fact, one can show that it is a relative divisor of N by the same argument

in [KR2011, Proposition 3.5]. We will use the same notation for the base change of Z (x) in N ′.

Let φ = φ0∞ ⊗ (⊗v∈Σfin

φv) such that

• φpfin := ⊗v∈Σfin−pφv is in S(V (ApF,fin))K

p

;

• φp = φ0p is the characteristic function of the a the self-dual lattice Λ+ of Vp;

• φ(0) = 0.

Let g = (gv) ∈ H ′(AF ) such that gp ∈ n(bp)K′p for some unipotent element n(bp) ∈ N ′(Fp). Consider

the generating series

Zφ(g) =∑

x∈K\V (AF,fin)−0

(ωχ(g)φ) (x)Z(x)K

=∑


ψp(bpT (x))(φ0p ⊗ (ωχ(gp)φp)

)(x)Z(x)Kp,0Kp .

Define

Zφ(g)0,Kp =∑



)(x)Z(x)0,Kp ,

whose generic fiber is the base change of Zφ(g) to Ep . Moreover, its completion at the F-point s is

[Zφ(g)0,Kp ]∧s =∑x∈V


)(x)Z (x). (5.8)

Here, φ0p is the characteristic function of Mor ((Y, jY), (X, iX)), and in particular, the notation Z (x)

makes sense since φ(0) = 0.

5.2.3 A formula for local intersection multiplicity

In this and the next subsections, we will assume that p is not divided by 2 and is inert in E. For every

pair (x1, x2) ∈ Mor ((Y, jY), (X, iX))2reg that are linearly independent, the formal divisors Z (x1) and

Z (x2) intersects properly at the unique closed point of N . We would like to calculate the intersection

number Z (x1) ·Z (x2). Assuming that (y1, y2) = (x1, x2)g for some g ∈ GL2(OEp ) such that (y1, y2)

122

has the moment matrix

T ((y1, y2)) =

$a

$b

with a nonnegative even and b positive odd. If we denote by Def(X[, (x1, x2)) the subring of Def(X[) =

OF 0

p[[t]] where (x1, x2) deforms, then

Z (x1) ·Z (x2) = lengthOF0p

Def(X[, (x1, x2)) = lengthOF0p

Def(X[, (y1, y2)) = Z (y1) ·Z (y2).

Therefore, we only need to study the intersection number Z (y1) ·Z (y2).

Let Yτ be the unique (up to isomorphism) formal OFp-module of dimension 1 and height 2 with

an OEp action jYτ , such that

• As OFp-modules, there is an isomorphism Y ∼= Yτ (and we fix such an isomorphism); and

• jYτ is given by the composition OEpτ−→ OEp

jY−−→ End(Y) ∼= End(Yτ ).

By [KR2011, Lemma 4.2], there is an isomorphism

ρX : Y ×Yτ → X

that commutes with OEp -actions, such that as elements of HomOEp(Y,Y ×Yτ ),

ρ−1X yα =

incα Πa α = 1

incα Πb α = 2,

where

• incα (α = 1, 2) denotes the inclusion of Y into the α-th component of Y ×Yτ ∼= Y ×Y; and

• Π is a fixed uniformizer of the division algebra End(Y).

In what follows, we will identify X with Y ×Yτ via the isomorphism ρX.

For an integer s ≥ 0, let Fs be a quasi-canonical lifting of level s, which is an OFp-module over

Spf OF sp

, unique up to the Galois action (cf. [Gro1986]). Therefore, it defines a morphism Spf OF sp→

N that is a closed immersion. Let Zs be the divisor of N defined by the image, which is independent

of Fs we choose. We have the following proposition generalizing [KR2011, Proposition 8.1] from Qp

to Fp.

123

Proposition 5.2.1. As divisors on N , we have

Z (y1) =

a∑s=0even

Zs; Z (y2) =

b∑s=1odd

Zs.

Proof. The original proof of [KR2011, Proposition 8.1] works again for one direction. Namely,

a∑s=0even

Zs ≤ Z (y1);

b∑s=1odd

Zs ≤ Z (y2).

To prove the other direction, we need only to prove that the intersection multiplicities of both sides

in two cases with the special fiber Nsp = Spf F[[t]] are the same. For the left-hand side, we have

a∑s=0even

Zs ·Nsp =

a∑s=0even

[O×Fp: UsFp

] =qa+1 − 1

q − 1;

b∑s=1odd

Zs ·Nsp =

b∑s=1odd

[O×Fp: UsFp

] =qb+1 − 1

q − 1,

where q is the cardinality of the residue field of Fp. Then the assertion follows from the following

proposition that generalizes [KR2011, Proposition 8.2].

Proposition 5.2.2. For y ∈ HomOEp(Y,Y ×Yτ ), the intersection multiplicity

Z (y) ·Nsp =qv+1 − 1

q − 1,

where v ≥ 0 is the valuation of (y, y)′, i.e., (y, y)′ ∈ $vO×Fp.

We keep the assumptions and notations in the above subsection. The results in [ARG2007] cited

in the proof of [KR2011, Proposition 8.4] also work for Fp, not just Qp. Therefor, for 0 < s ≤ b odd,

we have

Z (y1) ·Zs =

qa+1−1q−1 a < s;

qs−1q−1 + 1

2 (a+ 1− s)[O×Fp: UsFp

] a ≥ s.

Summing over s, we get the following local arithmetic Siegel-Weil formula at a good finite place.

Theorem 5.2.3. For every pair (x1, x2) ∈ Mor ((Y, jY), (X, iX))2reg that are linearly independent,

the intersection multiplicity Z (x1) · Z (x2) depends only on the GL2(OEp )-equivalence class of the

124

moment matrix T = T ((x1, x2)). Moreover, if

T ∼

$a

$b

0 ≤ a < b,

then we have

Hp(T ) := Z (x1) ·Z (x2) =1

2

a∑l=0

ql(a+ b+ 1− 2l),

where q is the cardinality of the residue field of Fp.

5.2.4 Proof of Proposition 5.2.2

We generalize the proof of [KR2011, Proposition 8.2] to the case Fp 6= Qp, still by using the theory of

windows and displays of p-divisible groups developed by T. Zink in [Zin2001, Zin2002]. In the proof,

we simply write F = Fp, E = Ep . Moreover, we let e and f be the ramification index and the

extension degree of residue fields of F/Qp, respectively. In particular, q = pf . Let R = F[[t]] and

A = W [[t]], where W = W (F) is the Witt ring. We extend the Frobenius automorphism σ on W to

A by letting σ(t) = tp. For any s ≥ 1, we let Rs = R/ts and As = A/ts. Then A (resp. As) is a

frame of R (resp. Rs). The category of formal p-divisible groups over R is equivalent to the category

of pairs (M,α) where

• M is a free A-module of finite rank; and

• α : M → M (σ) := Aσ ⊗AM is an A-linear injective homomorphism such that cokerα is a free

R-module.

The baby case would be the one where f = 1. Consider the p-divisible group Y over F of dimension

1 and (absolute) height 2e with action by OE . It corresponds to the pair (N, β), where

N = N0 ⊕N1 = OF 0n0 ⊕ O

F 0n1

is the Z/2-graded free OF 0 = OF ⊗Zp W -module of rank 2 (that is a free W -module of rank 2e),

and β(n0) = $n1, β(n1) = n0. We extend the Frobenius automorphism on W to OF 0 OF -linearly.

Similarly as in the proof of [KR2011, Proposition 8.2], the p-divisible group X = Y × Yτ over F

corresponds to (M,α) described there, and its universal deformation is (M,αt). The only difference

is that we should replace p by $. The rest of the proof follows in the same way.

125

Now we treat the general case and hence assume that f ≥ 2. Consider the p-divisible group Y

over F. It corresponds to the pair (N, β), where N is a Z/2-graded free OF ⊗Zp W -module of rank 2.

Since

OF ⊗Zp W =

f−1⊕j=0

O(σj)

F 0:=

f−1⊕j=0

OF ⊗W (k),σj W,

where k is the residue field of F , we can write

N =

f−1⊕j=0

O(σj)

F 0e0,j

⊕f−1⊕j=0

O(σj)

F 0e1,j

,

and

• β(ei,j) = ei,j+1 for i = 1, 2, 0 6 j < f − 1;

• β(e0,f−1) = e1,0;

• β(e1,f−1) = $e0,0.

Similarly, the p-divisible group Yτ corresponds to (Nτ , βτ ), which we write as

Nτ =

f−1⊕j=0

O(σj)

F 0eτ0,j

⊕f−1⊕j=0

O(σj)

F 0eτ1,j

,

and

• βτ (eτi,j) = eτi,j+1 for i = 0, 1, 0 6 j < f − 1;

• βτ (eτ1,f−1) = eτ0,0;

• βτ (eτ0,f−1) = $eτ1,0.

We extend (N, β) (resp. (Nτ , βτ )) to F[[t]] by scalar, which we still denote by the same notations.

The p-divisible group X corresponds the the direct sum (M,α) := (N, β) ⊕ (Nτ , βτ ). Under the

basis

e0,0, eτ1,0, . . . , e0,f−1, e

τ1,f−1; e1,0, e

τ0,0, . . . , e1,f−1, e

τ0,f−1,

126

the matrix of α is

α =

$

$

1

. . .

1

1

. . .

1

.

Let (M,αt) corresponds to the universal deformation of (X, iX) over F[[t]]. Then under the same

basis,

αt =

1 t

1 −t

1

. . .

1

1

. . .

1

· α.

Explicitly, we have

αt(ei,j) = ei,j+1 i = 0, 1 and j = 0, . . . , f − 2;

αt(e0,f−1) = e1,0 − teτ1,0;

αt(e1,f−1) = $e0,0;

αt(eτi,j) = eτi,j+1 i = 0, 1 and j = 0, . . . , f − 2;

αt(eτ1,f−1) = eτ0,0 + te0,0;

αt(eτ0,f−1) = $eτ1,0.

127

If we denote by σk(α) : M (σk) → M (σk+1) the induced homomorphism for k > 0. Then formally, we

have

σk(α)−1(ei,j) = ei,j−1 i = 1, 2 and j = 1, . . . , f − 1;

σk(α)−1(e0,0) =1

$e1,f−1;

σk(α)−1(e1,0) = e0,f−1 +tpk

$eτ0,f−1;

σk(α)−1(eτi,j) = eτi,j−1 i = 1, 2 and j = 1, . . . , f − 1;

σk(α)−1(eτ1,0) =1

$eτ0,f−1;

σk(α)−1(eτ0,0) = eτ1,f−1 −tpk

$e1,f−1.

Now let y correspond to the graded A1-linear homomorphism γ : N ⊗A A1 → M . Then the length

Z (y) ·Nsp of the deformation space of γ is the maximal number a such that there exists a diagram

N

γ

β // N (σ)

γ(σ)

M

αt // M (σ),

which commutes modulo ta, and γ lifts γ.

Case i: v = 2r is even. We may assume that γ = $rinc1 that is represented by the following

4f × 2f matrix

X(0) =

$r 0 · · · 0

0 0 · · · 0

0 $r · · · 0

0 0 · · · 0

......

. . ....

0 0 · · · $r

0 0 · · · 0

.

If r = 0, in order to lift γ to γ mod tp, we search for a 4f × 2f matrix X(1) with entries in Ap

such that X(1) ≡ X(0) in A1 and satisfies

αt X(1) = σ(X(1)) β.

128

But σ(X(1)) = σ(X(0)) = X(0). Therefore, we need to find the largest integer a ≤ p such that

α−1t X(0) β has integral entries mod ta. Since the entry at the place (e0,f−1, e

τ0,f−1) is t

$ , the

largest a is just 1. It follows that when v = r = 0, the proposition holds.

If r > 0, we first show that we can lift γ to γ mod tq2r

. By induction, we introduce X(k) for

k ≥ 1, i.e., the one satisfying X(k + 1) ≡ X(k) in Apk , and

αt X(k + 1) = σ(X(k + 1)) β.

Since σ(X(k + 1)) = σ(X(k)), we formally have

X(k + 1) = α−1t σ(X(k)) β.

We need to show that

X(2rf) = α−1t σ(αt)

−1 · · · σ2rf−1(αt)−1 X(0) β2rf

has integral entries. Let xi,j;i′,j′ (resp. xτi,j;i′,j′) be the entry of X(2rf) mod $ at the place (ei,j , ei′,j′)

(resp. (ei,j , eτi′,j′)). Then among all these terms, the only nonzero terms are

xτ0,j;0,j = (−1)r−1tpf−1−j(q2r−2+q2r−3+···+1) j = 0, . . . , f − 1;

x1,j;1,j = (−1)rtpf−1−j(q2r−1+q2r−2+···+1) j = 0, . . . , f − 1,

which implies that we can lift γ to γ mod tq2r

. Next, we consider the lift of γ to γ mod tpq2r

.

Therefore, we consider the matrix

X(2rf + 1) = α−1t σ(X(2rf)) β.

It has exactly one entry that is not integral: the place (e0,f−1, eτ0,f−1), whose non-integral part is

t

$(−1)rtp·p

f−1(q2r−1+q2r−2+···+1) =(−1)r

$tq

2r+q2r−1+···+1.

It turns out that the length Z (y) ·Nsp is exactly q2r+1−1q−1 = qv+1−1

q−1 .

Case ii: v = 2r+ 1 is odd. We may assume that γ = $sinc2 Π, where Π is the endomorphism

of Y determined by Π(e0,j) = e0,j and Π(e1,j) = $e0,j for j = 0, . . . , f − 1. Then γ is represented by

129

the following 4f × 2f matrix

0 · · · 0

$s+1 · · · 0

.... . .

...

0 · · · 0

0 · · · $s+1

0 · · · 0

$s · · · 0

.... . .

...

0 · · · 0

0 · · · $s

.

Similarly, we introduce the matrix Y (k) for k ≥ 0. We first show that γ can be lifted to γ

mod tq2s+1

, i.e., the matrix

Y ((2s+ 1)f) = α−1t σ(αt)

−1 · · · σ(2s+1)f−1(αt)−1 Y (0) β(2s+1)f

has integral entries. Let yi,j;i′,j′ (resp. yτi,j;i′,j′) be the entry of Y ((2s + 1)f) mod $ at the place

(ei,j , ei′,j′) (resp. (ei,j , eτi′,j′)). Then among all these terms, the only nonzero terms are

yτ0,j;0,j = (−1)stpf−1−j(q2s−1+q2s−2+···+1) j = 0, . . . , f − 1;

y1,j;1,j = (−1)s+1tpf−1−j(q2s+q2s−1+···+1) j = 0, . . . , f − 1,

which implies that we can lift γ to γ mod tq2s+1

. Next we consider the lift of γ to γ mod tpq2s+1

.

Therefore, we consider the matrix

Y ((2s+ 1)f + 1) = α−1t σ(Y ((2s+ 1)f)) β.

It has exactly one entry which is not integral: the place (e0,f−1, eτ0,f−1), whose non-integral part is

t

$(−1)s+1tp·p

f−1(q2s+q2s−1+···+1) =(−1)s+1

$tq

2s+1+q2s+···+1.

130

It turns out that the length Z (y) ·Nsp is exactly q2s+2−1q−1 = qv+1−1

q−1 . Therefore, the proposition is

proved.

5.3 Comparison

5.3.1 Non-archimedean Whittaker integrals

We calculate certain Whittaker integrals WT (s, g,Φ) and their derivatives (at s = 0) at a non-

archimedean place when T is of rank 2.

Assume that E/F is unramified and p > 2. We fix a selfdual OE-lattice Λ+ in V + and let

φ0+ ∈ S(V +) (resp. Φ0+ ∈ S((V +)2)) be the characteristic function of Λ+ (resp. (Λ+)2). Let ψ

be the unramified character of F . For T ∈ Her2(E) that is nonsingular, we consider the Whittaker

integral

WT (s, g,Φ0+) =

∫Her2(E)

ϕΦ0+,s(wn(u)g)ψT (n(u))−1du (5.9)

for Re s > 1, where du is the selfdual measure with respect to ψ. Write g = n(b)m(a)k under the

Iwasawa decomposition of H ′′. Then

(5.9) =

∫Her2(E)

(ω′′1(wn(u)n(b)m(a)k)Φ0+

)(0)λP (wn(u)n(b)m(a)k)sψ(− trTu)du

= ψ(trTb)

∫Her2(E)

λP (wn(u)m(a))sψ(− trTu)du

= ψ(trTb)|det a|1−sE WtaτTa(s, e,Φ0+).

Therefore, we only need to consider the integral WT (s, e,Φ0+). If T is not in Her2(OE), then

WT (s, e,Φ0+) is identically 0. For T ∈ Her2(OE), it is well-known (e.g., [Kud1997, Appendix]) that

for an integer r > 1, WT (r, e,Φ0+) = γV +αF (12+r, T ), where γV + is the Weil constant and αF is the

classical representation density (for hermitian matrices). By [Hir1999], we see that for r ≥ 0,

αF (12+r, T ) = PF (12, T ; (−q)−r)

for a polynomial PF (12, T ;X) ∈ Q[X]. By analytic continuation, we see that

WT (s, e,Φ0+) = γV +PF (12, T ; (−q)−s).

131

If ord(detT ) is odd, i.e., T can not be represented by V +, then WT (0, e,Φ0+) = PF (12, T ; 1) = 0.

Taking derivative at s = 0, we have

W ′T (0, e,Φ0+) = −γV + log q · d

dXPF (12, T ;X) |X=1 .

Moreover, we have the following results.

Proposition 5.3.1 (Hironaka, [Hir1999]). Suppose that T is GL2(OE)-equivalent to diag[$a, $b] with

0 ≤ a < b. Then

PF (12, T ;X) = (1 + q−1X)(1− q−2X)

a∑l=0

(qX)l

(a+b−2l∑k=0

(−X)k

).

Corollary 5.3.2. If a+ b is odd, then

W ′T (0, e,Φ0+) = γV +b2(0)−1 log q · 1

2

a∑l=0

ql(a+ b− 2l + 1).

5.3.2 Comparison on Shimura curves

In this subsection, we calculate the local height paring 〈Zφ1(g1), Zφ2(g2)〉p at a finite place p of E

that is good. Recall that we have a (compactified) Shimura curve MK constructed from the hermitian

spaceV, and assume that K is sufficiently small and decomposable. We also assume that φα (α = 1, 2)

are decomposable and K = KpKp-invariant.

Let S ⊂ Σfin be a finite subset of cardinality at least 2, such that for every place p ∈ Σfin − S, we

have

• p - 2, p is inert or split in E;

• ε(Vp) = 1;

• φα,p = φ0p (α = 1, 2) are the characteristic function of a selfdual lattice Λp = Λ+ ⊂ Vp;

• Kp = Kp,0 is the subgroup of U(Vp) stabilizing Λp, i.e., Kp is a hyperspecial maximal compact

subgroup;

• χ and ψ are both unramified at p.

We say a finite place p of E is good if it is not lying over some place in S. Assume that φα(0) = 0.

Consider the generating series Zφα(gα) for α = 1, 2. Write gα,p = n(bα,p)m(aα,p)kα,p in the Iwasawa

decomposition, and choose some element eα ∈ E× such that e−1α aα,p ∈ O×Ep

. Let gα = m(e−1α )gα, and

132

we have gα,p = n(bα,p)kα,p in the Iwasawa decomposition. Then Zφα(gα) = Zφα(gα). As in 5.2.2, we

have the series Zφα(gα)0,Kp for α = 1, 2. We let Zφα(gα) = Zφα(gα)0,Kp . The following is the main

theorem of this chapter.

Theorem 5.3.3. Suppose that φ1,v⊗φ2,v ∈ S(V2v)reg for at least one place v ∈ S, and gα ∈ P ′vH ′(AvF )

for α = 1, 2. Let p be a finite place that is not in S, MK;p = M0,Kp the smooth local model introduced

in 5.1.2, and Zφα(gα) the series introduced above. Then we have

1. If p is nonsplit in E, then

Ep(0, ı(g1, g∨2 ), φ1 ⊗ φ2) = −Vol(K)〈Zφ1

(g1), Zφ2(g2)〉p ,

where

• by definition,

〈Zφ1(g1), Zφ2

(g2)〉p = log q2(Zφ1(g1) · Zφ2

(g2));

• Ep(0, ı(g1, g∨2 ), φ1 ⊗ φ2) is defined in (2.14); and

• Vol(K) is the defined in Definition 4.3.3.

2. if p is split in E, then

Zφ1(g1) · Zφ2

(g2) = 0.

Combining this with Theorem 4.3.4, we have the following corollary.

Corollary 5.3.4. Assume that

• φα = φ0∞φα,fin (α = 1, 2) are decomposable as above;

• φ1,S ⊗ φ2,S is in S(V2S)reg, i.e., φ1,v ⊗ φ2,v ∈ S(V2

v)reg for every v ∈ S;

• φ1,v ⊗ φ2,v ∈ S(V2v)reg,dv for every v ∈ S that is nonsplit and some dv ≥ dψv (cf. 2.4.2);

• gα ∈ eSH ′(ASF ) (α = 1, 2);

• the local model MK;p is M0,Kp for all finite places p above p that are not in S.

Then

E′(0, ı(g1, g∨2 ), φ1 ⊗ φ2) = −Vol(K)

∑v|vv 6∈S

〈Zφ1(g1), Zφ2(g2)〉v ,

133

where the Green functions used in archimedean places are those defined in (4.55), which are not the

admissible Green functions in the sense in 3.4.1.

Proof of Theorem 5.3.3. 1. By Lemma 5.1.6, the special fiber of Zφα(gα) locates in the supersin-

gular locus [M0,Kp ]ss. If we denote by [Zφα(gα)]∧sp (α = 1, 2) the completion along the special

fiber, then

Zφ1(g1) · Zφ2

(g2) = [Zφ1(g1)]∧sp · [Zφ2

(g2)]∧sp = [Zφ1(g1)0,Kp ]∧sp · [Zφ2

(g2)0,Kp ]∧sp. (5.10)

Let hi (i = 1, . . . , l) be a set of representatives of the double coset [M0,Kp ]ss(F) ∼= H(Q)\Hpfin/K

p.

Then

(5.10) =

l∑i=1

[Zωχ(hi)φ1(g1)0,Kp ]∧s · [Zωχ(hi)φ2

(g2)0,Kp ]∧s , (5.11)

where we recall that(ωχ(hi)φα

)(x) = φα(h−1

i x). By (5.8),

(5.11) =

l∑i=1

∑x1∈V

ψp(b1,pT (x1))(φ0p ⊗

(ωχ(gp1 , hi)φ

p1

))(x1)Z (x1)

·

∑x2∈V

ψp(b2,pT (x2))(φ0p ⊗

(ωχ(gp2 , hi)φ

p2

))(x2)Z (x2)

=

∑(x1,x2)∈(V ∩Mor((Y,jY),(X,iX)))

2

ψp(tr bpT ((x1, x2)))

l∑i=1

(ω′′χ(ı(gp1 , g

p,∨2 )

) (φp1 ⊗ φ

p2

))(h−1i (x1, x2))Z (x1) ·Z (x2), (5.12)

where

bp =

b1,p 0

0 b2,p

.

Let xT be a representative of the pairs (x1, x2) such that T ((x1, x2)) = T . Then

(5.12) =∑

T∈Her2(Ep )∩GL2(OEp )

ψp(tr bpT )∑

h∈Hpfin/K

p


p,∨2 )

) (φp1 ⊗ φ

p2

))(h−1xT )Hp(T ).

By Theorem 5.2.3, Corollary 5.3.2, and following the same steps in the proof of Theorem 4.3.4,

134

we obtain that

−Vol(K) log q2(Zφ1(g1) · Zφ2(g2)) = Ep(0, ı(g1, g∨2 ), φ1 ⊗ φ2). (5.13)

Let

e =

e1

e2

.

By definition,

(5.13) =∑

Diff(T,V)=p

W ′T (0, ı(g1,p, g∨2,p), φ1,p ⊗ φ2,p)

∏v 6=p

WT (0, ı(g1,v, g∨2,v), φ1,v ⊗ φ2,v)

=∑

Diff(T,V)=p

W ′eτTe(0, ı(g1,p, g∨2,p), φ1,p ⊗ φ2,p)

∏v 6=p

WeτTe(0, ı(g1,v, g∨2,v), φ1,v ⊗ φ2,v)

=∑

Diff(T,V)=p

W ′T (0, ı(g1,p, g∨2,p), φ1,p ⊗ φ2,p)

∏v 6=p

WT (0, ı(g1,v, g∨2,v), φ1,v ⊗ φ2,v)

= Ep(0, ı(g1, g∨2 ), φ1 ⊗ φ2).

Therefore, (1) is proved.

2. It will be proved in a more general context in Lemma 6.1.1.

135

Chapter 6

Comparison at finite places: bad

reduction

In this chapter, we discuss the local height paring 〈Zφ1(g1), Zφ2

(g2)〉p on certain model MK at every

bad place p ∈ S. We will assume that

• φα = φ0∞φα,fin (α = 1, 2) are decomposable;

• φα,S ∈ S(VS)reg (α = 1, 2);

• φ1,S ⊗ φ2,S is in S(V2S)reg;

and gα ∈ eSH ′(ASF ) for α = 1, 2.

In 6.1, we discuss the contribution of the local height paring at a finite place p in S that is split in

E. In 6.2, we discuss the contribution of the local height paring at a finite place p in S that is nonsplit

in E and such that ε(Vp) = 1. In 6.3, we discuss the contribution of the local height paring at a finite

place p in S that is nonsplit in E and such that ε(Vp) = −1.

136

6.1 Split case

In this section, we discuss the contribution of the local height paring at a finite place p in S that is

split in E. Let p be a place in Σfin lying over p.

6.1.1 Integral models and ordinary reduction

Let K = KpKp be an open compact subgroup of H(Afin) with Kp sufficiently small and Kp = Kp,n

for n ≥ 0. Therefore, MK = Mn,Kp . In 5.1.2, we construct a smooth integral model M0,Kp for

M0,Kp;p on which there is a p-divisible group X. Then X[ → M0,Kp is an OFp-module of dimension

1 and height 2. Recall that a Drinfeld $n-structure for an OFp-module X of dimension 1 and height

2 over an OFp-scheme S is an OFp

-homomorphism

αn : (OFp/$nOFp

)2 → X[$n](S)

such that the image forms a full set of sections of X[$n] in the sense of [KM1985, 1.8]. Let Mn,Kp =

M0,Kp(n) be the universal scheme over M0,Kp of the Drinfeld $n-structure (cf. [HT2001, Lemma

II.2.1]). Then Mn,Kp is regular and finite over M0,Kp , whose generic fiber is Mn,Kp . Let M ′n,Kp =

Mn,Kp ×FpFnp be the base change, and M′n,Kp the normalization of Mn,Kp ×OFp

OFnp that is regular.

We denote by [M′n,Kp ]ord the ordinary locus, which is an open subscheme of the special fiber [M′n,Kp ]sp

and also the smooth locus.

The set of connected components of [M′n,Kp ]sp corresponds canonically to the set of geometric

connected components of Mn,Kp , hence to E×,1\A×,1E /ν(K). The set of irreducible components on

each connected component of [M′n,Kp ]sp, that is, the Igusa curves, corresponds to the set P(Vp)/Kp,n.

Here, P(Vp) is the set of all rank 1 Ep∼= Fp ⊕ Fp-submodules in Vp, where U(Vp) acts from right by

l.h = h−1l for l ∈ P(Vp) and h ∈ U(Vp). Together, the set of irreducible components of [M′n,Kp ]sp can

be identified with

Ign,Kp = P(Vp)/Kp,n ×(E×,1\A×,1E /ν(K)

).

We consider special cycles. We keep the same notations for the base change of special cycles Z(x)K

and the generating series Zφα(gα) on M ′n,Kp . We let Z(x)K (resp. Zφα(gα)) be the Zariski closure of

Z(x)K (resp. Zφα(gα)) in M′n,Kp . Since p is split in E, the special fiber [Zφα(gα)]sp is contained in

the ordinary locus [M′n,Kp ]ord. Let P(V )+ be the set of totally positive definite E-lines in V . Then

137

the set of geometric special points of Mn,Kp (also of Mn,Kp and M ′n,Kp) can be identified with

SpK = H(Q)\P(V )+ ×H(Afin)/K =∐

l∈H(Q)\P(V )+

Hl(Q)\H(Afin)/K,

and the set [M′n,Kp ]ord(F) can be identified with

∐l∈H(Q)\P(V )

Hl(Q)\((Nl\U(Vp)/Kp,n)×Hp

fin/Kp),

where Nl ⊂ U(Vp) is the unipotent radical of the parabolic subgroup stabilizing l. The reduction map

SpK → [M′n,Kp ]ord(F)→ Ign,Kp (6.1)

is given by

(l, h) 7→ (l, hp, hp) 7→ (h−1

p l, ν(hphp))

(cf. [Zha2001b, 5.4] for a discussion).

6.1.2 Coherence for intersection numbers

We compute the local height paring on the integral model M′n,Kp . Write Zφα = Zφα(gα) + Vφα(gα)

for some divisor Vφα(gα) supported on the special fiber as in 3.4.3. We have

(log q)−1〈Zφ1(g1), Zφ2

(g2)〉p

= (Zφ1(g1) + Vφ1(g1)) · (Zφ2(g2) + Vφ2(g2)− E(g2, φ2)ωK + E(g2, φ2)ωK)

= Zφ1(g1) · (Zφ2(g2) + Vφ2(g2)− E(g2, φ2)ωK) + E(g2, φ2)(Zφ1(g1) + Vφ1(g1)) · ωK)

= Zφ1(g1) · Zφ2

(g2) + Zφ1(g1) · Vφ2

(g2) + E(g2, φ2)Vφ1(g1) · ωK , (6.2)

where q is the cardinality of the residue field of Fp. First, we have the following simple lemma.

Lemma 6.1.1. Under the weaker hypotheses that φ1,v ⊗ φ2,v ∈ S(V2v)reg = S(V 2

v )reg and gα ∈

P ′vH′(AvF ) (α = 1, 2) for some finite place v other than p, Zφ1

(g1) and Zφ2(g2) do not intersect.

Proof. It follows immediately from the first map in (6.1).

Second, we define a function ν(•, φ2, g2) on Vp−0 in the following way. For any nonzero x ∈ Vp,

let lx be the line spanned by x that is an element in P(Vp). Set ν(x, φ2, g2) to be the coefficient of the

138

geometric irreducible component represented by (lx, 1) in Ign,Kp in Vφ2(g2). It is a locally constant

function and

ν(•, φ1,p;φ2, g2) =Vol(detK)

Vol(K)φ1,p ⊗ ν(•, φ2, g2)

extends to a function in S(Vp) such that ν(0, φ1,p;φ2, g2) = 0 since φ1,p(0) = 0. Then the intersection

number

Zφ1(g1) · Vφ2

(g2) =∑

x∈K\V⊗Afin

(ωχ(g1)φ1) (x)Z(x)K · Vφ2(g2) (6.3)

=∑

x∈K\V⊗Afin

T (x)∈F+

Vol(K)

Vol(K ∩H(Afin)x)

(ν(•, φ1,p;φ2, g2)⊗

(ωχ(gp1)φp1

))(x) (6.4)

since g1 ∈ epH ′(ApF ). If we let

E(s, g, ν(•, φ1,p;φ2, g2)⊗ φp1) =∑

γ∈P ′(F )\H′(F )

(ωχ(γg)

(ν(•, φ1,p;φ2, g2)⊗ φp1

))(0)λP ′(γg)s−

12

be an Eisenstein series that is holomorphic at s = 12 . Then we have

(6.3) = E(s, g1, ν(•, φ1,p;φ2, g2)⊗ φp1) |s= 12−W0(

1

2, g1, ν(•, φ1,p;φ2, g2)⊗ φp1)

by the standard Siegel–Weil argument, which is similar to Proposition 3.4.1. For simplicity, we write

E(p)(φ1, g1;φ2, g2) = log q

(E(s, g1, ν(•, φ1,p;φ2, g2)⊗ φp1) |s= 1

2−W0(

1

2, g1, ν(•, φ1,p;φ2, g2)⊗ φp1)

).

Finally, we let

A(p)(g1, φ1) = log q (Vφ1(g1) · ωK) .

Then we have the following proposition.

Proposition 6.1.2. Suppose that φα,S ∈ S(VS)reg (α = 1, 2), φ1,S ⊗ φ2,S is in S(V2S)reg, and gα ∈

eSH′(AS

F ) for α = 1, 2. Then

〈Zφ1(g1), Zφ2

(g2)〉p = E(p)(φ1, g1;φ2, g2) +A(p)(g1, φ1)E(g2, φ2).

139

6.2 Quasi-split case


nonsplit in E and such that ε(Vp) = 1. Let p be the only place in Σfin lying over p. We identify Vp

with Mat2(Fp) and the lattice Λ+ with Mat2(OFp).

6.2.1 Integral models and supersingular reduction

Let K = KpKp be an open compact subgroup of H(Afin) with Kp sufficiently small and Kp = Kp,n for

n large such that Fnp contains Ep . Therefore, MK = Mn,Kp . In 5.1.2, we construct a smooth integral

model M0,Kp for M0,Kp;p on which there is a p-divisible group X. Let Mn,Kp be the normalization

of M0,Kp in Mn,Kp , which is regular and finite over M0,Kp . Consider the base change M ′n,Kp =

Mn,Kp ×Ep Fnp . Let M′n,Kp be the normalization of Mn,Kp ×OEp

OFnp that is a regular mode of

M ′n,Kp . We have the following description of the supersingular locus

[M′n,Kp ]ss(F) ∼= H(Q)\(E×,1p /ν(Kp,n)× Hp

fin/Kp),

where H(Q) acts on the first factor through the determinant.

We denote by Xuniv the p-divisible group over N ∼= Spf RFp,2. Let RFp,2,n (cf. [HT2001, Lemma

II.2.2]) be such that SpecRFp,2,n = (SpecRFp,2)(n) is the universal scheme of the Drinfeld $n-

structure for Xuniv,[ that is in fact defined over SpecRFp,2. Let SpecR′n be the normalization of

Spec

(RFp,2 ⊗O

F0p

OFnp

)in Spec

(RFp,2,n ⊗O

F0p

Fnp

). The set of connected components of SpecR′n is

parameterized by coset O×Fp/ν(K†p,n). Let N ′

n be the neutral connected component of Spf R′n, which

is finite over N ′ ×OE0

pOFnp

. Its generic fiber N ′n,η is Galois over N ′

η = Spf RFp,2 ⊗OF0p

Fnp with the

Galois group

Kp,n := ker

(GL(Λ+,[/$nΛ+,[)

∧2

−−→ GL(OFp/$nOFp

)

),

which fits into the following exact sequence

1 // Kp,n// Kp,0/Kp,n

ν // E×,1p /ν(Kp,n) // 1 .

140

We define the universal p-divisible group X′n over N ′n by the following Cartesian diagram

X′n

// N ′n

Xuniv ×O

F0p

OFnp

// N ×OF0p

OFnp.

Moreover, we have the universal Drinfeld $n-structure

α′n,η : Λ+,[/$nΛ+,[ → X′[n,η[$n](N ′n,η)

for X′[n,η. In particular, we have the following p-adic uniformization for the completion of M′n,Kp along

the supersingular locus

[M′n,Kp ]∧ss∼= H(Q)\

(N ′n × E

×,1p /ν(Kp,n)× Hp

fin/Kp).

In 5.1.5, we have defined a 1-dimensional closed subscheme Z(x0)0,Kp of M0,Kp whose generic fiber

is Z(x0)0,Kp;p , for a Kp,0Kp-orbit Kp,0K

px0 in V (AF,fin) that is admissible. Define Z′(x0)n,Kp by

the following Cartesian diagram

Z′(x0)n,Kp

// M′n,Kp

Mn,Kp ×OEp

OFnp

Z(x0)0,Kp ×OEp

OFnp// M0,Kp ×OEp

OFnp .

It is easy to see that the Kp,nKp-orbits inside Kp,0K

px0 is parameterized by the finite group Kp,n. For

every such orbit Kp,nKpx, we define Z(x)n,Kp to be the union of irreducible components of Z′(x0)n,Kp

whose generic fiber contributes to the special divisor Z(x)n,Kp ×Ep Fnp on the generic fiber M ′n,Kp .

In 5.2.2, we have defined Z (x) that is a formal divisor of N , for x ∈ Mor ((Y, jY), (X, iX))reg.

141

Define Z ′(x)n by the following Cartesian diagram

Z ′(x)n

// N ′n

Z (x)×O

F0p

OFnp

// N ×OF0p

OFnp.

If we denote by Y the completion of Y along the special fiber, we have a universal homomorphism

[x] : Y×Spf OFp

Z ′(x)n → X′n |Z ′(x)n

of p-divisible groups over the formal scheme Z ′(x)n. Pick up any geometric point z of characteristic

0 on Z ′(x)n and choose a similitude X′n,z∼= Λ+ of OFp

-symplectic modules that induces the universal

Drinfeld $n-structure α′n,η at the point s. Then [x]∗(xp) defines a K†p,n-orbit in Λ+ ⊂ Vp such that

Ht(x) = Ht(x) for every x in the orbit. Here,

• xp is induced from x in the fixed unitary data;

• Ht(x) is the integer h (≥ 0) such that 12 TrEp/Fp

(x, x)′ ∈ $hO×Fp;

• Ht(x) is the largest integer h (≥ 0) such that detx ∈ $hO×Fp.

Therefore, we have the following decomposition

Z ′(x)n =⋃

x∈K†p,n\Λ+

Ht(x)=Ht(x)

Z (x, x)n

into (finitely many) formal divisors of N ′n .


φv) such that


p

;

• φp is in S(Vp)reg ∩ S(Vp)K†p,n , and its support is contained in Λ+.

In particular, φ(0) = 0. Consider the generating series

Zφ(epgp) =

∑x∈Kp,nKp\V (AF,fin)−0

(φp ⊗ (ωχ(gp)φp)) (x)Z(x)Kp,nKp .

142

Define

Zφ(epgp)n,Kp =


(φp ⊗ (ωχ(gp)φp)) (x)Z(x)n,Kp ,

whose generic fiber is the base change of Zφ(epgp) to Fnp . Moreover, its completion at the closed point

sn of N ′n is

[Zφ(epgp)n,Kp ]∧sn =

∑x∈V

∑x∈Kp,n\Λ+

(x,x)=(x,x)′

(ωχ(gp)φp) (x)φp(x)Z (x, x)n. (6.5)


Define the subset

Redqsn = (x1, x2; x1, x2) ⊂

(Λ+)2 ×Mor ((Y, jY), (X, iX))

2reg

by the conditions that

• Ht(xα) = Ht(xα) for α = 1, 2;

• K†p,nx1 and K†p,nx2 are linearly independent.

Then by definition, for (x1, x2; x1, x2) ∈ Redqsn , the formal divisors Z (x1, x1)n and Z (x2, x2)n will

intersect properly. We let

m(x1, x2; x1, x2) = Z (x1, x1)n ·Z (x2, x2)n

be the intersection multiplicity, which is a well-defined continuous function on Redqsn . The following

lemma is straightforward.

Lemma 6.2.1. Suppose that for α = 1, 2, φα,p ∈ S(Vp)reg ∩ S(Vp)K†p,n whose support is contained in

Λ+ and such that φ1,p ⊗ φ2,p is in S(V 2p )reg. Then the following function

µ(x1, x2;φ1,p ⊗ φ2,p) =∑

x1∈Kp,n\Λ+

(x1,x1)=(x1,x1)′

∑x2∈Kp,n\Λ+

(x2,x2)=(x2,x2)′

(φ1,p ⊗ φ2,p) (x1, x2)m(x1, x2; x1, x2)

that is a priori defined on Mor ((Y, jY), (X, iX))2reg is a Schwartz function in S(V 2

p ) via extension by

zero.

143

Let hp,j ∈ Kp,0 (j = 1, . . . ,m) be a set of representatives of the coset Kp,n\Kp,0/Kp,n, and define

µ(x1, x2;φ1,p ⊗ φ2,p) =

m∑j=1

µ(x1, x2;ω′′χ(hp,j) (φ1,p ⊗ φ2,p)

),

whose value does not depend on the representatives we choose.

We return to our original assumption for this chapter that φα,S ∈ S(VS)reg (α = 1, 2), φ1,S⊗φ2,S is

in S(V2S)reg, and gα ∈ eSH ′(AS

F ) for α = 1, 2. Choose some element eα ∈ E× such that ωχ(m(eα))φα,p

is supported on Λ+ for α = 1, 2. We have

Zφα(gα) =∑

xα∈K\Vfin

(ωχ(gα)φα) (xα)Z(xα)K

=∑

xα∈K\Vfin

(ωχ(gα)φα) (xαeα)Z(xαeα)K

=∑

xα∈K\Vfin

(ωχ(gα)φα) (xαeα)Z(xα)K .

Therefore, we can add one more assumption that φα,p is in S(Vp)reg ∩ S(Vp)K†p,n , and its support is

contained in Λ+. We let Zφα(gα) = Zφ(epgpα)n,Kp and denote by [Zφ(gα)]∧sp its completion along the

special fiber that is contained in the supersingular locus [M′n,Kp ]ss. We have a similar decomposition

as (6.2).

First, we consider

Zφ1(g1) · Zφ2(g2) = [Zφ1(g1)]∧sp · [Zφ2(g2)]∧sp = [Zφ1(epgp1)n,Kp ]∧sp · [Zφ2(epg

p2)n,Kp ]∧sp. (6.6)

Let hpi ∈ Hpfin (i = 1, . . . , l) be a set of representatives of the double coset H(Q)\Hp

fin/Kp.

Then

(6.6) =

l∑i=1

m∑j=1

[Zωχ(hp,j hpi )φ1

(epgp1)n,Kp ]∧sn · [Zωχ(hp,j h

pi )φ2

(epgp2)n,Kp ]∧sn . (6.7)

144

By (6.5),

(6.7) =

l∑i=1

m∑j=1

∑x1∈V

∑x1∈Kp,n\Λ+

(x1,x1)=(x1,x1)′

(ωχ(gp1)φp1

)(hp,−1i x1)φ1,p(h−1

p,jx1)Z (x1, x1)n

·

∑x2∈V

∑x2∈Kp,n\Λ+

(x2,x2)=(x2,x2)′

(ωχ(gp2)φp2

)(hp,−1i x2)φ2,p(h−1

p,jx2)Z (x2, x2)n

=

l∑i=1

∑(x1,x2)∈V 2


p,∨2 )

) (φp1 ⊗ φ

p2

))(hp,−1i (x1, x2))µ(x1, x2;φ1,p ⊗ φ2,p). (6.8)

We define

Φhor =

l∑i=1

(ω′′χ(hpi )

(φp1 ⊗ φ

p2

))⊗ µ(•;φ1,p ⊗ φ2,p)

that is a function in S(V (AF )2). Then

(6.8) =∑

(x1,x2)∈V 2


p,∨2 )

)Φhor

)((x1, x2)) .

We define the following theta series

θhor(p)(•;φ1, φ2) = log q

∑(x1,x2)∈V 2

(ω′′χ(•)Φhor

)((x1, x2)) ,

where q is the cardinality of the residue field of Ep . Then in summary, we have the following lemma.

Lemma 6.2.2. Under the previous assumption, we have

log q (Zφ1(g1) · Zφ2

(g2)) = θhor(p)(ı(g1, g

∨2 );φ1, φ2).

Second, we consider

Zφ1(g1) · Vφ2(g2) = [Zφ1(g1)]∧sp · Vφ2(g2) = [Zφ1(epgp1)n,Kp ]∧sp · Vφ2(g2). (6.9)

145

Let hpi ∈ Hpfin (i = 1, . . . , l) and hp,j ∈ Kp,0 (j = 1, . . . ,m) be as above. Then

(6.9) =

l∑i=1

m∑j=1


(epgp1)n,Kp ]∧sn · Vωχ(hp,j h

pi )φ2

(g2)

=

l∑i=1

m∑j=1

∑x1∈V

∑x1∈Kp,n\Λ+

(x1,x1)=(x1,x1)′

(ωχ(gp1)φp1

)(hp,−1i x1)φ1,p(h−1

p,jx1)Z (x1, x1)n · Vωχ(hp,j hpi )φ2

(g2).

(6.10)

The function

ν(x1;φ1,p, φ2, g2) =

m∑j=1

∑x1∈Kp,n\Λ+

(x1,x1)=(x1,x1)′

φ1,p(h−1p,jx1)Z (x1, x1)n · Vωχ(hp,j)φ2

(g2),

which is originally defined for x1 ∈ Mor ((Y, jY), (X, iX))reg, can be extended by zero to a function

in S(Vp). Then

(6.10) =

l∑i=1

∑x1∈V

(ωχ(gp1)φp1

)(hp,−1i x1)ν(x1;φ1,p, φ2, g2). (6.11)

We define

φver =

l∑i=1

(ωχ(hpi )φ

p1

)⊗ ν(•;φ1,p, φ2, g2)

that is a function in S(V (AF )). Then

(6.11) =∑x1∈V

(ωχ(gp1)φver

)(x1).


θver(p)(•;φ1, φ2, g2) = log q

∑x1∈V

(ωχ(•)φver) (x1).

Then in summary, we have the following lemma.


log q (Zφ1(g1) · Vφ2(g2)) = θver(p)(g1;φ1, φ2, g2)

that is a theta series for g1 ∈ epH ′(ApF ).

146

Finally, we let

A(p)(g1, φ1) = log q (Vφ1(g1) · ωK) .

Then in summary, we have the following proposition.

Proposition 6.2.4. For φ1,S ⊗ φ2,S ∈ S(V2S)reg and gα ∈ eSH ′(AS

F ) (α = 1, 2),

〈Zφ1(g1), Zφ2(g2)〉p = θhor(p)(ı(g1, g

∨2 );φ1, φ2) + θver

(p)(g1, φ1, φ2, g2) +A(p)(g1, φ1)E(g2, φ2).

6.3 Nonsplit case


nonsplit in E and such that ε(Vp) = −1. Let p be the only place in Σfin lying over p. We identify Vp

with Bp, the unique division quaternion algebra over Fp, and the lattice Λ− with OBp.

6.3.1 Integral models and Cerednik–Drinfeld uniformization: minimal lev-

el

Let K = KpKp be an open compact subgroup of H(Afin) with Kp sufficiently small and Kp = Kp,n.

In this subsection, we study the case where n = 0. In 5.1.2, we introduce the integral model M0,Kp

defined over SpecOEp , whose generic fiber is M0,Kp;p . We fix an F-point s on the common neutral

component of M†0,K†,pp ,K†,p

and M0,Kp , which corresponds to a quintuple (A, θA, iA, ηp, ηpp). Repeat

the process of 5.1.4, we obtain the E-hermitian space V with a fixed isometry γp = (γpp , γp) : V ⊗F

ApF,fin → V ⊗F Ap

F,fin, and the reductive group H = ResF/QU(V ).

Let X = Xs = (Ap∞)21 and iX : OBp

→ End(X) be the induced OBp-action. Then (X, iX)

is a special formal OBp-module over SpecF (cf. [Dri1976, BC1991, KR2000]). In particular X is of

dimension 2 and height 4. Then Aut(X, iX) acts on Mor ((Y, jY), (X, iX)) and can be identified with

GL2(OFp). For every integer h, we define a functor Ωh in the category of schemes over SpecOF 0

p

where $ is locally nilpotent: for every such scheme S, Ωh(S) is the set of equivalence classes of pairs

(X, ρX) where

• X is a special formal OBp-module over S;

• ρX : X×SpecFSsp → X×SpecFSsp is a quasi-isogeny of height h of special formal OBp-modules.

According to Drinfeld [Dri1976], this functor is represented by a formal scheme Ωh over Spf OF 0

p. Let

147

Ω = Ω01 and Ω′ = Ω×OF0p

OE0

p.

If we denote by [M†0,K†,pp ,K†,p

]∧sp (resp. [M0,Kp ]∧sp) the formal completion along the special fiber,

then we have the following Cerednik–Drinfeld uniformization (cf. [Dri1976,BC1991,RZ1996]):

[M†0,K†,pp ,K†,p

]∧sp ×OFpOF 0

p

∼= H†(Q)\

(∐h∈Z

Ωh ×r∏i=2


pfin)/K†,p

);

[M0,Kp ]∧sp ×OEpOE0

p∼= H(Q)\Ω′ × Hp

fin/Kp.

In fact, the above isomorphisms underly the corresponding uniformization for the universal p-divisible

groups. For example, let Xuniv → Ω and X′ → Ω′ be the universal special formal OBp-modules,

respectively. Then we have the following commutative diagram:

[X]∧sp ×OEpOE0

p

∼ // H(Q)\X′ × Hpfin/K

p

[M0,Kp ]∧sp ×OEp

OE0

p

∼ // H(Q)\Ω′ × Hpfin/K

p.

For every nonzero element x ∈ Mor ((Y, jY), (X, iX)), we define a subfunctor Z (x) of Ω0 as

follows: for every scheme S in the previously mentioned category, Z (x)(S) is the set of equivalence

classes of (X, ρX) ∈ Ω0(S) such that the following composed homomorphism

(Y×SpecO

F0p

SpecF

)×SpecF Ssp = Y ×SpecF Ssp

x−→ X×SpecF SspρX−−→ X ×S Ssp

extends to a homomorphism Y ×SpecOF0p

S → X. Then Z (x) is represented by a closed formal

subscheme Z (x) of Ω0. We denote by Z (x)hor the horizontal part of the associated divisor of Z (x),

which is empty if (x, x) = 0.

6.3.2 Integral models and Cerednik–Drinfeld uniformization: higher level

Let K = KpKp be an open compact subgroup of H(Afin) with Kp sufficiently small and Kp = Kp,n

for n large such that Fnp contains Ep . Therefore, MK = Mn,Kp .

Let Ωn,η = Xunivη [$n] − Xuniv

η [$n−1] be the etale covering over the generic fiber Ωη, viewed as

a rigid space, with the Galois group K†p,0/K†p,n = (OBp

/$nOBp)×. Let Ωn be the normalization

of Ω ×OF0p

OFnp

in Ωn,η ×F 0pFnp , whose set of connected components is parameterized by the coset

1This is not to be confused with the differential form Ω in Chapter 4

148

O×Fp/ν(K†p,n). Let Ω′n be the neutral connected component of Ωn, which is finite over Ω′ ×O

E0p

OFnp

.

The formal scheme Ω′n is not regular but has double points. We replace Ω′n by blowing up all its

double points and denote by the same notation. It is easy to see that

(Mn,Kp ×E Fnp

)rig ∼= H(Q)\(

Ω′n,η × E×,1p /ν(Kp,n)× Hp

fin/Kp).

The group H(Q) acts on Ω′n by the universal property of normalization and blowing-up of double

points. The quotient

H(Q)\(

Ω′n × E×,1p /ν(Kp,n)× Hp

fin/Kp)

is regular, flat and projective over Spf OFnp

(for sufficiently small Kp). By Grothendieck Existence

Theorem, we have a regular scheme Mn,Kp that is flat and projective over SpecOFnp , and a morphism

pn : Mn,Kp →M0,Kp ×OEpOFnp , such that the following diagram commutes:

[Mn,Kp ]∧sp

[pn]∧sp

∼ // H(Q)\(

Ω′n × E×,1p /ν(Kp,n)× Hp

fin/Kp)

[M0,Kp ]∧sp ×OEpOFnp

∼ //(H(Q)\Ω′ × Hp

fin/Kp)×O

E0p

OFnp.

Define Z ′(x)n by the following Cartesian diagram

Z ′(x)n

// Ω′n

Z (x)hor ×O

F0p

OFnp

// Ω×OF0p

OFnp.

By the similar argument as in 6.2.1, we have the following decomposition

Z ′(x)n =⋃

x∈K†p,n\Λ−

Ht(x)=Ht(x)

Z (x, x)n

into (finitely many) formal divisors of Ω′n.

In 5.1.5, we have defined a 1-dimensional closed subscheme Z(x0)0,Kp of M0,Kp whose generic fiber

is Z(x0)0,Kp;p , for a Kp,0Kp-orbit Kp,0K

px0 in V (AF,fin) that is admissible. Define Z′(x0)n,Kp by

149

the following Cartesian diagram

Z′(x0)n,Kp

// Mn,Kp

Z(x0)hor

0,Kp ×OEpOFnp

// M0,Kp ×OEpOFnp ,

where Z(x0)hor0,Kp denotes the horizontal part of the associated divisor. For every orbit Kp,nK

px inside

Kp,0Kpx0, we define Z(x)n,Kp to be the union of irreducible components of Z′(x0)n,Kp whose generic

fiber contributes to the special divisor Z(x)n,Kp ×Ep Fnp on the generic fiber Mn,Kp ×E Fnp .


φv) such that


p

;

• φp is in S(Vp)reg ∩ S(Vp)K†p,n , and its support is contained in Λ−.

In particular, φ(0) = 0. Consider the generating series

Zφ(epgp) =


(φp ⊗ (ωχ(gp)φp)) (x)Z(x)Kp,nKp .

Define

Zφ(epgp)n,Kp =


(φp ⊗ (ωχ(gp)φp)) (x)Z(x)n,Kp ,

whose generic fiber is the base change of Zφ(epgp) to Fnp . Moreover, its completion along the neutral

component is (the image of)

∑x∈V

∑x∈Kp,n\Λ+

(x,x)=(x,x)′

(ωχ(gp)φp) (x)φp(x)Z (x, x)n. (6.12)


Let Mor ((Y, jY), (X, iX))reg ⊂ Mor ((Y, jY), (X, iX)) be the subset consisting of x with (x, x) 6= 0.

Define the subset

Rednsn = (x1, x2; x1, x2) ⊂

(Λ−)2 ×Mor ((Y, jY), (X, iX))

2reg

by the conditions that

150

• Ht(xα) = Ht(xα) for α = 1, 2;

• K†p,nx1 and K†p,nx2 are linearly independent.

Then by definition, for (x1, x2; x1, x2) ∈ Rednsn , the formal divisors Z (x1, x1)n and Z (x2, x2)n will

intersect properly. We let

m(x1, x2; x1, x2) = Z (x1, x1)n ·Z (x2, x2)n

be the intersection multiplicity, which is a well-defined continuous function on Rednsn . The following

lemma is straightforward.

Lemma 6.3.1. Suppose that for α = 1, 2, φα,p ∈ S(Vp)reg ∩ S(Vp)K†p,n whose support is contained in

Λ− and such that φ1,p ⊗ φ2,p is in S(V 2p )reg. Then the following function

µ(x1, x2;φ1,p ⊗ φ2,p) =∑

x1∈Kp,n\Λ−(x1,x1)=(x1,x1)′

∑x2∈Kp,n\Λ−

(x2,x2)=(x2,x2)′

(φ1,p ⊗ φ2,p) (x1, x2)m(x1, x2; x1, x2)

that is a priori defined on Mor ((Y, jY), (X, iX))2reg is a Schwartz function in S(V 2

p ) via extension by

zero.

The remaining discussion follows similarly as in 6.2.2. We sketch the process. Let hp,j ∈ Kp,0

(j = 1, . . . ,m) be similarly defined previously. Let

µ(x1, x2;φ1,p ⊗ φ2,p) =

m∑j=1

µ(x1, x2;ω′′χ(hp,j) (φ1,p ⊗ φ2,p)

),

whose value does not depend on the representatives we choose.

We return to our original assumption for this chapter that φα,S ∈ S(VS)reg (α = 1, 2), φ1,S⊗φ2,S is

in S(V2S)reg, and gα ∈ eSH ′(AS

F ) for α = 1, 2. Choose some element eα ∈ E× such that ωχ(m(eα))φα,p

is supported on Λ+ for α = 1, 2. We have

Zφα(gα) =∑

xα∈K\Vfin

(ωχ(gα)φα) (xαeα)Z(xα)K .

Therefore, we can add one more assumption that φα,p is in S(Vp)reg ∩ S(Vp)K†p,n , and its support is

contained in Λ−. We let Zφα(gα) = Zφ(epgpα)n,Kp and denote by [Zφ(gα)]∧sp its completion along the

special fiber. We have a similar decomposition as (6.2).

151

First, we consider

Zφ1(g1) · Zφ2(g2) = [Zφ1(g1)]∧sp · [Zφ2(g2)]∧sp = [Zφ1(epgp1)n,Kp ]∧sp · [Zφ2

(epgp2)n,Kp ]∧sp. (6.13)

Let hpi ∈ Hpfin (i = 1, . . . , l) be a set of representatives of the double coset H(Q)\Hp

fin/Kp. We assume

that hp1 is the identity.

Then

(6.13) =

l∑i=1

m∑j=1


(epgp1)n,Kp ]∧h1

· [Zωχ(hp,j hpi )φ2

(epgp2)n,Kp ]∧h1

, (6.14)

where the subscript h1 means that we take completion along the component indexed by h1. By (6.12),

(6.14) =

l∑i=1

m∑j=1

∑x1∈V

∑x1∈Kp,n\Λ−

(x1,x1)=(x1,x1)′

(ωχ(gp1)φp1

)(hp,−1i x1)φ1,p(h−1

p,jx1)Z (x1, x1)n

·

∑x2∈V

∑x2∈Kp,n\Λ−

(x2,x2)=(x2,x2)′

(ωχ(gp2)φp2

)(hp,−1i x2)φ2,p(h−1

p,jx2)Z (x2, x2)n

=

l∑i=1

∑(x1,x2)∈V 2


p,∨2 )

) (φp1 ⊗ φ

p2

))(hp,−1i (x1, x2))µ(x1, x2;φ1,p ⊗ φ2,p). (6.15)

We define

Φhor =

l∑i=1

(ω′′χ(hpi )

(φp1 ⊗ φ

p2

))⊗ µ(•;φ1,p ⊗ φ2,p)

that is a function in S(V (AF )2). Then

(6.15) =∑

(x1,x2)∈V 2


p,∨2 )

)Φhor

)((x1, x2)) .


θhor(p)(•;φ1, φ2) = log q

∑(x1,x2)∈V 2

(ω′′χ(•)Φhor

)((x1, x2)) ,

where q is the cardinality of the residue field of Ep . Then in summary, we have the following lemma.

152


log q (Zφ1(g1) · Zφ2(g2)) = θhor(p)(ı(g1, g

∨2 );φ1, φ2).

Second, we consider

Zφ1(g1) · Vφ2

(g2) = [Zφ1(g1)]∧sp · Vφ2

(g2) = [Zφ1(epg

p1)n,Kp ]∧sp · Vφ2

(g2). (6.16)

Let hpi ∈ Hpfin (i = 1, . . . , l) and hp,j ∈ Kp,0 (j = 1, . . . ,m) be as above. Then

(6.16) =

l∑i=1

m∑j=1


(epgp1)n,Kp ]∧h1

· Vωχ(hp,j hpi )φ2

(g2)

=

l∑i=1

m∑j=1

∑x1∈V

∑x1∈Kp,n\Λ−

(x1,x1)=(x1,x1)′

(ωχ(gp1)φp1

)(hp,−1i x1)φ1,p(h−1

p,jx1)Z (x1, x1)n · Vωχ(hp,j hpi )φ2

(g2).

(6.17)

The function

ν(x1;φ1,p, φ2, g2) =

m∑j=1

∑x1∈Kp,n\Λ−

(x1,x1)=(x1,x1)′

φ1,p(h−1p,jx1)Z (x1, x1)n · Vωχ(hp,j)φ2

(g2),

which is originally defined for x1 ∈ Mor ((Y, jY), (X, iX))reg, can be extended by zero to a function

in S(Vp). Then

(6.17) =

l∑i=1

∑x1∈V

(ωχ(gp1)φp1

)(hp,−1i x1)ν(x1;φ1,p, φ2, g2). (6.18)

We define

φver =

l∑i=1

(ωχ(hpi )φ

p1

)⊗ ν(•;φ1,p, φ2, g2)

that is a function in S(V (AF )). Then

(6.18) =∑x1∈V

(ωχ(gp1)φver

)(x1).

153


θver(p)(•;φ1, φ2, g2) = log q

∑x1∈V

(ωχ(•)φver) (x1).

Then in summary, we have the following lemma.


log q (Zφ1(g1) · Vφ2

(g2)) = θver(p)(g1;φ1, φ2, g2)

that is a theta series for g1 ∈ epH ′(ApF ).

Finally, we let

A(p)(g1, φ1) = log q (Vφ1(g1) · ωK) .

Then in summary, we have the following proposition.

Proposition 6.3.4. For φ1,S ⊗ φ2,S ∈ S(V2S)reg and gα ∈ eSH ′(AS

F ) (α = 1, 2),

〈Zφ1(g1), Zφ2

(g2)〉p = θhor(p)(ı(g1, g

∨2 );φ1, φ2) + θver

(p)(g1, φ1, φ2, g2) +A(p)(g1, φ1)E(g2, φ2).

154

Chapter 7

Arithmetic inner product formula:

the main theorem

We finish the proof of the main theorem, i.e., the arithmetic inner product formula for n = 1 in this

chapter. In 7.1, we introduce the general theory of holomorphic projection for the group U(1, 1)F , and

compute such projection for the analytic kernel function. In particular, this process will relate the

archimedean local terms to admissible height parings (at archimedean places) and leaves local terms

at finite places unchanged. Based on all these previous results, we come to the final stage of the proof

in 7.2.

155

7.1 Holomorphic projection

In this section, we define and compute the holomorphic projection of the analytic kernel function

E′(0, ı(g1, g∨2 ), φ1 ⊗ φ2), and study its relation with the geometric kernel function when n = 1. We

follow the general theory in the case of GL2 in [GZ1986,Zha2001a,Zha2001b,YZZa]. In what follows,

n = 1 and H ′ = H1 in particular.

7.1.1 Holomorphic and quasi-holomorphic projection: generality

Let k = (kι)ι ∈ ZΣ∞ be a sequence of integers. We denote by A0(H ′) ⊂ A(H ′) the subspace of cusp

forms. Let Ak0(H ′) ⊂ A0(H ′) be the subspace consisting of those cusp forms of weight (1 + k, 1 − k)

(cf. Definition 2.1.2). Let Z ′ be the center of H ′, which is isomorphic to E×,1, as an F -torus. We

define a character ζk of Z ′∞ by letting ζkι (zι) = z2kιι . Let A(H ′, ζk) ⊂ A(H ′) be the subspace consisting

of the forms that have archimedean central character ζk. It is clear that Ak0(H ′) ⊂ A(H ′, ζk). Recall

that F+ ⊂ F is the subset of totally positive elements. For each fixed t ∈ F+, the t-th Whittaker

functions (with respect to the standard additive character) of all members of Ak0(H ′) are the same

one, namely, the function W kt defined by

W kt (n(b)m(a)[k1, k2]) =

∏ι∈Σ∞

exp (2πit (bι + iaιaι)) (aιaι) k1+kι1,ι k1−kι

2,ι ,

where a = (aι) ∈ E×∞, b = (bι) ∈ F∞, and [k1, k2] = ([k1,ι, k2,ι]) ∈ K′∞.

We denote by Ak0(H ′×H ′) ⊂ A0(H ′×H ′) the subspace consisting of cusp forms f(•, •) such that

for every g ∈ H ′(AF ), both f(g, •) and f(•, g) are in Ak0(H ′). The space A0(H ′ × H ′) is a Hilbert

space with norm given by the Petersson inner product 〈−,−〉H′×H′ .

Definition 7.1.1 (Holomorphic projection). We define a linear map

Pr : A(H ′ ×H ′, ζk)→ Ak0(H ′ ×H ′),

which sends f to the unique element Pr(f) in the later space satisfying 〈f , f ′〉H′×H′ = 〈Pr(f), f ′〉H′×H′

for every f ′ ∈ Ak0(H ′ ×H ′).

Let ψ′ be a character of F\AF such that ψ′∞(x) = ψ∞(tx) for t ∈ F×∞. We recall that ψ∞ is the

standard additive character at archimedean places as assumed in the beginning of Chapter 3. For an

156

automorphic form f ∈ A(H ′ ×H ′, ζk), we define

fψ′,s(g1, g2) = (4π)2dW kt (g1,∞)W k

t (g2,∞)

∫∫[Z′(F∞)N ′(F∞)\H′(F∞)]2

λP ′(h1)sλP ′(h2)s

fψ′(h1g1,fin, h2g2,fin)W kt (h1)W k

t (h2)dh1dh2.

Proposition 7.1.2. Suppose that f ∈ A(H ′×H ′, ζk) has the following asymptotic behavior: for some

ε > 0,

f(m(a1)g1,m(a2)g2) = Og1,g2(|a1a2|1−εAE

)as |a1a2|AE →∞ for aα ∈ A×E (α = 1, 2). Then the holomorphic projection Pr(f) has the ψ′-Whittaker

function

Pr(f)ψ′(g1, g2) = lims→0

fψ′,s(g1, g2).

Proof. Let ζα (α = 1, 2) be two automorphic characters of Z ′(AF ) such that ζα,∞ = ζk. Consider two

ψ′-Whittaker functions Wα(gα) = W kt (gα,∞)Wα

fin(gα,fin) (α = 1, 2) of H ′(AF ) with central character

ζα, such that Wαfin is compactly supported modulo Z ′(AF,fin)N ′(AF,fin). We define the Poincare series

to be

PWα(gα) = lims→0+

∑γ∈Z′(F )N ′(F )\H′(F )

Wα(γgα)λP ′(γ∞gα,∞)s.

For a function f ′ ∈ A(H ′ ×H ′, ζk), we let

f ′ζ1,ζ2(g1, g2) =

∫∫[Z′(F )\Z′(AF )]2

f ′(g1z1, g2z2)ζ−11 (z1)ζ−1

2 (z2)dz1dz2.

Assume that f has the asymptotic behavior as in the proposition. Then on one hand,

〈f , PW 1 ⊗ PW 2〉H′×H′

=

∫∫[Z′(AF )H′(F )\H′(AF )]2

fζ1,ζ2(g1, g2)PW 1(g1)PW 2(g2)dg1dg2

= lims→0+

∫∫[Z′(AF )N ′(F )\H′(AF )]2

fζ1,ζ2(g1, g2)W 1(g1)W 2(g2)λP ′(g1,∞)sλP ′(g2,∞)sdg1dg2

= lims→0+

∫∫[Z′(AF )N ′(AF )\H′(AF )]2

(fζ1,ζ2)ψ(g1, g2)W 1(g1)W 2(g2)λP ′(g1,∞)sλP ′(g2,∞)sdg1dg2. (7.1)

157

On the other hand, we have

〈Pr(f), PW 1 ⊗ PW 2〉H′×H′

=

∫∫[Z′(F∞)N ′(F∞)\H′(F∞)]2

W kt (g1)W k

t (g2)W kt (g1)W k

t (g2)dg1dg2∫∫[Z′(AF,fin)N ′(AF,fin)\H′(AF,fin)]2

(Pr(f)ζ1,ζ2)ψ′(g1,fin, g2,fin)W 1fin(g1,fin)W 2

fin(g2,fin)dg1,findg2,fin

= (4π)−2d

∫∫[Z′(AF,fin)N ′(AF,fin)\H′(AF,fin)]2

(Pr(f)ζ1,ζ2)ψ′(g1,fin, g2,fin)W 1fin(g1,fin)W 2

fin(g2,fin)dg1,findg2,fin.

(7.2)

Since, by definition, 〈Pr(f), PW 1 ⊗ PW 2〉H′×H′ = 〈f , PW 1 ⊗ PW 2〉H′×H′ , (7.1) and (7.2) are equal for

all (ζ1, ζ2) and (W 1fin,W

2fin). Therefore, the proposition follows.

For general f ∈ A(H ′×H ′, ζk) that may not have the asymptotic behavior as in Proposition 7.1.2,

we propose the following definition.

Definition 7.1.3. For f ∈ A(H ′ ×H ′, ζk), we define

Pr(f)ψ′(g1, g2) = consts=0 fψ′,s(g1, g2),

where consts=a denotes the constant term at s = a (possibly after meromorphic continuation around

a). We define the quasi-holomorphic projection of f to be

Pr(f)(g1, g2) =∑ψ′

Pr(f)ψ′(g1, g2),

where the sum is taken over all nontrivial characters of F\AF .

In fact, the same definition still makes sense if f is only left invariant under N ′(F ) × N ′(F ). In

some middle steps of later calculation, we need to apply Pr to such functions or even currents.

Then Proposition 7.1.2 amounts equivalently to say that if f satisfies the asymptotic behavior

there, we have Pr(f) = Pr(f).

158

7.1.2 Siegel–Fourier expansion of Eisenstein series on U(2, 2)

We study the (Siegel–)Fourier expansion of the Eisenstein series E(s, g,Φ) at g = e ∈ H ′′(AF ) =

H2(AF ) for Φ = φ1 ⊗ φ2 ∈ S(V2), where V is a hermitian space of rank 2 over AE . By definition

E(s, g, φ1 ⊗ φ2) =∑

γ∈P (F )\H′′(F )

(ωχ(γg)(φ1 ⊗ φ2)) (0)λP (γg)s,

which is absolutely convergent when Re s > 1. The following lemma is straightforward.

Lemma 7.1.4. The group H ′′(F ) is the disjoint union of the following cosets:

1.

P (F )w2

1 b11 b12

1 b21 b22

1

1

, b =

b11 b12

b21 b22

∈ Her2(E);

2.

P (F )w2,1

1 0 0

c 1 0 b22

1 −cτ

1

, c ∈ E, b22 ∈ F ;

3. P (F ),

where we recall that

w2 =

12

−12

; w2,1 =

1

1

1

−1

.

In particular, we have

E(s, g, φ1 ⊗ φ2) = E2(s, g, φ1 ⊗ φ2) + E0(s, g, φ1 ⊗ φ2) +∑c∈E

E1,c(s, g, φ1 ⊗ φ2),

159

where

E2(s, g, φ1 ⊗ φ2) =∑

b∈Her2(E)

(ωχ(w2n(b)g)(φ1 ⊗ φ2)) (0)λP (w2n(b)g)s;

E1,c(s, g, φ1 ⊗ φ2) =∑b22∈F

ωχw2,1m

1

c 1

n

0 0

0 b22

g

(φ1 ⊗ φ2)

(0)

λP

w2,1m

1

c 1

n

0 0

0 b22

g

s

E0(s, g, φ1 ⊗ φ2) = (ω(g)(φ1 ⊗ φ2)) (0)λP (g)s.

They are all invariant under the left translation by N(F ).

Let T be an element in Her2(E). We calculate

EβT (s, e, φ1 ⊗ φ2) :=

∫N(F )\N(AF )

Eβ(s, n(b′), φ1 ⊗ φ2)ψ(trTb′)−1dn

for β = 2, 0, and 1, c for c ∈ E, respectively.

For β = 2, we have

E2T (s, e, φ1 ⊗ φ2) =

∫N(AF )

(ωχ(w2n(b′))(φ1 ⊗ φ2)) (0)λP (w2n(b′))sψ(trTb′)−1dn. (7.3)

For β = 0, we have

E0T (s, e, φ1 ⊗ φ2) =

(φ1 ⊗ φ2) (0) if T = 02;

0 otherwise.

(7.4)

Now we calculate for β = 1, c with c ∈ F . To simplify the formulae, we ignore the term λP (•)s in

the following calculation. We have

E1,cT (s, e, φ1 ⊗ φ2)

=

∫b11∈F\AF

∫b12∈E\AE

∫b22∈AF

ωχw2,1m

1

c 1

n

b11 b12

bτ12 b22

(φ1 ⊗ φ2)

(0)

ψ

trT

b11 b12

bτ12 b22

−1

db11db12db22. (7.5)

160

Change variables b11 b12

bτ12 b22

7→ 1

c 1

b11 b12

bτ12 b22

1 cτ

1

.

Then

(7.5) =

∫b11∈F\AF

∫b12∈E\AE

∫b22∈AF

ωχw2,1n

b11 b12

bτ12 b22

m

1

c 1

(φ1 ⊗ φ2)

(0)

ψ

tr

1 −cτ

1

T

1

−c 1

b11 b12

bτ12 b22

−1

db11db12db22

=

∫b11∈F\AF

∫b12∈E\AE

∫b22∈AF

ωχw2,1n

0 0

0 b22

m

1

c 1

(φ1 ⊗ φ2)

(0)

ψ

tr

1 −cτ

1

T

1

−c 1

b11 b12

bτ12 b22

−1

db11db12db22.

Therefore, E1,cT (s, e, φ1 ⊗ φ2) = 0 unless

T =

1 cτ

1

0 0

0 t

1

c 1

for some t ∈ F . In the later case,

E1,cT (s, e, φ1 ⊗ φ2)

=

∫b∈AF

∫x∈V

ωχn

0 0

0 b

m

1

c 1

(φ1 ⊗ φ2)

(0, x)ψ(tb)−1dbdx

=

∫b∈AF

∫x∈V

ωχm

1

c 1

(φ1 ⊗ φ2)

(0, x)ψ ((q(x)− t)b) dbdx

=

∫b∈AF

∫x∈V

φ1(cx)φ2(x)ψ ((q(x)− t)b) dbdx. (7.6)

We remark that in the above calculation, all terms involving λP (•)s are ignored. In particular, when

c = 0,

E1,0T (s, e, φ1 ⊗ φ2) = φ1(0)×Wt(s+

1

2, e, φ2). (7.7)

161

7.1.3 Holomorphic projection of analytic kernel functions

We apply 7.1.1 to the analytic kernel function E′(0, ı(g1, g∨2 ), φ1 ⊗ φ2)). Here, φα = φ0

∞ ⊗ φα,fin is in

S(V) for a totally positive definite incoherent hermitian space over AE of rank 2. Let χ′ = χ | Z ′∞,

which is ζkχ

2 . Then the analytic kernel function is in A(H ′ ×H ′, χ′). Unfortunately, it does not have

the asymptotic behavior in Proposition 7.1.2. To find its holomorphic projection, we introduce the

following function

F (s; g1, g2;φ1, φ2) = E(s+1

2, g1, φ1)E(s+

1

2, g2, φ2) ∈ A(H ′ ×H ′, χ),

which is holomorphic at s = 0.

Proposition 7.1.5. The difference

E′(0, ı(g1, g∨2 ), φ1 ⊗ φ2)− F ′(0; g1, g2;φ1, φ2)

has the asymptotic behavior in Proposition 7.1.2.

Proof. Since the problem is symmetric in g1 and g2, we prove the asymptotic behavior only in g1.

Moreover, we can assume that a1 has 1 as its finite part. Let a = diag[a∞, 1] with a∞ ∈ E×∞. We

consider the behavior of the difference

E′(0,m(a), φ1 ⊗ φ2)− F ′(0;m(a∞), e; φ1, φ2) (7.8)

as |a∞| → ∞, where φα = ωχ(gα)φα for α = 1, 2. We apply the calculation in 7.1.2 to φ1 ⊗ φ2. We

see that the following terms

d

ds|s=0 E

2T (s,m(a), φ1 ⊗ φ2), T ∈ Her2(E);

d

ds|s=0 E

1,cT (s,m(a), φ1 ⊗ φ2), c 6= 0, T ∈ Her2(E)

are bounded by O(log |a|). Therefore, we only need to consider the difference

d

ds|s=0

(E0T (s,m(a), φ1 ⊗ φ2) +

∑t∈F

E1,0diag[0,t](s,m(a), φ1 ⊗ φ2)− F (s;m(a∞), e; φ1, φ2)

). (7.9)

162

By (7.4) and (7.7), we have on the one hand,

E0T (s,m(a), φ1 ⊗ φ2) +

∑t∈F

E1,0diag[0,t](s,m(a), φ1 ⊗ φ2)

=(ωχ(a∞)φ1

)(0)|a∞|s × φ2(0) +

∑t∈F

(ωχ(a∞)φ1

)(0)|a∞|s ×Wt(s+

1

2, e, φ2)

=(ωχ(a∞)φ1

)(0)|a∞|s × E(s+

1

2, e, φ2).

On the other hand,

F (s;m(a∞), e; φ1, φ2) =

((ωχ(a∞)φ1

)(0)|a∞|s +

∑t∈F

Wt(s+1

2,m(a∞), φ1)

)× E(s+

1

2, e, φ2).

Therefore,

(7.9) =∑t∈F

W ′t (1

2,m(a∞), φ1)× E(

1

2, e, φ2) +

∑t∈F

Wt(1

2,m(a∞), φ1)× E′(1

2, e, φ2),

which is bounded by O(log |a|). The proposition then follows.

By the above proposition, we have

Pr (E′(0, ı(g1, g∨2 ), φ1 ⊗ φ2))

= Pr (E′(0, ı(g1, g∨2 ), φ1 ⊗ φ2)− F ′(0; g1, g2;φ1, φ2)) + Pr (F ′(0; g1, g2;φ1, φ2))

= Pr (E′(0, ı(g1, g∨2 ), φ1 ⊗ φ2)− F ′(0; g1, g2;φ1, φ2)) + Pr (F ′(0; g1, g2;φ1, φ2)) . (7.10)

Since

F ′(0; g1, g2;φ1, φ2) = E′(1

2, g1, φ1)E(

1

2, g2, φ2) + E(

1

2, g1, φ1)E′(

1

2, g2, φ2),

its holomorphic projection vanishes. Therefore,

(7.10) = Pr (E′(0, ı(g1, g∨2 ), φ1 ⊗ φ2)− F ′(0; g1, g2;φ1, φ2))

= Pr (E′(0, ı(g1, g∨2 ), φ1 ⊗ φ2))− Pr (F ′(0; g1, g2;φ1, φ2))

= Pr (E′(0, ı(g1, g∨2 ), φ1 ⊗ φ2))− Pr

(E′(

1

2, g1, φ1)E(

1

2, g2, φ2)

)− Pr

(E(

1

2, g1, φ1)E′(

1

2, g2, φ2)

).

163

But for α, α′ = 1, 2,

Pr

(E′(

1

2, gα, φα)E(

1

2, gα′ , φα′)

)= Pr

(E′(

1

2, gα, φα)

)Pr

(E(

1

2, gα′ , φα′)

)= Pr

(E′(

1

2, gα, φα)

)E∗(

1

2, gα′ , φα′),

where

E∗(1

2, gα, φα) =

∑t∈F

Wt(1

2, gα, φα) = E(

1

2, gα, φα)− (ωχ(gα)φα) (0).

In summary, we have the following proposition.

Proposition 7.1.6. Assume that there is a finite place p such that φα,p(0) = 0 for α = 1, 2. Then

for gα ∈ P ′pH ′(ApF ),

Pr (E′(0, ı(g1, g∨2 ), φ1 ⊗ φ2))

= Pr (E′(0, ı(g1, g∨2 ), φ1 ⊗ φ2))− Pr

(E′(

1

2, g1, φ1)

)E(

1

2, g2, φ2)− E(

1

2, g1, φ1)Pr

(E′(

1

2, g2, φ2)

).

7.1.4 Quasi-holomorphic projection of analytic kernel functions

We calculate Pr (E′(0, ı(g1, g∨2 ), φ1 ⊗ φ2)) for gα ∈ eSH ′(AS

F ) under the following assumptions:



• φ1,v ⊗ φ2,v ∈ S(V2v)reg,dv for every v ∈ S that is nonsplit and some dv ≥ dψv (cf. 2.4.2).

Then we recall the formula (2.22)

E′(0, ı(g1, g∨2 ), φ1 ⊗ φ2) =

∑v 6∈S

E′v(0, ı(g1, g∨2 ), φ1 ⊗ φ2).

If we apply the formula defining Pr to the above summation, all terms except E′ι(0, ı(g1, g∨2 ), φ1⊗φ2)

for ι ∈ Σ∞ will not change. Therefore, we fix a ι ∈ Σ∞ and consider Pr (E′ι(0, ı(g1, g∨2 ), φ1 ⊗ φ2)). By

Theorem 4.3.4, it amounts to consider

Pr (−2 Vol(K)〈(Zφ1(g1),Ξφ1

(g1)ι′) , (Zφ2(g2),Ξφ2

(g2)ι′)〉ShK ) .

164

For simplicity, we will omit the term −2 Vol(K) in the following computation. Recall (4.56),

〈(Zφ1(g1),Ξφ1(g1)ι′) , (Zφ2(g2),Ξφ2(g2)ι′)〉ShK

=

l∑i=1

∫(H(Q)∩K)\D


(ωχ(g1)φ1) (T (x1), h−1i x1)Ξx1a(g1)

∗


(ωχ(g2)φ2) (T (x2), h−1i x2)Ξx2a(g2)

,

where a(gα) is denoted as aα before. Since it is symmetric in g1 and g2, we only need to compute

Pr

∑x∈V n,T (x)∈Her+n (E)

(ωχ(•)φ) (T (x), x)Ξxa(•)

. (7.11)

By Definition 7.1.3,

(7.11) =∑t∈F+

Pr

∑T (x)=t

(ωχ(•)φ) (t, x)Ξxa(•)

ψt

.

For each term,

Pr

∑T (x)=t

(ωχ(•)φ) (t, x)Ξxa(•)

ψt

(g)

= consts→0(4πt)Wkχ

2t (gι)

∑T (x)=t

∫Z′ιN

′ι\H′ι

λP ′(h)s (ωχ(gιh)φ) (t, x)Ξxa(h)dh. (7.12)

Taking substitution y = aa, we have

(7.12) = consts→0(4πt)Wkχ

2t (gι)

∑T (x)=t

(ωχ(gι)φι) (t, x)

∫ ∞0

Ξx√yys exp(−4πty)dy

= consts→0(4πt)∑

T (x)=t

(ωχ(g)φ) (t, x)

∫ ∞0

Ξx√yys exp(−4πty)dy. (7.13)

Let

δx(z) =R(x, z)

2t= − (xz, xz)

(x, x).

165

Then

(7.13) = consts→0(4πt)∑

T (x)=t

(ωχ(g)φ) (t, x)

∫ ∞0

(∫ ∞1

exp(−4πtyuδx(z))

udu

)ys exp(−4πty)dy


T (x)=t

(ωχ(g)φ) (t, x)t−1−s∫ ∞

0

∫ ∞1

exp(−4πyuδx(z))

uys exp(−4πty)dudy


T (x)=t

(ωχ(g)φ) (t, x)t−1−s∫ ∞

1

1

u

(∫ ∞0

exp (−4πy(1 + uδx(z)))

)du


T (x)=t

(ωχ(g)φ) (t, x)Γ(1 + s)

(4πt)1+s

∫ ∞1

du

u(1 + uδx(z))1+s

= consts→1

∑T (x)=t

(ωχ(g)φ) (t, x)

∫ ∞1

du

u(1 + uδx(z))s. (7.14)

Following [GZ1986], we introduce the Legendre function of the second type as follows

Qs−1(t) =

∫ ∞0

(t+√t2 − 1 coshu

)−sdu, t > 1, s > 0.

Then the admissible Green function attach to the divisor∑T (x)=t (ωχ(g)φ) (t, x)Zx is

Ξadmφ (g)t = consts→1 2

∑T (x)=t

(ωχ(g)φ) (t, x)Qs−1(1 + 2δx(z)).

By [GZ1986], we have

∫ ∞1

du

u(1 + uc)s= 2Qs−1(1 + 2c) +O(c−s−1), c→ +∞. (7.15)

Combining (7.14), (7.15), Corollary 5.3.4 and Proposition 7.1.6, we have the following proposition.

Proposition 7.1.7. Assume the three assumptions on φα at the beginning of the subsection. Then

the following identity

Pr (E′(0, ı(g1, g∨2 ), φ1 ⊗ φ2))

= −Vol(K)∑

v|v,v 6∈S

〈Zφ1(g1), Zφ2(g2)〉v − Pr

(E′(

1

2, g1, φ1)

)E(

1

2, g2, φ2)− E(

1

2, g1, φ1)Pr

(E′(

1

2, g2, φ2)

)

holds for gα ∈ eSH ′(ASF ). In particular, the archimedean local height paring is computed via admissible

Green functions.

166

7.2 Proof of the main theorem

We prove the arithmetic inner product formula for n = 1.

7.2.1 Difference of kernel functions

We assume that


• φα,S ∈ S(VS)reg (α = 1, 2);


• φ1,v ⊗ φ2,v ∈ S(V2v)reg,dv for every v ∈ S that is nonsplit and some dv ≥ dψv (cf. 2.4.2).

Let

E(g1, g2;φ1 ⊗ φ2) = Pr (E′(0, ı(g1, g∨2 ), φ1 ⊗ φ2))−E(g1, g2;φ1 ⊗ φ2),

which is a function in A(H ′ ×H ′, χ′). By (3.15), Propositions 6.1.2, 6.2.4, 6.3.4 and 7.1.7, we have

that for gα ∈ eSH ′(ASF ) (α = 1, 2), E(g1, g2;φ1 ⊗ φ2) is the sum of the following terms:

EI(g1, g2;φ1 ⊗ φ2) = −E(g1, φ1)A(g2, φ2)−A(g1, φ1)E(g2, φ2)− CE(g1, φ1)E(g2, φ2);

EII(g1, g2;φ1 ⊗ φ2) =∑p|pp∈S

A(p)(g1, φ1)E(g2, φ2);

EIII(g1, g2;φ1 ⊗ φ2) =∑

p|p splitp∈S

E(p)(g1, φ1; g2, φ2);

EIV(g1, g2;φ1 ⊗ φ2) =∑

p|p non-splitp∈S

θhor(p)(ı(g1, g

∨2 );φ1, φ2) + θver

(p)(g1, φ1; g2, φ2);

EV(g1, g2;φ1, φ2) = −Pr

(E′(

1

2, g1, φ1)

)E(

1

2, g2, φ2)− E(

1

2, g1, φ1)Pr

(E′(

1

2, g2, φ2)

).

Let π be an irreducible cuspidal automorphic representation of H ′ such that π∞ is a discrete series of

weight (1− kχ

2 , 1 + kχ

2 ), and ε(π, χ) = −1. Then for every f ∈ π and f∨ ∈ π∨, the integral

∫∫[eSH′(AS

F )]2f(g1)f∨(g∨2 )χ−1(det g2)E♥(g1, g2;φ1 ⊗ φ2) = 0,

for ♥ = I, II, III, IV, V. This is due to the facts that each term inside E♥ involves only Eisenstein

series, automorphic characters or theta series in gα ∈ eSH ′(ASF ) for at least one α, and that eSH

′(ASF )

167

is a dense subset of H ′(F )\H ′(AF ).

7.2.2 The final step

From the above subsection, we obtain the following identity.

∫∫[H′(F )\H′(AF )]2

f(g1)f∨(g2)χ−1(det g2)E′(0, ı(g1, g2), φ⊗ φ∨)

=

∫∫[H′(F )\H′(AF )]2

f(g1)f∨(g2)χ−1(det g2)E(g1, g∨2 ;φ⊗ φ∨)

= Vol(K)

∫∫[H′(F )\H′(AF )]2

f(g1)f∨(g2)〈Θφ(g1),Θφ∨(g2)〉KNT

= 〈Θfφ,Θ

f∨

φ∨〉NT, (7.16)

where (φ, φ∨) satisfy the assumptions in the previous subsection.

Theorem 7.2.1 (Arithmetic inner product formula). Let π be an irreducible cuspidal automorphic

representation of H1(AF ), χ a character of E×A×F \A×E, such that π∞ is a discrete series representa-

tion of weight (1− kχ

2 , 1 + kχ

2 ), ε(π, χ) = −1. Let V be a totally positive definite incoherent hermitian

space over AE of rank 2. For each f ∈ π and φ ∈ S(V)U∞ , we have the arithmetic theta lifting Θfφ.

Then

1. If V is not isometric to V(π, χ), then Θfφ is always trivial.

2. If V ∼= V(π, χ), then for every f ∈ π, f∨ ∈ π∨ and every φ, φ∨ ∈ S(V)U∞ that are decomposable,

we have

〈Θfφ,Θ

f∨

φ∨〉NT =L′( 1

2 , π, χ)

LF (2)L(1, εE/F )

∏v

Z∗(0, χv, fv, f∨v , φv ⊗ φ∨v ),

where in the last product, almost all factors are 1.

In other words, Conjecture 3.3.6 holds when n = 1.

Proof. We prove for (2) first, and then for (1).

1. On the one hand, in (2.10), we define the functional

α(f, f∨, φ, φ∨) =∏v

Z∗(0, χv, fv, f∨v , φv ⊗ φ∨v )

in⊗

v∈Σ HomH′v×H′v (R(Vv, χv), π∨v χvπv), which is nonzero sinceV ∼= V(π, χ). By Proposition

2.4.10 and the assumption that π∞ is a discrete series representation of weight (1− kχ

2 , 1 + kχ

2 ),

168

we can choose local components fv, f∨v for all v ∈ Σ and φv, φ

∨v for v ∈ Σfin such that (φ, φ∨)

satisfy the assumptions at the beginning of the previous subsection, and α(f, f∨, φ, φ∨) 6= 0.

On the other hand, we define another functional

γ(f, f∨, φ, φ∨) := 〈Θfφ,Θ

f∨

φ∨〉NT = Vol(K)〈Θfφ,Θ

f∨

φ∨〉KNT,

which is also in⊗

v∈Σ HomH′v×H′v (R(Vv, χv), π∨v χvπv). By Proposition A.2.1, the space⊗

v∈Σ HomH′v×H′v (R(Vv, χv), π∨v χvπv) has dimension 1. In other words, γ/α is a constant.

Applying (7.16) to the data we choose previously, and by (2.13), we have

γ

α=

L′( 12 , π, χ)

LF (2)L(1, εE/F ).

Therefore, (2) of the theorem follows.

2. The functional γ defined above is zero since V is not isometric to V(π, χ). If we take φ∨ = φ

and f∨ = f , then Θfφ = 0 since the paring 〈−,−〉NT is positive definite.

169

Appendix

There are three appendix. In A.1, we prove the theta dichotomy for unitary groups over non-

archimedean fields. For symplectic-orthogonal pairs, this is proved by S. Kudla and S. Rallis. In

A.2, we prove a multiplicity one result in the theory of local theta correspondence in a special case.

In A.3, we study the theta correspondence of unramified representations for unitary groups, following

the method of S. Rallis.

170

A.1 Theta dichotomy for unitary groups

In this section, we prove the theta dichotomy for unitary groups over non-archimedean fields. For

symplectic-orthogonal pairs, this is proved by Kudla–Rallis in [KR2005]. We will follow the same

line1.

Let F be a non-archimedean local field whose characteristic is not 2, and a quadratic field extension

E/F with Gal(E/F ) = 1, τ. Let OF (resp. OE) be the ring of integers, $ (resp. $E) a uniformizer

of F (resp. E), and q = |OF /$OF | (resp. qE = |OE/$EOE |).

Let n ≥ 1 be an integer. We denote by V ±n the (left) hermitian spaces over E of dimension n such

that ε(V ±n ) = ±1, and H±n = U(V ±n ). Let (W, 〈−,−〉) be a (right) skew-hermitian space over E of the

same dimension n, and H ′ = U(W ).

As in 2.2.1, we have the doubling space W ⊕ (−W ), and H ′′ = U(W ⊕ (−W )), where (−W ) =

(W,−〈−,−〉). We fix a character χ : E× → C× such that χ|F× = εnE/F . Let In(s, χ) be the space

of degenerate principal series, which is an admissible representation of H ′′. We have In(0, χ) =

R(V +n , χ)⊕R(V −n , χ). For any irreducible admissible representation π of H ′,

HomH′×H′ (In(0, χ), π χπ∨) 6= 0.

In fact, we have the following result of theta dichotomy:

Proposition A.1.1. There is exactly one between the following two spaces

HomH′×H′(R(V +n , χ), π χπ∨); HomH′×H′(R(V −n , χ), π χπ∨),

which is nonzero.

Proof. From the discussion above, we know that there is at least one space that is nonzero. If they are

both nonzero, then HomH′(S((V +n )n), π) 6= 0 and HomH′(S((V −n )n), π) 6= 0, where H ′ acts on both

spaces of Schwartz functions through the Weil representation ωχ.

Let g0 ∈ GLF (W ) be a τ -linear automorphism such that

〈w1g0, w2g0〉 = 〈w1, w2〉1During the preparation of this article, Gong and Grenie [GG2011] also prove the same results independently. Our

methods are similar and we refer to their article for more details

171

for all w1, w2 ∈W . By [MVW1987],

π Ad g0∼= π∨,

and we have (ωχ Ad g0, S((V −n )n)

) ∼= (ωχ−1 , S((−V −n )n)).

Therefore,

HomH′((ωχ, S((V +

n )n)), π)6= 0;

HomH′((ωχ−1 , S((−V −n )n)), π∨

)6= 0.

Taking product and the paring between π and π∨, we have

HomH′((ω1, S((V +

n ⊕ (−V −n ))n)),1)6= 0,

where 1 is the trivial representation of H ′. It is easy to see that V +n ⊕ (−V −n ) ∼= V −2n. We have

HomH′(S((V −2n)n),1) 6= 0. By Lemma A.1.2 below, we have

HomH′×H′(Rn(V −2n, 1),1) 6= 0.

Then by [KS1997, Theorem 1.2 (4)], Rn(V −2n, 1) is the unique irreducible submodule of In(n2 , 1), which

is also the image of the intertwining map

M∗(−n2, 1) : In(−n

2, 1)→ Rn(V −2n, 1).

By [KS1997, Proposition 4.3 (2)], ker(M∗(−n2 , 1)) = C · ϕ0, where ϕ0 ∈ In(−n2 , 1) is the constant

function with value 1. Therefore, there is a nonzero functional

ζ ∈ HomH′×H′(In(−n

2, 1),1

)

such that ζ(ϕ0) = 0. By Lemma A.1.3 below,

HomH′×H′(In(−n

2, 1),1

)= C · Z(−n

2, 1, 1,−)

and Z(−n2 , 1, 1, ϕ0) 6= 0, which is a contradiction. The proposition is proved.

172

Lemma A.1.2. For any hermitian space V over E of dimension m and χ such that χ|F× = εmE/F ,

the following are equivalent:

• HomH′ ((ωχ, S(V n)), π) 6= 0;

• HomH′×H′(Rn(V, χ), π χπ∨) 6= 0.

Proof. Since HomH′ ((ωχ, S(V n)), π) 6= 0 is equivalent to Θχ(π, V ) 6= 0, and Θχ(π, V ) admits an

irreducible quotient by [MVW1987, Chapitre 3, IV, Theoreme 4 (2-a)], the lemma follows by [HK-

S1996, Proposition 3.1].

Before we proceed, let us recall some notations from [HKS1996, Section 4]. Let Y ⊂W ⊕ (−W ) be

the graph of the identity map and PY ⊂ H ′′ the parabolic subgroup that stabilizes Y . Let r0 be the

Witt index of W . Then PY \H ′′ can be canonically identified with the set of isotropic n-dimensional

subspaces of W ⊕ (−W ). The H ′ ×H ′-orbit of Z in PY \H ′′ is uniquely determined by

d = dim(Z ∩W ) = dim(Z ∩ (−W )).

Therefore,

H ′′ =∐

0≤d≤r0

Ωd :=∐

0≤d≤r0

PY δd(H′ ×H ′)

for some representative δd ∈ H ′′. The (topological) closure

Ωr =∐r′≥r

Ωr′ .

Let

In(s, χ) = I(r0)n (s, χ) ⊃ I(r0−1)

n (s, χ) ⊃ · · · ⊃ I(0)n (s, χ)

be the filtration given by support, and

Q(r)n = I(r)

n (s, χ) ⊃ I(r−1)n (s, χ) ∼= IndH

′×H′Pr×Pr

(χ| • |s+

r2

E χ| • |s+r2

E ⊗ S(H ′n−2r)),

where the induction is normalized. Here Pr is the parabolic subgroup of H ′ with the Levi quotient

Mr isomorphic to GLr(E) × H ′n−2r, and the maximal unipotent subgroup Nr, where H ′n−2r is the

unitary group of some skew-hermitian space of dimension n− 2r.

173

For every section ϕs ∈ In(s, χ) and matrix coefficient φ of π∨, we define the zeta integral

Z(s, χ, φ, ϕ) =

∫H′φ(g)ϕs((g, 1))dg.

If the section is standard, it has a meromorphic continuation to the entire complex. If for some s0 ∈ C

at which every such continuation is holomorphic, then the zeta integral defines a nonzero element

Z(s0, χ,−,−) ∈ HomH′×H′(In(s0, χ), π χπ∨).

Moreover, if HomH′×H′(Qr(s0, χ), πχπ∨) = 0 for every r ≥ 1, then all zeta integrals are holomorphic

at s0, and

HomH′×H′(In(s0, χ), π χπ∨) = C · Z(s0, χ,−,−).

For the trivial representation, we have the following lemma.

Lemma A.1.3. For χ = 1 and s0 = −n2 , HomH′×H′(Qr(s0, 1),1) = 0 for every r ≥ 1. In particular,

Z(s0, 1, 1, ϕ0) 6= 0.

Proof. For r ≥ 1 and every irreducible admissible representation π of H ′,

HomH′×H(Qr(s, 1), π π∨)

= HomH′×H′(π∨ π, IndH

′×H′Pr×Pr (| • |−s−

r2

E | • |−s−r2

E ⊗ C∞(H ′n−2r)))

= HomMr×Mr

((π∨)Nr πNr , | • |

−s+n2−r

E | • |−s+n2−r

E ⊗ C∞(H ′n−2r)).

In particular, when π = 1, we have

HomH′×H(Qr(s, 1),1)

= HomMr×Mr

(1, | • |−s+

n2−r

E | • |−s+n2−r

E ⊗ C∞(H ′n−2r))

= HomGLr(E)×GLr(E)

(1, | • |−s+

n2−r

E | • |−s+n2−r

E

).

Therefore, if s0 = −n2 , then HomH′×H′(Qr(s0, 1),1) = 0 for every r ≥ 1. The fact Z(s0, 1, 1, ϕ0) 6= 0

is due to the following lemma.

Lemma A.1.4. Let ϕ0s be the standard section such that ϕ0

s|K = 1, where K is the maximal compact

subgroup of H ′′. Let bn(s) =∏n−1i=0 L(2s+ n− i, εiE/F ). Then we have three cases:

174

1. If n = 2m (m ≥ 1) and r0 = m,

Z(s, 1, 1, ϕ0) =1

bn(s)

m−1∏i=−m

(1− qi−sE

)−1.

2. If n = 2m (m ≥ 1) and r0 = m− 1,

Z(s, 1, 1, ϕ0) =1

bn(s)

(1− q−sE

) m−1∏i=−m

(1− qi−sE

)−1.

3. If n = 2m+ 1 (m ≥ 0) and r0 = m,

Z(s, 1, 1, ϕ0) =1

bn(s)

m∏i=−m

(1− qi−s−

12

E

)−1

.

Proof. For (1) and (3), if E/F is unramified, they are [GPSR1987, Part A, Proposition 6.2] (for n

even) and [Li1992, Theorem 3.1] (for n odd). The calculation for the integral

∫H′ϕ0s((g, 1))dg

in these papers works also for E/F ramified, since the only thing we need to check is the ratio cw(χ)

of the intertwining operator (cf. [Li1992, Page 189]). The formula for such ratio holds more generally,

in particular, when E/F is ramified, by [Cas1980, Theorem 3.1].

For (2), one can prove similarly as in [Li1992, Section 3]. In fact, (1) is enough for our use.

A.2 Uniqueness of local invariant functionals

In this section, we fix a place v ∈ Σ and suppress it from notations. We prove that the space

HomH′×H′(I2(0, χ), π∨χπ) is of dimension 1, following [HKS1996]. Here, χ is a character of E×. If

χ is trivial on F×, i.e., χ−1 = χτ , then we define a character πχ of H ′ as follows. For every element

g ∈ H ′, det g is in E×,1. In particular, det g = eg/eτg for some eg ∈ E× by Hilbert’s Theorem 90. Set

πχ(g) = χ(eg) that is independent of eg we choose.

Proposition A.2.1. Let χ be a character of E×, and π an irreducible admissible representation of

H ′ that is not π−1χ if χ−1 = χτ . Then we have

dimCHomH′×H′(I2(0, χ), π∨ χπ) = 1,

175

and L(s, π, χ) is holomorphic at s = 12 .

In fact, the proposition holds for arbitrary representation π. Since we do not need such generality

in this article, we put this condition for simplicity. In what follows, we also assume that v is finite for

simplicity. The case for archimedean places is similar, and actually will not be used in the article.

Recall from 2.2.1, we have, as a special case, the following decomposition of double cosets H ′′ =

Ω0

∐Ω1, where Ω0 = Pγ0ı(H

′ × H ′) is open and Ω = Pı(H ′ × H ′) is closed. Therefore, we have a

subspace I(0)2 (0, χ) ⊂ I2(0, χ), where

I(0)2 (0, χ) = ϕ ∈ I2(0, χ) | Suppϕ ⊂ Ω0

that is invariant under the action of H ′ ×H ′ by right translation via ı. As H ′ ×H ′-representations,

we denote

Q(0)2 (0, χ) = I

(0)2 (0, χ); Q

(1)2 (0, χ) = I2(0, χ)/I

(0)2 (0, χ).

We have a linear isomorphism

Q(0)2 (0, χ)→ S(H ′)(1⊗ χ)

ϕ 7→ Ψ(g) = ϕ (γ0ı(g,12)) ,

where S(H) denotes the space the Schwartz functions on H ′ that is viewed as a representation of

H ′ ×H ′. Since

ϕ (γ0ı(g,12)ı(g1, g2)) = ϕ(γ0ı(g2, g2)ı(g−1

2 gg1,12))

= χ(det g2)ϕ(γ0ı(g

−12 gg1,12)

),

the above isomorphism is H ′ ×H ′-equivariant. There is a unique, up to constant, H ′ ×H ′-invariant

functional on S(H ′)⊗ (π π∨) given by

Ψ⊗ (f ⊗ f∨) 7→∫H′〈π(g)f, f∨〉Ψ(g)dg.

Since

HomH′×H′ (S(H ′)⊗ (π π∨) ,C) = HomH′×H′ (S(H ′), π∨ π)

= HomH′×H′ (S(H ′)⊗ (1 χ) , π∨ χπ) = HomH′×H′(Q

(0)2 (0, χ), π∨ χπ

),

176

we have

dimCHomH′×H′(Q

(0)2 (0, χ), π∨ χπ

)= 0.

For Q(1)2 (0, χ), we have the following lemma.

Lemma A.2.2. If χ−1 6= χτ , or χ−1 = χτ but π 6= π−1χ , then

HomH′×H′(Q

(1)2 (0, χ), π∨ χπ

)= 0.

Proof. It is easy to see that the following map

Q(1)2 (0, χ)→ I1(

1

2, χ) I1(

1

2, χ)

ϕ 7→ ((g1, g2) 7→ ϕ (ı(g1, g2)))

is H ′ ×H ′-equivariant isomorphism. Therefore,

HomH′×H′(Q

(1)2 (0, χ), π∨ χπ

)= HomH′×H′

(I1(

1

2, χ) I1(

1

2, χ), π∨ χπ

)= HomH′×H′

(π χ−1π∨, I1(−1

2, χ−1) I1(−1

2, χ−1)

).

By [KS1997, Theorem 1.2] for v finite and nonsplit, [KS1997, Theorem 1.3] for v finite and split (and

[Lee1994, Theorem 6.10 (1-b)] for v infinite), the only (possible) irreducible H ′-submodule properly

contained in I1(− 12 , χ−1) is isomorphic to π−1

χ . Therefore, the lemma follows by our assumption.

Proof of Proposition A.2.1. The normalized zeta integral (2.7) has already defined a nonzero ele-

ment in HomH′×H′ (I2(0, χ), π∨ χπ). Therefore, the dimension is at least 1. If it is greater than

one, we can find a nonzero element in HomH′×H′ (I2(0, χ), π∨ χπ) whose restriction to I(0)2 (0, χ)

is zero since dimCHomH′×H′(Q

(0)2 (0, χ), π∨ χπ

)= 1. Then it defines a nonzero element in

HomH′×H′(Q

(1)2 (0, χ), π∨ χπ

)that is 0 by the above lemma. Therefore,

dimCHomH′×H′ (I2(0, χ), π∨ χπ) = 1.

For the L-factor, the restriction of the normalized zeta integral to I(0)2 (0, χ) is nonzero. Since the

original zeta integral is absolutely convergent at s = 0 if ϕ ∈ I(0)2 (0, χ), L(s, π, χ) can not have a pole

at s = 12 , by realizing that b2(s) is holomorphic and nonzero at s = 0.

177

A.3 Theta correspondence of unramified representations

In this section, we study the theta correspondence of unramified representations for unitary groups.

Let F be a p-adic local field with p 6= 2, E/F an unramified quadratic field extension with

Gal(E/F ) = 1, τ. Let OF (resp. OE) be the ring of integers of F (resp. E), $ a uniformizer of OF ,

and q the cardinality of OF /$OF . Let ψ be an unramified additive character of F , which determines

an unramified additive character of E by composing with 12 TrE/F . Let dx be the selfdual Haar

measure of E with respect to ψ (

12 TrE/F

), and d×x = dx

|x|E the Haar measure of E×, normalized

such that |$|E = q−2. We will use slightly different notations from 2.1.1.

Let n,m ≥ 1 be two integers, and r = minm,n. Let (Wn, 〈−,−〉) be a skew-hermitian space

over E whose skew hermitian form is given by

1n

−1n

under a basis e1, . . . , en; e∗1, . . . , e

∗n. Let (Vm, (−,−)) be a hermitian space over E whose hermitian

form is given by 1m

1m

under a basis f1, . . . , fm; f∗1 , . . . , f

∗m. Let H ′n = U(Wn) (resp. Hm = U(Vn)) be the corresponding

group of isometries, and

K ′n = H ′n ∩GL(OE〈e1, . . . , en; e∗1, . . . , e∗n〉) (resp. Km = Hm ∩GL(OE〈f1, . . . , fm; f∗1 , . . . , f

∗n〉))

be a hyperspecial maximal subgroup. We have a Weil representation ω = ωχ=1,ψ on the space S(V nm)

of Schwartz functions, whose formulae are given in 2.1.1.

Let W ∗n,i = spanEe∗i+1, . . . , e∗n for 0 ≤ i ≤ n, and V ∗m,j = spanEf∗j+1, . . . , f

∗m for 0 ≤ j ≤ m.

Then we have filtration of the maximal isotropic subspaces W ∗n,0 and V ∗m,0, respectively as

W ∗n,0 ⊃W ∗n,1 ⊃ · · · ⊃W ∗n,n = 0; V ∗m,0 ⊃ V ∗m,1 ⊃ · · · ⊃ V ∗m,m = 0.

Up to conjugacy, the maximal parabolic subgroups ofH ′n×Hm are precisely those subgroups P ′n,i×Pm,j

that consists of elements (h′, h) stabilizing the subspace W ∗n,i⊗V ∗m,j ⊂Wn⊗Vm, for i = 0, 1, . . . , n and

j = 0, 1, . . . ,m. Let N ′n,i ×Nm,j be its unipotent radical. Then the Levi quotient P ′n,i × Pm,j/N ′n,i ×

178

Nm,j is isomorphic to (GLn−i(E) ×H ′i) × (GLm−j(E) ×Hj). For 0 ≤ t ≤ n, we define an algebraic

closed subsets Σt to be

Σt = x = (x1, . . . , xn) ∈ V nm | (xi, xj) = 0 for t+ 1 ≤ j ≤ n.

We say that a function φ ∈ S(V nm) is spherical if it is invariant under the action of K ′n × Km. The

same proof of [Ral1984, Proposition 2.2] implies the following lemma.

Lemma A.3.1. Let φ be a spherical function in S(V nm) such that for every h′ ∈ H ′n, ω(h′)φ vanishes

on the subset Σ0. Then ω(h′)φ vanishes identically.

We identify V nm with Mat2m,n(E) via the basis f1, . . . , fm; f∗1 , . . . , f∗m. Then the action of

GLn(E) × Hm is given by (A, h).X = hXA−1. We have the following version of [Ral1984, Lem-

ma 3.1],

Lemma A.3.2. Let Σ(i)0 = X ∈ Σ0 | rankX = i. Then Σ

(i)0 , if nonempty, is an orbit under

GLn(E)×Hm; and Σ0 is a disjoint union of orbits of the form Σ(i)0 for i = 0, 1, . . . , r, in which Σ

(r)0

is the unique open one.

Let

B′n =

A

tAτ,−1

1n B

1n

|A is lower triangular and B ∈ Hern(E)

,

whose Levi decomposition is B′n = T ′nU′n with

T ′n = diag[t1, . . . , tn; tτ,−11 , . . . , tτ,−1

n ] | ti ∈ E×,

and U ′n begin the unipotent radical. Let

Bm,r =

A

tAτ,−1

1m B

1m

|A =

A1 A2

A3

,

whereA1 ∈ Matr,r(E) is lower triangular, A3 ∈ Matm−r,m−r(E) is upper triangular, A2 ∈ Matr,m−r(E),

and B is skew-hermitian. We have the Levi decomposition Bm,r = TmUm,r with Tm = T ′m and Um,r

begin the unipotent radical. Then B′n ×Bm,r is a minimal parabolic subgroup of H ′n ×Hm.

Review some facts about spherical representations of H ′n ×Hm. For ν = (ν1, . . . , νn) ∈ Cn, define

179

the space I ′(ν) consisting of all locally constant functions ϕ : H ′n → C satisfying

ϕ(h′t′u′) = δ′− 1

2n (t′)

n∏i=1

|t′i|νiE ϕ(h′)

for all h′ ∈ H ′n, t ∈ T ′n, and u′ ∈ U ′n. Here,

δ′n =

n∏i=1

|t′i|2i−1E

is the modulus function of B′n. These I ′(ν) provide all spherical principal series of H ′n. Let S(H ′n//K′n)

be the spherical Hecke algebra of H ′n. Then we have the Fourier–Satake isomorphism

FS : S(H ′n//K′n)→ C[X1, X

−11 , . . . , Xn, X

−1n ]W (H′n),

such that for every f ′ ∈ S(H ′n//K′n),

FS(f ′)(q2ν1 , q−2ν1 , . . . , q2νn , q−2νn) = traceI′(ν)(f′).

For µ = (µ1, . . . µm) ∈ Cm, define the space I(µ) consisting of all locally constant functions ϕ : Hm →

C satisfying

ϕ(htu) = δ− 1

2m,r(t)

m∏j=1

|tj |µjE ϕ(h)

for all h ∈ Hm, t ∈ Tm, and u ∈ Um,r. Here,

δm,r(t) =

r∏j=1

|tj |2m−2r+2j−1E

m∏j=r+1

|tj |2m−2j+1E

is the modulus function of Bm,r. These I(µ) provide all spherical principal series of Hm. Let

S(Hm//Km) be the spherical Hecke algebra of Hm. Then we have the Fourier–Satake isomorphis-

m

FS : S(Hm//Km)→ C[X1, X−11 , . . . , Xm, X

−1m ]W (Hm),

such that for every f ∈ S(Hm//Km),

FS(f)(q2µ1 , q−2µ1 , . . . , q2µm , q−2µm) = traceI(µ)(f).

Let Yr ⊂ GLr(E) be the group of lower-triangular matrices, which has the Levi decomposition

180

Yr = ArLr. Here, Ar = diag[a1, . . . , ar]|ai ∈ E× and Lr is the unipotent radical. We write elements

in Lr in the form (ljk)j>k. The group Yr has the following right invariant measure

dyr =

r∏i=1

|ai|2i−(r+1)E d×ai

∏1≤k<j≤r

dljk,

where dljk is certain measure on Lr normalized as in [Ral1982, Page 490]. Let σ = (σ1, . . . , σr) ∈ Cr

such that Reσi 0. For every function φ ∈ S(V nm), the following integral

Zσ(φ) =

∫Yr

φ

yr 0

0 0

r∏i=1

|ai|σE dyr

is absolutely convergent. Define a map Zσ sending φ ∈ S(V nm) to the function

(h′, h) 7→ Zσ(ω(h′−1, h−1)φ

)on H ′n ×Hm. It is a nonzero H ′n ×H ′m-equivariant map from S(V nm) to S(H ′n ×Hm).

Lemma A.3.3. For σ = (σ1, . . . , σr) ∈ Cr such that Reσi 0, the image of the above intertwining

map Zσ lies in I ′(ν)⊗ I(µ), where

ν =

(2 + σ1 −m−

3

2, . . . , 2r + σr −m−

3

2, (r + 1)−m− 1

2, . . . , n−m− 1

2

);

µ =

(−2− σ1 +m+

3

2, . . . ,−2r + σr +m+

3

2,−(r + 1) +m+

1

2, . . . ,

1

2

).

Proof. We have

Zσ(φ)(h′t′u′, htu) =

∫Yr

(ω(u′−1t′−1h′−1, u−1t−1h−1

)φ) yr 0

0 0

r∏i=1

|ai|σEdyr

=

∫Yr

(ω(t′−1h′−1, t−1h−1

)φ) yr 0

0 0

r∏i=1

|ai|σEdyr

=

∫Yr

|det t′|−mE(ω(h′−1, h−1

)φ)t

yr 0

0 0

t′−1

r∏i=1

|ai|σEdyr. (A.1)

181

Changing variable yr 7→ y′′r = diag[t1, . . . , tr]yr, we have

r∏i=1

|ai|σEdyr =

r∏i=1

|ti|r−3i−σi+2E

r∏i=1

|tiai|σEdy′′r .

Changing variable yr 7→ y′r = diag[t′−11 , . . . , t′−1

r ], we have

r∏i=1

|ai|σEdyr =

r∏i=1

|ti|i+σi−1E

r∏i=1

∣∣t′−1i ai

∣∣σE

dy′r.

Therefore,

(A.1) =

r∏i=1

|t′i|i+σi−m−1E

n∏i=r+1

|t′i|−mE

r∏j=1

|tj |r−3j−σj+2E

∫Yr

(ω(h′−1, h−1

)φ) yr 0

0 0

r∏i=1

|ai|σEdyr

=

r∏i=1

|t′i|i+σi−m−1E

n∏i=r+1

|t′i|−mE

r∏j=1

|tj |r−3j−σj+2E (Zσ(φ)) (h′, h).

The lemma follows immediately.

The above lemma implies the following facts. If m ≥ n = r, then there is a surjective homomor-

phism

Φm,n : S(Hm//Km)→ S(H ′n//K′n)

satisfying that

Zσ (Φm,n(f)− f) = 0

for all f ∈ S(Hm//Km) and Reσi 0. Using the Fourier–Satake isomorphism, the map Φm,n is given

by

C[X1, X−11 , . . . , Xm, X

−1m ]W (Hm) → C[X1, X

−11 , . . . , Xn, X

−1n ]W (H′n),

where

logqXj 7→ logqXj , j = 1, . . . , n;

logqXj 7→ 2m− 2j + 1, j = n+ 1, . . . ,m.

In particular, when m = n, Φm,n is the identity map.

If n > m = r, similarly there is a surjective homomorphism

Φ′n,m : S(H ′n//K′n)→ S(Hm//Km)

182

satisfying that

Zσ (f ′ − Φ′n,m(f ′)) = 0

for all f ′ ∈ S(H ′n//K′n) and Reσi 0. Using the Fourier–Satake isomorphism, the map Φ′n,m is given

by

C[X1, X−11 , . . . , Xn, X

−1n ]W (H′n) → C[X1, X

−11 , . . . , Xm, X

−1m ]W (Hm),

where

logqXi 7→ logqXi, i = 1, . . . ,m;

logqXi 7→ 2m− 2i+ 1, i = m+ 1, . . . , n.

Lemma A.3.4. Suppose that φ ∈ S(V nm) is spherical, and Zσ(φ) = 0 for all σ ∈ Cr such that

Reσi 0. Then ω(h′)φ vanishes on Σ0 for all h′ ∈ H ′n.

Proof. It suffices to show that ω(h′)φ vanishes on Σ(r)0 since it is dense open in Σ0. Since Σ

(r)0 is

transitive under the action of GLn(E)×Hm, we need only to show that

(ω(h′, h)φ)

1r 0

0 0

= 0

for all (h′, h) ∈ GLn(E) × Hm. We can write h′ = b′k′ with b′ ∈ B′n, k′ ∈ K ′n, and h = bk with

b ∈ B′m,r, k ∈ Km. Since φ is spherical by assumption, we have

(ω(h′, h)φ)

1r 0

0 0

= (ω(b′, b)φ)

1r 0

0 0

= φ

X 0

0 0

with X ∈ Matr,r(E). Therefore, the lemma follows from [Ral1982, Lemma 5.2] for k = E.

Combining Lemmas A.3.1, A.3.3 and A.3.4, we have the following proposition.

Proposition A.3.5. The ideal

In,m = f ∈ S(H ′n//K′n)⊗ S(Hm//Km) | ω(f) = 0

is generated by

Φm,n(f)− f | f ∈ S(Hm//Km) (resp. f ′ − Φ′n,m(f ′) | f ′ ∈ S(H ′n//K′n))

183

if m ≥ n (resp. m < n).

When E = F ⊕F , the corresponding unitary group H ′n (resp. Hm) can be identified with GLn(F )

(resp. GLm(F )). The Weil representation ω, which realizes on the space S(Matm,n(F )), is simply given

by the formula (ω(g′, g)φ) (x) = φ(g−1xg′) for g′ ∈ GLn(F ), g ∈ GLm(F ) and φ ∈ S(Matm,n(F ))

(cf. [Ral1982, Section 6]). Without lost of generality, we assume that n ≥ m. Then the ideal

Jn,m = f ∈ S(GLn(F )//GLn(OF ))⊗ S(GLm(F )//GLm(OF )) | ω(f) = 0

is generated by

f −Ψn,m(f) | f ∈ S(GLn(F )//GLn(OF )).

In terms of the Fourier–Satake isomorphism, the surjective homomorphism Ψn,m is given by

C[X1, X−11 , . . . , Xn, X

−1n ]W (GLn(F )) → C[X1, X

−11 , . . . , Xm, X

−1m ]W (GLm(F )),

where

logqXi 7→ − logqXi +n−m

2, i = 1, . . . ,m;

logqXi 7→ −i+n+ 1

2, i = m+ 1, . . . , n.

Proposition A.3.5 and its analogue in the above case imply the following corollary.

Corollary A.3.6. Assume that m = n. Then the groups H ′n and Hn are isomorphic.

1. Let π be an unramified irreducible admissible representation of H ′n. Then the theta correspon-

dence of π to Hn, with respect to the Weil representation ω, is nontrivial and isomorphic to

π.

2. Let π be an unramified irreducible admissible representation of GLn(F ), and χ an unramified

character of F×. Then the theta correspondence of π to GLn(F ), with respect to the Weil

representation ωχ defined by (ωχ(g′, g)φ) (x) = χ(det g′)φ(g−1xg′) for φ ∈ S(Matn,n(F )), is

nontrivial and isomorphic to π∨ ⊗ χ.

184

Bibliography

[Ada] J. Adams, Discrete series and characters of the component group, available at http://fa.institut.math.

jussieu.fr/node/44. preprint. ↑58

[ARG2007] ARGOS, Argos Seminar on Intersections of Modular Correspondences, Asterisque 312 (2007). Held at

the University of Bonn, Bonn, 2003–2004. MR2340365 ↑123

[Art1984] J. Arthur, On some problems suggested by the trace formula, Lie group representations, II (College

Park, Md., 1982/1983), Lecture Notes in Math., vol. 1041, Springer, Berlin, 1984, pp. 1–49. MR748504

(85k:11025) ↑55

[Art1989] , Unipotent automorphic representations: conjectures, Asterisque 171-172 (1989), 13–71. Orbites

unipotentes et representations, II. MR1021499 (91f:22030) ↑55

[Art] , The endoscopic classification of representations: orthogonal and symplectic groups, available at

http://www.claymath.org/cw/arthur/. preprint. ↑56

[AMRT2010] A. Ash, D. Mumford, M. Rapoport, and Y.-S. Tai, Smooth compactifications of locally symmetric vari-

eties, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2010. With the

collaboration of Peter Scholze. MR2590897 (2010m:14067) ↑51

[BBD1982] A. A. Beılinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces,

I (Luminy, 1981), Asterisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR751966

(86g:32015) ↑55

[Beı1987] A. Beılinson, Height pairing between algebraic cycles, Current trends in arithmetical algebraic geome-

try (Arcata, Calif., 1985), Contemp. Math., vol. 67, Amer. Math. Soc., Providence, RI, 1987, pp. 1–24.

MR902590 (89g:11052) ↑5, 59

[Blo1984] S. Bloch, Height pairings for algebraic cycles, Proceedings of the Luminy conference on algebraic K-theory

(Luminy, 1983), 1984, pp. 119–145, DOI 10.1016/0022-4049(84)90032-X. MR772054 (86h:14015) ↑5, 59

[BW2000] A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive

groups, 2nd ed., Mathematical Surveys and Monographs, vol. 67, American Mathematical Society, Provi-

dence, RI, 2000. MR1721403 (2000j:22015) ↑58

[BC1991] J.-F. Boutot and H. Carayol, Uniformisation p-adique des courbes de Shimura: les theoremes de Cerednik

et de Drinfel’d, Asterisque 196-197 (1991), 7, 45–158 (1992) (French, with English summary). Courbes

modulaires et courbes de Shimura (Orsay, 1987/1988). MR1141456 (93c:11041) ↑111, 146, 147

http://fa.institut.math.jussieu.fr/node/44

http://fa.institut.math.jussieu.fr/node/44

http://www.ams.org/mathscinet-getitem?mr=2340365





http://www.claymath.org/cw/arthur/













185

[BY2009] J. H. Bruinier and T. Yang, Faltings heights of CM cycles and derivatives of L-functions, Invent. Math.

177 (2009), no. 3, 631–681, DOI 10.1007/s00222-009-0192-8. MR2534103 (2011d:11146) ↑6

[Car1986] H. Carayol, Sur la mauvaise reduction des courbes de Shimura, Compositio Math. 59 (1986), no. 2,

151–230 (French). MR860139 (88a:11058) ↑106, 107, 108, 112, 117

[Cas1980] W. Casselman, The unramified principal series of p-adic groups. I. The spherical function, Compositio

Math. 40 (1980), no. 3, 387–406. MR571057 (83a:22018) ↑174

[DM1969] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Etudes

Sci. Publ. Math. 36 (1969), 75–109. MR0262240 (41 #6850) ↑111

[Dri1973] V. G. Drinfel’d, Two theorems on modular curves, Funkcional. Anal. i Prilozen. 7 (1973), no. 2, 83–84

(Russian). MR0318157 (47 #6705) ↑66

[Dri1976] , Coverings of p-adic symmetric domains, Funkcional. Anal. i Prilozen. 10 (1976), no. 2, 29–40

(Russian). MR0422290 (54 #10281) ↑146, 147

[DS2005] F. Diamond and J. Shurman, A first course in modular forms, Graduate Texts in Mathematics, vol. 228,

Springer-Verlag, New York, 2005. MR2112196 (2006f:11045) ↑63

[GG2011] Z. Gong and L. Grenie, An inequality for local unitary theta correspondence, Ann. Fac. Sci. Toulouse

Math. (6) 20 (2011), no. 1, 167–202 (English, with English and French summaries). MR2830396 ↑26, 170

[GPSR1987] S. Gelbart, I. Piatetski-Shapiro, and S. Rallis, Explicit constructions of automorphic L-functions, Lecture

Notes in Mathematics, vol. 1254, Springer-Verlag, Berlin, 1987. MR892097 (89k:11038) ↑2, 24, 174

[GZ1986] B. H. Gross and D. B. Zagier, Heegner points and derivatives of L-series, Invent. Math. 84 (1986), no. 2,

225–320, DOI 10.1007/BF01388809. MR833192 (87j:11057) ↑6, 155, 165

[Gro1986] B. H. Gross, On canonical and quasicanonical liftings, Invent. Math. 84 (1986), no. 2, 321–326, DOI

10.1007/BF01388810. MR833193 (87g:14051) ↑122

[Har1993] M. Harris, L-functions of 2× 2 unitary groups and factorization of periods of Hilbert modular forms, J.

Amer. Math. Soc. 6 (1993), no. 3, 637–719, DOI 10.2307/2152780. MR1186960 (93m:11043) ↑24

[HKS1996] M. Harris, S. S. Kudla, and W. J. Sweet, Theta dichotomy for unitary groups, J. Amer. Math. Soc. 9

(1996), no. 4, 941–1004, DOI 10.1090/S0894-0347-96-00198-1. MR1327161 (96m:11041) ↑3, 17, 23, 24, 26,

172, 174

[HT2001] M. Harris and R. Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of

Mathematics Studies, vol. 151, Princeton University Press, Princeton, NJ, 2001. With an appendix by

Vladimir G. Berkovich. MR1876802 (2002m:11050) ↑136, 139

[Hir1999] Y. Hironaka, Spherical functions and local densities on Hermitian forms, J. Math. Soc. Japan 51 (1999),

no. 3, 553–581, DOI 10.2969/jmsj/05130553. MR1691493 (2000c:11064) ↑130, 131

[How1989] R. Howe, Transcending classical invariant theory, J. Amer. Math. Soc. 2 (1989), no. 3, 535–552, DOI

10.2307/1990942. MR985172 (90k:22016) ↑27

[Ich2004] A. Ichino, A regularized Siegel–Weil formula for unitary groups, Math. Z. 247 (2004), no. 2, 241–277,

DOI 10.1007/s00209-003-0580-5. MR2064052 (2005g:11082) ↑3, 19, 64

[Ich2007] , On the Siegel–Weil formula for unitary groups, Math. Z. 255 (2007), no. 4, 721–729, DOI

10.1007/s00209-006-0045-8. MR2274532 (2007m:11063) ↑3, 19




































186

[Kar1979] M. L. Karel, Functional equations of Whittaker functions on p-adic groups, Amer. J. Math. 101 (1979),

no. 6, 1303–1325, DOI 10.2307/2374142. MR548883 (81b:10021) ↑29

[KM1985] N. M. Katz and B. Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol. 108,

Princeton University Press, Princeton, NJ, 1985. MR772569 (86i:11024) ↑119, 136

[Kot1992] R. E. Kottwitz, Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), no. 2,

373–444, DOI 10.2307/2152772. MR1124982 (93a:11053) ↑108

[Kud1997] S. S. Kudla, Central derivatives of Eisenstein series and height pairings, Ann. of Math. (2) 146 (1997),

no. 3, 545–646, DOI 10.2307/2952456. MR1491448 (99j:11047) ↑1, 17, 86, 88, 130

[Kud2002] , Derivatives of Eisenstein series and generating functions for arithmetic cycles, Asterisque 276

(2002), 341–368. Seminaire Bourbaki, Vol. 1999/2000. MR1886765 (2003a:11070) ↑1

[Kud2003] , Modular forms and arithmetic geometry, Current developments in mathematics, 2002, Int. Press,

Somerville, MA, 2003, pp. 135–179. MR2062318 (2005d:11086) ↑1, 56

[KM1986] S. S. Kudla and J. J. Millson, The theta correspondence and harmonic forms. I, Math. Ann. 274 (1986),

no. 3, 353–378, DOI 10.1007/BF01457221. MR842618 (88b:11023) ↑84, 87

[KM1990] , Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomor-

phic modular forms in several complex variables, Inst. Hautes Etudes Sci. Publ. Math. 71 (1990), 121–172.

MR1079646 (92e:11035) ↑49, 56

[KR1994] S. S. Kudla and S. Rallis, A regularized Siegel–Weil formula: the first term identity, Ann. of Math. (2)

140 (1994), no. 1, 1–80, DOI 10.2307/2118540. MR1289491 (95f:11036) ↑17, 28, 29, 34, 35, 64

[KR2005] , On first occurrence in the local theta correspondence, Automorphic representations, L-functions

and applications: progress and prospects, Ohio State Univ. Math. Res. Inst. Publ., vol. 11, de Gruyter,

Berlin, 2005, pp. 273–308. MR2192827 (2007d:22028) ↑22, 24, 170

[KR2000] S. S. Kudla and M. Rapoport, Height pairings on Shimura curves and p-adic uniformization, Invent.

Math. 142 (2000), no. 1, 153–223, DOI 10.1007/s002220000087. MR1784798 (2001j:11042) ↑146

[KR2011] , Special cycles on unitary Shimura varieties I. Unramified local theory, Invent. Math. 184 (2011),

no. 3, 629–682, DOI 10.1007/s00222-010-0298-z. MR2800697 ↑121, 122, 123, 124

[KRY2006] S. S. Kudla, M. Rapoport, and T. Yang, Modular forms and special cycles on Shimura curves, Annals of

Mathematics Studies, vol. 161, Princeton University Press, Princeton, NJ, 2006. MR2220359 (2007i:11084)

↑1, 6, 56

[KS1997] S. S. Kudla and W. J. Sweet Jr., Degenerate principal series representations for U(n, n), Israel J. Math.

98 (1997), 253–306, DOI 10.1007/BF02937337. MR1459856 (98h:22021) ↑14, 25, 171, 176

[Lar1992] M. J. Larsen, Arithmetic compactification of some Shimura surfaces, The zeta functions of Picard modular

surfaces, Univ. Montreal, Montreal, QC, 1992, pp. 31–45. MR1155225 (93d:14037) ↑50

[Lee1994] S. T. Lee, On some degenerate principal series representations of U(n, n), J. Funct. Anal. 126 (1994),

no. 2, 305–366, DOI 10.1006/jfan.1994.1150. MR1305072 (95j:22023) ↑25, 176

[Li1992] J.-S. Li, Nonvanishing theorems for the cohomology of certain arithmetic quotients, J. Reine Angew.

Math. 428 (1992), 177–217, DOI 10.1515/crll.1992.428.177. MR1166512 (93e:11067) ↑2, 24, 174


































187

[LST2011] J.-S. Li, B. Sun, and Y. Tian, The multiplicity one conjecture for local theta correspondences, Invent.

Math. 184 (2011), no. 1, 117–124, DOI 10.1007/s00222-010-0287-2. MR2782253 (2012b:22023) ↑26, 27

[Man1972] Ju. I. Manin, Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat. 36

(1972), 19–66 (Russian). MR0314846 (47 #3396) ↑66

[Mın2008] A. Mınguez, Correspondance de Howe explicite: paires duales de type II, Ann. Sci. Ec. Norm. Super. (4)

41 (2008), no. 5, 717–741 (French, with English and French summaries). MR2504432 (2010h:22024) ↑27

[MVW1987] C. Mœglin, M.-F. Vigneras, and J.-L. Waldspurger, Correspondances de Howe sur un corps p-adique,

Lecture Notes in Mathematics, vol. 1291, Springer-Verlag, Berlin, 1987 (French). MR1041060 (91f:11040)

↑171, 172

[Pau1998] A. Paul, Howe correspondence for real unitary groups, J. Funct. Anal. 159 (1998), no. 2, 384–431, DOI

10.1006/jfan.1998.3330. MR1658091 (2000m:22016) ↑26

[PSR1987] I. Piatetski-Shapiro and S. Rallis, Rankin triple L functions, Compositio Math. 64 (1987), no. 1, 31–115.

MR911357 (89k:11037) ↑35

[Ral1982] S. Rallis, Langlands’ functoriality and the Weil representation, Amer. J. Math. 104 (1982), no. 3, 469–515,

DOI 10.2307/2374151. MR658543 (84c:10025) ↑180, 182, 183

[Ral1984] S. Rallis, Injectivity properties of liftings associated to Weil representations, Compositio Math. 52 (1984),

no. 2, 139–169. MR750352 (86d:11038) ↑2, 178

[Ral1987] S. Rallis, L-functions and the oscillator representation, Lecture Notes in Mathematics, vol. 1245, Springer-

Verlag, Berlin, 1987. MR887329 (89b:11046) ↑28, 64, 101

[RZ1996] M. Rapoport and Th. Zink, Period spaces for p-divisible groups, Annals of Mathematics Studies, vol. 141,

Princeton University Press, Princeton, NJ, 1996. MR1393439 (97f:14023) ↑120, 147

[Shi1982] G. Shimura, Confluent hypergeometric functions on tube domains, Math. Ann. 260 (1982), no. 3, 269–302,

DOI 10.1007/BF01461465. MR669297 (84f:32040) ↑72, 73, 74, 75, 76, 77, 80

[Sou1992] C. Soule, Lectures on Arakelov geometry, Cambridge Studies in Advanced Mathematics, vol. 33, Cam-

bridge University Press, Cambridge, 1992. With the collaboration of D. Abramovich, J.-F. Burnol and J.

Kramer. MR1208731 (94e:14031) ↑60, 83

[Tan1999] V. Tan, Poles of Siegel Eisenstein series on U(n, n), Canad. J. Math. 51 (1999), no. 1, 164–175, DOI

10.4153/CJM-1999-010-4. MR1692899 (2000e:11073) ↑14, 29, 34

[Wal1990] J.-L. Waldspurger, Demonstration d’une conjecture de dualite de Howe dans le cas p-adique, p 6= 2,

sixtieth birthday, Part I (Ramat Aviv, 1989), Israel Math. Conf. Proc., vol. 2, Weizmann, Jerusalem,

1990, pp. 267–324 (French). MR1159105 (93h:22035) ↑26, 27

[Wal1988] N. R. Wallach, Lie algebra cohomology and holomorphic continuation of generalized Jacquet integrals,

Representations of Lie groups, Kyoto, Hiroshima, 1986, Adv. Stud. Pure Math., vol. 14, Academic Press,

Boston, MA, 1988, pp. 123–151. MR1039836 (91d:22014) ↑29

[YZZ2009] X. Yuan, S.-W. Zhang, and W. Zhang, The Gross–Kohnen–Zagier theorem over totally real fields, Compos.

Math. 145 (2009), no. 5, 1147–1162, DOI 10.1112/S0010437X08003734. MR2551992 (2011e:11109) ↑4,

46, 48, 49, 50

































188

[YZZa] , Gross–Zagier formula on Shimura curves, Annals of Mathematics Studies. to appear. ↑6, 8, 27,

38, 106, 155

[YZZb] , Triple product L-series and Gross–Schoen cycles I: split case, available at http://www.math.

columbia.edu/~szhang/papers/Preprints.htm. preprint. ↑27, 32

[Zha2001a] S.-W. Zhang, Heights of Heegner points on Shimura curves, Ann. of Math. (2) 153 (2001), no. 1, 27–147,

DOI 10.2307/2661372. MR1826411 (2002g:11081) ↑106, 155

[Zha2001b] , Gross–Zagier formula for GL2, Asian J. Math. 5 (2001), no. 2, 183–290. MR1868935

(2003k:11101) ↑106, 137, 155

[Zha2009] W. Zhang, Modularity of generating functions of special cycles on Shimura varieties, ProQuest LLC, Ann

Arbor, MI, 2009. Thesis (Ph.D.)–Columbia University. MR2717745 ↑4, 48

[Zin2001] T. Zink, Windows for displays of p-divisible groups, Moduli of abelian varieties (Texel Island, 1999),

Progr. Math., vol. 195, Birkhauser, Basel, 2001, pp. 491–518. MR1827031 (2002c:14073) ↑124

[Zin2002] , The display of a formal p-divisible group, Asterisque 278 (2002), 127–248. Cohomologies p-adiques

et applications arithmetiques, I. MR1922825 (2004b:14083) ↑124

[Zuc1982] S. Zucker, L2 cohomology of warped products and arithmetic groups, Invent. Math. 70 (1982/83), no. 2,

169–218, DOI 10.1007/BF01390727. MR684171 (86j:32063) ↑57

http://www.math.columbia.edu/~szhang/papers/Preprints.htm

http://www.math.columbia.edu/~szhang/papers/Preprints.htm












189

Index

L -admissible Green function, 61

0-dimensional Shimura variety, 106

admissible (E-subspace), 43

analytic kernel function, 31

archimedean local arithmetic Siegel–Weil for-

mula, 95

arithmetic Chow group, 60

arithmetic divisor, 60

flat, 60

principal, 60

arithmetic theta lifting, 56, 67

arithmetic theta series, 63

basic unitary datum, 115

Beilinson–Bloch height paring, 59

coherent hermitian space, 16

cohomologically trivial cycle, 60

compactified generating series, 54

positive partial, 54

compactified special cycle, 53

degenerate principal series, 14

flat arithmetic divisor, 60

generating series, 45

compactified, 54

positive partial compactified, 54

geometric kernel function, 67

good section, 23

Green function, 60

L -admissible, 61

height function on D, 86

height paring

Beilinson–Bloch, 59

Neron–Tate, 61

modified, 61

hermitian space, 9

hermitian line bundle, 61

hermitian space

nearby, 42

nondegenerate (skew-), 10

skew-, 9

hermitian space (over adeles)

coherent, 16

incoherent, 16

nondegenerate, 16

holomorphic projection, 155

quasi-, 157

incoherent hermitian space, 16

kernel function

analytic, 31

geometric, 67

local zeta integral, 23

modified height pairing, 61

moment matrix, 17

Neron–Tate height paring, 61

nearby hermitian space, 42

nearby quadratic space, 45

nondegenerate (skew-)hermitian space, 10

190

nondegenerate hermitian space (over adeles),

16

normalized zeta integral, 24

of weight (a,b), 16

ordinary (point), 112

positive partial compactified generating series,

54

principal arithmetic divisor, 60

quadratic space

nearby, 45

quasi-holomorphic projection, 157

regular test function, 32

section

good, 23

standard, 14

unramified, 14

Shimura variety

0-dimensional, 106

of orthogonal group, 45

of unitary group, 42

Siegel–Weil formula, 19

archimedean local arithmetic, 95

skew-hermitian space, 9

special cycle, 44

compactified, 53

standard section, 14

supersingular (point), 112

test function, 31

of discriminant d, 36

regular, 32

theta dichotomy, 25

theta function, 18

theta lifting, 21

unramified section, 14

Weil representation, 17

Whittaker integral, 28

zeta integral

local, 23

normalized, 24

191

Index of Notations

(−],−[), 111

A0(H ′), 155

Ak0(H ′ ×H ′), 155

Afin,A, 9

AK ,AK,fin,K∞, 9

A(g, φ), 68

A(H), 29

A(H ′, ζk), 155

Ak0(H ′), 155

α(f, f∨, φ, φ∨), 27

am(s), 34

A(p)(φ1, g1;φ2, g2), 138

(A, θ, i), 111

β(f, f∨, φ, φ∨), 27

bm(s), 24

CH1fin(M)C, 60

CH1(M)0C, 60

CH1(M)C, 63

CH1

C(M), 60

CH1

C(M)0, 60

CHr(Sh(H))C, 45

CHr(Sh∼)C, 54

CHr(X), 10

consts=a, 157

dc, 60

D(g, φ), 63

diag[a1, . . . , ar], 10

Diff(T,V), 32

D(ι), 42

(E, ϑE, jE), 115

E(g1, g2; Φ), 67

E(g, φ), 66

E(g, ϕs), 14

E(p)(φ1, g1;φ2, g2), 138

ε(π, χ), 26

ε(V ), 16

ε(V), 16

E(s, g, φ), 18

ET (s, g,Φ), 30

(E, ϑ, j; x), 115

fψ′,s, 156

γV , 17

H, 116

H†, 107

h†, 107

Herr, 13

Hpfin, 106

h(ι), 42

Hp(T ), 124

Hr, 13

H(x)∞, H(T )∞, 86

ı, 20

ı0, 20

ı, 17

ι(p), 104

Ir(s, χ), 14

ıχ,1, 17

IV (g, φ), 18

192

K±0 , 104

kχ, 15

Kp,n, 106

Kr, 13

Λ±, 104

λPr , 18

L†0,K†,pp ,K†,p

, 108

L†K†;p

, 108

LK , 106

LK ,L , 43

LK;p , 108

L ∼K ,L∼, 53

Ln,Kp , 106

Ls, 105

Ls, 105

L(s, π, χ), 24

M0,Kp , 111

[M0,Kp ]ss, 112

[M0,Kp ]∧ss, 120

MK , 67

MK;p , 68

Mn,Kp , 136, 139, 148

M′n,Kp , 136, 139


, 111


, 108

[M†0,K†,pp ,K†,p

]ss, 112

M†K†

, 107

M†K†;p

, 108

MK , 62

MK;p , 108

Mn,Kp , 106

Mor ((Y, jY), (X, iX))reg, 120

Mr, 13

N , 21

Nr, 13

N ,N ′, 120

OT , 28

Ω,Ω′, 147

ωχ, 17

ω′′χ, 21

ωK , 67

P , 21

φ, 17

φ0∞, 44

ϕφ,s, 18

Pic(M), 61

Pic(M)Q, 62

Pic(Sh(H)), 43

Pic(Sh∼), 53

Pic(X), 11

πK′

K , 43

$, 104

pM, 61

Pr, 13

Pr, 155

Pr, 157

ψT , 28

RFp,2, 120

R(V, χ), 25

Sh(H), 43

ShK(H), 42

Sh∼K ,Sh∼, 53

193

Σ, 13

Σfin, 13

Σ∞, 13

Σ, 13

Σfin, 13

Σ∞, 13

Σ(V), 16

S, 42

Std, 22

S(V 2n)d, 37

S(V 2n)reg,d, 37

S(V r), 17, 18

S(V r)reg, 32

S(V2nS )reg, 32

S(Vr)U∞ , 44

S(Vr)U∞K , 44

T 1F ′ , 104

TF , 104

Θfφ, 56

θfφ, 21

θ(g, h;φ), 18

Θφ(g), 67

θhor(p)(•;φ1, φ2), 144, 151

Θφ(g), 63

θver(p)(•;φ1, φ2, g2), 145, 153

TpA, 110

T (x), 17

UsL, 105

V , 116

V (ι), 42

Vol(K), 101

V ±, 16

Vs, 16

V(π, χ), 26

(−W ), 20

W ′′, 20

W ′,W′, 20

Wr, 13

WT (g, f), 29

WT (g, ϕs), 28

W kt , 155

WT (s, g,Φ), 29

wr, 10

wr,d, 10

(X, iX), 120

X, 111

X†, 107

X†, 111

Ξφ(g)ι′ , 102

Ξx, 86

(Y, jY), 115

(Yτ , jYτ ), 122

Y, 115

Z†(x†, h†)0,K†,pp ,K†,p , 117

Zφ(g)0,Kp , 121

[Zφ(g)0,Kp ]∧s , 121

Z(x)0,Kp , 117

Z(χ, f, f∨, ϕs), 23

ζk, 155

Z∗(χ, f, f∨, ϕs), 24

Z∗(s, χ, f, f∨,Φ), 40

Z∗(s, χ, f, f∨, φ⊗ φ∨), 26

194

(Z, gι), 60

Z1

C(M), 60

Zφ(g), 67

Zφ(g), 45

Z∼φ (g), 54

Z∼,+φ (g), 54

Z (x), 121

Z(V1)K , 44

Z(x)0,Kp;p , 116

Z(x)K , 44

Z(x)∼K , 53

Arithmetic inner product formula for unitary groupsyl2269/thesis_d.pdf · 2012-09-06 · Arithmetic...

Documents

Transcript of Arithmetic inner product formula for unitary groupsyl2269/thesis_d.pdf · 2012-09-06 · Arithmetic...