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© 2012Yifeng Liu

All rights reserved

Abstract

Arithmetic inner product formulafor unitary groups

Yifeng Liu

We study central derivatives of L-functions of cuspidal automorphic representationsfor unitary groups of even variables defined over a totally real number field, and theirrelation with the canonical height of special cycles on Shimura varieties attached tounitary groups of the same size. We formulate a precise conjecture about an arithmeticanalogue of the classical Rallis’ inner product formula, which we call arithmetic innerproduct 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 tocertain unitary groups of two variables.

For an irreducible cuspidal automorphic representation of a quasi-split unitarygroup, we can associate it an ε-factor, which is either 1 or −1, via the dichotomyphenomenon 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 modularitytheorem 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 considerthe 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 asa part of the local comparison of the conjectural arithmetic inner product formula.

Contents

Table of Contents i

Acknowledgments iv

1 Introduction 1

2 Doubling method and analytic kernel functions 82.1 Siegel–Weil formula and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Degenerate principal series and Eisenstein series . . . . . . . . . . . . . . . . . 92.1.2 Real quasi-split unitary groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.3 Weil representations and theta functions . . . . . . . . . . . . . . . . . . . . . . 112.1.4 Siegel–Weil formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Doubling integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.1 Decomposition of global period integrals . . . . . . . . . . . . . . . . . . . . . . 132.2.2 Local zeta integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Central special values of L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.1 Theta lifting and central L-values . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.2 Vanishing of central L-values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 Analytic kernel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.1 Regular test functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.2 Test functions of higher discriminant . . . . . . . . . . . . . . . . . . . . . . . . 242.4.3 Density of test functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Arithmetic theta lifting and arithmetic kernel functions 283.1 Modularity of generating series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.1 Shimura varieties of unitary groups . . . . . . . . . . . . . . . . . . . . . . . . . 293.1.2 Special cycles and generating series . . . . . . . . . . . . . . . . . . . . . . . . . 303.1.3 Pullback formula and modularity theorem . . . . . . . . . . . . . . . . . . . . . 31

3.2 Smooth compactification of unitary Shimura varieties . . . . . . . . . . . . . . . . . . 343.2.1 Compactified unitary Shimura varieties . . . . . . . . . . . . . . . . . . . . . . 343.2.2 Compactified generating series . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3 Arithmetic theta lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3.1 Cohomological triviality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3.2 Arithmetic inner product formula: the general conjecture . . . . . . . . . . . . 40

3.4 Arithmetic kernel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.4.1 Neron–Tate height pairing on curves . . . . . . . . . . . . . . . . . . . . . . . . 403.4.2 Degree of generating series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4.3 Decomposition of arithmetic kernel functions . . . . . . . . . . . . . . . . . . . 45

i

4 Comparison at infinite places 474.1 Archimedean Whittaker integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.1.1 Elementary reduction steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.1.2 Analytic continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.1.3 First-order derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2 Archimedean local height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.2.1 Green currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.2.2 Invariance under U(2): an exercise in Calculus . . . . . . . . . . . . . . . . . . 59

4.3 An archimedean local Siegel–Weil formula . . . . . . . . . . . . . . . . . . . . . . . . . 654.3.1 Comparison on the hermitian domain . . . . . . . . . . . . . . . . . . . . . . . 654.3.2 Proof of Lemma 4.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.3.3 Comparison on Shimura varieties . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5 Comparison at finite places: good reduction 725.1 Integral models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.1.1 Change of Shimura data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.1.2 Moduli interpretations and integral models: minimal level . . . . . . . . . . . . 745.1.3 Basic abelian scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.1.4 The nearby space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.1.5 Integral special subschemes: minimal level . . . . . . . . . . . . . . . . . . . . . 805.1.6 Remark on the case F = Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2 Local intersection numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.2.1 p-adic uniformization of supersingular locus . . . . . . . . . . . . . . . . . . . . 825.2.2 Special formal subschemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.2.3 A formula for local intersection multiplicity . . . . . . . . . . . . . . . . . . . . 845.2.4 Proof of Proposition 5.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.3 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.3.1 Non-archimedean Whittaker integrals . . . . . . . . . . . . . . . . . . . . . . . 895.3.2 Comparison on Shimura curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6 Comparison at finite places: bad reduction 936.1 Split case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.1.1 Integral models and ordinary reduction . . . . . . . . . . . . . . . . . . . . . . 946.1.2 Coherence for intersection numbers . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.2 Quasi-split case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.2.1 Integral models and supersingular reduction . . . . . . . . . . . . . . . . . . . . 966.2.2 Coherence for intersection numbers . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.3 Nonsplit case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.3.1 Integral models and Cerednik–Drinfeld uniformization: minimal level . . . . . . 1016.3.2 Integral models and Cerednik–Drinfeld uniformization: higher level . . . . . . . 1026.3.3 Coherence for intersection numbers . . . . . . . . . . . . . . . . . . . . . . . . . 103

7 Arithmetic inner product formula: the main theorem 1067.1 Holomorphic projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.1.1 Holomorphic and quasi-holomorphic projection: generality . . . . . . . . . . . . 1077.1.2 Siegel–Fourier expansion of Eisenstein series on U(2, 2) . . . . . . . . . . . . . . 1087.1.3 Holomorphic projection of analytic kernel functions . . . . . . . . . . . . . . . . 1107.1.4 Quasi-holomorphic projection of analytic kernel functions . . . . . . . . . . . . 112

7.2 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147.2.1 Difference of kernel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147.2.2 The final step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

ii

Appendix 116A.1 Theta dichotomy for unitary groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117A.2 Uniqueness of local invariant functionals . . . . . . . . . . . . . . . . . . . . . . . . . . 120A.3 Theta correspondence of unramified representations . . . . . . . . . . . . . . . . . . . 121

Bibliography 126Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Index of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

iii

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 subjectincluding the original problem, and his consistent encouragement and interest in this work. I amindebted to Xinyi Yuan, Shou-Wu Zhang and Wei Zhang who were kind enough to share with mesome of the ideas in their recent joint work, which are crucial to this one. I am also thankful to WeeTeck 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 usefulconversations during the long-term preparation.

I would like to thank Stephen S. Kudla and Joachim Schwermer for inviting me to the workshopAutomorphic Forms: New Directions at Mathematisches Forschungsinstitut Oberwolfach in March2011 and to present this work there; YoungJu Choie and Sug Woo Shin for inviting me to give a minicourse on this subject in the Theta Festival : The 4th ILJU School of Mathematics held at PohangUniversity of Science and Technology in August 2011. Without their wonderful organization andhospitality, I would have missed very good opportunities to communicate with other mathematicianson 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 summersof last four years. Especially in July 2008, I was able to attend a workshop on Arithmetic Geometryheld at the Center, where Jianshu Li and Shou-Wu Zhang gave two talks that directly motivated meto 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 Departmentof Mathematics, Columbia University for making it enjoyable.

Last but not least, I thank my family and my friends for their everlasting encouragement andsupport in every aspects of my life.

iv

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 theviewpoint 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 specialclass of algebraic varieties, called Shimura varieties, for which one can systematically construct a largesupply 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 specialcycles. 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 thetheta 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 productformula, which we call arithmetic inner product formula, and confirm it for unitary groups of twovariables. In particular, we calculate the Neron–Tate height of special points on Shimura curvesattached 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, whichfirst appeared in [Ral1984]. The original formula is to calculate the Petersson inner product of twoautomorphic forms on an orthogonal group that are lifted from a symplectic group through thetalifting. It turns out, using the Siegel–Weil formula, that the inner product is related to a diagonalintegral on the doubling symplectic group of the original automorphic forms with certain Eisensteinseries. This doubling method was later generalized to other cases by S. Gelbart, I. Piatetski-Shapiroand S. Rallis [GPSR1987]. This diagonal integral is in fact Eulerian. In other words, it decomposesinto so-called local zeta integrals. These local zeta integrals are directly related to the L-factors of thecorresponding 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 parallelto the arithmetic theory developed later. Let F be a totally real field, and E/F a totally imaginaryquadratic extension. For an integer n ≥ 1, let H ′ = U(n, n)F be the unique quasi-split unitary groupof a skew-hermitian space over E (with respect to the Galois involution of E/F ) of rank 2n. Let Hbe the unitary group of a hermitian space V over E of rank 2n. Both H ′ and H are reductive groupsover F . We have a Weil representation 1 ω of H ′(AF )×H(AF ), realizing on S(V (AE)n): the spaceof Schwartz functions on V (AE)n. For such an element φ ∈ S(V (AE)n), we have the following thetaseries

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 ′) bean irreducible representation of H ′(AF ) contained in the space of cusp forms of H ′. For every f ∈ πand 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[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 fromthe Satake parameter and the ramified part LS(π) is defined in [HKS1996] as a greatest commondivisor;

• 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.

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.

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

3

1. ε(π) = 1. Then we can choose a suitable hermitian space V such that the set of theta lifting θfφfor φ ∈ S(V (AE)n) and f ∈ π contains a nonzero function if and only if L( 1

2 , 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, wepropose an arithmetic theory where the central derivative of the L-function relates to the lifting onthe level of cycles. Such formulation will be elaborated in the next subsection.

Arithmetic inner product formula

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.From such π, we can construct, instead of the hermitian space V over E, a hermitian space (or rathera hermitian module) V over AE of rank 2n. Such V is incoherent in the sense that there does notexist 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 ofisometry ofV, which is a reductive group overAF . By the theory of Shimura variety, we attach toH (aprojective system of) Shimura varieties (Sh(H)K)K for open compact subgroups K ⊆ H(AF,fin). Theyare smooth quasi-projective varieties over SpecE. Let S(V(AE)n)U∞ ⊂ S(V(AE)n) be the subspaceof those Schwartz functions whose archimedean components are essentially the Gaussian. FollowingS. Kudla, for φ ∈ S(V(AE)n)U∞ , we define the generating series Zφ(g), which is a “function” onH ′(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 viewedas the arithmetic analogue of the classical theta series θ(g, •;φ), where the later is a “function” onH ′(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 onCHn(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 caseof 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 fromthe idea of W. Zhang [Zha2009]. Moreover, the proof for the case where the generating series hascodimension 1 reduces to the result in [YZZ2009].

Recall that in the case of classical theta lifting, we construct θfφ simply by taking the inner productof f and the theta series. In view of the above theorem, we have the following parallel definition. Wedefine 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 expressionby 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, weinvoke the conjectural Beilinson–Bloch height paring [Beı1987, Blo1984], which is, in this particularcase, a hermitian paring

〈−,−〉BB : CHn(Sh(H))0C × CHn(Sh(H))0

C → C.

4

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 Θf

φ ∈ CHn(Sh(H))0C. The first

issue will be discussed in 3.2.1, where we propose some constructions as well as some conjectures. Tosimplify the discussion in the introduction, we assume that Sh(H)K is already proper, which is thecase 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 productformula.

Conjecture (Arithmetic inner product formula, Conjecture 3.3.6). Let π be an irreducible cuspidalautomorphic 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 productformula (1.1).

The main result of the article is the following theorem, which justifies the above conjecture in thecase n = 1. We point out that when n = 1, Sh(H)K is a curve. Therefore, the Beilinson–Bloch heightparing 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 irreduciblecuspidal 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 ),

where S is a finite set of primes, and the local factors Z∗ are same to those in the Rallis inner productformula (1.1).

The first appearance of such arithmetic analogue of Rallis’ inner product formula is the mainresult of [KRY2006]. The authors studied the case where f is a new form of PGL2(Q) of weight 2 andsquare-free level. In particular, the corresponding Shimura curve where the height paring is taken onis 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 thatis 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 yearsago 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 therepresentation theory of the so-called restriction problem. As for this work, we generalize the formulasof 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 dealwith the problem for general n, not just 1, once we are able to do so.

5

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 principalseries 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 suchthat E′(0, g, φ1 ⊗ φ2) can be expressed as a sum of local terms indexed by (almost all) places of Fthat 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 actualderivative L′( 1

2 , 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 exceptin 2.4.3.

We introduce the Shimura varieties attached to certain unitary groups, their special cycles andgenerating series in Chapter 3. Moreover, we discuss the case of non-proper Shimura varieties, wherewe 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 functionsE(g1, g2;φ1 ⊗ φ2) that compute the Neron–Tate height paring of the arithmetic theta lifting. Thenmodulo certain volume factors and terms that are perpendicular to f and f∨, we have the followingdecomposition 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 maintheorem, we need to compare the analytic and arithmetic kernel functions place by place. There arethree 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 beformulated for general dimensions. We prove a formula which we call the archimedean local arithmeticSiegel–Weil formula. Let m ≥ 2 be an integer. Let T be a (nondegenerate) hermitian matrix inGLm(C) of signature (m − 1, 1). Let Φ0 be the Gaussian on V m, where V is the standard positivedefinite complex hermitian space of dimension m. On the one hand, we have the Whittaker integralWT (s, e,Φ0) that is a holomorphic function in s. It is not hard to see that WT (0, e,Φ0) = 0. Onthe other hand, we define an archimedean local intersection number H(T )∞, which is the volume ofthe open unit ball D in Cm−1 with respect to a star product of Green currents constructed fromKudla–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 )∞,

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 bythe 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 isnonsplit 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

6

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 theShimura curve at p. Let V + be the 2-dimensional hermitian space over E with a selfdual latticeΛ+, 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 Corollary5.3.2). 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 takinginner product with f and f∨. This is compatible with the analytic side since the corresponding termsare all zero. We are only able to make such argument when n = 1. Now we come to the final stageof 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 partcoincide with the corresponding part of the arithmetic kernel function. To prove the arithmetic innerproduct formula in the full generality, we apply the result of multiplicity one proved in A.2. The laststep is extremely crucial to make us able to avoid explicit computations at all bad places. We wouldlike to remark that such idea originally comes from [YZZa].

Conventions and notations

Convention 1.0.1. All rings will have a unit.

Notation 1.0.2. • We denote by Z the ring of (rational) integers. We let Q, R and C be thefields 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 changeR-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. componentaway from S) for the corresponding (decomposable) adelic object; −∞ (resp. −fin) means theinfinite/archimedean (resp. finite) part.

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

• The symbol Tr and Nm stand for the trace (resp. reduced trace) and norm (resp. reducednorm), respectively, if they are applied to fields or rings of adeles (resp. simple algebras). Thesymbol 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 ofm-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.

7

Definition 1.0.4. • Let R be a commutative ring and R′ an etale (commutative) algebra over Rof rank 2. Let τ : r 7→ rτ for r ∈ R′ be the nontrivial automorphism of R′ over R. For n ≥ 0, ahermitian (resp. skew-hermitian) space of rank n over R′ with respect to τ is a free module Vover 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′).

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 basisv1, . . . , 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 choiceof the basis.

• In practice, R will be a field and R′/R a (possibly split) extension of degree 2, or R the ring ofadeles of a number field and R′ that of a quadratic field extension. The involution τ will alwaysbe 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−d1d

−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 moduluscharacter of G. Precisely, for g ∈ G, we have the adjoint action Adg on the Lie algebra LieG. Take anonzero Haar measure dx on LieG. Then λG(g) = Ad∗g dx/dx.

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. Inpractice, X will be smooth over K. Therefore, CHr(X) can be canonically identified with theabelian 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, whichshould not be understood as the Picard scheme or Picard stack of X.

8

Chapter 2

Doubling method and analytickernel functions

The goal of this chapter is to introduce the analytic kernel functions. In 2.1, we review the Siegel–Weilformula and some of its generalization that are related to our problem. In 2.2, we review the theory ofI. Piatetski-Shapiro and S. Rallis on the doubling method. In particular, we deduce the Rallis’ innerproduct formula in certain cases from the doubling method and the Siegel–Weil formula. We alsointroduce the (normalized) local zeta integrals that will serve as the local terms in both classical andarithmetic inner product formula. In 2.3, we introduce the L-function and the formula representingit. We show that the central L-value of an automorphic representation vanishes if its global epsilonfactor equals −1. Then we introduce the analytic kernel functions that compute the L-derivatives. In2.4, 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 unramifiednonsplit finite places of F .

9

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 byτ the nontrivial element in Gal(E/F ) and εE/F : A×F /F

× → ±1 the quadratic character associatedby the class field theory. Let Σ (resp. Σfin, Σ∞) be the set of all places (resp. finite places, infiniteplaces) 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 respectto the involution τ , which is equipped with a skew-hermitian form 〈−,−〉 such that there is an E-basise1, . . . , 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 subgroupPr(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 characteristiczero; 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

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 toU(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 principalseries 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 ifits restriction to Kr is independent of s. It is called unramified if it takes value 1 on Kr.

10

Now we view F and E as number fields. For a (continuous) character χ of A×E that is trivial onE×, 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 astandard 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 quadraticextension C/R) of rank 2r with a basis e1, . . . , er; er+1, . . . , e2r, under which the skew-hermitianform 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 areductive 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 ofU(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 aninteger. 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×Ewhose 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). Leta, b be integers. We say π is of weight (a, b) if the minimal Kr-type of π is the characterdetadetb, i.e., its sends [k1, k2] to (det k1)a(det k2)b.

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∞.

11

2.1.3 Weil representations and theta functions

Let us review the classification of (nondegenerate) hermitian spaces. Let m ≥ 1 be an integer andv ∈ 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 hermitianspaces 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 overEv 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 classifiedby 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 basisunder 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 afinite set. Finally, we let ε(V) =

∏v∈Σ ε(Vv).

Definition 2.1.3 (Coherent/incoherent hermitian spaces). We say a (nondegenerate) hermitian spaceV over AE is coherent (resp. incoherent) if the cardinality of Σ(V) is even (resp. odd), i.e., ε(V) = 1(resp. −1).

By the Hasse principle, there is a hermitian space V over E such that V ∼= V ⊗F AF if and only ifV is coherent. These two terminologies are inspired from the coherent/incoherent collections of localquadratic 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 dimensionm 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 itsC×-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. Givena 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 havea Weil representation (with respect to ψ) ωχ = ωχ,ψ of Hr × H on the space S(V r). Explicitly, forφ ∈ 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),

12

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

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

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 moduluscharacter

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

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).

13

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 seriesis holomorphic at s0.

Remark 2.1.6. In Theorem 2.1.5 (3), if V is isotropic, we still have a (regularized) Siegel–Weilformula. However, since the theta integral IV (g, φ) is not necessarily convergent, a regularizationprocess must be applied. The inner product introduced in the next section also requires a regularizationprocess. Since the classical inner product formula is not the purpose of this article, we will alwaysassume that V is anisotropic for simplicity, or pretend that the regularization process has been appliedfor 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 )contained in A0(H ′): the space of cuspidal automorphic forms on H ′(AF ). We will not distinguish πwith its underlying space. Let π∨ be the contragredient representation which is realized on the spaceof 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 naturalembedding

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 b1c1 d1

), g2 =

(a2 b2c2 d2

), g∨ =

(1n

−1n

)g

(1n

−1n

)−1

14

and

ı0(g1, g2) =

a1 b1

a2 b2c1 d1

c2 d2

.

Notation 2.2.1. Let

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

W′

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

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 thelatter one is realized on the space S(V n) ⊗ S(V n). Here we realize the contragredient representationω∨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′vgets volume 1 for every v ∈ Σ. Similarly, for φ∨ ∈ S(V n) and f∨ ∈ π∨, we have θf

∨

φ∨ . The reader shouldbe careful that in the contragredient side, the Weil representation used to form the theta functionshould 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 respectto 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

15

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

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).

16

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 sectionsis 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 continuationto 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. Infact, 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 appliedto the archimedean case.

Now suppose E/F is unramified (including split) at v and ψ, χ, π are also unramified. Letf0 ∈ πK′ , f∨0 ∈ (π∨)K

′such that 〈f0, f

∨0 〉 = 1, and ϕ0

s be the unramified standard section. Then thecalculation 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].

17

For further discussion, we need to recall a result on degenerate principal series. In what follows, wewill use the notation H ′′ instead of U(W ′′) for short, and recall our embedding ı : H ′×H ′ → H ′′. LetV be a hermitian space of dimension 2n over E. Then ϕφ(g) = ωχ(g)φ(0) defines an H ′′-intertwiningmap S(V 2n) → I2n(0, χ) whose image R(V, χ) is isomorphic to S(V 2n)H . Recall in 2.1.3 that wedenote 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 theL-function of the representation π.

Recall that we have an irreducible unitary cuspidal automorphic representation π of H ′ = Hn anda hermitian space V over E of dimension 2n. One key question in the theory of theta lifting is whetherθfφ is nonvanishing. A sufficient condition is to look at the local invariant functional as follows. Wehave 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).

Proof. If v is (real) archimedean, it is due to [Pau1998, Theorem 2.9]. If v is non-archimedean, it isdue 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 holomorphicat the point s = 0. Namely, we have

〈θfφ, θf∨

φ∨〉H =L( 1

2 , π, χ)

2∏2ni=1 L(i, εiE/F )

∏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 theunramified calculation. In particular, for v 6∈ S, Vv ∼= V (πv, χv). Then one necessary condition for θfφto 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 overAE 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

18

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 isheuristic for the arithmetic case. For this purpose, let us assume the following conjecture proposedby 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-archimedeanand has odd residue characteristic; by Li–Sun–Tian [LST2011] for every non-archimedean placev; 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 forevery 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 hasresidue 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)

2∏2ni=1 L(i, εiE/F )

.

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].

19

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.

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

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

20

Lemma 2.3.6. Assume that v is a finite place; E/F , ψ and χ are all unramified; and V = V +. LetΦ0 be the characteristic function of (Λ+)2n for a selfdual OE-lattice Λ+, and T ∈ Her2n(OF ) withdetT ∈ 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 formsof 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)

for s near 0. It is an H ′′(AF )-intertwining map exactly when s = 0. Then for T nonsingular (and snear 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 vanishesfor 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) withnonzero projection in R(V, χ) such that Φv,0 ∈ S(V2n

v )reg (cf. Definition 2.4.1).For every gv ∈ evH

′′(AvF ), the functional Φv 7→ WT (0, gv,ΦvΦv) 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 anonsingular 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, whichis a contradiction.

2. Assume that E and E ′ are both nontrivial intertwining maps. By Lemma 2.3.7, there is anonsingular T such that WT (g, •) E does not vanish. By Lemma 2.3.4 (1-b)(2), there existsc ∈ C such that WT (g, •) E ′ = cWT (g, •) E . Furthermore, c is independent of nonsingular Tsince all of those that can be represented by V are in a single M(F )-orbit under the conjugationaction 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 choosea nonsingular T ∈ Her2n(F ) that is representable by V. By (2.12), Lemma 2.3.5 (2) and Lemma2.3.6, we can find a suitable Φ such that WT (0, e,Φ) 6= 0. Therefore, E0 6= 0.

21

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

[H′(F )\H′(AF )]2f(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( 1

2 , 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)

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′).

22

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 ∈ Σsuch that T is not represented by Vv, i.e., there does not exist x1, . . . , x2n ∈ Vv whose moment matrixis 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 gin 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 testfunction 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 thefollowing proposition.

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

w2n,d =

1d

12n−d1d

−12n−d

(2.17)

23

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

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

v . 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 allinfinite 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 )

and

bm,v(s) =

m−1∏i=0

Lv(2s+m− i, εiE/F )

24

as (2.6). Therefore, WT (e, ϕS′

s ) has a meromorphic continuation to the entire complex plane. Forv ∈ 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 tothe entire complex plane. By Lemma 2.3.5 (1), WT (e, ϕv,s) and hence W ∗T (e, ϕv,s) have meromorphiccontinuation to the entire complex plane. Together with the meromorphic continuation of WT awayfrom 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 ofLemma 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

is a nontrivial N -intertwining map from I2n(0, χ) to CN,ψT . Now if v ∈ S, ϕv,0 = ϕΦv,0 for aregular test function Φv ∈ S(V2n

v )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 allnonsingular 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 thissubsection. 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

25

• 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 = ∅,

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 wewant to find such a T . Take an element T1 ∈ H with detT1 6= 0. Since H is open, we can find aneighborhood 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 thecharacter ψT is trivial on N(OF ) ∼= Her2n(OE) for all T ∈ Her2n(p−dE ). We slightly abuse notationsince dψ depends also on 2n. We also need to mention that this is not the usual discriminant ordifferent of a p-adic additive character. However, the difference between them depends only on n andthe 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,

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 definitionsand results above can be applied to arbitrary dimensions.

26

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

v )reg for v ∈ S and Φv ∈ S(V2nv )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 kernelfunction. However, for our proof of the main theorem, not arbitrary choices will work. We now showthat there are “sufficiently many” test functions satisfying these choices we have made, in the senseof 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 fromnotations. 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 onS(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 Her0

2n(E)\H′d is open and⋃d

(Her0

2n(E)\H′d)

= Her02n(E)\

⋂d

H′d = Her02n(E)\H′ = H,

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 andA : S(V )→ π a surjective H ′-intertwining map, where H ′ acts on S(V ) through a Weil representationω. 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 everyn ∈ 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 ofdimension 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 elementΦ = φ1 ⊗ φ2 ∈ S(V 2)reg such that φα ∈ S(V )reg (α = 1, 2) and B(Φ) 6= 0.

27

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. ByLemma 2.4.8, we can also assume that φ1 ∈ S(V )reg. For x ∈ Supp(φ1), let Vx be the subspace of Vgenerated by x and V x its orthogonal complement. They are both nondegenerate hermitian spaces ofdimension 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 asH ′-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)

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 conditionthat 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 ofH ′ 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 2

v )reg,d (resp. S(V 2v )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.

28

Chapter 3

Arithmetic theta lifting andarithmetic kernel functions

In this chapter, we study the geometric part of the arithmetic inner product. In 3.1, we introducethe Shimura varieties of unitary groups which we work with, and on them the special cycles andgenerating series. We prove the theorem of modularity of the generating series. For the case whereShimura varieties are not proper, we need to consider their compactifications, which will be discussedin 3.2. In 3.3, we define the arithmetic theta lifting and prove its cohomological triviality, which enableus to formulate the explicit conjecture of the arithmetic inner product formula. Finally, we restrictourselves 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 arithmetickernel 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.

29

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 beintegers. 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 compactsubgroup K of H(Afin), there is a Shimura variety ShK(H) of dimension m− 1 defined over the reflexfield 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 identifyingH(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 theprincipal 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

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 anelement

L ∈ Pic(Sh(H))Q := lim−→K

Pic(ShK(H))Q.

30

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 elementh ∈ 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 admissibleand fix arbitrary ι. Take v1 ∈ V1 with nonzero norm. Then q(v1) = 1

2 (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 Theoremto find an element h1 ∈ U(Vfin) = H(Afin) such that h1v1 = v as elements in Vfin. Choose avector v′ ∈ 〈v〉⊥ ⊂ V (ι) with nonzero norm. Let h′ ∈ H(Afin) which stabilizes 〈v′〉⊥ and act onthe Afin,E-line spanned by v′ through the multiplication by (deth1)−1. Then h′h1v1 = h′v = v forh = h′h1 ∈ SU(Vfin) = Hder(Afin).

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

∼= V ⊥1 ⊂ Vfin. Let H1 be the corresponding unitary group, we have a finite morphism ofShimura varieties

ςV1 : ShK1(H1)→ ShK(H) (3.2)

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

fin, 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(Vr

ι )Uι ⊂ S(Vr

ι ) 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).

31

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 denoteφ(T (x), x) = φ∞(y)φfin(x) for every y ∈ Vr

∞ with T (y) = T (x), whose value does not depend onthe choice of y. This makes sense since Z(x)K 6= 0 only for Vx admissible and hence T (x) is totallysemi-positive definite. It is easy to see that Zφ(g) is compatible under the pullback of πK

′

K , hencedefines a series with values in CHr(Sh(H))C := lim−→K

CHr(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 -moduleV 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 forevery 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 identifyingG(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 ∈ Vrfin, and Z ′φ(g′) for φ ∈ S(Vr)O∞K′ and g′ ∈ Gr(Afin),

respectively (cf. [YZZ2009]). Here we introduce the standard symplectic F -space W ′r (comparing tothe 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, weuse 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 nearbyobjects like V = V (ι), H = H(ι), D = D(ι), etc. Therefore, we have the usual notion of Shimuravariety ShK(H,X) (resp. ShK′(G,X

′) with a connected component X+ of X ′) which is defined overι(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 isdefined over EK (resp. EK′): a finite abelian extension of ι(F ) in C. The canonical embeddingHder → Gder (cf. Remark 3.1.4 (1) below) of reductive groups and the embedding D → D′ by

32

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 theclasses 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 compatibilityof 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) andh ∈ γG(Afin)x′1gk

′ for some k′ ∈ K ′, where G(Afin)x′1 is the subgroup of G(Afin) fixing x′1. Wenow show that γG(Afin)x′1gk

′ ∩ Hder(Afin) = G(Afin)γx′1γgk′ ∩ Hder(Afin) 6= ∅, i.e., G(Afin)γx′1 ∩

Hder(Afin)k′−1g−1γ−1 6= ∅, which is true by Lemma 3.1.2. Therefore, (z, h) represents a C-point inthe 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 holdson 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 imageis actually contained in Gder.

2. In the proof of Proposition 3.1.3, we can still use the adelic description of the C-points ofShK(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). Thefollowing 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, thearchimedean 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).

33

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 ofl(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 provefor 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)

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

(ωχ(wr,r−1)ωχ(g)φ

y1,y2)(T (x, y1 + y2), (x, y1 + y2))ςVx∗ Z(y1)Kx

=∑

x∈K\Vr−1fin

∑y1∈Kx\Vxfin

∑y2∈Vx

(ωχ(wr,r−1)ωχ(g)φ

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).

34

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 suppressthem as in the previous discussion. It is clear that we only need to prove Zφ(w1g) = Zφ(g) forg ∈ G1(AF ) since G1(A∞,F )K1,∞ = H1(A∞,F ). As before, we assume that φ∞ is the Gaussianand K is sufficiently small. Recall that π0(ShK(H,X)ι,C) ∼= T (Q)\T (Afin)/ det(K). We have thefollowing 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 theprojection to CH1(ShK(H,X)t)C. Here, Zφ(g) is the generating series on ShKh(Hder, X). Nowshrinking Kh if necessary such that we can apply Corollary 3.1.5, we have Zφ(g) = i∗K′Z

′φ(g) for g ∈

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 notproper, 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 positiveinteger D, OE its ring of integers and τ the nontrivial Galois involution on E. Let (V, (−,−)) be ahermitian space of dimension m over E of signature (m − 1, 1). If m = 2, we further assume thatdetV ∈ 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

. . .

1z/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. Whenm = 2, it is easy since we only need to add cusps. When m = 3 and H is quasi-split, a canonicalsmooth compactification (even of the integral model) has been constructed in [Lar1992]. In fact, thesame 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 thata ≥ b > 1 or V is over a totally real field but not Q and indefinite at every archimedean place, thenwe are not clear whether there exists a canonical smooth compactification (in a suitable sense).

35

Now let us assume that m > 2. In order to to use modular interpretations, we should work withthe group of unitary similitude. For every v, w ∈ V , the paring

(v, w)′ = TrE/Q(j(v, w))

defines a symplectic form of V satisfying (ev, w)′ = (v, eτw)′ for every e ∈ E. Let GH = GU(V ) suchthat 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

. . .

zz

.

For every sufficiently small open compact subgroup K of GH(Afin), we have the Shimura varietyShK(GH,Gh) which is smooth, quasi-projective but non-proper over E of dimension m−1. Althoughwe do not have a map of Shimura data, ShK∩H(Afin)(H,h) and ShK(GH,Gh) have the same neutralcomponent for sufficiently small K. Therefore, it is same to give a canonical smooth compactificationof 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 constantsection 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 ofS, η is a π1(S, s)-invariant K-class of OE⊗Z Z-linear symplectic similitude η : V

Z→ 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.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 andeach geometric component is isomorphic to an extension of an abelian variety of dimension m−2by a finite group.

36

The boundary part GYK parameterizes the degeneration of abelian varieties with above PELdata. We consider a semiabelian variety G over an algebraically closed field of characteristic 0 withi : OE → End(G) such that tr(i(e); Lie(G)) = (m − 1)e + eτ . For every e ∈ OE , we have followingcommutative 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 andabelian 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 onlyone possibility, namely a = b = 1 and r = 2. Then T is of rank 2 and A is an abelian variety ofdimension m− 2 such that OE acts on A and tr(i(e); Lie(A)) = (m− 2)e. Let A1 be an elliptic curveof CM type (OE , e 7→ e). Then A is isogenous to Am−2

1 . Each geometric point s of GYK correspondsto a semiabelian variety Gs = (Ts → Gs → As) as above with certain level structure which will bedefined later. For two geometric points s, s′ in the same geometric connected component, the abelianvariety part As ∼= As′ and the rank 1 OE-modules X(Ts) and X(Ts′) are isomorphic. It is easy to seethat if A and T are fixed, then the set of such G, up to isomorphism, is parameterized by X(T )⊗OE Awhich is an abelian variety of dimension m− 2.

To include the level structure, we consider only one geometric component since it is same forothers. This means that we fix T and A with OE-actions but, of course, not G. Let us fix a maximalisotropic 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 onthe 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 andVW,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 GYKis isomorphic to (a connected component of) an extension of X(T ) ⊗OE A by VW /VW,H for some Tand A as above. There is a universal object π : G→ Sh∼K(GH,Gh) which is a semiabelian scheme ofrelative 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 andYK = Sh∼K −ShK . They have the same properties as above. We apply the above notations also tothe trivial case m = 2. Let L ∼K be the line bundle on Sh∼K induced from

∧mπ∗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 thecompactification, (L ∼K )K defines an element in Pic(Sh∼) := lim−→K

Pic(Sh∼K). We also need to extendspecial cycles and the generating series. For 1 ≤ r < m and x ∈ V rfin := V r ⊗Q Afin, we define thecompactified special cycle to be

Z(x)∼K =

Z(Vx)∼Kc1(L ∼,∨K )r−dimE Vx if Vx is admissible

0 otherwise

where Z(Vx)∼K is just the Zariski closure of Z(Vx)K in Sh∼K . We define the compactified generatingseries 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,

37

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 followingpositive 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 proposethe 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 anautomorphic 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 wepoint 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),

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 smoothdivisor, 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 byIH•(Sh#,C) = lim−→K

IH•(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).

38

3.3 Arithmetic theta lifting

We assume Conjecture 3.2.1 and the following assumptions on A-packets, which are a certain part ofthe 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.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 thesame unramified representation, then Π1 = Π2.

• Let U(m)∗ be the quasi-split unitary group. Then we have the correspondence of A-packets ofinner 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 bookof 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 Shimuravarieties.

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 constructedthe 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 acocycle on the canonical model of Shimura variety, while theirs (just in the case of Shimura curves) isan Arakelov divisor on certain integral model, which is not canonically defined. It is more natural todefine arithmetic theta lifting just as a cocycle on the generic fiber in order to consider the canonicalheight.

In the following discussion, let m = 2n and r = n. Let π be an irreducible cuspidal automorphicrepresentation 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 (underωχ) is the trivial representation of U(2n, 0)R for every archimedean place ι. Therefore, V(π, χ) isa totally positive definite incoherent hermitian space over AE . Now we fix an incoherent hermitianspace 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 theclass 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.

39

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(Θf

φ) 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

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 writex = e

eτ 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 byCorollary A.3.6, we have that for v as above, JL(Σ)v = JL(Σv) = Σv and Πv ⊗ χv contain the sameunramified 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 weprove 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 andφ as above.

When n = 1, this follows from Corollary 3.4.2 by just computing the degree of the generatingseries 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.

40

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) forsufficiently 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 anonzero 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 conjecturethat the Beilinson–Bloch height pairing is non-degenerate, every arithmetic theta lifting Θf

φ = 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 automorphicrepresentation 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∨ andevery 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 classicalNeron–Tate height paring which will be recalled in 3.4.1, hence is defined unconditionally.

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 andsome 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)0

C be the group of cohomologically trivial cycleswhich 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 andprojective 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 anonzero rational function f on M, we define the associated principal arithmetic divisor to be

div(f) =(div(f),− log |fι,C|2

).

41

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 theorthogonal 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) consistingof 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 independentof the choices of preimages and also the regular model M.

In practice, the cycles we are interested in are not automatically cohomologically trivial. We needto make some modifications with respect to some auxiliary data. This is quite easy if we are workingover a curve. Let Pic(M) be the abelian group of isomorphism classes of hermitian line bundles onM . Recall that a hermitian line bundle is the data L = (L , ‖•‖ι), where L ∈ Pic(M) and ‖•‖ι isa (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 Ziwhen computing via (3.10). We say that the Green function gι of Z is L -admissible if the followingequalities 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 holomorphicline bundle (Lι,C, ‖ • ‖ι) which is a (1, 1)-form.

42

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 totallypositive definite hermitian space over AE of rank 2 and we write H ′ = H1. Let H = ResAF /AU(V) bethe associated unitary group over A. For every (sufficiently small; Assumption 3.1.1) open compactsubgroup K of H(Afin), there is a Shimura curve ShK(H), smooth over the reflex field E. For everyembedding 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, wecan compactify them by add cusps. We denote by MK the compactified (resp. original) Shimura curveif ShK(H) is non-proper (resp. proper), and M the projective system (MK)K with respect to theprojections 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−→K

Pic(MK)Q.

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 ), whereW0(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 beingnon-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 computeare the generating series Z∼φ (g) which are not necessarily cohomologically trivial. We use the dualHodge 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 andg ∈ 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′)

43

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

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 onA×,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 Whittakerintegral 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) = bis totally positive. Without lost of generality, let us assume that x is contained in the image ofsome (rational) nearby hermitian space V (ι) → Vfin and K ′ is sufficiently small. The isomorphism

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)

44

for b totally positive, and Db(g, φ) = 0 otherwise.On the other hand, Eb(s, g, φ) is holomorphic at s = 1

2 for b 6= 0. For b not totally positive,Eb(s, g, φ)|s= 1

2= 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),

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 thatD = 1.

• If ShK(H) is not proper, we calculate the degree of the Hodge bundle as the classical way onmodular 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).

ThenΘφ(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 combinationof an Eisenstein series and an automorphic character.

Proposition 3.4.1 implies the following corollary on the modularity of the generating series in thecompactified case.

Corollary 3.4.2. Let ` be a linear functional of CH1(M)C. Then `(Z∼φ )(g) and hence `(Θφ)(g) areabsolutely convergent and are automorphic forms of H ′.

Proof. Assume that φ is invariant under K ⊆ H(Afin). We only need to prove that the identityZ∼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

45

supported on the set of cusps. By a theorem of Manin–Drinfeld (cf. [Man1972,Dri1973]) which positsthat every two cusps are same in CH1(MK)C, we have an exact sequence

C→ CH1(MK)C → CH1(ShK(H))C → 0.

Therefore, we only need to prove that degZ∼φ (γg) = degZ∼φ (g), which is true by the Proposition3.4.1.

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 thefollowing 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′ forsome 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).

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 notrequire any canonicality of p−1

MK. 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)

46

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 andgα (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)

47

Chapter 4

Comparison at infinite places

We compare local terms of analytic and arithmetic kernel functions at an archimedean place. Section4.1 is dedicated to the computation on the analytic side. We calculate certain Whittaker integral andits derivative, following the method of G. Shimura. In 4.2, we introduce a local height function on thehermitian domain in terms of the Kudla–Millson form, and prove an important invariance propertyof such height. The actual comparison of the derivative of Whittaker integrals and the local heightfunction is accomplished in 4.3.

48

4.1 Archimedean Whittaker integrals

In this section, we calculate the Whittaker integral WT (s, g,Φ) and its derivative (at s = 0) at anarchimedean place. In particular, we fix archimedean places ι : F → C and ι′ : E → C over ι thatwill 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 positivedefinite 2n-dimensional complex hermitian space, and Φ0 ∈ S(V 2n) the Gaussian. We have also acharacter 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 theadditive 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)

Sincewn(u)m(a) = wm(a)n(a−1u ta−1) = m(ta−1)wn(a−1u ta−1),

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 thecase of even dimension. In other words, V will be the standard positive definite complex hermitianspace of dimension m > 0. For T ∈ Herm(C), the Whittaker integral WT (s, e,Φ0) is absolutelyconvergent for Re s > m

2 .

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)

49

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 theselfdual 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)

(i1m1m

)=

(1m

−1m

)(1m u

1m

)(i1m1m

)=

(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 fromthose 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

50

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

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)

51

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,

whereXp,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).

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 theproduct 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, andsatisfies 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),

we haveords=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 ).

52

4.1.3 First-order derivatives

By Proposition 4.1.5 (1), the T -th coefficient will not contribute to the analytic kernel functionE′(0, g,Φ) if signT = (p, q) with q ≥ 2. Therefore, we focus on the case where q = 1, and studythe 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 ztz 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−mdet

(x ztz y + 1

)β−mdxdydz,

(4.14)

where we apply the selfdual measure dx on Herm−1(C), the Lebesgue measure dy on R, and themeasure dz that is 2m−1 times the Lebesgue measure on Matm−1,1(C). We make change of variablesas 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 opensubset. Since

det

(x+ 1m−1 z

tz y

)= det(u+ 1m−1 + w tw)v det(1m−1w

tw)−1;

det

(x ztz y + 1

)= (v + 1 + tww)(detu) det(1m−1w

tw)−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)

1Two letters g appearing here stand for different meanings.

53

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

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).

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)

54

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

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 ,

55

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

are both polynomials in s, we need only to prove for s < 0. We have(1n −

√−sc1

)(1n

√−sc√

−s tc 1

)(1n

−√−s tc 1

)=

(1n + sc tc

1

),

and (1n

−√−s tc 1

)(1n

√−sc√

−s tc 1

)(1n −

√−sc1

)=

(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) ,

where the sum is taken over all subsets of 1, . . . ,m− 1.

56

Proof. Letujk = − (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 inK. 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 subsetJ = j1 < · · · < jt ⊂ 1, . . . ,m−1, let εJ(σ) ∈ ±1 be a factor that depends only on J and σ. Thisfactor comes from the combinatorics in taking successive partial derivatives. In later calculation, weonly need to know its value in the case where σ maps J to itself. Then, if we let σJ be the restrictionof 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

∣∣∣ .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.

57

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 contributeto the local height pairing at an archimedean place. We also prove some properties of such height. Abasic reference for archimedean Green currents and height paring is [Sou1992, Chapter II]. We keepthe notations in 4.1. We fix an integer m ≥ 2.

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 ′), introducedin 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 andonly if T (x) is positive definite. Suppose r = 1, for z ∈ Dm−1, let x = xz + xz be the orthogonaldecomposition with respect to the line z, i.e., xz ∈ z and xz ⊥ z. Let R(x, z) = −(xz, xz) that isnonnegative since z is negative definite, and R(x, z) = 0 if and only if x = 0 or z ∈ Dx. Explicitly, letx = (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 logarithmicgrowth 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)]

as currents on Dm−1.

58

We will only give a proof for m = 2, and the proof for general m is similar but involves tediouscomputations.

Proof. We start from showing that ddcξ(x) = ω(x) holds away from Dx. Let x = (x1, x2) andz ∈ 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).

The remaining discussion is same as in the proof of [Kud1997, Proposition 11.1], from Lemma 11.2on 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 weconsider x = (x1, . . . , xr) ∈ V ′r with nonsingular moment matrix T (x). Then using the star productof 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)].

59

Definition 4.2.2 (Height functions (on D)). For x = (x1, . . . , xm) ∈ V ′m with nonsingular momentmatrix 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 thusdepends only on the (nonsingular) moment matrix T (x). Sometimes we simply write H(T )∞ for thisfunction.

The following proposition claims that H(T )∞ is in fact invariant under the conjugation action ofU(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 ≥ 3and the proposition holds for m − 1. Since U(m)R is generated by diagonal matrices, permutationmatrices, 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

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 mm d2

)with d1, d2 ∈ R and m ∈ C. Choose a complex number ε with norm 1 such that ε2m ∈ R. Then(

εε−1

)(d1 mm d2

) t(ε

ε−1

)=

(d1 ε2mε2m 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 ofProposition 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 )∞.

60

Suppose T = diag[a,−b] with a, b > 0,

T [θ] =

(d1,θ mθ

mθ d2,θ

)∈ Sym2(R).

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 havedxθdθ

= −yθ;dyθdθ

= xθ; H(T [θ])∞ = H((xθ, yθ))∞.

Letzx,θ =

x2,θ

x1,θ; zy,θ =

y2,θ

y1,θ.

ThenDxθ = [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)).

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 .

61

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 MM 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))

),

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. Inother 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 .

62

By (4.32) and (4.33), the integral I ′(T [θ]) is independent of θ. There we only need to consider thesecond 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 ;

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 haved

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) pointszx,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;

63

• 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 thefact that exp(−2π(R1 +R2)) decays rapidly as |z| goes to 1, we have∫

[θ0,θ1]×Dεα∗(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, theintegral is zero.

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.

64

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 forany 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)

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)) ,

65

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), otherthan (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.

4.3 An archimedean local Siegel–Weil formula

In this section, we set up a relation between derivatives of Whittaker integrals and the height functionsdefined in the previous section. Furthermore, we prove a local arithmetic analogue of the Siegel–Weilformula 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)

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)

66

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)

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 onDm−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. Thetheorem 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 followingequality 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.

67

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 .

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;

andσj ∧ 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∣∣ = zjzkand

∣∣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−3

Ω 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.

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)

68

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 correspondingterms 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 qinside 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.

69

4.3.3 Comparison on Shimura varieties

We use previous results to compute the archimedean local height paring on the unitary Shimuravarieties with respect to suitable Green currents. We recall some notations from 3.1.1 and 3.1.2. Letn ≥ 1 be an integer. We have a totally positive definition incoherent hermitian space V over AEof 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∈Σfin

Kp is decomposable. Let

φα = φ0∞⊗φα,fin (α = 1, 2) be decomposable Schwartz functions with φα,fin ∈ S(Vn

fin)K . Suppose thatφ1,p ⊗ φ2,p ∈ S(V2n

p )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 ).

Definition 4.3.3 (Volume of open compact). For every finite place p of F , we define a measure dhon 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(V2n

p ),

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, thevolume 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 allelements in VxT . In other words, if we identify D with D2n−1, then z is in DxT . Choose a set ofrepresentatives 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. Thenit 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 Iwasawadecomposition 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 translationunder hi (i = 2, . . . , l), we obtain the a Green current Ξφ(g)ι′ for Zφ(g)ι′ . The following is our maintheorem of this chapter.

70

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 placeι′.

Proof. It is clear that we can assume that gα,ι = m(aα) with aα ∈ GLn(Eι′) for α = 1, 2. Then wehave

〈(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

∗

∑x2∈V n,T (x2)∈Her+n (E)

(ωχ(g2)φ2) (T (x2), h−1i x2)Ξx2a2

=

l∑i=1

∑x1∈V n,T (x1)∈Her+n (E)

∑x2∈V n,T (x2)∈Her+n (E)

(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

).

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)

71

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.

72

Chapter 5

Comparison at finite places: goodreduction

We compare local terms of analytic and arithmetic kernel functions at an unramified finite place. In5.1, we introduce the integral models for the Shimura curves, and extend the special cycles and thegenerating series to the models. At a good place, we need to calculate certain intersection multiplicityon the smooth model. This is done in 5.2. The computation of the derivative of Whittaker integralsat 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 afinite 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-hermitianspace with ε(V ±) = ±1, which is unique up to isometry and H± = U(V ±). Let Λ± be a maximalOE-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 extensionF ′/F , where F ′×,1 = ker (Nm : F ′× → F×). If F is totally real, let F+ be the set of all totallypositive 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 unramifiedextension 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 byOLs

.

• We fix an algebraic closure of Fp to be F.

73

5.1 Integral models

In the next four subsections, we will assume that F 6= Q. The rest case is slightly different and infact 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 assumethat the embedding ι(p) ι1 : E → Cp induces the place p. As before, we suppress ι1 and ι1 for thenearby objects. We have the hermitian space V over E of dimension 2 whose signature is (1, 1) atι1 and (2, 0) elsewhere, the unitary group H over Q, and the Shimura curve MK = ShK(H,X) for asufficiently small open compact subgroup K ⊂ H(Afin), which is a smooth projective curve definedover 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 determinantmap. Then we have the 0-dimensional Shimura variety LK = Shν(K)(T

1E , ν(X)), and a smooth

morphism (also denoted by) ν : MK → LK of ι1(E)-schemes such that the fiber of each geometricpoint is connected.

Let us define a subgroup Kp,n of U(Vp) for every integer n ≥ 0. Since Vp is (isometric to) eitherV + or V −, we have the lattice Λ± (if p is split, we only have the positive one). We define Kp,n to bethe subgroup of K±0 consisting of elements that have trivial action on Λ±/$nΛ±. Then Kp,0 = K±0is 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 datain order to obtain the moduli interpretations and integral models. This is analogous to the caseconsidered in [Car1986, YZZa, Zha2001a, Zha2001b], and we refer the detailed proof of various factsto [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 thequadratic form TrE/F (−,−), where (−,−) is the hermitian form on V . More precisely, when v isfinite, 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 henceV has both left and right multiplication by B. We fix an embedding E → B, through which theaction of E induced from the left multiplication of B coincides with the E-vector space structure ofV . Let G = ResF/QB

× with center T ∼= TF and

G† = G×T TF †ν†−→ T × T 1

F † ,

whereν†(g × z) =

(Nm g · zzτ

†,z

zτ†

).

74

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 spaceof dimension 2 that is isometric to its original complex hermitian space structure inherited from theE-hermitian space V . In such a way, X† can be identified with the set of negative definite complexlines 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 canonicalinvolution on B and the conjugation on F †. Consider the underlying Q-vector space V † of B†. Definea 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 groupQ×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 definet†(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

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;

75

• 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 ofS, η is a π1(S, s)-invariant K†-orbit of B† ⊗Afin-linear symplectic similitude η : V † ⊗Afin →Het

1 (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 θto 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

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

• A is an abelian scheme over S of dimension 4d;

• 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 symplecticsimilitude ηp : V † ⊗Apfin → Het

1 (As,Apfin);

• 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 .

76

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 overSpecOFp

to 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 .

Two quintuples (A, θ, i, ηp, ηpp) and (A′, θ′, i′, (ηp)′, (ηpp)′) are equivalent if there exists a prime-to-pisogeny A → A′ satisfying the same requirements in the last moduli problem. For sufficiently smallK†,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 needsto impose that (Ap∞)2

1 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 toMat2(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 thecontext). Let m > 0 be an integer. We denote by M ] = Mm (arranged in a column) as a leftMatm(R)-module in the natural way. Conversely, for any left Matm(R)-module N , we denote byN [ = eN the (left) R-module, where e = diag[1, 0, . . . , 0] ∈ Matm(R), and the action is given byr.(en) = (e × diag[r, . . . , r]).n for r ∈ R and n ∈ N . It is easy to see that the pair of adjointfunctors (−],−[) induce an equivalence between the category of left R-modules and the category ofleft 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)

77

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 universaldeformation 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

M†0,K†,pp ,K†,p

(resp. M0,Kp).

5.1.3 Basic abelian scheme

In order to obtain the moduli interpretation of special cycles, we will construct construct a specialabelian scheme, which we name the basic abelian scheme. We fix an imaginary element µ in E, thatis, an element µ 6= 0 such that µτ = −µ. Since we are interested only in the place p, we identify thefollowing 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

withE // 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 subfieldcontained 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 theusual multiplication. We choose a CM type

Φ = ι1 ⊗ ι11, ι1 ⊗ ι21; ιi ⊗ ι1i , ι•i ⊗ ι1i | i = 2, . . . , d

78

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 isfinite 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 canidentify 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†.

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

(e11). (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

79

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

• A is an abelian scheme over S of dimension 2d;

• 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 symplecticsimilitude ηp : V ‡ ⊗Apfin → Het

1 (As,Apfin).

The notion of being equivalent is similarly defined as before. Moreover, we can extend this modularfunctor 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 SpecOE0

pis 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. LetY = (Ep∞)2

1 and iY : OEp → End(Y) be the induced OEp -action. Therefore, Y is the special fiberof 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 thesupersingular locus of the common neutral component of M†

0,K†,pp ,K†,pand 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−1

E ϑ−1E y

∨ θA x

induces a E†-hermitian form on V †. If we let (A0, θA0 , iA0) be the isogeny class of (A, θA, iA), thenB† = End(A0, iA0) is a quaternion algebra over F †. Moreover, the underlying F †-quadratic spacesof V † and B† are isometric. The hermitian form (−,−)′ induces a Q-symplectic form on V †. If welet H† be the corresponding group of symplectic similitude, then Aut(A0, θA0 , iA0) can be identified

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 Vvis 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

1Here (A, iA) really means (A, iA | O‡), and we apply the similar convention in what follows.

80

such that%∗(x

pp) ∈ 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 cycleintroduced in 3.1.2. For K = Kp,0K

p, let us consider the curve M0,Kp;p , which is the base changeMK ×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 definitenorm in E. Here, K† = Z×p × O×Bp

× K†,pp × K†,p and K = Kp,0Kp 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;

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 theGalois group Gal(OE0

p/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, forevery 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.

81

From now on, we assume that (p is nonsplit in E) and ε(Vp) = 1. Following [Car1986, 11], we havethe 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 thefollowing 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.

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 thatι(p) ι1 : E → Cp induces the place p. We identify E with a subfield of C via the embedding ι. Ifp is split in E, we denote p• the other place of E above p. We have the hermitian space V over E ofdimension 2 and signature (1, 1), the unitary group H over Q. For a sufficiently small open compactsubgroup K ⊂ H(Afin), the Shimura curve ShK(H,X) is a smooth and quasi-projective curve definedover 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 thatB, as an Q-quadratic space (of dimension 4), is isometric to V as an F -quadratic space with thequadratic form TrE/Q(−,−), where (−,−) is the hermitian form on V . We identify two quadraticspaces 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 ofB coincides with the E-vector space structure of V . We let H† = B×, which is different from thecase F 6= Q. Then similarly, we have the Shimura curve ShK†(H

†, X†) defined over Q. We can viewV 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 neutralcomponents Sh(H)0,Kp;p and Sh(H†)0,K†,p;p are defined and isomorphic over E0

p .

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 isomorphicto 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;

82

• ηp is a K†,p-level structure, that is, a π1(S, s)-invariant K†,p-orbit of B⊗Apfin-linear symplecticsimilitude ηp : V ⊗Apfin → Het

1 (As,Apfin).

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 ellipticcurve. Moreover, we have the canonical polarization θ : A → A∨ and the OB = Mat2(Z)-actioni : 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 isZ(x)0,Kp;p . It only depends on the orbits Kp,0K

px. In the isotropic case, Z(x)0,Kp is disjoint fromthe 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 specialfiber [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, pwill 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)

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 0

pwhere $ 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 ×SpecO

F0p

SpecF.

Two pairs (G, ρG) and (G′, ρG′) are equivalent if there is an isomorphism G′ → G sending ρG toρG′ . Then N is represented by the formal scheme N , which is isomorphic to Spf RFp,2, whereRFp,2 = O

F 0p[[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

83

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

B×pi/K†,pp × H†(A

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 mentionedcategory, Z (x)(S) is the set of equivalence classes of (G, ρG) ∈ N (S) such that the following composedhomomorphism(

Y×SpecOF0p

SpecF

)×SpecF Ssp = Y ×SpecF Ssp

x−→ X×SpecF Sspρ]G−−→ G] ×S Ssp

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 argumentin [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). Considerthe generating series

Zφ(g) =∑

x∈K\V (AF,fin)−0

(ωχ(g)φ) (x)Z(x)K

=∑

x∈K\V (AF,fin)−0

ψp(bpT (x))(φ0p ⊗ (ωχ(gp)φp)

)(x)Z(x)Kp,0Kp .

Define

Zφ(g)0,Kp =∑

x∈K\V (AF,fin)−0

ψp(bpT (x))(φ0p ⊗ (ωχ(gp)φp)

)(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

ψp(bpT (x))(φ0p ⊗ (ωχ(gp)φp)

)(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.

84

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 everypair (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 intersectionnumber Z (x1) ·Z (x2). Assuming that (y1, y2) = (x1, x2)g for some g ∈ GL2(OEp ) such that (y1, y2)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 withan 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 overSpf O

F sp, unique up to the Galois action (cf. [Gro1986]). Therefore, it defines a morphism Spf O

F sp→

N that is a closed immersion. Let Zs be the divisor of N defined by the image, which is independentof Fs we choose. We have the following proposition generalizing [KR2011, Proposition 8.1] from Qp

to Fp.

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).

85

To prove the other direction, we need only to prove that the intersection multiplicities of both sidesin 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 followingproposition 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] citedin 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 themoment 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 ofwindows 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 theextension degree of residue fields of F/Qp, respectively. In particular, q = pf . Let R = F[[t]] andA = W [[t]], where W = W (F) is the Witt ring. We extend the Frobenius automorphism σ on W toA by letting σ(t) = tp. For any s ≥ 1, we let Rs = R/ts and As = A/ts. Then A (resp. As) is aframe of R (resp. Rs). The category of formal p-divisible groups over R is equivalent to the categoryof 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 freeR-module.

86

The baby case would be the one where f = 1. Consider the p-divisible group Y over F of dimension1 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 Fcorresponds to (M,α) described there, and its universal deformation is (M,αt). The only differenceis that we should replace p by $. The rest of the proof follows in the same way.

Now we treat the general case and hence assume that f ≥ 2. Consider the p-divisible group Yover 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

basise0,0, e

τ1,0, . . . , e0,f−1, e

τ1,f−1; e1,0, e

τ0,0, . . . , e1,f−1, e

τ0,f−1,

the matrix of α is

α =

$$

1. . .

11

. . .

1

.

87

Let (M,αt) corresponds to the universal deformation of (X, iX) over F[[t]]. Then under the samebasis,

αt =

1 t1 −t

1. . .

11

. . .

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.

If we denote by σk(α) : M (σk) → M (σk+1) the induced homomorphism for k > 0. Then formally, wehave

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 lengthZ (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 · · · 00 0 · · · 00 $r · · · 00 0 · · · 0...

.... . .

...0 0 · · · $r

0 0 · · · 0

.

88

If r = 0, in order to lift γ to γ mod tp, we search for a 4f × 2f matrix X(1) with entries in Apsuch that X(1) ≡ X(0) in A1 and satisfies

αt X(1) = σ(X(1)) β.

But σ(X(1)) = σ(X(0)) = X(0). Therefore, we need to find the largest integer a ≤ p such thatα−1t X(0) β has integral entries mod ta. Since the entry at the place (e0,f−1, e

τ0,f−1) is t

$ , thelargest 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) fork ≥ 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 , e

τi′,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 bythe following 4f × 2f matrix

0 · · · 0$s+1 · · · 0

.... . .

...0 · · · 00 · · · $s+1

0 · · · 0$s · · · 0...

. . ....

0 · · · 00 · · · $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

89

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 , e

τi′,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.

It turns out that the length Z (y) ·Nsp is exactly q2s+2−1q−1 = qv+1−1

q−1 . Therefore, the proposition isproved.

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 letφ0+ ∈ S(V +) (resp. Φ0+ ∈ S((V +)2)) be the characteristic function of Λ+ (resp. (Λ+)2). Let ψbe the unramified character of F . For T ∈ Her2(E) that is nonsingular, we consider the Whittakerintegral

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 theIwasawa 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), thenWT (s, e,Φ0+) is identically 0. For T ∈ Her2(OE), it is well-known (e.g., [Kud1997, Appendix]) thatfor an integer r > 1, WT (r, e,Φ0+) = γV +αF (12+r, T ), where γV + is the Weil constant and αF is theclassical 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).

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.

90

Proposition 5.3.1 (Hironaka, [Hir1999]). Suppose that T is GL2(OE)-equivalent to diag[$a, $b] with0 ≤ 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 Ethat is good. Recall that we have a (compactified) Shimura curve MK constructed from the hermitianspaceV, and assume that K is sufficiently small and decomposable. We also assume that φα (α = 1, 2)are decomposable and K = KpK

p-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 compactsubgroup;

• 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 Iwasawadecomposition, and choose some element eα ∈ E× such that e−1

α aα,p ∈ O×Ep. Let gα = m(e−1

α )gα, and

we have gα,p = n(bα,p)kα,p in the Iwasawa decomposition. Then Zφα(gα) = Zφα(gα). As in 5.2.2, wehave the series Zφα(gα)0,Kp for α = 1, 2. We let Zφα(gα) = Zφα(gα)0,Kp . The following is the maintheorem 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 introducedin 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, thenZφ1

(g1) · Zφ2(g2) = 0.

Combining this with Theorem 4.3.4, we have the following corollary.

91

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.

ThenE′(0, ı(g1, g

∨2 ), φ1 ⊗ φ2) = −Vol(K)

∑v|vv 6∈S

〈Zφ1(g1), Zφ2

(g2)〉v ,

where the Green functions used in archimedean places are those defined in (4.55), which are not theadmissible 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 specialfiber, 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

(ω′′χ(ı(gp1 , g

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,we obtain that

−Vol(K) log q2(Zφ1(g1) · Zφ2

(g2)) = Ep(0, ı(g1, g∨2 ), φ1 ⊗ φ2). (5.13)

92

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.

93

Chapter 6

Comparison at finite places: badreduction

In this chapter, we discuss the local height paring 〈Zφ1(g1), Zφ2

(g2)〉p on certain model MK at everybad 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 inE. In 6.2, we discuss the contribution of the local height paring at a finite place p in S that is nonsplitin E and such that ε(Vp) = 1. In 6.3, we discuss the contribution of the local height paring at a finiteplace p in S that is nonsplit in E and such that ε(Vp) = −1.

94

6.1 Split case

In this section, we discuss the contribution of the local height paring at a finite place p in S that issplit 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 forM0,Kp;p on which there is a p-divisible group X. Then X[ → M0,Kp is an OFp

-module of dimension1 and height 2. Recall that a Drinfeld $n-structure for an OFp

-module X of dimension 1 and height2 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, LemmaII.2.1]). Then Mn,Kp is regular and finite over M0,Kp , whose generic fiber is Mn,Kp . Let M ′n,Kp =Mn,Kp ×Fp

Fnp be the base change, and M′n,Kp the normalization of Mn,Kp ×OFpOFnp 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 ]spand 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 oneach 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 byl.h = h−1l for l ∈ P(Vp) and h ∈ U(Vp). Together, the set of irreducible components of [M′n,Kp ]sp canbe 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)Kand the generating series Zφα(gα) on M ′n,Kp . We let Z(x)K (resp. Zφα(gα)) be the Zariski closure ofZ(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 . Thenthe 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, h

p) 7→ (h−1p l, ν(hph

p))

(cf. [Zha2001b, 5.4] for a discussion).

95

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 thegeometric irreducible component represented by (lx, 1) in Ign,Kp in Vφ2

(g2). It is a locally constantfunction 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 intersectionnumber

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 letA(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).

96

6.2 Quasi-split case

In this section, we discuss the contribution of the local height paring at a finite place p in S that isnonsplit in E and such that ε(Vp) = 1. Let p be the only place in Σfin lying over p. We identify Vpwith 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 integralmodel M0,Kp for M0,Kp;p on which there is a p-divisible group X. Let Mn,Kp be the normalizationof M0,Kp in Mn,Kp , which is regular and finite over M0,Kp . Consider the base change M ′n,Kp =Mn,Kp ×Ep F

np . 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 .

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 alongthe 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 fiberis Z(x0)0,Kp;p , for a Kp,0K

p-orbit Kp,0Kpx0 in V (AF,fin) that is admissible. Define Z′(x0)n,Kp by

97

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. Forevery such orbit Kp,nK

px, 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.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 characteristic0 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 thatHt(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 .

Let φ = φ0∞ ⊗ (⊗v∈Σfin

φv) such that

• φpfin := ⊗v∈Σfin−pφv is in S(V (ApF,fin))K

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 .

98

Define

Zφ(epgp)n,Kp =

∑x∈Kp,nKp\V (AF,fin)−0

(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)

6.2.2 Coherence for intersection numbers

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 byzero.

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,pis 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 .

99

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 decompositionas (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)

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

(ω′′χ(ı(gp1 , g

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

(ω′′χ(ı(gp1 , g

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)

100

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

[Zωχ(hp,j hpi )φ1

(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).

We define the following theta series

θver(p)(•;φ1, φ2, g2) = log q

∑x1∈V

(ωχ(•)φver) (x1).

Then in summary, we have the following lemma.

Lemma 6.2.3. Under the previous assumption, we have

log q (Zφ1(g1) · Vφ2(g2)) = θver(p)(g1;φ1, φ2, g2)

that is a theta series for g1 ∈ epH ′(ApF ).

Finally, we letA(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

In this section, we discuss the contribution of the local height paring at a finite place p in S that isnonsplit in E and such that ε(Vp) = −1. Let p be the only place in Σfin lying over p. We identify Vpwith Bp, the unique division quaternion algebra over Fp, and the lattice Λ− with OBp

.

101

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 withGL2(OFp

). For every integer h, we define a functor Ωh in the category of schemes over SpecOF 0p

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

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

B×pi/K†,pp × H†(A

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-divisiblegroups. 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 asfollows: 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.

1This is not to be confused with the differential form Ω in Chapter 4

102

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

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 itsdouble 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 doublepoints. 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 morphismpn : Mn,Kp →M0,Kp ×OEp

OFnp , 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 bythe following Cartesian diagram

Z′(x0)n,Kp

// Mn,Kp

Z(x0)hor

0,Kp ×OEpOFnp

// M0,Kp ×OEpOFnp ,

103

where Z(x0)hor0,Kp denotes the horizontal part of the associated divisor. For every orbit Kp,nK

px insideKp,0K

px0, we define Z(x)n,Kp to be the union of irreducible components of Z′(x0)n,Kp whose genericfiber contributes to the special divisor Z(x)n,Kp ×Ep F

np on the generic fiber Mn,Kp ×E Fnp .

Let φ = φ0∞ ⊗ (⊗v∈Σfin

φv) such that

• φpfin := ⊗v∈Σfin−pφv is in S(V (ApF,fin))K

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 .

Define

Zφ(epgp)n,Kp =

∑x∈Kp,nKp\V (AF,fin)−0

(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)

6.3.3 Coherence for intersection numbers

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

• 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 byzero.

104

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,pis 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).First, we consider

Zφ1(g1) · Zφ2

(g2) = [Zφ1(g1)]∧sp · [Zφ2

(g2)]∧sp = [Zφ1(epg

p1)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

[Zωχ(hp,j hpi )φ1

(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

(ω′′χ(ı(gp1 , g

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

(ω′′χ(ı(gp1 , g

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.

105

Lemma 6.3.2. 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(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

[Zωχ(hp,j hpi )φ1

(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).

We define the following theta series

θver(p)(•;φ1, φ2, g2) = log q

∑x1∈V

(ωχ(•)φver) (x1).

Then in summary, we have the following lemma.

Lemma 6.3.3. Under the previous assumption, we have

log q (Zφ1(g1) · Vφ2(g2)) = θver(p)(g1;φ1, φ2, g2)

that is a theta series for g1 ∈ epH ′(ApF ).

Finally, we letA(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).

106

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 thischapter. In 7.1, we introduce the general theory of holomorphic projection for the group U(1, 1)F , andcompute such projection for the analytic kernel function. In particular, this process will relate thearchimedean local terms to admissible height parings (at archimedean places) and leaves local termsat finite places unchanged. Based on all these previous results, we come to the final stage of the proofin 7.2.

107

7.1 Holomorphic projection

In this section, we define and compute the holomorphic projection of the analytic kernel functionE′(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 cuspforms. Let Ak

0(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. Wedefine a character ζk of Z ′∞ by letting ζkι (zι) = z2kι

ι . Let A(H ′, ζk) ⊂ A(H ′) be the subspace consistingof the forms that have archimedean central character ζk. It is clear that Ak

0(H ′) ⊂ A(H ′, ζk). Recallthat F+ ⊂ F is the subset of totally positive elements. For each fixed t ∈ F+, the t-th Whittakerfunctions (with respect to the standard additive character) of all members of Ak

0(H ′) are the sameone, namely, the function W k

t 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 Ak

0(H ′×H ′) ⊂ A0(H ′×H ′) the subspace consisting of cusp forms f(•, •) such thatfor every g ∈ H ′(AF ), both f(g, •) and f(•, g) are in Ak

0(H ′). The space A0(H ′ × H ′) is a Hilbertspace 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 ′ ∈ Ak

0(H ′ ×H ′).

Let ψ′ be a character of F\AF such that ψ′∞(x) = ψ∞(tx) for t ∈ F×∞. We recall that ψ∞ is thestandard additive character at archimedean places as assumed in the beginning of Chapter 3. For anautomorphic 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 someε > 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′-Whittakerfunction

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 twoψ′-Whittaker functions Wα(gα) = W k

t (gα,∞)Wαfin(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 bePWα(gα) = lim

s→0+

∑γ∈Z′(F )N ′(F )\H′(F )

Wα(γgα)λP ′(γ∞gα,∞)s.

108

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)

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 forall (ζ1, ζ2) and (W 1

fin,W2fin). 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 arounda). 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 behaviorthere, we have Pr(f) = Pr(f).

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.

109

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

11

, b =

(b11 b12

b21 b22

)∈ Her2(E);

2.

P (F )w2,1

1 0 0c 1 0 b22

1 −cτ1

, c ∈ E, b22 ∈ F ;

3. P (F ),

where we recall that

w2 =

(12

−12

); w2,1 =

1

11

−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),

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

(ωχ

(w2,1m

((1c 1

))n

((0 00 b22

))g

)(φ1 ⊗ φ2)

)(0)

λP

(w2,1m

((1c 1

))n

((0 00 b22

))g

)sE0(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)

110

Now we calculate for β = 1, c with c ∈ F . To simplify the formulae, we ignore the term λP (•)s inthe following calculation. We have

E1,cT (s, e, φ1 ⊗ φ2)

=

∫b11∈F\AF

∫b12∈E\AE

∫b22∈AF

(ωχ

(w2,1m

((1c 1

))n

((b11 b12

bτ12 b22

)))(φ1 ⊗ φ2)

)(0)

ψ

(trT

(b11 b12

bτ12 b22

))−1

db11db12db22. (7.5)

Change variables (b11 b12

bτ12 b22

)7→(

1c 1

)(b11 b12

bτ12 b22

)(1 cτ

1

).

Then

(7.5) =

∫b11∈F\AF

∫b12∈E\AE

∫b22∈AF

(ωχ

(w2,1n

((b11 b12

bτ12 b22

))m

((1c 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

(ωχ

(w2,1n

((0 00 b22

))m

((1c 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 00 t

)(1c 1

)for some t ∈ F . In the later case,

E1,cT (s, e, φ1 ⊗ φ2)

=

∫b∈AF

∫x∈V

(ωχ

(n

((0 00 b

))m

((1c 1

)))(φ1 ⊗ φ2)

)(0, x)ψ(tb)−1dbdx

=

∫b∈AF

∫x∈V

(ωχ

(m

((1c 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, whenc = 0,

E1,0T (s, e, φ1 ⊗ φ2) = φ1(0)×Wt(s+

1

2, e, φ2). (7.7)

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 inS(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 havethe asymptotic behavior in Proposition 7.1.2. To find its holomorphic projection, we introduce thefollowing 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.

111

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×∞. Weconsider 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. Wesee 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)

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),

112

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)

).

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. Thenfor gα ∈ P ′pH ′(A

pF ),

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:

• φα = φ0∞φα,fin (α = 1, 2) are decomposable;

• φ1,S ⊗ φ2,S is in S(V2S)reg;

• 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 ) .

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

∑x1∈V n,T (x1)∈Her+n (E)

(ωχ(g1)φ1) (T (x1), h−1i x1)Ξx1a(g1)

∗

∑x2∈V n,T (x2)∈Her+n (E)

(ωχ(g2)φ2) (T (x2), h−1i x2)Ξx2a(g2)

,

113

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).

Then

(7.13) = consts→0(4πt)∑

T (x)=t

(ωχ(g)φ) (t, x)

∫ ∞0

(∫ ∞1

exp(−4πtyuδx(z))

udu

)ys exp(−4πty)dy

= consts→0(4πt)∑

T (x)=t

(ωχ(g)φ) (t, x)t−1−s∫ ∞

0

∫ ∞1

exp(−4πyuδx(z))

uys exp(−4πty)dudy

= consts→0(4πt)∑

T (x)=t

(ωχ(g)φ) (t, x)t−1−s∫ ∞

1

1

u

(∫ ∞0

exp (−4πy(1 + uδx(z)))

)du

= consts→0(4πt)∑

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)).

114

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. Thenthe 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 ′(AS

F ). In particular, the archimedean local height paring is computed via admissibleGreen functions.

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 = 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;

• φ1,v ⊗ φ2,v ∈ S(V2v)reg,dv for every v ∈ S that is nonsplit and some dv ≥ dψv (cf. 2.4.2).

LetE(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 havethat for gα ∈ eSH ′(AS

F ) (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 ofweight (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 Eisensteinseries, automorphic characters or theta series in gα ∈ eSH ′(AS

F ) for at least one α, and that eSH′(AS

F )is a dense subset of H ′(F )\H ′(AF ).

115

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 automorphicrepresentation 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 ),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.

116

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. InA.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, followingthe method of S. Rallis.

117

A.1 Theta dichotomy for unitary groups

In this section, we prove the theta dichotomy for unitary groups over non-archimedean fields. Forsymplectic-orthogonal pairs, this is proved by Kudla–Rallis in [KR2005]. We will follow the sameline1.

Let F be a non-archimedean local field whose characteristic is not 2, and a quadratic field extensionE/F with Gal(E/F ) = 1, τ. Let OF (resp. OE) be the ring of integers, $ (resp. $E) a uniformizerof F (resp. E), and q = |OF /$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 suchthat ε(V ±n ) = ±1, and H±n = U(V ±n ). Let (W, 〈−,−〉) be a (right) skew-hermitian space over E of thesame 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 areboth nonzero, then HomH′(S((V +

n )n), π) 6= 0 and HomH′(S((V −n )n), π) 6= 0, where H ′ acts on bothspaces of Schwartz functions through the Weil representation ωχ.

Let g0 ∈ GLF (W ) be a τ -linear automorphism such that

〈w1g0, w2g0〉 = 〈w1, w2〉

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), whichis also the image of the intertwining map

M∗(−n2, 1) : In(−n

2, 1)→ Rn(V −2n, 1).

1During the preparation of this article, Gong and Grenie [GG2011] also prove the same results independently. Ourmethods are similar and we refer to their article for more details

118

By [KS1997, Proposition 4.3 (2)], ker(M∗(−n2 , 1)) = C · ϕ0, where ϕ0 ∈ In(−n2 , 1) is the constantfunction 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.

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 anirreducible 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 ) bethe graph of the identity map and PY ⊂ H ′′ the parabolic subgroup that stabilizes Y . Let r0 be theWitt index of W . Then PY \H ′′ can be canonically identified with the set of isotropic n-dimensionalsubspaces 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′ .

LetIn(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 quotientMr isomorphic to GLr(E) × H ′n−2r, and the maximal unipotent subgroup Nr, where H ′n−2r is theunitary group of some skew-hermitian space of dimension n− 2r.

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 ∈ Cat 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 holomorphicat s0, and

HomH′×H′(In(s0, χ), π χπ∨) = C · Z(s0, χ,−,−).

For the trivial representation, we have the following lemma.

119

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, ϕ

0) 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:

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 neven) 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.

120

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 spaceHomH′×H′(I2(0, χ), π∨χπ) is of dimension 1, following [HKS1996]. Here, χ is a character of E×. Ifχ is trivial on F×, i.e., χ−1 = χτ , then we define a character πχ of H ′ as follows. For every elementg ∈ H ′, det g is in E×,1. In particular, det g = eg/e

τg 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 ofH ′ that is not π−1

χ if χ−1 = χτ . Then we have

dimCHomH′×H′(I2(0, χ), π∨ χπ) = 1,

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 generalityin this article, we put this condition for simplicity. In what follows, we also assume that v is finite forsimplicity. 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 ofH ′ ×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 ′-invariantfunctional 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

),

we havedimCHomH′×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.

121

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 properlycontained in I1(− 1

2 , 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.

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 determinesan unramified additive character of E by composing with 1

2 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〉))

122

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,jthat consists of elements (h′, h) stabilizing the subspace W ∗n,i⊗V ∗m,j ⊂Wn⊗Vm, for i = 0, 1, . . . , n andj = 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 ×Nm,j is isomorphic to (GLn−i(E) ×H ′i) × (GLm−j(E) ×Hj). For 0 ≤ t ≤ n, we define an algebraicclosed 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. Thesame 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 vanisheson 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,rbegin 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, definethe 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

123

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. LetS(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 decompositionYr = ArLr. Here, Ar = diag[a1, . . . , ar]|ai ∈ E× and Lr is the unipotent radical. We write elementsin 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) ∈ Crsuch that Reσi 0. For every function φ ∈ S(V nm), the following integral

Zσ(φ) =

∫Yr

φ

((yr 00 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 intertwiningmap 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

).

124

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 00 0

)t′−1

) r∏i=1

|ai|σEdyr. (A.1)

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 thatZσ (Φm,n(f)− f) = 0

for all f ∈ S(Hm//Km) and Reσi 0. Using the Fourier–Satake isomorphism, the map Φm,n is givenby

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)

satisfying thatZσ (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

byC[X1, X

−11 , . . . , Xn, X

−1n ]W (H′n) → C[X1, X

−11 , . . . , Xm, X

−1m ]W (Hm),

125

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 thatReσi 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 00 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 withb ∈ B′m,r, k ∈ Km. Since φ is spherical by assumption, we have

(ω(h′, h)φ)

((1r 00 0

))= (ω(b′, b)φ)

((1r 00 0

))= φ

((X 00 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))

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 givenby 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 byf −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 toπ.

2. Let π be an unramified irreducible admissible representation of GLn(F ), and χ an unramifiedcharacter of F×. Then the theta correspondence of π to GLn(F ), with respect to the Weilrepresentation ωχ defined by (ωχ(g′, g)φ) (x) = χ(det g′)φ(g−1xg′) for φ ∈ S(Matn,n(F )), isnontrivial and isomorphic to π∨ ⊗ χ.

126

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129

Index

L -admissible Green function, 610-dimensional Shimura variety, 106

admissible (E-subspace), 43analytic kernel function, 31archimedean local arithmetic Siegel–Weil for-

mula, 95arithmetic Chow group, 60arithmetic divisor, 60

flat, 60principal, 60

arithmetic theta lifting, 56, 67arithmetic theta series, 63

basic unitary datum, 115Beilinson–Bloch height paring, 59

coherent hermitian space, 16cohomologically trivial cycle, 60compactified generating series, 54

positive partial, 54compactified special cycle, 53

degenerate principal series, 14

flat arithmetic divisor, 60

generating series, 45compactified, 54positive partial compactified, 54

geometric kernel function, 67good section, 23Green function, 60

L -admissible, 61

height function on D, 86height paring

Beilinson–Bloch, 59Neron–Tate, 61

modified, 61hermitian space, 9hermitian line bundle, 61hermitian space

nearby, 42nondegenerate (skew-), 10skew-, 9

hermitian space (over adeles)coherent, 16incoherent, 16nondegenerate, 16

holomorphic projection, 155quasi-, 157

incoherent hermitian space, 16

kernel functionanalytic, 31geometric, 67

local zeta integral, 23

modified height pairing, 61moment matrix, 17

Neron–Tate height paring, 61nearby hermitian space, 42nearby quadratic space, 45nondegenerate (skew-)hermitian space, 10nondegenerate hermitian space (over adeles),

16normalized zeta integral, 24

of weight (a,b), 16ordinary (point), 112

positive partial compactified generating series,54

principal arithmetic divisor, 60

quadratic spacenearby, 45

quasi-holomorphic projection, 157

regular test function, 32

sectiongood, 23standard, 14unramified, 14

Shimura variety0-dimensional, 106of orthogonal group, 45of unitary group, 42

Siegel–Weil formula, 19archimedean local arithmetic, 95

skew-hermitian space, 9special cycle, 44

compactified, 53standard section, 14supersingular (point), 112

test function, 31of discriminant d, 36regular, 32

theta dichotomy, 25theta function, 18

130

theta lifting, 21

unramified section, 14

Weil representation, 17Whittaker integral, 28

zeta integrallocal, 23normalized, 24

131

Index of Notations

(−],−[), 111

A0(H ′), 155Ak

0(H ′ ×H ′), 155Afin,A, 9AK ,AK,fin,K∞, 9A(g, φ), 68A(H), 29A(H ′, ζk), 155Ak

0(H ′), 155α(f, f∨, φ, φ∨), 27am(s), 34A(p)(φ1, g1;φ2, g2), 138(A, θ, i), 111

β(f, f∨, φ, φ∨), 27bm(s), 24

CH1fin(M)C, 60

CH1(M)0C, 60

CH1(M)C, 63

CH1

C(M), 60

CH1

C(M)0, 60CHr(Sh(H))C, 45CHr(Sh∼)C, 54CHr(X), 10consts=a, 157

dc, 60D(g, φ), 63diag[a1, . . . , ar], 10Diff(T,V), 32D(ι), 42

(E, ϑE, jE), 115E(g1, g2; Φ), 67E(g, φ), 66E(g, ϕs), 14E(p)(φ1, g1;φ2, g2), 138ε(π, χ), 26ε(V ), 16ε(V), 16E(s, g, φ), 18ET (s, g,Φ), 30(E, ϑ, j; x), 115

fψ′,s, 156

γV , 17

H, 116H†, 107

h†, 107Herr, 13Hp

fin, 106h(ι), 42Hp(T ), 124Hr, 13H(x)∞, H(T )∞, 86

ı, 20ı0, 20ı, 17ι(p), 104Ir(s, χ), 14ıχ,1, 17IV (g, φ), 18

K±0 , 104kχ, 15Kp,n, 106Kr, 13

Λ±, 104λPr , 18

L†0,K†,pp ,K†,p

, 108

L†K†;p

, 108LK , 106LK ,L , 43LK;p , 108L ∼K ,L

∼, 53Ln,Kp , 106Ls, 105Ls, 105L(s, π, χ), 24

M0,Kp , 111[M0,Kp ]ss, 112[M0,Kp ]∧ss, 120MK , 67MK;p , 68Mn,Kp , 136, 139, 148M′n,Kp , 136, 139

M†0,K†,pp ,K†,p

, 111

M†0,K†,pp ,K†,p

, 108

[M†0,K†,pp ,K†,p

]ss, 112

M†K†

, 107

M†K†;p

, 108MK , 62MK;p , 108Mn,Kp , 106

132

Mor ((Y, jY), (X, iX))reg, 120Mr, 13

N , 21Nr, 13N ,N ′, 120

OT , 28Ω,Ω′, 147ωχ, 17ω′′χ, 21ωK , 67

P , 21φ, 17φ0∞, 44ϕφ,s, 18

Pic(M), 61Pic(M)Q, 62Pic(Sh(H)), 43Pic(Sh∼), 53Pic(X), 11πK′

K , 43$, 104pM, 61Pr, 13Pr, 155Pr, 157ψT , 28

RFp,2, 120R(V, χ), 25

Sh(H), 43ShK(H), 42Sh∼K ,Sh∼, 53Σ, 13Σfin, 13Σ∞, 13Σ, 13Σfin, 13Σ∞, 13Σ(V), 16S, 42Std, 22S(V 2n)d, 37S(V 2n)reg,d, 37S(V r), 17, 18S(V r)reg, 32S(V2n

S )reg, 32S(Vr)U∞ , 44S(Vr)U∞K , 44

T 1F ′ , 104TF , 104

Θfφ, 56

θfφ, 21θ(g, h;φ), 18

Θφ(g), 67θhor

(p)(•;φ1, φ2), 144, 151

Θφ(g), 63θver

(p)(•;φ1, φ2, g2), 145, 153TpA, 110T (x), 17

UsL, 105

V , 116V (ι), 42Vol(K), 101V ±, 16Vs, 16V(π, χ), 26

(−W ), 20W ′′, 20

W ′,W′, 20

Wr, 13WT (g, f), 29WT (g, ϕs), 28W kt , 155

WT (s, g,Φ), 29wr, 10wr,d, 10

(X, iX), 120X, 111X†, 107X†, 111Ξφ(g)ι′ , 102Ξx, 86

(Y, jY), 115(Yτ , jYτ ), 122Y, 115

Z†(x†, h†)0,K†,pp ,K†,p , 117

Zφ(g)0,Kp , 121[Zφ(g)0,Kp ]∧s , 121Z(x)0,Kp , 117Z(χ, f, f∨, ϕs), 23ζk, 155Z∗(χ, f, f∨, ϕs), 24Z∗(s, χ, f, f∨,Φ), 40Z∗(s, χ, f, f∨, φ⊗ φ∨), 26(Z, gι), 60

Z1

C(M), 60

Zφ(g), 67Zφ(g), 45

133

Z∼φ (g), 54

Z∼,+φ (g), 54Z (x), 121Z(V1)K , 44Z(x)0,Kp;p , 116Z(x)K , 44Z(x)∼K , 53