Arithmetic inner product formula
for unitary groups
Yifeng Liu
Submitted in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
in the Graduate School of Arts and Sciences
COLUMBIA UNIVERSITY
2012
c 2012
Yifeng Liu
All rights reserved
Abstract
Arithmetic inner product formula
for unitary groups
Yifeng Liu
We study central derivatives of L-functions of cuspidal automorphic representations
for unitary groups of even variables defined over a totally real number field, and their
relation with the canonical height of special cycles on Shimura varieties attached to
unitary groups of the same size. We formulate a precise conjecture about an arithmetic
analogue of the classical Rallis’ inner product formula, which we call arithmetic inner
product formula, and confirm it for unitary groups of two variables. In particular,
we calculate the Néron–Tate height of special points on Shimura curves attached to
certain unitary groups of two variables.
For an irreducible cuspidal automorphic representation of a quasi-split unitary
group, we can associate it an -factor, which is either 1 or −1, via the dichotomy
phenomenon of local theta lifting. If such factor is −1, the central L-value of the representation always vanishes and the Rallis’ inner product formula is not interesting.
Therefore, we are motivated to consider its central derivative, and propose the arithmetic inner product formula. In the course of such formulation, we prove a modularity
theorem of the generating series on the level of Chow groups. We also show the cohomological triviality of the arithmetic theta lifting, which is a necessary step to consider
the canonical height. As evidence, we also prove an arithmetic local Siegel–Weil formula at archimedean places for unitary groups of arbitrary sizes, which contributes as
a part of the local comparison of the conjectural arithmetic inner product formula.
Contents
Table of Contents
i
Acknowledgments
v
1 Introduction
1
2 Doubling method and analytic kernel functions
2.1
2.2
2.3
2.4
Siegel–Weil formula and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.1.1
Degenerate principal series and Eisenstein series . . . . . . . . . . . . . . . . .
13
2.1.2
Real quasi-split unitary groups . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.1.3
Weil representations and theta functions . . . . . . . . . . . . . . . . . . . . . .
16
2.1.4
Siegel–Weil formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
Doubling integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.2.1
Decomposition of global period integrals . . . . . . . . . . . . . . . . . . . . . .
19
2.2.2
Local zeta integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
Central special values of L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.3.1
Theta lifting and central L-values . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.3.2
Vanishing of central L-values . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
Analytic kernel functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
2.4.1
Regular test functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
2.4.2
Test functions of higher discriminant . . . . . . . . . . . . . . . . . . . . . . . .
36
2.4.3
Density of test functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
3 Arithmetic theta lifting and arithmetic kernel functions
3.1
12
41
Modularity of generating series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
3.1.1
42
Shimura varieties of unitary groups . . . . . . . . . . . . . . . . . . . . . . . . .
i
3.2
3.3
3.4
3.1.2
Special cycles and generating series . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.1.3
Pullback formula and modularity theorem . . . . . . . . . . . . . . . . . . . . .
45
Smooth compactification of unitary Shimura varieties
. . . . . . . . . . . . . . . . . .
50
3.2.1
Compactified unitary Shimura varieties . . . . . . . . . . . . . . . . . . . . . .
50
3.2.2
Compactified generating series . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
Arithmetic theta lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
3.3.1
Cohomological triviality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
3.3.2
Arithmetic inner product formula: the general conjecture . . . . . . . . . . . .
59
Arithmetic kernel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
3.4.1
Néron–Tate height pairing on curves . . . . . . . . . . . . . . . . . . . . . . . .
60
3.4.2
Degree of generating series . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
3.4.3
Decomposition of arithmetic kernel functions . . . . . . . . . . . . . . . . . . .
67
4 Comparison at infinite places
4.1
4.2
4.3
69
Archimedean Whittaker integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
4.1.1
Elementary reduction steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
4.1.2
Analytic continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
4.1.3
First-order derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
Archimedean local height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
4.2.1
Green currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
4.2.2
Invariance under U(2): an exercise in Calculus . . . . . . . . . . . . . . . . . .
87
An archimedean local Siegel–Weil formula . . . . . . . . . . . . . . . . . . . . . . . . .
95
4.3.1
Comparison on the hermitian domain . . . . . . . . . . . . . . . . . . . . . . .
95
4.3.2
Proof of Lemma 4.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
4.3.3
Comparison on Shimura varieties . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5 Comparison at finite places: good reduction
5.1
104
Integral models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.1.1
Change of Shimura data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.1.2
Moduli interpretations and integral models: minimal level . . . . . . . . . . . . 108
5.1.3
Basic abelian scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.1.4
The nearby space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.1.5
Integral special subschemes: minimal level . . . . . . . . . . . . . . . . . . . . . 116
5.1.6
Remark on the case F =
Q
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
ii
5.2
5.3
Local intersection numbers
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.2.1
p-adic uniformization of supersingular locus . . . . . . . . . . . . . . . . . . . . 119
5.2.2
Special formal subschemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.2.3
A formula for local intersection multiplicity . . . . . . . . . . . . . . . . . . . . 121
5.2.4
Proof of Proposition 5.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.3.1
Non-archimedean Whittaker integrals . . . . . . . . . . . . . . . . . . . . . . . 130
5.3.2
Comparison on Shimura curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6 Comparison at finite places: bad reduction
6.1
6.2
6.3
Split case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.1.1
Integral models and ordinary reduction . . . . . . . . . . . . . . . . . . . . . . 136
6.1.2
Coherence for intersection numbers . . . . . . . . . . . . . . . . . . . . . . . . . 137
Quasi-split case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.2.1
Integral models and supersingular reduction . . . . . . . . . . . . . . . . . . . . 139
6.2.2
Coherence for intersection numbers . . . . . . . . . . . . . . . . . . . . . . . . . 142
Nonsplit case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.3.1
Integral models and Čerednik–Drinfeld uniformization: minimal level . . . . . . 146
6.3.2
Integral models and Čerednik–Drinfeld uniformization: higher level . . . . . . . 147
6.3.3
Coherence for intersection numbers . . . . . . . . . . . . . . . . . . . . . . . . . 149
7 Arithmetic inner product formula: the main theorem
7.1
7.2
135
154
Holomorphic projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.1.1
Holomorphic and quasi-holomorphic projection: generality . . . . . . . . . . . . 155
7.1.2
Siegel–Fourier expansion of Eisenstein series on U(2, 2) . . . . . . . . . . . . . . 158
7.1.3
Holomorphic projection of analytic kernel functions . . . . . . . . . . . . . . . . 161
7.1.4
Quasi-holomorphic projection of analytic kernel functions . . . . . . . . . . . . 163
Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
7.2.1
Difference of kernel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
7.2.2
The final step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Appendix
169
A.1 Theta dichotomy for unitary groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
A.2 Uniqueness of local invariant functionals . . . . . . . . . . . . . . . . . . . . . . . . . . 174
iii
A.3 Theta correspondence of unramified representations
Bibliography
. . . . . . . . . . . . . . . . . . . 177
184
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
Index of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
iv
Acknowledgments
Without the advise and encouragement of many people, this work would never have come into existence. First of all, I am grateful to my advisor, Shou-Wu Zhang for introducing me this subject
including the original problem, and his consistent encouragement and interest in this work. I am
indebted to Xinyi Yuan, Shou-Wu Zhang and Wei Zhang who were kind enough to share with me
some of the ideas in their recent joint work, which are crucial to this one. I am also thankful to Wee
Teck Gan, Atsushi Ichino, Aise Johan de Jong, Luis Garcia Martinez, Frans Oort, Michael Rapoport,
Mingmin Shen, Ye Tian, Chenyan Wu, Shunsuke Yamana, Tonghai Yang and Weizhe Zheng for useful
conversations during the long-term preparation.
I would like to thank Stephen S. Kudla and Joachim Schwermer for inviting me to the workshop
Automorphic Forms: New Directions at Mathematisches Forschungsinstitut Oberwolfach in March
2011 and to present this work there; YoungJu Choie and Sug Woo Shin for inviting me to give a mini
course on this subject in the Theta Festival : The 4th ILJU School of Mathematics held at Pohang
University of Science and Technology in August 2011. Without their wonderful organization and
hospitality, I would have missed very good opportunities to communicate with other mathematicians
on this work and related subjects.
I also deeply appreciate the Morningside Center of Mathematics, Chinese Academy of Sciences,
in Beijing for providing me a perfect research environment with full hospitality during the summers
of last four years. Especially in July 2008, I was able to attend a workshop on Arithmetic Geometry
held at the Center, where Jianshu Li and Shou-Wu Zhang gave two talks that directly motivated me
to work on this problem in the following years.
For my Ph.D. life from 2007 to 2012, I must thank all the staff, faculty and students at Department
of Mathematics, Columbia University for making it enjoyable.
Last but not least, I thank my family and my friends for their everlasting encouragement and
support in every aspects of my life.
v
1
Chapter 1
Introduction
A central question in Number Theory is to solve Diophantine equations, that is, to study the polynomial equations in the field of rational numbers, or more generally, in number fields. From the
viewpoint of algebraic geometry, the zero locus of a set of polynomial equations in an affine or projective space defines naturally some geometric object, which is called an algebraic variety. Therefore,
it is important to study geometry objects defined over number fields. Among them, there is a special
class of algebraic varieties, called Shimura varieties, for which one can systematically construct a large
supply of points (i.e., solutions), or more generally, cycles (i.e., families of solutions).
In this article, we study Shimura varieties associated to certain unitary groups and their special
cycles. Moreover, we relate the arithmetic of such cycles to L-functions of automorphic representations,
which are analytic objects. The method we use to set up such a relation is an arithmetic analogue of the
theta lifting in the classical theory of automorphic representation. This is first observed by S. Kudla
[Kud1997, Kud2002, Kud2003] and later developed by S. Kudla, M. Rapport and T. Yang [KRY2006].
We formulate a precise conjecture about an arithmetic analogue of the classical Rallis’ inner product
formula, which we call arithmetic inner product formula, and confirm it for unitary groups of two
variables. In particular, we calculate the Néron–Tate height of special points on Shimura curves
attached to certain unitary groups of two variables.
2
Rallis’ inner product formula
Let us briefly review the Rallis’ inner product formula in the classical theory of theta lifting, which
first appeared in [Ral1984]. The original formula is to calculate the Petersson inner product of two
automorphic forms on an orthogonal group that are lifted from a symplectic group through theta
lifting. It turns out, using the Siegel–Weil formula, that the inner product is related to a diagonal
integral on the doubling symplectic group of the original automorphic forms with certain Eisenstein
series. This doubling method was later generalized to other cases by S. Gelbart, I. Piatetski-Shapiro
and S. Rallis [GPSR1987]. This diagonal integral is in fact Eulerian. In other words, it decomposes
into so-called local zeta integrals. These local zeta integrals are directly related to the L-factors of the
corresponding representations. In fact, they prove in many cases that when everything is unramified,
the local zeta integral coincides with the local Langlands L-factor, modified by some Tate L-factors.
Later, J. Li [Li1992] extended such results to unitary groups.
In the introduction, we only look at a special case of Rallis’ inner product formula, which is parallel
to the arithmetic theory developed later. Let F be a totally real field, and E/F a totally imaginary
quadratic extension. For an integer n ≥ 1, let H 0 = U(n, n)F be the unique quasi-split unitary group
of a skew-hermitian space over E (with respect to the Galois involution of E/F ) of rank 2n. Let H
be the unitary group of a hermitian space V over E of rank 2n. Both H 0 and H are reductive groups
AF ) × H(AF ), realizing on S(V (AE )n): the space
of Schwartz functions on V (AE )n . For such an element φ ∈ S(V (AE )n ), we have the following theta
over F . We have a Weil representation
1
ω of H 0 (
series
θ(g, h; φ) =
X
(ω(g, h)φ) (x),
x∈V n (E)
which is a smooth, slowly increasing function on H 0 (F )\H 0 (
an irreducible representation of H 0 (
and φ ∈ S(V (
AF ) × H(F )\H(AF ). Let π ⊂ A0(H 0) be
AF ) contained in the space of cusp forms of H 0. For every f ∈ π
AE )n), we define the theta lifting to be
θφf (h)
Z
=
A
H 0 (F )\H 0 (
θ(g, h; φ)f (g)dg.
F)
AE )n), viewed as the underlying
∨
Similarly, we have θφf ∨ for f ∨ ∈ π ∨ = f | f ∈ π and φ∨ ∈ S(V (
space of the contragredient representation ω ∨ . Applying a regularized Siegel–Weil formula of A. Ichino
1 It depends on the choice of a
×
×
F \ E . For simplicity, we will
E×
A A
AF ,
nontrivial additive character ψ of F \
assume that χα are both trivial.
and two characters χα (α = 1, 2) of
3
[Ich2004, Ich2007], we obtain the following formula2
∨
hθφf , θφf ∨ iH =
Y
L( 1 , π)
Z ∗ (0, fv , fv∨ , φv ⊗ φ∨
Q2n 2
v ),
i
2 i=1 L(i, E/F ) v∈S
(1.1)
where
• h−, −iH denotes the Petersson inner product on H (for a suitable Haar measure);
• S is a finite set of primes of F containing all archimedean places;
• L(s, π) = LS (s, π)LS (s, π) is the L-function of π, whose unramified part LS (π) is defined from
the Satake parameter and the ramified part LS (π) is defined in [HKS1996] as a greatest common
divisor;
A×F associated to the quadratic extension E/F ; and
• E/F is the quadratic character of F × \
• Z ∗ (0, fv , fv∨ , 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 (πv ) ∈ {±1}, which is 1 if v 6∈ S,
n
and such that Z ∗ (0, fv , fv∨ , φv ⊗ φ∨
v ) is not always zero if and only if (πv ) = ηEv /Fv ((−1) det Vv ).
Q
We let (π) = v (πv ). There are two cases.
1. (π) = 1. Then we can choose a suitable hermitian space V such that the set of theta lifting θφf
for φ ∈ S(V (
AE )n) and f ∈ π contains a nonzero function if and only if L( 12 , π) 6= 0.
2. (π) = −1. Then whatever V we choose, all theta lifting θφf is trivial. In this case, Rallis’ inner
product formula is not interesting. In fact, as we will see in Theorem 2.3.9, L( 21 , π) = 0 for such
π.
Therefore, it is natural in case (2) to ask the information about L0 ( 12 , π). Parallel to the classical
theory where the central value of the L-function relates to the lifting on the level of functions, we
propose an arithmetic theory where the central derivative of the L-function relates to the lifting on
the level of cycles. Such formulation will be elaborated in the next subsection.
Arithmetic inner product formula
We now consider the second case, that is, π has the factor (π) = −1. Therefore, L( 21 , π) = 0. We also
assume that the archimedean component π∞ of π is a discrete series representation of certain type.
2 Here,
we assume that V is anisotropic for simplicity to avoid the regularization process.
4
From such π, we can construct, instead of the hermitian space V over E, a hermitian space (or rather
V over AE of rank 2n. Such V is incoherent in the sense that there does not
exist a hermitian space V over E making V ∼
= V ⊗E AE , which is parallel to the fact that (π) = −1.
Moreover, by our assumption on π∞ , V is totally positive definite. Let H = U(V) be the group of
isometry of V, which is a reductive group over AF . By the theory of Shimura variety, we attach to H (a
projective system of) Shimura varieties (Sh(H)K )K for open compact subgroups K ⊆ H(AF,fin ). They
are smooth quasi-projective varieties over Spec E. Let S(V(AE )n )U ⊂ S(V(AE )n ) be the subspace
a hermitian module)
∞
of those Schwartz functions whose archimedean components are essentially the Gaussian. Following
V AE )n)U
S. Kudla, for φ ∈ S( (
∞
, we define the generating series Zφ (g), which is a “function” on
AF ) whose values are formal series in CHn(Sh(H))C, the injective limit of groups of Chow cycles
(with coefficients in C) of codimension n on Sh(H)K for all K. The series Zφ (g) should be viewed
H 0(
as the arithmetic analogue of the classical theta series θ(g, •; φ), where the later is a “function” on
AF ) whose values are automorphic forms of H. Parallel to the automorphy property of θ(g, •; φ)
(as a “function” on H 0 (AF )), we have the following result.
H 0(
Theorem (Modularity of the generating series, Theorem 3.1.6). Let l be a linear functional on
H
CHn (Sh( ))C . Then
1. If l(Zφ )(g) is absolutely convergent, it is an automorphic form of H 0 .
2. If n = 1, l(Zφ )(g) is absolutely convergent for every l.
In fact, the above theorem holds for all codimensions, not just n. There is also a version in the case
of symplectic-orthogonal pairs, which is proved by X. Yuan, S.-W. Zhang and W. Zhang [YZZ2009].
The proof for both cases use the induction process on the codimension, which originally comes from
the idea of W. Zhang [Zha2009]. Moreover, the proof for the case where the generating series has
codimension 1 reduces to the result in [YZZ2009].
Recall that in the case of classical theta lifting, we construct θφf simply by taking the inner product
of f and the theta series. In view of the above theorem, we have the following parallel definition. We
define the arithmetic theta lifting to be
Θfφ =
Z
A
H 0 (F )\H 0 (
f (g)Zφ (g)dg,
F)
for f ∈ π. Rigorously speaking, the above integration is formal and we should justify such expression
by applying a linear functional l as in the above theorem. Nevertheless, the (Betti) cohomology class
cl(Θfφ ) of Θfφ is always well-defined. To find an arithmetic analogue of the Petersson inner product, we
5
invoke the conjectural Beilinson–Bloch height paring [Beı̆1987, Blo1984], which is, in this particular
case, a hermitian paring
H
H
h−, −iBB : CHn (Sh( ))0C × CHn (Sh( ))0C →
H
C.
H
Here, CHn (Sh( ))0C ⊂ CHn (Sh( ))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.
H
H
First, we need Sh( )K to be proper. Second, we would like to have Θfφ ∈ CHn (Sh( ))0C . The first
issue will be discussed in 3.2.1, where we propose some constructions as well as some conjectures. To
H
simplify the discussion in the introduction, we assume that Sh( )K is already proper, which is the
case when, for example, F 6=
Q. Then we have the following result concerning the second issue.
H
Proposition (Proposition 3.3.4). Assume that Sh( )K is proper for all K. Then the cohomology
class cl(Θfφ ) is trivial.
In view of the above result, we formulate the following conjecture of the arithmetic inner product
formula.
Conjecture (Arithmetic inner product formula, Conjecture 3.3.6). Let π be an irreducible cuspidal
AF ) as above.
automorphic representation of H 0 (
V AE )n)U
f ∨ ∈ π ∨ and every φ, φ∨ ∈ S( (
∞
In particular, (π) = −1. Then for every f ∈ π,
, we have
Y
∨
L0 ( 1 , π)
hΘfφ , Θfφ∨ iBB = Q2n 2 i
Z ∗ (0, fv , fv∨ , φv ⊗ φ∨
v ),
L(i,
)
i=1
E/F v∈S
where S is a finite set of primes, and the local factors Z ∗ are same to those in the Rallis inner product
formula (1.1).
The main result of the article is the following theorem, which justifies the above conjecture in the
H
case n = 1. We point out that when n = 1, Sh( )K is a curve. Therefore, the Beilinson–Bloch height
paring is simply the well-known Néron–Tate height paring, denoted by h−, −iNT .
Theorem (Arithmetic inner product formula, Theorem 7.2.1). Let n = 1. Let π be an irreducible
cuspidal automorphic representation of H 0 (
V AE )n)U
f ∨ ∈ π ∨ and every φ, φ∨ ∈ S( (
∨
hΘfφ , Θfφ∨ iNT =
∞
AF ) as in the above conjecture.
Then for every f ∈ π,
, we have
Y
L0 ( 12 , π)
Z ∗ (0, fv , fv∨ , φv ⊗ φ∨
v ),
LF (2)L(1, E/F )
v∈S
6
where S is a finite set of primes, and the local factors Z ∗ are same to those in the Rallis inner product
formula (1.1).
The first appearance of such arithmetic analogue of Rallis’ inner product formula is the main
Q
result of [KRY2006]. The authors studied the case where f is a new form of PGL2 ( ) of weight 2 and
square-free level. In particular, the corresponding Shimura curve where the height paring is taken on
is the one attached to a division quaternion algebra over
Q. More recently, J. Bruinier and T. Yang
[BY2009] studied the case where the Shimura curve is the modular curve. They obtain a formula that
is very similar to ours here, and from which they deduce certain cases of the Gross–Zagier formula.
In fact, the study of derivative of L-functions was initiated by Gross and Zagier about thirty years
ago in the pioneer paper [GZ1986]. The recent work of X. Yuan, S.-W. Zhang and W. Zhang [YZZa]
generalizes this formula in an extremely broad and conceptual form, based on the connection with the
representation theory of the so-called restriction problem. As for this work, we generalize the formulas
of Kudla et al. in a uniform and explicit form, by exploring the theory of local theta correspondence,
as we will see in the next subsection where we outline the idea of the proof.
Outline of the proof
The proof of the main theorem consists of six parts, which occupy the following six chapters respectively. Throughout the process, we make our argument as general as possible. In other words, we deal
with the problem for general n, not just 1, once we are able to do so.
In Chapter 2, we introduce analytic kernel functions that compute the derivative of L-functions.
Such kernel functions are derivatives of Siegel Eisenstein series associated to degenerate principal
V AE )n)
series on the doubling unitary group. Precisely, given a pair of Schwartz functions φα ∈ S( (
(α = 1, 2), we have a kernel function E 0 (0, g, φ1 ⊗ φ2 ). We show that one can choose φα carefully such
that E 0 (0, g, φ1 ⊗ φ2 ) can be expressed as a sum of local terms indexed by (almost all) places of F
that are nonsplit in E, for g in an open dense subset. Precisely, we have the following decomposition
(2.22)
E 0 (0, ι(g1 , g2∨ ), φ1 ⊗ φ2 ) =
X
Ev (0, ι(g1 , g2∨ ), φ1 ⊗ φ2 ),
v6∈S
where S is a finite set of finite places of F at which ramification occurs. To compute the actual
derivative L0 ( 21 , π), we only need to take the inner product of E 0 (0, •, φ1 ⊗φ2 ) with f and f ∨ . Therefore,
we should study Ev (0, g, φ1 ⊗ φ2 ). Throughout this chapter, all discussions work for general n except
in 2.4.3.
7
We introduce the Shimura varieties attached to certain unitary groups, their special cycles and
generating series in Chapter 3. Moreover, we discuss the case of non-proper Shimura varieties, where
we need to compactify everything we introduce above. As mentioned previously, we prove the modularity of the generating series, and the cohomological triviality of the arithmetic theta lifting. Finally,
in 3.4, where we restrict ourselves to the case n = 1, we introduce the arithmetic kernel functions
E(g1, g2; φ1 ⊗ φ2) that compute the Néron–Tate height paring of the arithmetic theta lifting.
Then
modulo certain volume factors and terms that are perpendicular to f and f ∨ , we have the following
decomposition of the arithmetic kernel function (3.15)
E(g1, g2; φ1 ⊗ φ2) =
X
Ev (g1, g2; φ1 ⊗ φ2),
◦
v ◦ ∈Σ◦
where Σ◦ is the set of places of E, and
Ev (g1, g2; φ1 ⊗ φ2) = hZbφ (g1), Zbφ (g2)iv
◦
1
2
◦
is a local height paring on a certain model of the Shimura curve. It is clear that to prove the main
theorem, we need to compare the analytic and arithmetic kernel functions place by place. There are
three cases. First, v is an archimedean place. Second, v is a finite nonsplit place that is not in S.
Third, v is either finite split or in S.
The first case is treated in Chapter 4. We reduce the comparison to a local question, which can be
formulated for general dimensions. We prove a formula which we call the archimedean local arithmetic
Siegel–Weil formula. Let m ≥ 2 be an integer. Let T be a (nondegenerate) hermitian matrix in
C
GLm ( ) of signature (m − 1, 1). Let Φ0 be the Gaussian on V m , where V is the standard positive
definite complex hermitian space of dimension m. On the one hand, we have the Whittaker integral
WT (s, e, Φ0 ) that is a holomorphic function in s. It is not hard to see that WT (0, e, Φ0 ) = 0. On
the other hand, we define an archimedean local intersection number H(T )∞ , which is the volume of
the open unit ball D in
Cm−1 with respect to a star product of Green currents constructed from
Kudla–Millson forms. We prove the following result.
Theorem (Archimedean local arithmetic Siegel–Weil formula, Theorem 4.3.1). Let T be a (nonde-
C
generate) hermitian matrix in GLm ( ) of signature (m − 1, 1). Then we have
WT0 (0, e, Φ0 ) = Cm exp(−2π tr T )H(T )∞ ,
8
where Cm is some nonzero constant depending only on m, not on T .
The above theorem specified to m = 2 will pose a close relation between Ev (0, ι(g1 , g2∨ ), φ1 ⊗ φ2 )
and
Ev (g1, g2; φ1 ⊗ φ2) for an archimedean place v dividing v◦.
◦
More precisely, they are related by
the process of holomorphic projection, which we discuss in 7.1.
The second case is treated in Chapter 5. We prove that for a finite place v = p outside S that is
nonsplit in E, Ep (0, ι(g1 , g2∨ ), φ1 ⊗ φ2 ) and
Ep (g1, g2; φ1 ⊗ φ2) are equal, where p◦ is the unique place
◦
of E over p. This comparison is again reduced to a local question as follows. For a 2-by-2 hermitian
matrix T with entries in OEp◦ , the ring of integers of Ep◦ , such that its determinant has odd valuation,
we define a number Hp (T ) that is some intersection multiplicity on the smooth integral model of the
Shimura curve at p◦ . Let V + be the 2-dimensional hermitian space over E with a selfdual lattice
Λ+ , and Φ0+ the characteristic function of Λ+ . We have a p-adic Whittaker integral WT (s, e, Φ0+ ),
holomorphic in s. Since det T has odd valuation, WT (0, e, Φ0+ ) = 0. We prove the following result.
Theorem (Non-archimedean local arithmetic Siegel–Weil formula, Theorem 5.2.3 and Corollary
5.3.2). Let T be a 2-by-2 hermitian matrix with entries in OEp◦ such that det T has odd valuation.
Then we have
WT0 (0, e, Φ0+ ) = Cp · Hp (T ),
where Cp is some nonzero constant depending only on the place p, not on T .
In the entire chapter, we restrict ourselves to the case n = 1.
The third case is treated in Chapter 6. We prove that under careful choices of φα (α = 1, 2), there
is no contribution of terms
Ev (g1, g2; φ1 ⊗ φ2) for a finite place v either split in E or in S, after taking
◦
inner product with f and f ∨ . This is compatible with the analytic side since the corresponding terms
are all zero. We are only able to make such argument when n = 1. Now we come to the final stage
of the proof of the main theorem, which is accomplished in Chapter 7. As we have mentioned before,
we apply the holomorphic projection to the analytic kernel function to make its archimedean part
coincide with the corresponding part of the arithmetic kernel function. To prove the arithmetic inner
product formula in the full generality, we apply the result of multiplicity one proved in A.2. The last
step is extremely crucial to make us able to avoid explicit computations at all bad places. We would
like to remark that such idea originally comes from [YZZa].
Conventions and notations
Convention 1.0.1. All rings will have a unit.
9
Notation 1.0.2.
• We denote by
Z the ring of (rational) integers.
We let
Q, R and C be the
fields of rational, real and complex numbers, respectively.
• For a
Z-module M
and a commutative ring R, we denote MR = M ⊗Z R the base change
R-module.
• We denote by
Afin = Zb ⊗Z Q =
limN
←−
Z/N Z
⊗Z
Q the ring of finite adèles; A=R × Afin the
ring of full adèles.
• For a number field K, we let
AK = A ⊗Q K, AK,fin = Afin ⊗Q K and K∞ = R ⊗Q K.
• As usual, for a subset S of places, −S (resp. −S ) means the S-component (resp. component
away from S) for the corresponding (decomposable) adèlic object; −∞ (resp. −fin ) means the
infinite/archimedean (resp. finite) part.
• For a field K, we fix a separable closure K sep of K and denote ΓK = Gal(K sep /K) the Galois
group of K.
• The symbol Tr and Nm stand for the trace (resp. reduced trace) and norm (resp. reduced
norm), respectively, if they are applied to fields or rings of adèles (resp. simple algebras). The
symbol tr stands the trace for matrix and linear transforms.
Notation 1.0.3.
• For a ring R and integers m, n > 0, we denote by Matm,n (R) the ring of
m-by-n matrix with entries in R. We also set Matn (R) = Matn,n (R).
• We denote by 1n (resp. 0n ) the identity (resp. zero) matrix of rank n.
• We denote by t g the transpose of a matrix g.
Definition 1.0.4.
• Let R be a commutative ring and R0 an étale (commutative) algebra over R
of rank 2. Let τ : r 7→ rτ for r ∈ R0 be the nontrivial automorphism of R0 over R. For n ≥ 0, a
hermitian (resp. skew-hermitian) space of rank n over R0 with respect to τ is a free module V
over R0 of rank n equipped with a map
(−, −) : V × V → R0
satisfying that
1. It is R0 -linear in the first variable, i.e., for r ∈ R and v, v 0 ∈ V , (rv, v 0 ) = r(v, v 0 ).
2. It is (R0 , τ )-linear in the second variable, i.e., for r ∈ R and v, v 0 ∈ V , (v, rv 0 ) = rτ (v, v 0 ).
10
3. It is τ -symmetric (resp. τ -antisymmetric), i.e., for v, v 0 ∈ V , (v, v 0 ) = (v 0 , v)τ (resp. (v, v 0 ) =
−(v 0 , v)τ ).
In fact, under the assumption (3), assumptions (1) and (2) imply each other.
• A hermitian or skew-hermitian space V over R0 of rank n is nondegenerate if their is a basis
{v1 , . . . , vn } of V over R0 such that the matrix ((vi , vj ))1≤i,j≤n has determinant in R0× : the
group of invertible elements in R0 . It is clear that this property does not depend on the choice
of the basis.
• In practice, R will be a field and R0 /R a (possibly split) extension of degree 2, or R the ring of
adèles of a number field and R0 that of a quadratic field extension. The involution τ will always
be clear in the context and hence we will not say with respect τ in general.
• In the main part of the article, all hermitian spaces will be of finite rank and nondegenerate.
Notation 1.0.5. Let r ≥ 1 be an integer, we set


wr = 

−1r
1r 
.
For 0 ≤ d ≤ r, we set

wr,d




=




1d
1r−d
1d
−1r−d




.



For r elements a1 , . . . , ar in a ring, we set

 a1

diag[a1 , . . . , ar ] = 



..
.
ar





to be the diagonal matrix.
Notation 1.0.6. Let G be a Lie group over a local field. We denote by λG : G →
C× the modulus
character of G. Precisely, for g ∈ G, we have the adjoint action Adg on the Lie algebra Lie G. Take a
nonzero Haar measure dx on Lie G. Then λG (g) = Ad∗g dx/dx.
11
Notation 1.0.7. Let K be a field and X a scheme of finite type over Spec K,
• For an integer r ≥ 0, we denote by CHr (X) the Chow cohomology group of codimension r. In
practice, X will be smooth over K. Therefore, CHr (X) can be canonically identified with the
abelian group of Chow cycles of X of codimension r over K.
• If r = 1, we set Pic(X) = CH1 (X). In other words, Pic(X) is the Picard group of X, which
should not be understood as the Picard scheme or Picard stack of X.
12
Chapter 2
Doubling method and analytic
kernel functions
The goal of this chapter is to introduce the analytic kernel functions. In 2.1, we review the Siegel–Weil
formula and some of its generalization that are related to our problem. In 2.2, we review the theory of
I. Piatetski-Shapiro and S. Rallis on the doubling method. In particular, we deduce the Rallis’ inner
product formula in certain cases from the doubling method and the Siegel–Weil formula. We also
introduce the (normalized) local zeta integrals that will serve as the local terms in both classical and
arithmetic inner product formula. In 2.3, we introduce the L-function and the formula representing
it. We show that the central L-value of an automorphic representation vanishes if its global epsilon
factor equals −1. Then we introduce the analytic kernel functions that compute the L-derivatives. In
2.4, we study these analytic kernel functions. We prove that for certain nice choices of test functions,
the analytic kernel function can be decomposed into terms indexed by archimedean and unramified
nonsplit finite places of F .
13
2.1
2.1.1
Siegel–Weil formula and generalizations
Degenerate principal series and Eisenstein series
Let F be a totally real number field and E a totally imaginary quadratic extension of F . We denote by
τ the nontrivial element in Gal(E/F ) and E/F :
A×F /F × → {±1} the quadratic character associated
by the class field theory. Let Σ (resp. Σfin , Σ∞ ) be the set of all places (resp. finite places, infinite
places) of F , and Σ◦ , Σ◦fin , Σ◦∞ those of E. We fix a nontrivial additive character ψ of
a continuous character ψ :
AF → C×, which is trivial on F .
AF /F , that is,
For a positive integer r, we denote by Wr the standard skew-hermitian space over E with respect
to the involution τ , which is equipped with a skew-hermitian form h−, −i such that there is an E-basis
{e1 , . . . , e2r } satisfying
• hei , ej i = 0;
• her+i , er+j i = 0;
• hei , er+j i = δij for 1 ≤ i, j ≤ r.
Let Hr = U(Wr ) be the unitary group of Wr , which is a reductive group over F . The group Hr (F ),
in which F can be itself or its completion at some place, is generated by the parabolic subgroup
Pr (F ) = Nr (F )Mr (F ) and the element wr . Precisely,
Nr (F ) =





Mr (F ) =





and

1r
n(b) = 




b
|
b
∈
Her
(E)
;

r


1r


a
m(a) = 
t τ,−1
a


wr = 
−1r

 | a ∈ GLr (E)



;



1r 

as in Notation 1.0.5. Here, Herr (E) = b ∈ Matr (E) | bτ = t b .
We fix a place v ∈ Σ and suppress it from notations. Thus F = Fv is a local field of characteristic
zero; E = Ev := E ⊗F Fv is a quadratic extension of F which could be split; and Hr = Hr,v := Hr (Fv )
is a reductive Lie group over the local field. We define Kr to be a maximal compact subgroup of Hr
14
in the following way:
• If v is finite, then
Kr = Hr ∩ GL(OE he1 , . . . , e2n i) ⊂ GL(Wr ).
• If v is (real) infinite, then
Kr = Hr ∩ U(2r)R ⊂ GL(2r)C ∼
= GL(Wr )
where the isomorphism is determined by the basis {e1 , . . . , e2n }. Therefore, Kr is isomorphic to
U(r)R × U(r)R (2.1).
For s ∈
C and a character χ of E ×, we denote by Ir (s, χ) = sIndHP (χ| • |s+
E
r
r
r
2
) the degenerate principal
series representation (cf. [KS1997]) of Hr , where sInd stands for the (non-normalized) smooth Kr finite induction. Precisely, it is realized on the space of smooth Kr -finite functions ϕs on Hr satisfying
s+ r2
ϕs (n(b)m(a)g) = χ(det a)| det a|E
ϕs (g)
for all g ∈ Hr , m(a) ∈ Mr and n(b) ∈ Nr . A (holomorphic) section ϕs of Ir (s, χ) is called standard if
its restriction to Kr is independent of s. It is called unramified if it takes value 1 on Kr .
A
Now we view F and E as number fields. For a (continuous) character χ of ×
E that is trivial on
N
0
E × , and s ∈ , we have an admissible representation Ir (s, χ) = v∈Σ Ir (s, χv ) of Hr ( F ), where
C
A
the restriction in the restricted tensor product refers to the collection of unramified sections. For a
standard section ϕs = ⊗ϕs,v ∈ Ir (s, χ), we define the Eisenstein series
X
E(g, ϕs ) =
ϕs (γg).
γ∈Pr (F )\Hr (F )
The series is absolutely convergent if Re s >
r
2,
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
extension
C (with respect to the quadratic
C/R) of rank 2r with a basis {e1, . . . , er ; er+1, . . . , e2r }, under which the skew-hermitian
15
form is given by the matrix


wr = 
−1r

1r 
.
Let U(r, r)R be the subgroup of ResC/R GL(WR,r ) preserving the skew-hermitian form, which is a
reductive group over
R. We also let
C
U(r)R = {g ∈ ResC/R GLr ( ) | t gg = 1r }1
(2.1)
C
be a subgroup of ResC/R GLr ( ). We have the following embedding
U(r)R × U(r)R → U(r, r)R

1  k1 + k2
(k1 , k2 ) 7→ [k1 , k2 ] := 
2
ik1 − ik2

−ik1 + ik2 
.
k1 + k2
Moreover, the above embedding identifies U(r)R × U(r)R as a maximal compact subgroup Kr of
U(r, r)R .
Notation 2.1.1. We introduce the following notation.
C, whose image is contained in R, and ι◦i ,
R with C through the embedding ι◦i . In particular,
1. We let ιi (i = 1, . . . , d) be all embeddings of F into
ι•i those of E above ιi . We identify E ⊗F,ιi
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 Eι× , which is identified with
via ι◦ , such that χι | Fι× ∼
=
C×
R× equals sgnm. Here, sgn denotes the sign character and m is an
integer. In particular, we can write
χι (z) = p
zk
χι
k χι
,
|zz|
for a unique integer kχι that has the same parity as m. If χ is an automorphic character of
whose restriction to
m
A×F equals ηE/F
, we set kχ = (kχ
ι1
A×E
, . . . , kχιd ).
Definition 2.1.2 (Weights). We define the notions of weights as follows.
1. Let π be an irreducible (Lie U(r, r)R , Kr )-module (or its Casselman–Wallach globalization). Let
a, b be integers. We say π is of weight (a, b) if the minimal Kr -type of π is the character
deta detb , i.e., its sends [k1 , k2 ] to (det k1 )a (det k2 )b .
16
A
2. Let π be an irreducible automorphic representation of Hr ( F ). For two d-tuples of integers a =
Nd
(a1 , . . . , ad ), b = (b1 , . . . , bd ), we say π∞ = 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 of Hr (
AF ) is of weight (a, b) if f (g[k1,ι, k2,ι]) = (det k1,ι)a (det k2,ι)b f (g)
ι
ι
for every ι ∈ Σ∞ .
2.1.3
Weil representations and theta functions
Let us review the classification of (nondegenerate) hermitian spaces. Let m ≥ 1 be an integer and
v ∈ Σ be a place of F . For a hermitian space V over Ev of rank m, we define
m(m−1)
(V ) = Ev /Fv (−1) 2 det V ∈ {±1}.
There are three cases:
• If v ∈ Σfin such that E is nonsplit at v, then up to isometry, there are two different hermitian
spaces over Ev of dimension m ≥ 1: V ± determined by (V ± ) = ±1.
• If v ∈ Σfin such that E is split at v, then up to isometry, there is only one hermitian space V +
over Ev of dimension m.
• If v ∈ Σ∞ (which is real), then up to isometry, there are m + 1 different hermitian spaces over
Ev of dimension m: Vs with signature (s, m − s) where 0 ≤ s ≤ m.
In the global situation, up to isometry, all hermitian spaces V over E of dimension m are classified
by signatures at infinite places and det V ∈ F × / Nm E × . 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.
V is nondegenerate if there is a basis
under which the matrix representing the hermitian form is invertible in GLm (AE ). For a place v ∈ Σ,
we let Vv = V ⊗A Fv , Vfin = V ⊗A AF,fin ; and define Σ(V) = {v ∈ Σ | (Vv ) = −1}, which is a
Q
finite set. Finally, we let (V) = v∈Σ (Vv ).
Recall in Definition 1.0.4 that in this case, a hermitian space
F
F
Definition 2.1.3 (Coherent/incoherent hermitian spaces). We say a (nondegenerate) hermitian space
V over AE is coherent (resp. incoherent) if the cardinality of Σ(V) is even (resp. odd), i.e., (V) = 1
(resp. −1).
17
By the Hasse principle, there is a hermitian space V over E such that
V ∼= V ⊗F AF if and only if
V is coherent. These two terminologies are inspired from the coherent/incoherent collections of local
quadratic spaces introduced by S. Kudla in the orthogonal case in [KR1994, Kud1997].
We fix a place v ∈ Σ and suppress it from notations. For a hermitian space V of dimension
m with hermitian form (−, −) and a positive integer r, we can construct a symplectic space W =
ResE/F Wr ⊗E V of dimension 4rm over F with the skew-symmetric form TrE/F h−, −i ⊗ (−, −),
where Res stands for the Weil restriction. Let Sp(W) be the symplectic group and Mp(W) be its
C×-metaplectic cover fitting into the following exact sequence:
1
/
C×
/ Mp(W)
/ Sp(W)
/ 1.
We let H = U(V ) be the unitary group of V and S(V r ) the space of Schwartz functions on V r . Given
a character χ of E × satisfying χ|F × = m
E/F , we have a splitting homomorphism
ı̃(χ,1) : Hr × H → Mp(W)
lifting the natural homomorphism ı : Hr × H → Sp(W) (cf. [HKS1996, Section 1]). We thus have
a Weil representation (with respect to ψ) ωχ = ωχ,ψ of Hr × H on the space S(V r ). Explicitly, for
φ ∈ S(V r ) and h ∈ H,
• (ωχ (n(b))φ) (x) = ψ(tr bT (x))φ(x);
m
• (ωχ (m(a))φ) (x) = | det a|E2 χ(det a)φ(xa);
b
• (ωχ (wr )φ) (x) = γV φ(x);
• (ωχ (h)φ) (x) = φ(h−1 x),
where
T (x) =
1
((xi , xj ))1≤i,j≤r
2
is the moment matrix of x; γV is the Weil constant associated to the underlying quadratic space of V
(and also ψ); φb is the Fourier transform
Z
b
φ(x)
=
φ(y)ψ
Vr
1
t
TrE/F (x, y) dy
2
18
using the selfdual measure dy on V r with respect to ψ.
In the global situation where F is the number field, by taking the restricted tensor product over
AF ) × H(AF ) on the space S(V r ) :=
all local Weil representations, we obtain a representation of Hr (
N0
r
v∈Σ S(Vv ).
Warning 2.1.4. We abuse notation by denoting S(V r ) for spaces of both local and adèlic Schwartz
AE )r ) in the adèlic case.
functions. More precisely, we should use S(V (
We feel that there is little
danger of confusion since in the adèlic 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
r
= m
E/F and φ ∈ S(V ), we define the theta
function
X
θ(g, h; φ) =
(ωχ (g, h)φ) (x),
x∈V r (E)
AF ) × H(F )\H(AF ).
which is a smooth, slowly increasing function on Hr (F )\Hr (
Consider the
integral
Z
IV (g, φ) =
A
H(F )\H(
θ(g, h; φ)dh,
F)
AF )) = 1.
if it is absolutely convergent. Here we normalize the measure dh such that Vol(H(F )\H(
It is well-known that IV (g, φ) is absolutely convergent for all φ if m > 2r or V is anisotropic.
2.1.4
Siegel–Weil formulae
It is immediate to see that
ϕφ,s (g) = (ωχ (g)φ) (0)λPr (g)s−
m−r
2
is a standard section in Ir (s, χ) for every φ ∈ S(V r ). Recall the following formula for the modulus
character
λPr (g) = λPr (n(b)m(a)k) = | det a|AE ,
AF ).
if g = n(b)m(a)k under the Iwasawa decomposition with respect to the parabolic subgroup Pr (
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−r
2 .
Then we have
19
1. If m > 2r, E(s0 , g, φ) is absolutely convergent and
E(s0 , g, φ) = IV (g, φ).
2. If r < m ≤ 2r and V is anisotropic, E(s, g, φ) is holomorphic at s0 and
E(s, g, φ)|s=s0 = IV (g, φ).
3. if m = r and V is anisotropic, E(s, g, φ) is holomorphic at s0 = 0 and
E(s, g, φ)|s=0 = 2IV (g, φ).
Proof.
1. It is the classical Siegel–Weil formula.
2. It is a generalized Siegel–Weil formula proved in [Ich2007, Theorem 1.1].
3. It is a generalized Siegel–Weil formula proved in [Ich2004, Theorem 4.2].
In what follows, we simply write E(s0 , g, φ) for E(s, g, φ)|s=s0 for simplicity if the Eisenstein series
is holomorphic at s0 .
Remark 2.1.6. In Theorem 2.1.5 (3), if V is isotropic, we still have a (regularized) Siegel–Weil
formula. However, since the theta integral IV (g, φ) is not necessarily convergent, a regularization
process must be applied. The inner product introduced in the next section also requires a regularization
process. Since the classical inner product formula is not the purpose of this article, we will always
assume that V is anisotropic for simplicity, or pretend that the regularization process has been applied
for general V in the following discussion.
2.2
2.2.1
Doubling integrals
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 0
instead of Hn ; P 0 instead of Pn ; N 0 instead of Nn and K0 instead of Kn . Therefore, χ|A× = 1 is the
F
N0
trivial character. Let π = v∈Σ πv be an irreducible cuspidal automorphic representation of H 0 ( F )
A
20
AF ). We will not distinguish π
contained in A0 (H 0 ): the space of cuspidal automorphic forms on H 0 (
with its underlying space. Let π ∨ be the contragredient representation which is realized on the space
of complex conjugation of functions in π.
We denote by (−W ) the skew-hermitian space over E with the skew-hermitian form −h−, −i.
−
Therefore, we have a basis {e−
1 , . . . , e2n } satisfying
−
• he−
i , ej i = 0;
−
• he−
r+i , er+j i = 0;
−
• he−
i , en+j i = −δij for 1 ≤ i ≤ n.
Let W 00 = W ⊕ (−W ) be the direct sum of the two skew-hermitian spaces. There is a natural
embedding
ı : H 0 × H 0 ,→ U(W 00 )
(2.2)
which is, under the basis
• {e1 , . . . , e2n } of W and
−
−
−
00
• {e1 , . . . , en ; e−
1 , . . . , en ; en+1 , . . . , e2n ; −en+1 , . . . , −e2n } of W ,
given by ı(g1 , g2 ) = ı0 (g1 , g2∨ ) where

a1
g1 = 
c1


b1 
a2
 , g2 = 
d1
c2


b2  ∨ 1n
, g = 
d2
−1n
 
 1n
g
and

a
 1



ı0 (g1 , g2 ) = 

 c1


b1
a2
d1
c2


b2 

.



d2
Notation 2.2.1. Let
−
W 0 = spanE {e1 , . . . , en ; e−
1 , . . . , en }
0
−
W = spanE {en+1 , . . . , e2n ; −e−
n+1 , . . . , −e2n },
−1
−1n


21
be a pair of complimentary Lagrangian subspaces of W 00 . Let P be the parabolic subgroup of U(W 00 )
stabilizing W 0 and N the unipotent radical of P .
0
For the complete polarization W 00 = W 0 ⊕ W , we can realize the Weil representation of U(W 00 ),
∨
denoted by ωχ00 (with respect to ψ), on the space S(V 2n ), such that ı∗ ωχ00 ∼
, where the
= ωχ,ψ χωχ,ψ
latter one is realized on the space S(V n ) ⊗ S(V n ). Here we realize the contragredient representation
∨
ωχ,ψ
on the space S(V n ) through the bilinear pairing
hφ, φ∨ iV =
Z
A
V n(
φ(x)φ∨ (x)dx
E)
∨
for φ, φ∨ ∈ S(V n ). Then ωχ,ψ
is identified with ωχ−1 ,ψ−1 .
For φ ∈ S(V n ) and f ∈ π, we define thetheta lifting
θφf (h) =
Z
A
H 0 (F )\H 0 (
θ(g, h; φ)f (g)dg.
F)
AF ), where dg = ⊗v∈Σdgv such that K0v
It is a well-defined, slowly increasing function on H(F )\H(
∨
gets volume 1 for every v ∈ Σ. Similarly, for φ∨ ∈ S(V n ) and f ∨ ∈ π ∨ , we have θφf ∨ . The reader should
be careful that in the contragredient side, the Weil representation used to form the theta function
should also be the contragredient one, i.e., ωχ∨ . We have
∨
hθφf , θφf ∨ iH
Z
:=
∨
A
H(F )\H(
F)
Z
=
Z
A
H(F )\H(
F)
Z
=
θφf (h)θφf ∨ (h)dh
A
[H 0 (F )\H 0 (
θ(g1 , h; φ)f (g1 )θ(g2 , h; φ∨ )f ∨ (g2 )dg1 dg2 dh
F
)]2
Z
A
H(F )\H(
F)
A
[H 0 (F )\H 0 (
Z
=
A
[H 0 (F )\H 0 (
θ(ı(g1 , g2 ), h; φ ⊗ φ∨ )f (g1 )f ∨ (g2 )χ−1 (det g2 )dg1 dg2 dh
F )]
2
∨
−1
f (g1 )f (g2 )χ
F )]
2
Z
(det g2 )
A
H(F )\H(
θ(ı(g1 , g2 ), h; φ ⊗ φ∨ )dhdg1 dg2 .
(2.3)
F)
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
Z
A
[H 0 (F )\H 0 (
f (g1 )f ∨ (g2 )χ−1 (det g2 )E(0, ı(g1 , g2 ), φ ⊗ φ∨ )dg1 dg2 .
2
F )]
We should mention that the Eisenstein series on U(W 00 ) appearing above is formed with respect
22
to the parabolic subgroup P (see Notation 2.2.1), i.e.,
X
E(s, g, Φ) = E(g, ϕΦ,s ) =
ωχ00 (γg)Φ(0)λP (γg)s
γ∈P (F )\ U(W 00 )(F )
for g ∈ U(W 00 )(
AF ), Φ ∈ S(V 2n) and Re s > n.
The coset P (F )\ U(W 00 )(F ) can be canonically
identified with the space of isotropic n-planes in W 00 . Under the right action of H 0 (F ) × H 0 (F )
through ı, the orbit of an n-plane Z is determined by the invariant d = dim Z ∩ W = dim Z ∩ (−W ).
Let γd be a representative of the corresponding double coset where 0 ≤ d ≤ n. In particular, we take


1n




γ0 = 

−1n

1n
1n
1n
1n




 and γn = 14n



(cf. [KR2005]). Let Std be the stabilizer of P γd ı(H 0 × H 0 ) in H 0 × H 0 . In particular St0 = ∆(H 0 ) is
the diagonal subgroup. Therefore, for a standard section ϕs ∈ I2n (s, χ) and Re s > n,
Z
[H 0 (F )\H 0 (
A
f (g1 )f ∨ (g2 )χ−1 (det g2 )E(ı(g1 , g2 ), ϕs )dg1 dg2
F
)]2
Z
=
A
[H 0 (F )\H 0 (
=
(f ⊗ f ∨ χ−1 )(g)
F )]
2
A
d=0
ϕs (γı(g))dg
γ∈P (F )\ U(W 00 )(F )
n Z
X
Std (F )\H 0 (
X
(f ⊗ f ∨ χ−1 )(g)ϕs (γd ı(g))dg.
(2.4)
2
F)
When d > 0, Std has nontrivial unipotent radical. Since f and f ∨ are cuspidal, we have
Z
(2.4) =
∆(H 0 (F ))\H 0 (
Z
=
A
H 0 (F )\H 0 (
AZ )
F)
Z
=
F
(f ⊗ f ∨ χ−1 )(g)ϕs (γ0 ı(g))dg
2
A
H0(
f (g1 g2 )f ∨ (g1 )χ−1 (det g1 )ϕs (γ0 ı(g1 g2 , g1 ))dg1 dg2
F)
Z
A
H 0 (F )\H 0 (
F)
A
H0(
π(g2 )f (g1 )f ∨ (g1 )χ−1 (det g1 )ϕs (p(g1 )γ0 ı(g2 , 1))dg1 dg2
(2.5)
F)
AF ), we have
where p(g1 )γ0 = γ0 ı(g1 , g1 ), and under the Levi decomposition p(g1 ) = n(b)m(a) ∈ P (
23
det a = det g1 . Therefore,
Z
Z
(2.5) =
H0(
A
Z
=
H0(
=
A
π(g2 )f (g1 )f ∨ (g1 )dg1 ϕs (γ0 ı(g2 , 1))dg2
F)
hπ(g)f, f ∨ iϕs (γ0 ı(g, 1))dg
F)
YZ
v∈Σ
A
H 0 (F )\H 0 (
F)
Hv0
hπv (gv )fv , fv∨ iϕs,v (γ0 ı(gv , 1))dgv
where we assume that f , f ∨ and ϕs are all decomposable. In summary, we have the following
proposition.
Proposition 2.2.2. Let f , f ∨ and ϕs be as above. Then for Re s > n, the integral
Z
A
[H 0 (F )\H 0 (
=
YZ
v∈Σ
Hv0
f (g1 )f ∨ (g2 )χ−1 (det g2 )E(ı(g1 , g2 ), ϕs )dg1 dg2
2
F )]
hπv (gv )fv , fv∨ iϕs,v (γ0 ı(gv , 1))dgv
defines an element in the space
HomH 0 (AF )×H 0 (AF ) (I2n (s, χ), π ∨ χπ) =
O
HomHv0 ×Hv0 (I2n (s, χv ), πv∨ χv πv ).
v∈Σ
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
Z(χ, f, f , ϕs ) =
hπ(g)f, f ∨ iϕs (γ0 ı(g, 1))dg,
H0
which is absolutely convergent when Re s > 2n. In [HKS1996, Section 6], the family of good sections
is introduced. For every good section ϕs , the zeta integral Z(χ, f, f ∨ , ϕs ) is a rational function in q −s ,
where q is the cardinality of the residue field of F . In particular, it has a meromorphic continuation
to the entire complex plane. Consider the family of zeta integrals
{Z(χ, f, f ∨ , ϕs ) | f ∈ π, f ∨ ∈ π ∨ , ϕs is good}
and the fractional ideal I of the ring
C[qs, q−s] in its fraction field generated by the above family. In
24
fact, I is generated by P (q −s )−1 , for a unique polynomial P (X) ∈
C[X] such that P (0) = 1. We let
1
1
L(s + , π, χ) =
2
P (q −s )
be the local doubling L-series of Piatetski-Shapiro and Rallis. Same construction can also be applied
to the archimedean case.
Now suppose E/F is unramified (including split) at v and ψ, χ, π are also unramified. Let
0
0
f0 ∈ π K , f0∨ ∈ (π ∨ )K such that hf0 , f0∨ i = 1, and ϕ0s be the unramified standard section. Then the
calculation in [GPSR1987] and [Li1992] (see [Li1992, Theorem 3.1]) shows that
Z(χ, f0 , f0∨ , ϕ0s ) =
L(s + 21 , BC(π) ⊗ χ)
b2n (s)
where
bm (s) =
m−1
Y
L(2s + m − i, iE/F )
(2.6)
i=0
is a product of local Tate factors. For the general case,
b2n (s)Z(χ, f, f ∨ , ϕs )
L(s + 21 , 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 )
|s=0
L(s + 12 , π, χ)
(2.7)
defines a nonzero element in HomH 0 ×H 0 (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 representations π of H 0 and characters χ of E × , we have
L(s, π, χ) = L(s, BC(π) ⊗ χ).
This is known when E/F , χ and π are all unramified due to (the same method of) Kudla–Rallis
[KR2005, Section 5]. It is also known when n = 1 due to [Har1993].
For further discussion, we need to recall a result on degenerate principal series. In what follows, we
25
will use the notation H 00 instead of U(W 00 ) for short, and recall our embedding ı : H 0 × H 0 ,→ H 00 . Let
V be a hermitian space of dimension 2n over E. Then ϕφ (g) = ωχ (g)φ(0) defines an H 00 -intertwining
map S(V 2n ) → I2n (0, χ) whose image R(V, χ) is isomorphic to S(V 2n )H . Recall in 2.1.3 that we
denote by V ± the two non-isometric hermitian spaces of dimension 2n when v is finite nonsplit; V +
the only hermitian space (up to isometry) of dimension 2n when v is finite split; and Vs (0 ≤ s ≤ 2n)
the 2n + 1 non-isometric hermitian spaces of dimension 2n when v is infinite.
Proposition 2.2.4. Let notations be as above. We have
1. If v is finite nonsplit, then R(V + , χ) and R(V − , χ) are irreducible, inequivalent, and I2n (0, χ) =
R(V + , χ) ⊕ R(V − , χ).
2. If v is finite split, then R(V, χ) is irreducible, and I2n (0, χ) = R(V + , χ).
3. If v is infinite, then R(Vs , χ) are irreducible, inequivalent, and I2n (0, χ) =
Proof.
L2n
s=0
R(Vs , χ).
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
2.3.1
Central special values of L-functions
Theta lifting and central L-values
We study the relation between the theta lifting θφf defined in 2.2.1 and the central special value of the
L-function of the representation π.
Recall that we have an irreducible unitary cuspidal automorphic representation π of H 0 = Hn and
a hermitian space V over E of dimension 2n. One key question in the theory of theta lifting is whether
θφf is nonvanishing. A sufficient condition is to look at the local invariant functional as follows. We
have the following proposition, which is usually referred as theta dichotomy.
Proposition 2.3.1. For every nonsplit place v ∈ Σ, HomHv0 ×Hv0 (R(Vv , χv ), πv∨ χv πv ) 6= 0 for exactly
one hermitian space Vv (up to isometry) over Ev of dimension 2n.
Such Vv will be denoted as V (πv , χv ).
26
Proof. If v is (real) archimedean, it is due to [Pau1998, Theorem 2.9]. If v is non-archimedean, it is
due to Proposition A.1.1 or [GG2011, Theorem 2.10], and the nonvanishing of Z ∗ .
In Proposition 2.3.1, if we let ϕs = ϕφ⊗φ∨ ,s and set
∗
∨
),
Z ∗ (s, χv , fv , fv∨ , φv ⊗ φ∨
v ) = Z (χv , fv , fv , ϕφv ⊗φ∨
v ,s
then both sides have meromorphic continuation to the entire complex plane, which are holomorphic
at the point s = 0. Namely, we have
L( 21 , π, χ)
∨
hθφf , θφf ∨ iH =
2
Q2n
Y
i
i=1 L(i, E/F ) v∈S
Z ∗ (0, χv , fv , fv∨ , φv ⊗ φ∨
v ),
(2.8)
in which the product of normalized zeta integrals can actually be taken over a finite set S by the
f
unramified calculation. In particular, for v 6∈ S, Vv ∼
= V (πv , χv ). Then one necessary condition for θφ
to be nonvanishing for some f and φ is that each local (normalized) zeta integral is not identically zero.
In other words, we should have Vv ∼
= V (πv , χv ) for all v ∈ Σ. Let
V(π, χ) be the hermitian space over
AE such that V(π, χ)v ∼= V (πv , χv ), and let (πv , χv ) = (V (πv , χv )), (π, χ) = Qv∈Σ (πv , χv ). If
(π, χ) = −1. Then V(π, χ) is incoherent, and hence for every V , the theta lifting θφf always vanishes.
If (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 θφf 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 (π, χ) = 1, which is
heuristic for the arithmetic case. For this purpose, let us assume the following conjecture proposed
by S. Kudla and S. Rallis (in the symplectic-orthogonal case).
Conjecture 2.3.2 (cf. [HKS1996]). We have
dim HomHv0 ×Hv0 (I2n (0, χv ), πv∨ χv πv ) = 1
(2.9)
for every irreducible admissible representation πv of Hv0 .
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, Theorem A]) and the Local Howe Duality Conjecture.
3. The multiplicity preservation has been proved by Waldspurger [Wal1990] when v is non-archimedean
27
and has odd residue characteristic; by Li–Sun–Tian [LST2011] for every non-archimedean place
v; and by Howe [How1989] for every archimedean place v.
4. The Local Howe Duality Conjecture has been proved by Waldspurger [Wal1990] when v is nonarchimedean and has odd residue characteristic; by Minguez [Mı́n2008] for type II dual pairs for
every non-archimedean place v; and by Howe [How1989] for every archimedean place v.
5. In particular, the above conjecture is known if v does not have residue characteristic 2; or v has
residue characteristic 2 but is split in E.
Let V = V (π, χ) and R(V, χ) =
N0
R(Vv , χv ). On the one hand, the functional
∨
β(f, f ∨ , φ, φ∨ ) = hθφf , θφf ∨ iH
defines an element in
HomH 0 (AF )×H 0 (AF ) (R(V, χ), π ∨ χπ) =
O
HomHv0 ×Hv0 (R(Vv , χv ), πv∨ χv πv ).
v∈Σ
On the other hand, the functional
α(f, f ∨ , φ, φ∨ ) =
Y
Z ∗ (0, χv , fv , fv∨ , φv ⊗ φ∨
v)
(2.10)
v∈Σ
(when everything is decomposable, otherwise we take the linear combination) defines also an eleN
∨
0
0
ment in
v∈Σ HomHv ×Hv (R(Vv , χv ), πv χv πv ), which is nonzero. However, by (2.9), the space
HomHv0 ×Hv0 (R(Vv , χv ), πv∨ χv πv ) is of dimension one. Therefore, β is a constant multiple of α. This
constant, by (2.8), is equal to
L( 1 , π, χ)
β
= Q2n 2
.
α
2 i=1 L(i, iE/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].
2.3.2
Vanishing of central L-values
We prove that the central L-value L( 12 , π, χ) vanishes when (π, χ) = −1.
28
AF )-admissible representation
By Proposition 2.2.4, we have a decomposition of H 00 (
I2n (0, χ) =
M
V
V
R( , χ) =
M O0
V
v∈Σ
Vv , χv ),
R(
where the direct sum is taken over all (isometry classes of) hermitian spaces over
V
AE
of rank 2n.
We note that each R( , χ) is irreducible. Recall the group H 00 = U(W 00 ) and its standard parabolic
subgroup P with is unipotent radical N as in Notation 2.2.1. First, we need some lemmas on local
representations.
Fix a place v and suppress it from notations. For T ∈ Her2n (E), let OT = {x ∈ V 2n | T (x) = T }.
Define a character ψT of N ∼
= Her2n (E) by ψT (n(b)) = ψ(tr T b).
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
dim R(V, χ)N,ψT =
2. Suppose v is infinite, i.e., E/F =



1
if OT 6= ∅;


0
otherwise.
C/R and T is nonsingular. The space of H-invariant tempered
distribution T on S(V 2n ) such that
T (ωχ (X)Φ) = dψT (X)T (Φ)
for X ∈ Lie N is of dimension 1 (resp. 0) if OT 6= ∅ (resp. OT = ∅).
Proof.
1. It is [Ral1987, Lemma 4.2].
2. It is [Ral1987, Lemma 4.2] and [KR1994, Proposition 2.9].
We now construct the twisted Jacquet module R(V, χ)N,ψT or the invariant distribution explicitly
29
if it is not trivial. For a standard section ϕs ∈ I2n (s, χ), define the Whittaker integral
Z
WT (g, ϕs ) =
ϕs (wng)ψT (n)−1 dn,
N
where w = w2n and dn is selfdual with respect to ψ. The integral WT (g, ϕs ) is absolutely convergent
when Re s > n. It is easy to see that WT (e, •) : I2n (s, χ) →
CN,ψ
T
is an N -intertwining map. Let
WT (s, g, Φ) = WT (g, ϕΦ,s ) for Φ ∈ S(V 2n ). We have the following lemmas.
Lemma 2.3.5. Assume that T is nonsingular. Then
1. WT (g, ϕs ) is entire.
2. The map Φ 7→ WT (0, e, Φ) realizes the surjective N -intertwining map S(V 2n ) → R(V, χ) →
R(V, χ)N,ψT or the invariant distribution in Lemma 2.3.4 (2).
Proof.
1. It is [Kar1979, Corollary 3.6.1] for v finite and [Wal1988, Theorem 8.1] for v infinite.
2. It is [KR1994, Proposition 2.7].
Lemma 2.3.6. Assume that v is a finite place; E/F , ψ and χ are all unramified; and V = V + . Let
Φ0 be the characteristic function of (Λ+ )2n for a selfdual OE -lattice Λ+ , and T ∈ Her2n (OF ) with
det T ∈ O×
F . Then we have
WT (s, e, Φ0 ) = b2n (s)−1 .
Proof. This is [Tan1999, Proposition 3.2].
Now suppose we are in the global situation. We denote by A(G) the space of automorphic forms
of G for a reductive group G. For T ∈ Her2n (F ), define the T -th Fourier coefficient of f (g) ∈ A(H 00 )
as
Z
WT (g, f ) =
A
N (F )\N (
For a hermitian space
f (ng)ψT (n)−1 dn.
F)
V over AE of rank 2n, we have a family of linear maps
V
Es : R( , χ) → A(H 00 )
Φ 7→ E(s, g, Φ) = E(g, ϕΦ,s )
(2.11)
30
AF )-intertwining map exactly when s = 0. Then for T nonsingular (and s
for s near 0. It is an H 00 (
near 0), we have
ET (s, g, Φ) := WT (g, Es (Φ)) =
Y
WT (s, gv , Φv ).
(2.12)
v∈Σ
AF )-intertwining map E : R(V, χ) → A(H 00), if WT (g, •) ◦ E
Lemma 2.3.7. For every H 00 (
vanishes
for all nonsingular T , then E = 0.
V2n) with
Proof. Fix a finite place v, by Lemma 2.3.4 (1), we can find a section Φ0 = Φv,0 Φv ∈ S(
V
Vv2n)reg (cf. Definition 2.4.1).
For every g v ∈ ev H 00 (AvF ), the functional Φv 7→ WT (0, g v , Φv Φv ) factors through the twisted
Jacquet module S(Vv2n )N ,ψ . If T is singular, then by our choice of Φv,0 and Lemma 2.3.4 (1-a),
WT (0, g v , Φv,0 Φv ) = 0. Similarly, WT (0, g, Φv,0 Φv ) = 0 for all g ∈ Pv H 00 (AvF ) since the action of
Pv stabilizes the subspace S(Vv2n )reg . For T nonsingular, WT ≡ 0 by the assumption. Therefore,
E (Φ0 )(g) = 0 for g ∈ Pv H 00 (AvF ). It follows that E (Φ0 ) = 0. Therefore, E = 0 by our choice of Φ0
and the irreducibility of R(V, χ).
nonzero projection in R( , χ) such that Φv,0 ∈ S(
v
T
Proposition 2.3.8. We have
1. If
V is incoherent, then dim HomH
2. If
V is coherent, then dim HomH
Proof.
00 (
A
V
(R( , χ), A(H 00 )) = 0.
00 (
F)
A
(R( , χ), A(H 00 )) = 1 and E0 (2.11) is a basis.
F)
V
1. Assume that E is a nontrivial intertwining map. Then by Lemma 2.3.7, there is a
nonsingular T ∈ Her2n (F ) such that WT (g, •) ◦ E does not vanish. By Lemma 2.3.4 (1-b) and
(2), T is representable by
Vv for every v ∈ Σ, i.e., OT 6= ∅. However, V will be coherent, which
is a contradiction.
2. Assume that E and E 0 are both nontrivial intertwining maps. By Lemma 2.3.7, there is a
nonsingular T such that WT (g, •) ◦ E does not vanish. By Lemma 2.3.4 (1-b)(2), there exists
C such that WT (g, •) ◦ E 0 = cWT (g, •) ◦ E . Furthermore, c is independent of nonsingular T
since all of those that can be represented by V are in a single M (F )-orbit under the conjugation
action on N (F ). By Lemma 2.3.7, E 0 − cE = 0, i.e., dim HomH (A ) (R(V, χ), A(H 00 )) ≤ 1.
For the rest, we need to prove that E0 is actually nontrivial. Since V is coherent, we can choose
a nonsingular T ∈ Her2n (F ) that is representable by V. By (2.12), Lemma 2.3.5 (2) and Lemma
c∈
00
F
2.3.6, we can find a suitable Φ such that WT (0, e, Φ) 6= 0. Therefore, E0 6= 0.
31
Now we can state the following main result.
Theorem 2.3.9. If (π, χ) = −1, then L( 12 , π, χ) = 0.
Proof. Let
V = V(π, χ). Then it is incoherent. We can choose suitable fv , fv∨, φv and φ∨v when some
∨
∨
of E, ψ, χ, π is ramified at v, such that Z ∗ (0, χv , fv , fv∨ , φv ⊗ φ∨
v ) 6= 0. Let f , f , φ and φ be adèlic
vectors with these subscribed local components and unramified ones at the places where E, ψ, χ, π are
unramified. From Proposition 2.2.2 (after analytic continuation), we have
Z
A
[H 0 (F )\H 0 (
L( 21 , π, χ)
= Q2n
f (g1 )f ∨ (g2 )χ−1 (det g2 )E(0, ı(g1 , g2 ), φ ⊗ φ∨ )dg1 dg2
F
)]2
Y
i
i=1 L(i, E/F ) v∈S
Z ∗ (0, χv , fv , fv∨ , φv ⊗ φ∨
v ).
V
However, E0 is zero on R( , χ) by Proposition 2.3.8 (1). We have E(0, ı(g1 , g2 ), φ⊗φ∨ ) ≡ 0. Therefore,
L( 21 , 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 12 . In fact, we have
Z
[H 0 (F )\H 0 (
d
=
|s=0
ds
=
=
A
f (g1 )f ∨ (g2 )χ−1 (det g2 )
F
)]2
d
|s=0 E(s, ı(g1 , g2 ), φ ⊗ φ∨ )dg1 dg2
ds
Z
f (g1 )f ∨ (g2 )χ−1 (det g2 )E(s, ı(g1 , g2 ), φ ⊗ φ∨ )dg1 dg2
[H 0 (F )\H 0 ( F )]2
Y
L(s + 12 , π, χ)
d
|s=0 Q2n
Z ∗ (s, χv , fv , fv∨ , φv ⊗ φ∨
v)
i
ds
i=1 L(2s + i, E/F ) v∈S
Q
∗
∨
Y
L0 ( 12 , π, χ)
1
d
v∈S Z (s, χv , fv , fv , φv ⊗
∗
∨
∨
Z
(0,
χ
,
f
,
f
,
φ
⊗
φ
)
+
L(
,
π,
χ)
|
Q2n
Q
v
v
v
s=0
v
v
2n
i
i
2
ds
i=1 L(i, E/F ) v∈S
i=1 L(2s + i, E/F )
A
Y
L0 ( 1 , π, χ)
= Q2n 2
Z ∗ (0, χv , fv , fv∨ , φv ⊗ φ∨
v ).
i
L(i,
)
i=1
E/F v∈S
(2.13)
Definition 2.3.10 (Analytic kernel functions). We call E 0 (0, g, Φ) =
V2n).
kernel function associated to the test function Φ ∈ S(
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)) =
φ∨
v)
Y
v∈Σ
WT (s, gv , Φv )
d
ds |s=0
E(s, g, Φ) the analytic
32
if Φ = ⊗Φv is decomposable. Therefore,
X
E(s, g, Φ) =
X
ET (s, g, Φ) +
T sing.
Y
WT (s, gv , Φv ).
T nonsing. v∈Σ
Taking derivative at s = 0, we have
E 0 (0, g, Φ) =
X
T sing.
=
X
Q
v 0 6=v
X
ET0 (0, g, Φ) +
X
X
Vv
0
WT (0, gv0 , Φv0 )
v 0 6=v
Y
WT0 (0, gv , Φv )
WT (0, gv0 , Φv0 ).
v 0 6=v
v∈Σ T nonsing.
WT (0, gv , Φv ) 6= 0 only if
Y
WT0 (0, gv , Φv )
T nonsing. v∈Σ
T sing.
However,
X
ET0 (0, g, Φ) +
represents T for all v 0 6= v by Lemma 2.3.4 (1-b).
V is incoherent, Vv can not represent T . For T nonsingular, there are only finitely many v ∈ Σ
such that T is not represented by Vv , i.e., there does not exist x1 , . . . , x2n ∈ Vv whose moment matrix
is T . We denote the set of such v by Diff(T, V). Then
Since
E 0 (0, g, Φ) =
X
ET0 (0, g, Φ) +
T sing.
X
Ev (0, g, Φ),
v∈Σ
where
Ev (0, g, Φ) =
X
V
WT0 (0, gv , Φv )
Diff(T, )={v}
Y
WT (0, gv0 , Φv0 ).
(2.14)
v 0 6=v
In fact, the summation in (2.14) is taken only over those v that are nonsplit in E.
2.4
2.4.1
Analytic kernel functions
Regular test functions
We prove that the summation of ET0 (0, g, Φ) over singular T ’s vanishes for certain choice of Φ, and g
AF ). We follow the idea in [YZZb].
in a suitable subset of H 00 (
Definition 2.4.1 (Regular test functions). Let v be a place and V a hermitian space over Ev . A test
function φ ∈ S(V r ) is regular, if φ(x) = 0 for x with degenerate moment matrix, i.e., det T (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(
following proposition.
VS2n)reg = Nv∈S S(Vv2n)reg . We have the
33
VS2n)reg ⊗ S(VS,2n), ords=0ET (s, g, Φ) ≥ k for T
Proposition 2.4.2. For Φ = ΦS ΦS ∈ S(
AF,S )H 00(ASF ).
singular
and g ∈ P (
We can assume that Φ = ⊗v∈Σ Φv is decomposable with Φv ∈ S(
Vv2n)reg for v ∈ S and rank T =
2n − r < 2n. Choose a ∈ GL2n (E) such that

 0r
aT t aτ = 

Te
(2.15)


with Te ∈ Her2n−r (E). Then
ET (s, g, Φ) = EaT t aτ (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
Z
X
ET (s, g, Φ) =
A
N (F )\N (
F)
ϕΦ,s (γng)ψT (n)−1 dn
γ∈P (F )\H 00 (F )
Z
X
=
A
N (F )\N (
(r(g)ϕΦ,s ) (γn)ψT (n)−1 dn
(2.16)
F ) γ∈P (F )\H 00 (F )
where r(g) standards for the action of H 00 on I2n (s, χ) by right translation. We need to unfold this
summation. Recall that for 0 ≤ d ≤ 2n,

w2n,d
in Notation 1.0.5.

1d




=



1d
−12n−d


12n−d 





(2.17)
Then {w2n,d | 0 ≤ d ≤ 2n} is a set of representatives of the double coset
WP/N \WH 00 /WP/N of Weyl groups. In particular, w2n,0 = w2n = w. We have a Bruhat decomposition
H 00 (F ) =
2n
a
P (F )w2n,d P (F ),
d=0
where F can be a global field or its local completions.
Lemma 2.4.3. For v ∈ S and gv ∈ Pv , the support of r(gv )ϕΦv ,s is contained in P (Fv )wN (Fv ).
34
Proof. It suffices to prove that ϕΦv ,s vanishes on P (Fv )w2n,d P (Fv ) for d > 0 since gv ∈ P (Fv ). For
g = n(b1 )m(a1 )w2n,d n(b2 )m(a2 ) ∈ P (Fv )w2n,d P (Fv ), we have
ϕΦv ,s (g) = (ωχv (g)Φ) (0)λ(g)s
=
Z
χv (det a1 a2 )| det a1 a2 |nEv λ(g)s
V
2n−d
v
ψb2 (T (x))Φv (xa2 )dx
Vv2n−d is viewed as a subset of Vv2n by the map (x1, . . . , x2n−d) 7→ (0, . . . , 0, x1, . . . , x2n−d).
Since Φv is regular and d > 0, Φv (xa2 ) = 0 for x ∈ Vv2n−d . Therefore, the lemma follows.
where
AF,S )H 00(ASF ),
By the above lemma, we have for g ∈ P (
Z
X
(2.16) =
A
N (F )\N (
Z
X
=
A
N (F )\N (
Z
=
A
N(
=
Y
v∈Σ
γ∈wN (F )
(r(g)ϕΦ,s ) (wn)ψT (n)−1 dn
ϕs (wn)ψT (n)−1 dn
AZ
N(
F)
(r(g)ϕΦ,s ) (γn)ψT (n)−1 dn
F)
Z
=
(r(g)ϕΦ,s ) (γn)ψT (n)−1 dn
F ) γ∈P (F )\P (F )wP (F )
F)
ϕv,s (wnv )ψT (nv )−1 dnv
(2.18)
Nv
where we denote by ϕs instead of r(g)ϕΦ,s for simplicity. Let S 0 ⊂ Σ be a finite subset containing all
infinite places such that for all v 6∈ S 0 , v, χv and ψv are unramified; ϕv,s is the (unique) unramified
section in I2n (s, χv ) (hence S 0 ⊃ S) and det Te ∈ O×
Fv . Then
!
(2.18) =
Y
0
WT (e, ϕv,s ) WT (e, ϕSs ).
v∈S 0
By [KR1994, Page 36] and [Tan1999, Proposition 3.2],
0
0
WT (e, ϕSs ) =
aS2n (s)
0
a2n−r (s − 2r )bS2n (s)
S0
where
am,v (s) =
m−1
Y
i=0
Lv (2s + i − m + 1, iE/F )
(2.19)
35
and
bm,v (s) =
m−1
Y
Lv (2s + m − i, iE/F )
i=0
0
as (2.6). Therefore, WT (e, ϕSs ) has a meromorphic continuation to the entire complex plane. For
v ∈ S 0 , we normalize the Whittaker integral to be
WT∗ (e, ϕv,s ) =
a2n−r,v (s − 2r )b2n,v (s)
WT (e, ϕv,s ).
a2n,v (s)
From the argument (and also notations) in [KR1994, Page 35],
WT (e, ϕv,s ) = WTe (e, i∗ ◦ Ur,v (s)ϕv,s ).
By [PSR1987, Section 4], the (local) intertwining operator Ur,v (s) has a meromorphic continuation to
the entire complex plane. By Lemma 2.3.5 (1), WT (e, ϕv,s ) and hence WT∗ (e, ϕv,s ) have meromorphic
continuation to the entire complex plane. Together with the meromorphic continuation of WT away
from S 0 and WT∗ in S 0 , (2.19) has a meromorphic continuation which equals
Y
a2n (s)
WT∗ (e, ϕv,s ).
r
a2n−r (s − 2 )b2n (s)
0
v∈S
Proof of Proposition 2.4.2. At the point s = 0, we have b2n (0) =
r−1
Y
a2n (0)
=
L(−i, i+1
r
E/F ) ∈
a2n−r (− 2 )
i=0
Q2n
i=1
L(i, iE/F ) ∈
C× and
C× .
Let κv = ords=0 WT∗ (e, •) be the order of the functional at s = 0 for v ∈ S 0 and
κ0v = ords=0 WT∗ (s, e, •)|S(Vv2n )reg
for v ∈ S. Since ET (e, ϕΦ ) = 0 if Φ = ⊗Φv for at least one Φv regular, by (2.19) and the proof of
P
Lemma 2.3.7, we have κ0v0 + v0 6=v∈S 0 κv ≥ 1 for every v0 in S. Also by the definition of WT , we see
that
ϕv,0 7→ s−κv WT∗ (e, ϕv,s )|s=0
36
is a nontrivial N -intertwining map from I2n (0, χ) to
Vv2n)reg .
regular test function Φv ∈ S(
CN,ψ
T
. Now if v ∈ S, ϕv,0 = ϕΦv ,0 for a
By Lemma 2.3.4 (1-a), ϕv,0 goes to 0 under the above map,
i.e., κ0v ≥ κv + 1 for v ∈ S. Therefore,
ords=0
Y
WT∗ (e, ϕv,s ) ≥
v∈S 0
X
X
κ0v +
v∈S
κv ≥ k − 1 + κ0v0 +
v∈S 0 −S
X
κv ≥ k.
v0 6=v∈S 0
The proposition follows.
VS2n)reg ⊗ S(VS,2n) which is
In conclusion, if we choose S such that |S| ≥ 2, and Φ = ΦS ΦS ∈ S(
AF,S )H 00(ASF ), we have
decomposable, then for g ∈ P (
E 0 (0, g, Φ) =
X
Ev (0, g, Φ).
(2.20)
v∈Σ
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 WT0 (0, e, Φv ) = 0 for all
nonsingular T that are not representable by
Vv .
Since the issue is local, we fix a nonsplit place v ∈ S and suppress it from notations in this
subsection. Let V be one of V ± and V 0 the other one which is not isometric to V . Let
Her02n (E) = {T ∈ Her2n (E) | det T 6= 0};
H = {b ∈ Her02n (E) | b = T (x) for some x ∈ V 2n };
H0 = {b0 ∈ Her02n (E) | b0 = T (x0 ) for some x0 ∈ V 02n };
0
Hd0 = {b0 + b00 | b0 ∈ H0 and b00 ∈ Her2n (p−d
E )} ∩ Her2n (E),
where pE is the maximal ideal of OE . Then
• Her02n (E) = H
• ···
•
T
•
S
0
H−1
`
H00
H0 ;
H10
d
Hd0 = H0 ;
d
Hd0 = Her02n (E).
···;
We say a test function Φ ∈ S(V 2n ) is of discriminant d if
{T (x) | x ∈ Supp(Φ)} ∩ Hd0 = ∅,
Z
d∈ ,
37
and denote by S(V 2n )d the space of such functions. Set S(V 2n )reg,d = S(V 2n )reg ∩ S(V 2n )d .
Z
Lemma 2.4.4. For every d ∈ , S(V 2n )reg,d is not empty.
Z
Proof. Fix an element d ∈ . We only need to prove that there exists T 6∈ Hd0 such that det T 6= 0.
0
Then T + Her2n (p−d
E ) ∩ H = ∅. Any test function with support whose elements have moment
0
2n
matrices contained in T + Her2n (p−d
)reg,d . Now we
E ) ∩ Her2n (E), which is open, will be in S(V
want to find such a T . Take an element T1 ∈ H with det T1 6= 0. Since H is open, we can find a
neighborhood T1 + Her2n (pνE ) ⊂ H for some ν ∈
Z.
If ν ≤ −d, then we are done. Otherwise, let $
be the uniformizer of F . Then $−ν−d (T1 + Her2n (pνE )) ⊂ H. However, $−ν−d (T1 + Her2n (pνE )) =
−ν−d
$−ν−d T1 + Her2n (p−d
T1 will serve for our purpose.
E ) . Therefore, T = $
Since ψ is nontrivial, we can define its discriminant dψ to be the largest integer d such that the
character ψT is trivial on N (OF ) ∼
= Her2n (OE ) for all T ∈ Her2n (p−d
E ). We slightly abuse notation
since dψ depends also on 2n. We also need to mention that this is not the usual discriminant or
different of a p-adic additive character. However, the difference between them depends only on n and
the ramification of E/F . We have the following proposition.
Proposition 2.4.5. Let d ≥ dψ be an integer and Φ ∈ S(V 2n )reg,d . Then WT (s, e, Φ) ≡ 0 for T ∈ H0
nonsingular.
Proof. For Re s > n,
Z
WT (s, e, Φ) =
(ωχ (wn)Φ) (0)λ(wn)s ψT (n)−1 dn
N
is absolutely convergent. Therefore, it equals
Z
Z
V 2n
Her2n (E)
Z
ψ(tr bT (x))Φ(x)dx λ(wn(b))s ψ(− tr T b)db
Z
=
λ(wn(b))s ψ(tr(T (x) − T )b)db
Φ(x)dx
V
2n
Her2n (E)
Z
Z
=
λ(wn(b))s ψT (x)−T (n(b))db.
Φ(x)dx
V
2n
(2.21)
Her2n (E)
Since λ(wn(b)n(b1 )) = λ(wn(b)) for b1 ∈ Her2n (OE ),
Z
(2.21) =
Z
Φ(x)dx
V 2n
s
Z
λ(wn(b)) ψT (x)−T (n(b))db
Her2n (E)/ Her2n (OE )
ψT (x)−T (n(b1 ))db1 ,
Her2n (OE )
38
in which the last integral is zero for all x ∈ Supp(Φ) by our assumption on Φ. Therefore, WT (s, e, Φ) ≡
0 after continuation. In particular, WT0 (0, e, Φ) = 0.
Remark 2.4.6. Obviously, it is not necessary to assume the dimension of V to be even. All definitions
and results above can be applied to arbitrary dimensions.
V2n) with Φv ∈
In conclusion, let S be a finite subset of Σfin with |S| ≥ 2, and Φ = ⊗Φv ∈ S(
Vv2n)reg for v ∈ S and Φv ∈ S(Vv2n)reg,d
S(
v
for v ∈ S nonsplit with dv ≥ dψv . Then combining (2.20),
we have
E 0 (0, g, Φ) =
X
Ev (0, g, Φ)
(2.22)
v6∈S
for g ∈ eS H 00 (
2.4.3
ASF ).
Density of test functions
We have made particular choices of test functions to simplify the formula of the analytic kernel
function. However, for our proof of the main theorem, not arbitrary choices will work. We now show
that there are “sufficiently many” test functions satisfying these choices we have made, in the sense
of Proposition 2.4.10. We follow the idea in [YZZa].
We keep our notations in 2.4.1 and 2.4.2. In particular, v will be a place in S and suppressed from
notations. Recall that we have an H 00 -intertwining map
S(V 2n ) S(V 2n )H ∼
= R(V, χ) ,→ I2n (0, χ)
through the Weil representation ωχ00 . Therefore, we obtain an H 0 × H 0 admissible representation on
S(V 2n ) through the embedding ı (2.2).
Lemma 2.4.7. If v is nonsplit, then for every d ∈
Z we have
S(V 2n )reg = ωχ00 (m(F × 12n ))S(V 2n )reg,d .
Proof. Fix an element d ∈
Z.
For every function Φ ∈ S(V 2n )reg , Supp(Φ) is a compact subset of H.
Since Her02n (E)\Hd0 is open and
[
d
\
Her02n (E)\Hd0 = Her02n (E)\ Hd0 = Her02n (E)\H0 = H,
d
39
the family Her02n (E)\Hd
0
there exists d0 ∈
Z is an open covering of Supp(Φ), hence has a finite subcover.
d∈
Z such that Supp(Φ) ∩ Hd0
0
Therefore,
= ∅. If d0 ≥ d, we are done. Otherwise, consider
Φ0 = ωχ00 (m($d0 −d 12n ))Φ. Then Supp(Φ0 ) ∩ Hd0 = ∅. The lemma follows.
In the rest of this subsection, let n = 1. Then H 0 = U(W1 ).
Lemma 2.4.8. Let π be an irreducible admissible representation of H 0 which is not of dimension 1 and
A : S(V ) → π a surjective H 0 -intertwining map, where H 0 acts on S(V ) through a Weil representation
ω. Then for every φ with A(φ) 6= 0, there is φ0 ∈ S(V )reg such that A(φ0 ) 6= 0 and Supp(φ0 ) ⊂ Supp(φ).
Proof. Let f = A(φ), if there exists n ∈ N 0 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 φ0 = ω(n)φ − φ ∈ S(V )reg and Supp(φ0 ) ⊂ Supp(φ). If for every
n ∈ N 0 , π(n)f = f , then f will be fixed by an open subgroup of H 0 containing N since π is smooth.
However, any such subgroup will contain SU(W1 ). Therefore, π factors through H 0 / 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 0 which are not of
dimension 1. Then for every surjective H 0 × H 0 -intertwining map B : S(V ) ⊗ S(V ) = S(V 2 ) → π1 π2
where H 0 × H 0 acts on S(V ) ⊗ S(V ) by a pair of Weil representations ω1 ω2 , there is an element
Φ = φ1 ⊗ φ2 ∈ S(V 2 )reg such that φα ∈ S(V )reg (α = 1, 2) and B(Φ) 6= 0.
Proof. Let Φ0 ∈ S(V 2 ) be an element such that B(Φ0 ) 6= 0, write Φ0 =
P
φi,1 ⊗ φi,2 as an element
in S(V ) ⊗ S(V ). Therefore, we can assume that there is φ1 ⊗ φ2 such that B(φ1 ⊗ φ2 ) 6= 0. By
Lemma 2.4.8, we can also assume that φ1 ∈ S(V )reg . For x ∈ Supp(φ1 ), let Vx be the subspace of V
generated by x and V x its orthogonal complement. They are both nondegenerate hermitian spaces of
P
dimension 1. As H 0 -representation, S(V ) = S(Vx ) ⊗ S(V x ). Write φ2 =
φi,x ⊗ φxi according to this
decomposition. We can assume that there is one φx ⊗ φx such that B(φ1 ⊗ (φx ⊗ φx )) 6= 0. Since as
H 0 -representation, S(V x ) is generated by the subspace S(V x )reg . We can then write
φx ⊗ φ x =
X
ω2 (gj ) ω2−1 (gj )φx ⊗ φxj
40
with φxj ∈ S(V x )reg . So we can further assume that B(φ1 ⊗ (φx ⊗ φx )) 6= 0 with φx ∈ S(V x )reg ,
i.e., Supp(φx ⊗ φx ) ∩ Vx = ∅. Applying Lemma 2.4.8 again, we can further assume that there
(x)
exists φ2
(x)
(x)
∈ S(V )reg such that Supp(φ2 ) ⊂ Supp(φx ⊗ φx ) and B(φ1 ⊗ φ2 ) 6= 0. The condition
that Supp(φ2 ) ∩ V x = ∅ is open for x. Therefore, we can find a neighborhood Ux of x such that
(x)
(φ1 | Ux ) ⊗ φ2
∈ S(V 2 )reg . Since Supp(φ1 ) is compact, we can find Φ of this kind such that B(Φ) 6=
0.
Recall the (normalized) zeta integrals (2.7), and that for Φ ∈ S(V 2n ), we set Z ∗ (s, χ, f, f ∨ , Φ) =
Z ∗ (χ, f, f ∨ , ϕΦ,s ). Combining Lemma 2.4.7 and Lemma 2.4.9, we have the following proposition.
Proposition 2.4.10. Let n = 1, v ∈ Σfin , π be an irreducible cuspidal automorphic representation of
H 0 and Vv = V (πv , χv ). For every d ∈
Z, we can find fv ∈ πv , fv∨ ∈ πv∨ and φα ∈ S(V )reg (α = 1, 2)
with the property that φ1,v ⊗ φ2,v ∈ S(Vv2 )reg,d (resp. S(Vv2 )reg ) if v is nonsplit (resp. split) in E, such
that the (normalized) zeta integral Z ∗ (0, χv , fv , fv∨ , φ1,v ⊗ φ2,v ) 6= 0.
41
Chapter 3
Arithmetic theta lifting and
arithmetic kernel functions
In this chapter, we study the geometric part of the arithmetic inner product. In 3.1, we introduce
the Shimura varieties of unitary groups which we work with, and on them the special cycles and
generating series. We prove the theorem of modularity of the generating series. For the case where
Shimura varieties are not proper, we need to consider their compactifications, which will be discussed
in 3.2. In 3.3, we define the arithmetic theta lifting and prove its cohomological triviality, which enable
us to formulate the explicit conjecture of the arithmetic inner product formula. Finally, we restrict
ourselves to the situation where n = 1 and hence the Shimura varieties are unitary Shimura curves,
in 3.4. We review the theory of Néron–Tate height paring, based on which we define the arithmetic
kernel functions and decompose them into local terms.
AF → C× such that ψι is the standard one, i.e., t 7→ exp(2πit)
We fix an additive character ψ : F \
(t ∈ Fι =
R) for any ι ∈ Σ∞ until the end of this article.
42
3.1
Modularity of generating series
3.1.1
Shimura varieties of unitary groups
We recall the notion attached to Shimura varieties of unitary groups. Let m ≥ 2 and 1 ≤ r < m be
V be a totally positive definite incoherent hermitian space over AE of rank m. Let H =
ResA /A U(V) be the unitary group which is a reductive group over A, and Hder = ResA /A SU(V)
×,1
its derived subgroup. Let T ∼
= ResA /A AE be the maximal abelian quotient of H, which is also
isomorphic to the center of H. Let T ∼
= ResF/Q E ×,1 be the unique (up to isomorphism) Q-torus such
that T ×Q A ∼
= T. Then T has the property that T (Q) is discrete in T (Afin ). For every open compact
subgroup K of H(Afin ), there is a Shimura variety ShK (H) of dimension m − 1 defined over the reflex
field E. For every embedding ι◦ : E ,→ C over ι ∈ Σ∞ , we have the following ι◦ -adic uniformization
integers. Let
F
F
F
H
Q
∼ (ι) ( )\ D(ι◦ ) ×
ShK ( )an
ι◦ = H
H(Afin)/K
.
We briefly explain the notations and meanings above:
H
H
◦
• ShK ( )an
ι◦ denotes the complex analytification of ShK ( ) via ι .
• Let V (ι) be the nearby E-hermitian space of
(ι)
(up to isometry) such that Vv
∼
=
Vv
V at ι, i.e., V (ι) is the unique E-hermitian space
(ι)
for v 6= ι but Vι
is of signature (m − 1, 1). Then
H (ι) = ResF/Q U(V (ι) ).
◦
• D(ι
)
is the symmetric hermitian domain consisting of all negative
ι◦
structure is given by the action of Fι ⊗F E −→
R ⊗R C ∼= C.
C-lines in Vι(ι) whose complex
)
Q
by conjugation, and on H(Afin )/K by identifying
H (ι) (Afin ) and H(Afin ) through the corresponding hermitian spaces.
◦
• The group H (ι) ( ) diagonally acts on D(ι
◦
In fact, the underlying real symmetric domain of D(ι
class of the Hodge map h(ι) :
)
R
can be identified with the H ι ( )-conjugacy
S = ResC/R Gm,C → HR(ι) ∼= U(m − 1, 1)R × U(m, 0)Rd−1 given by

 1m−1
h(ι) (z) = 

z/z



 , 1m , . . . , 1m  .
Assumption 3.1.1. From now on, we will assume that K is sufficiently small, e.g., contained in the
principal congruence subgroup for N ≥ 3 for the natural embedding into the general linear group.
H
Then ShK ( ) is a quasi-projective nonsingular E-scheme. It is proper if and only if
43
1. F 6=
Q; or
2. F =
Q, m = 2 and Σ(V) ! Σ∞.
H
Q
Afin)/ det K.
The set of geometric connected components of ShK ( ) can be identified with T ( )\T (
For every other open compact subgroup K 0 ⊂ K, we have an étale covering map
H
0
H
K
πK
: ShK 0 ( ) → ShK ( ).
H
Pic(ShK (H))Q which is ample.
H
(3.1)
H
Let Sh( ) be the projective system {ShK ( )}K . On each ShK ( ), we have a Hodge bundle LK ∈
0
K
They are compatible under the pullback of πK
, hence define an
element
H
H
L ∈ Pic(Sh( ))Q := lim Pic(ShK ( ))Q .
−→
3.1.2
K
Special cycles and generating series
Let V1 be an E-subspace of
Vfin, we say V1 is admissible if (−, −)|V
1
takes values in E and for every
nonzero x ∈ V1 , (x, x) is totally positive. We have the following lemma.
Lemma 3.1.2. The E-subspace V1 is admissible if and only if for every ι ∈ Σ∞ , there is an element
h∈
Hder(Afin) such that hV1 ⊂ V (ι) ⊂ Vfin and is totally positive definite.
Proof. The “if” direction is obvious. For the “only if” direction, let us assume that V1 is admissible
and fix arbitrary ι. Take v1 ∈ V1 with nonzero norm. Then q(v1 ) =
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 Theorem
Vfin)
H(Afin) such that h1v1 = v as elements in Vfin. Choose a
vector v 0 ∈ hvi⊥ ⊂ V (ι) with nonzero norm. Let h0 ∈ H(Afin ) which stabilizes hv 0 i⊥ and act on
the Afin,E -line spanned by v 0 through the multiplication by (det h1 )−1 . Then h0 h1 v1 = h0 v = v for
h = h0 h1 ∈ SU(Vfin ) = Hder (Afin ).
Replacing V1 by hV1 , we can assume that v1 = v ∈ Vfin . Since dim V1 < m, we can use induction
on r by considering the orthogonal complement of v in V1 and V (ι) to find an element h ∈ Hder (Afin )
such that hV1 ⊂ V (ι) ⊂ Vfin .
to find an element h1 ∈ U(
=
V1 be a totally positive definite (incoherent) hermitian space over AE such
that V1,fin ∼
= V1⊥ ⊂ Vfin . Let H1 be the corresponding unitary group, we have a finite morphism of
For admissible V1 , let
44
Shimura varieties
H1) → ShK (H)
ς V1 : ShK1 (
(3.2)
H1(Afin), such that the image of the map is represented, under the uniformization
at some ι◦ , by the points (z, h1 h) ∈ D(ι ) × H(Afin ) where
where K1 = K ∩
◦
• h is as in Lemma 3.1.2 (with respect to the ι dividing ι◦ );
• z ⊥ hV1 ;
• h1 fixes all elements in hV1 .
H
The image defines a special cycle Z(V1 )K ∈ CHr (ShK ( ))Q . It depends only on the class KV1 .
For x ∈
r
, let Vx be the E-subspace of Vfin generated by the components of x. We define
Vfin
Z(x)K =



Z(Vx )K c1 (LK∨ )r−dimE Vx
if Vx is admissible


0
otherwise.
Vιr ) of the Weil represen-
To introduce the generating series, we need a restriction on the space S(
tation when ι is infinite.
Vιr )U
We define a subspace S(
ι
Vιr ) consisting of functions of the form
⊂ S(
P (T (x)) exp(−2π tr T (x))
C
Vιr ) as the corresponding (Lie Hr,ι, Kr,ι)-
where P is a polynomial function on Herr ( ). If we view S(
module, then S(
Vιr )U
ι
is the submodule generated by the Gaussian
φ0∞ (x) = exp(−2π tr T (x)).
Let
V)
S(
r U∞
=
O
V
S(
!
r Uι
ι)
V
⊗ S(
r
fin );
ι∈Σ∞
for an open compact subgroup K of
S(
V)
r U∞ K
=
O
ι∈Σ∞
H(Afin).
V
S(
r Uι
ι)
!
r K
Vfin
)
⊗ S(
45
V r )U
For φ ∈ S(
∞
K
, we define the generating series to be
X
Zφ (g) =
x∈K\
V
(ωχ (g)φ) (T (x), x)Z(x)K
r
fin
H
AF ). Here for φ = φ∞φfin, we denote
r
φ(T (x), x) = φ∞ (y)φfin (x) for every y ∈ V∞
with T (y) = T (x), whose value does not depend on
as a formal series with values in CHr (ShK ( ))C for g ∈ Hr (
the choice of y. This makes sense since Z(x)K 6= 0 only for Vx admissible and hence T (x) is totally
0
K
semi-positive definite. It is easy to see that Zφ (g) is compatible under the pullback of πK
, hence
H
H
defines a series with values in CHr (Sh( ))C := limK CHr (ShK ( ))C .
−→
3.1.3
Pullback formula and modularity theorem
We briefly recall the notion of Shimura varieties attached to orthogonal groups. The
V is also a totally positive definite quadratic space over AF
AF -module
of rank 2m with the quadratic for-
m TrAE /AF (−, −). Then it is incoherent with discriminant in F × /F ×2 ⊂
V
A×F /A×2
F .
Let
G
=
V with the adjoint (quotient) group Gadj =
ResA /A SO(V) and the derived subgroup Gder = ResA /A Spin(V). For every open compact subgroup K 0 of G(Afin ), there is a Shimura variety ShK (G) defined over the reflex field F such that for
every embedding ι : F ,→ C, we have the following ι-adic uniformization
ResAF /A GSpin( ) be the special Clifford group of
F
F
0
G
Q
(ι)
∼
ShK 0 ( )an
G
(
)\
D0(ι) ×
=
ι
G(Afin)/K 0
.
We briefly explain the notations and meanings above:
G
G
• ShK 0 ( )an
ι denotes the complex analytification of ShK 0 ( ) via ι.
• Let V (ι) be the nearby F -quadratic space of
(ι)
(up to isometry) such that Vv
∼
=
Vv
V at ι, i.e., V (ι) is the unique F -quadratic space
(ι)
for v 6= ι but Vι
is of signature (2m − 2, 2). Then
G(ι) = ResF/Q GSpin(V (ι) ).
• D0(ι) is the symmetric hermitian domain consisting of all oriented negative definite 2-dimensional
(ι)
subspaces of Vι .
Q
G(Afin)/K 0 by identifying
G(ι) (Afin ) and G(Afin ) through the corresponding quadratic spaces.
We have similar notations of Hodge bundles, special cycles, and generating series on ShK (G),
r
which are denoted by LK0 , Z 0 (x)K for x ∈ Vfin
, and Zφ0 (g 0 ) for φ ∈ S(Vr )O K and g 0 ∈ Gr (Afin ),
• The group G(ι) ( ) diagonally acts on D0(ι) by conjugation, and on
0
0
0
∞
0
46
respectively (cf. [YZZ2009]). Here we introduce the standard symplectic F -space Wr0 (comparing to
the space Wr in 2.1.1) which has the basis {e1 , . . . , e2r } with the symplectic form
• hei , ej i = 0;
• her+i , er+j i = 0;
• hei , er+j i = δij for 1 ≤ i, j ≤ r.
The group Gr = Sp(Wr0 ) is an F -reductive group. Similarly, when defining the generating series, we
AF ) × G(A) on S(Vr ).
use the Weil representation ω (with respect to ψ) of Gr (
Now we fix an embedding ι◦ : E ,→
C over ι and suppress them from the notations of nearby
◦
objects like V = V (ι) , H = H (ι) , D = D(ι ) , etc. Therefore, we have the usual notion of Shimura
variety ShK (H, X) (resp. ShK 0 (G, X 0 ) with a connected component X + of X 0 ) which is defined over
ι◦ (E) (resp. ι(F )). Its neutral component is the connected Shimura variety Sh◦K (Hder , X) (resp.
Sh◦K 0 (Gder , X + )) attached to the connected Shimura datum (Hder , X) (resp. (Gder , X + )), which is
defined over EK (resp. EK 0 ): a finite abelian extension of ι(F ) in
C.
The canonical embedding
Hder ,→ Gder (cf. Remark 3.1.4 (1) below) of reductive groups and the embedding D ,→ D0 by
forgetting the E-action define an injective map of connected Shimura data (Hder , X) → (Gder , X + )
which gives an embedding
iK 0 : Sh◦K (Hder , X) ,→ Sh◦K 0 (Gder , X + )
which is defined over EK providing that K ∩ Hder (
Afin) = K 0 ∩ Hder(Afin) and K 0 is sufficiently small.
Let Z(x)◦K (resp. Z 0 (x)◦K 0 , Zφ (g)◦ , Zφ0 (g 0 )◦ ) be the restriction of Z(x)K (resp. Z 0 (x)K 0 , Zφ (g), Zφ0 (g 0 ))
to the neutral component.
Afin) = K 0 ∩ Hder(Afin).
Proposition 3.1.3. Assume that K 0 is sufficiently small and K ∩ Hder (
Then for x ∈
Vfin, the pullback of the special divisor i∗K Z 0(x)◦K
0
0
is the sum of Z(x1 )◦K indexed by the
classes x1 in K\K 0 x, both considered as elements in Chow groups.
Proof. If x = 0, then the only class in K\K 0 x is x1 = 0; the proposition follows from the compatibility
of Hodge bundles under pullbacks induced by maps of (connected) Shimura data.
Now we assume that hx, xi ∈ E and is totally positive. Suppose that (z, h) ∈ D × Hder (
Afin)
C-point in the scheme-theoretic intersection Sh◦K (Hder, X) ∩ Z 0(x1)◦K for some x1 ∈
K 0 x. Let g ∈ G(Afin ) such that gx1 = x01 ∈ V ⊂ Vfin . Then z ⊥ γx01 for some γ ∈ G(Q) and
h ∈ γG(Afin )x gk 0 for some k 0 ∈ K 0 , where G(Afin )x is the subgroup of G(Afin ) fixing x01 . We
now show that γG(Afin )x gk 0 ∩ Hder (Afin ) = G(Afin )γx γgk 0 ∩ Hder (Afin ) 6= ∅, i.e., G(Afin )γx ∩
represents a
0
0
1
0
1
0
1
0
1
0
1
47
C-point in
the special cycle Z(h−1 Ehγgx1 i)◦K of Sh◦K (Hder , X). If we write h = g1 γgk 0 with some g1 ∈ G(Afin )γx ,
Hder (
Afin)k0−1g−1γ −1 6= ∅, which is true by Lemma 3.1.2.
Therefore, (z, h) represents a
0
1
then
h−1 Ehγgx1 i = Ehh−1 γgx1 i = Ehk 0−1 g −1 γ −1 g1−1 γgx1 i = Ehk 0−1 x1 i.
Therefore, the scheme-theoretic intersection is indexed by the classes x1 in K\K 0 x. This also holds
on the level of Chow groups since the intersection is proper.
Remark 3.1.4. We have the following two remarks.
1. The canonical embedding Hder ,→ Gder is given by the following way. First, we have an embedding Hder ,→ H ,→ Gadj by forgetting the E-action on V = V (ι) . Since Hder is simply connected,
we have a canonical lifting Hder ,→ G. Since Hder has no nontrivial abelian quotient, the image
is actually contained in Gder .
C-points of
Sh◦K (Hder , X) (resp. Sh◦K (Gder , X + )) which is compatible with that of ShK (H) (resp. ShK (G))
2. In the proof of Proposition 3.1.3, we can still use the adèlic description of the
0
0
since Hder (resp. Gder ) is semisimple, of noncompact type and simply-connected.
The group Gr is canonically embedded in Hr by identifying the bases he1 , . . . , e2r i of Wr0 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 0 be as in Proposition 3.1.3. Then i∗K 0 Zφ0 (g 0 )◦ = Zφ (g 0 )◦ for
g 0 ∈ G1 (
AF ) and φ ∈ S(V)U
∞
K0
.
H
For a linear functional l ∈ CHr (Sh( ))∗C , we have a complex valued series
l(Zφ )(g) =
X
x∈K\
V
(ωχ (g)φ) (T (x), x)l(Z(x)K )
r
fin
for every K such that φ is invariant under K (which is of course independent of such choice). The
following is the main theorem.
Theorem 3.1.6 (Modularity of the generating series). We have
1. If l(Zφ )(g) is absolutely convergent, then it is an automorphic form of Hr . Moreover, the
archimedean component of (the representation generated by) l(Zφ ) is a discrete series represenχ
χ
m−k
tation of weight ( m+k
2 ,
2 ) (cf. Definition 2.1.2 and Notation 2.1.1).
48
2. If r = 1, then l(Zφ )(g) is absolutely convergent for every l.
Proof.
1. We proceed as in [YZZ2009, Section 4]. First, we can assume that φ = φ0∞ ⊗ φf since
other cases will follow from the (Lie Hr,ι , Kr,ι )-action. Assuming the absolute convergence of
l(Zφ )(g), we only need to check the automorphy, i.e., invariance under left translation of Hr (F ).
The weight part is clear.
It is easy to check the invariance under n(b) and m(a). For b ∈ Herr (E), the matrix bT (x) is F -
AF ). For a ∈ GLr (E),
rational if Z(x)K 6= 0. Therefore, l(Zφ )(n(b)g) = l(Zφ )(g) for all g ∈ Hr (
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−1 g) =
AF ).
l(Zφ )(g) for all g ∈ Hr (
Assuming this for r = 1 (cf. Lemma 3.1.7 below), we now prove
for general r > 1, following [YZZ2009] and [Zha2009].
We suppress l from notations for simplicity. Then for K sufficiently small, we have
X
Zφ (wr,r−1 g) =
V
x∈K\
(ωχ (wr,r−1 g)φ) (T (x), x)Z(x)K
r
fin
X
=
x∈K\
V
X
V
r−1
fin
y∈Kx \
(ωχ (wr,r−1 g)φ) (T (x, y), (x, y))Z((x, y))K ,
(3.3)
fin
where Kx is the stabilizer of x in K. Then
(3.3) =
X
V
x∈K\
X
r−1
fin
y1 ∈Kx \
X
V
x
fin
(ωχ (wr,r−1 g)φ) (T (x, y1 + y2 ), (x, y1 + y2 ))Z((x, y1 + y2 ))K ,
y2 ∈Vx
(3.4)
where
x
Vfin
is the orthogonal complement of Vx = Ehxi in Vfin . Recalling the morphism ς V
x
in
(3.2), we have
(3.4) =
X
X
x∈K\
V
r−1
fin
X
V
y1 ∈Kx \
x
fin
(ωχ (wr,r−1 )(ωχ (g)φ)) (T (x, y1 + y2 ), (x, y1 + y2 ))ς∗Vx Z(y1 )Kx .
y2 ∈Vx
(3.5)
Applying the case r = 1 to the special cycle Z(Vx )Kx , we have
(3.5) =
X
x∈K\
V
X X
r−1
fin
y1 ∈Kx \
V
x
fin
ωχ (wr,r−1 )ω\
χ (g)φ
y1 (T (x, y1 + y2 ), (x, y1 + y2 ))ς∗Vx Z(y1 )Kx ,
y2 ∈Vx
(3.6)
49
where the superscript y1 means taking the partial Fourier transformation along the y1 direction.
Applying the Poisson Summation Formula (recall that φ∞ = φ0∞ is the Gaussian), we have
(3.6) =
X
x∈K\
=
V
V
r−1
fin
V
V
x
fin
y1 ∈Kx \
V
r−1
fin
y1 ∈Kx \
V
ωχ (wr,r−1 )ω\
χ (g)φ
y1 ,y2 (T (x, y1 + y2 ), (x, y1 + y2 ))ς∗Vx Z(y1 )Kx
y2 ∈Vx
X y
ωχ (wr,r−1 )ω\
(T (x, y1 + y2 ), (x, y1 + y2 ))ς∗Vx Z(y1 )Kx
χ (g)φ
x
fin
X
X
x∈K\
y1 ∈Kx \
X
X
x∈K\
=
r−1
fin
X
x∈K\
=
V
X X
y2 ∈Vx
X
x
fin
(ωχ (g)φ) (T (x, y1 + y2 ), (x, y1 + y2 ))ς∗Vx Z(y1 )Kx
y2 ∈Vx
(ωχ (g)φ) (T (x), x)Z(x)K
r
fin
= Zφ (g).
2. It follows from the argument in Lemma 3.1.7, Corollary 3.1.5 and [YZZ2009, Theorem 1.3],
which uses the result in [KM1990].
Lemma 3.1.7. If r = 1, then l(Zφ )(w1 g) = l(Zφ )(g) for all g ∈ H1 (
AF ).
Proof. We suppress l from notations. Further, we fix an element ι◦ ∈ Σ◦∞ over ι ∈ Σ∞ and suppress
them as in the previous discussion. It is clear that we only need to prove Zφ (w1 g) = Zφ (g) for
AF ) since G1(A∞,F )K1,∞ = H1(A∞,F ).
g ∈ G1 (
As before, we assume that φ∞ is the Gaussian
Q
and K is sufficiently small. Recall that π0 (ShK (H, X)ι,C ) ∼
= T ( )\T (
Afin)/ det(K).
We have the
following inclusion
CH1 (ShK (H, X))C ,→
CH1 (ShK (H, X)t )C
M
Q
A
t∈T ( )\T (
(3.7)
fin )/ det K
where ShK (H, X)t is the (canonical model of the) corresponding (geometric) connected component.
Afin) such that det h ∈ t and Th be the Hecke operator.
Let h ∈ H(
Then Th : ShK h (H, X) →
ShK (H, X) induces
∼
Th◦ : Sh◦K h (Hder , X) = ShK h (H, X)1 −
→ ShK (H, X)t ,→ ShK (H, X),
where K h = hKh−1 . We have Th◦,∗ Zφ (g) = Zφ (g)◦ which is the image of (3.7) composed with the
projection to CH1 (ShK (H, X)t )C . Here, Zφ (g)◦ is the generating series on Sh◦K h (Hder , X). Now
shrinking K h if necessary such that we can apply Corollary 3.1.5, we have Zφ (g)◦ = i∗K 0 Zφ0 (g)◦ for g ∈
50
G1 (
Afin).
Applying [YZZ2009, Theorem 1.2 or Theorem 1.3], we conclude that Zφ (w1 g)◦ = Zφ (g)◦ .
The lemma follows by (3.7).
3.2
Smooth compactification of unitary Shimura varieties
We introduce the canonical smooth compactification of the unitary Shimura varieties if they are not
proper, and the compactified generating series on them.
3.2.1
Compactified unitary Shimura varieties
Let m ≥ 2 be an integer, E =
Q(j) ⊂ C with Im j > 0 and j2 = −D for some square-free positive
integer D, OE its ring of integers and τ the nontrivial Galois involution on E. Let (V, (−, −)) be a
hermitian space of dimension m over E of signature (m − 1, 1). If m = 2, we further assume that
det V ∈ Nm E × . Let H = U(V ) be the unitary group, we have the Hodge map h :
S → HR ∼=
U(m − 1, 1)R given by





h(z) = 




1
..




.



.
1
z/z
Afin).
Then we have the notion of Shimura variety ShK (H, h) for every open compact subgroup K of H(
For K sufficiently small, it is smooth, quasi-projective but non-proper over E of dimension m − 1.
Therefore, we need to construct a smooth compactification of ShK (H, h) to do height paring. When
m = 2, it is easy since we only need to add cusps. When m = 3 and H is quasi-split, a canonical
smooth compactification (even of the integral model) has been constructed in [Lar1992]. In fact, the
same construction works in more general cases (just for compactification of the canonical model),
namely for every H as above. We should mention that, if the signature of V is (a, b) such that
a ≥ b > 1 or V is over a totally real field but not
Q and indefinite at every archimedean place, then
we are not clear whether there exists a canonical smooth compactification (in a suitable sense).
Now let us assume that m > 2. In order to to use modular interpretations, we should work with
the group of unitary similitude. For every v, w ∈ V , the paring
(v, w)0 = TrE/Q (j(v, w))
51
defines a symplectic form of V satisfying (ev, w)0 = (v, eτ w)0 for every e ∈ E. Let GH = GU(V ) such
that for every
Q-algebra R,
GH(R) = {h ∈ GLm (E ⊗Q R) | (hv, hw)0 = λ(h)(v, w)0 for some λ(h) ∈ R× }
and the Hodge map Gh :
S → GHR ∼= GU(m − 1, 1)R is given by


z




Gh(z) = 



..




.



.
z
z
Afin), we have the Shimura variety
For every sufficiently small open compact subgroup K of GH(
ShK (GH, Gh) which is smooth, quasi-projective but non-proper over E of dimension m − 1. Although
we do not have a map of Shimura data, ShK∩H(Afin ) (H, h) and ShK (GH, Gh) have the same neutral
component for sufficiently small K. Therefore, it is same to give a canonical smooth compactification
of ShK (GH, Gh) instead of the original one. In fact, ShK (GH, Gh) is the moduli space of abelian
Z
b
varieties of certain PEL type. We fix a lattice VZ of V such that VZ ⊂ VZ⊥ and let VZ
b = VZ ⊗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, η̄),
where
• A is an abelian scheme over S of dimension m.
• θ : A → A∨ is a polarization.
• i : OE ,→ EndS (A) is a monomorphism of rings with units, such that tr(i(e); LieS (A)) =
(m − 1)e + eτ and θ ◦ i(e) = i(eτ )∨ ◦ θ for all e ∈ OE . Here, we view (m − 1)e + eτ as a constant
section of OS via the structure map E → OS .
• η̄ is a K-level structure, that is, for chosen geometric point s on each connected component of
Z
Z
ét
b
S, η̄ is a π1 (S, s)-invariant K-class of OE ⊗Z b -linear symplectic similitude η : VZ
b → H1 (As , ),
where the pairing on the latter space is the θ-Weil paring (hence the degree of θ is [VZ⊥ : 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 polyhedral cone decomposition. However, in our case, we have only one choice, namely the torus in a line.
52
We claim that there is a scheme Sh∼
K (GH, Gh) such that
• Sh∼
K (GH, Gh) is smooth and proper over E.
• eiK : ShK (GH, Gh) ,→ Sh∼
K (GH, Gh) is an open immersion.
0
K
• For K 0 ⊂ K, there is a morphism π
eK
such that the following diagram commutes
e
iK 0
ShK 0 (GH, Gh)
K
πK
/ Sh∼
K 0 (GH, Gh)
0
K
π
eK
ShK (GH, Gh)
e
iK
0
/ Sh∼
(GH,
Gh).
K
• The boundary GYK := Sh∼
K (GH, Gh) − ShK (GH, Gh) is a smooth divisor defined over E and
each geometric component is isomorphic to an extension of an abelian variety of dimension m−2
by a finite group.
The boundary part GYK parameterizes the degeneration of abelian varieties with above PEL
data. We consider a semiabelian variety G over an algebraically closed field of characteristic 0 with
i : OE ,→ End(G) such that tr(i(e); Lie(G)) = (m − 1)e + eτ . For every e ∈ OE , we have following
commutative diagram
0
/T
t
/G
0
/T
t
/G
α
/A
/0
α
/A
/ 0.
i(e)
Then the composition α ◦ i(e) ◦ t is trivial. Thus i induces actions of OE on both torus part T and
abelian variety part A. Suppose X(T ) ∼
=
Zr with r > 0. Then r is even since E is quadratic imaginary.
Further assuming that tr(i(e); Lie(T )) = ae + beτ , then a + b = r and a = b. Therefore, there is only
one possibility, namely a = b = 1 and r = 2. Then T is of rank 2 and A is an abelian variety of
dimension m − 2 such that OE acts on A and tr(i(e); Lie(A)) = (m − 2)e. Let A1 be an elliptic curve
of CM type (OE , e 7→ e). Then A is isogenous to Am−2
. Each geometric point s of GYK corresponds
1
to a semiabelian variety Gs = (Ts ,→ Gs → As ) as above with certain level structure which will be
defined later. For two geometric points s, s0 in the same geometric connected component, the abelian
variety part As ∼
= As0 and the rank 1 OE -modules X(Ts ) and X(Ts0 ) are isomorphic. It is easy to see
that if A and T are fixed, then the set of such G, up to isomorphism, is parameterized by X(T ) ⊗OE A
which is an abelian variety of dimension m − 2.
53
To include the level structure, we consider only one geometric component since it is same for
others. This means that we fix T and A with OE -actions but, of course, not G. Let us fix a maximal
isotropic subspace W of VZ . Then W is of rank 1. We have a filtration 0 ⊂ W ⊂ W ⊥ ⊂ VZ .
Let BW be the subgroup of H(
Afin) that preserves this filtration, NW
⊂ BW that acts trivially on
the associated graded modules, UW ⊂ NW that acts trivially on W ⊥ and VW = NW /UW . We fix
Z
Z
b
also a generator w of W . On the other hand, we fix an OE ⊗Z b generator wT of Hét
1 (T, ) and a
Z
Z
b
b
polarization θA : A → A∨ such that there exists a symplectic similitude of Hét
1 (A, ) and VW ⊗Z . For
Afin), let NW,K = NW ∩ K, UW,K = UW ∩ K and
a sufficiently small open compact subgroup K ⊂ H(
VW,K = NW,K /UW,K . Then the level structure of (G, T ,→ G → A) with respect to K is a VW,K -class
Z
Z
b
of isomorphisms W ⊥ ⊗Z b → Hét
1 (G, ) which sends w to wT and induces a symplectic similitude of
Z
Z
Z
ét
b
b
VW ⊗Z b and Hét
1 (A, ) = H1 (G, )/OE · wT . We conclude that every geometric component of GYK
is isomorphic to (a connected component of) an extension of X(T ) ⊗OE A by VW /VW,H for some T
and A as above. There is a universal object π : G → Sh∼
K (GH, Gh) which is a semiabelian scheme of
relative dimension m.
3.2.2
Compactified generating series
We come back to the Shimura variety ShK := ShK (H, h). The above canonical smooth compactification induces a canonical smooth compactification for ShK which we will denote by Sh∼
K and
YK = Sh∼
K − ShK . They have the same properties as above. We apply the above notations also to
Vm
the trivial case m = 2. Let LK∼ be the line bundle on Sh∼
π∗ ΩG/ Sh∼
on
K induced from
K (GH,Gh)
Sh∼
K (GH, Gh) which is an extension of the Hodge bundle LK on ShK . By the canonicality of the
compactification, (LK∼ )K defines an element in Pic(Sh∼ ) := limK Pic(Sh∼
K ). We also need to extend
−→
r
special cycles and the generating series. For 1 ≤ r < m and x ∈ Vfin
:= V r ⊗Q fin , we define the
A
compactified special cycle to be
Z(x)∼
K =


∼,∨ r−dimE Vx

Z(Vx )∼
K c1 (LK )
if Vx is admissible


0
otherwise
54
∼
where Z(Vx )∼
K is just the Zariski closure of Z(Vx )K in ShK . We define the compactified generating
series to be
Zφ∼ (g) =


P


r
x∈K\Vfin

P

 x∈K\V
fin
A
(ωχ (g)φ) (T (x), x)Z(x)∼
K
m > 2;
∼,∨
1
(ωχ (g)φ) (T (x), x)Z(x)∼
K + W0 ( 2 , g, φ)c1 (LK ) r = 1, m = 2,
V
for g ∈ Hr ( ) and φ ∈ S( r )U∞ . It is a formal series in CHr (Sh∼ )C := limK CHr (Sh∼
K )C . Here,
−→
Q
W0 (s, g, φ) = v W0 (s, gv , φv ) which is holomorphic at s = 21 . Moreover, we define the following
positive partial compactified generating series as
Zφ∼,+ (g) =
X
(ωχ (g)φ) (T (x), x)Z(x)∼
K,
r
x∈K\Vfin
T (x)0r
where the sum is taken over all x such that T (x) is totally positive definite. We would like to propose
the following conjecture on the modularity of the compactified generating series.
Conjecture 3.2.1. Let l be a linear functional on CHr (Sh∼ )C such that l(Zφ∼ )(g) is absolutely convergent. Then
AF ).
1. If 1 ≤ r ≤ m − 2, l(Zφ∼ )(g) is a holomorphic automorphic form of Hr (
2. If r = 1, m = 2, l(Zφ∼ )(g) is an automorphic form of H1 (
AF ), not necessarily holomorphic.
3. In general, if r = m − 1, l(Zφ∼,+ )(g) is the sum of the positive definite Fourier coefficients of an
AF ).
automorphic form of Hm−1 (
The case (2) will be proved in Corollary 3.4.2 and is actually not far from Theorem 3.1.6 as we
point out there.
Fix a rational prime `, there are class maps
∼
2r
ac
clK : CHr (Sh∼
K )C → Hét (ShK ×E E ,
Z`(r))Γ
E
⊗Z`
C,
0
K
that are compatible under π̃K
. They induce
2r
cl : CHr (Sh∼ )C → Hét
(Sh∼ ×E E ac ,
Z`(r))Γ
E
⊗Z `
C ⊂ H2rBet(Sh∼, C),
55
where
2•
Hét
(Sh∼ ×E E ac ,
ac
Z`(•)) = −lim
H2• (Sh∼
K ×E E , Z` (•));
→ ét
K
H•Bet (Sh∼ ,
C) = lim
H• (Sh∼
K (C), C),
−→ Bet
K
and the latter one is the Betti cohomology. Let
C
C C
H•Y (Sh∼ , ) = lim H•YK (Sh∼
K ( ), )
−→
K
be the inductive limit of cohomology groups with support in YK as K varies. Then since Y is a smooth
divisor, we have
C
C
C C
H•Y (Sh∼ , ) ∼
H•−2 (Y ( ), ).
= H•−2
Bet (Y, ) := lim
−→ Bet K
K
Let us denote by Sh#
K the Baily–Borel compactification of ShK . Therefore, we have the following
commutative diagram
Sh∼
K
jK
b
i#
K
e
iK
/ Sh# ,
< K
0 P -
ShK
which is compatible and more importantly, Hecke equivariant when K varies. We denote also by
C
C C
IH• (Sh# , ) = limK IH• (Sh#
K ( ), ) the inductive limit of the intersection cohomology groups. Then
−→
by [BBD1982, Théorème 6.2.5], we have the following exact sequence
C
H•Y (Sh∼ , )
C
/ H• (Sh∼ , )
Bet
j∗
C
/ IH• (Sh# , )
/ 0.
(3.8)
C
Let H•∂ (Sh∼ ) be the image of the first map, which is isomorphic to a quotient of H•−2
Bet (Y, ).
3.3
Arithmetic theta lifting
We assume Conjecture 3.2.1 and the following assumptions on A-packets, which are a certain part of
the Langlands–Arthur conjecture (see [Art1984, Art1989]).
• A-packets are defined for all unitary groups U(m) defined by a hermitian space over E of rank m.
56
We denote by AP(U(m)AF ) the set of A-packets of U(m) and AP(U(m)AF )disc ⊂ AP(U(m)AF )
the subset of discrete A-packets.
• If Π1 and Π2 are in AP(U(m)AF )disc such that for almost all v ∈ Σ, Π1,v and Π2,v contain the
same unramified representation, then Π1 = Π2 .
• Let U(m)∗ be the quasi-split unitary group. Then we have the correspondence of A-packets of
inner forms: JL : AP(U(m)AF )disc → AP(U(m)∗AF )disc .
Remark 3.3.1. The similar assumptions for orthogonal groups will be proved in the upcoming book
of Arthur [Art]. The similar approach should be possible to handle the case of unitary groups as well.
3.3.1
Cohomological triviality
We will fix an incoherent hermitian space
V as above and suppress H from the notations of Shimura
varieties.
Definition 3.3.2 (Arithmetic theta lifting). Let π be an irreducible cuspidal automorphic represen-
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
tation of Hr (
∞
Θfφ =


R

A
R



A
Hr (F )\Hr (
Hr (F )\Hr (
F)
f (g)Zφ (g)dg ∈ CHr (Sh)C
Sh is proper;
F)
f (g)Zφ∼ (g)dg ∈ CHr (Sh∼ )C
Sh is non-proper,
is called the arithmetic theta lifting of f , which is a (formal integral of) codimension r cycle on certain
(compactified) Shimura variety of dimension m − 1. The cohomology class of Θfφ | Sh is well-defined
due to [KM1990]. The original idea of this construction comes from S. Kudla, see . He constructed
the arithmetic theta series as an Arakelov divisor on certain integral model of a Shimura curve.
Remark 3.3.3. The original idea of the arithmetic theta lifting comes from the work of Kudla
[Kud2003, Section 8] and Kudla–Rapoport–Yang [KRY2006, Section 9.1]. However, there is an essential difference between our definition and theirs, in the sense that our arithmetic theta lifting is a
cocycle on the canonical model of Shimura variety, while theirs (just in the case of Shimura curves) is
an Arakelov divisor on certain integral model, which is not canonically defined. It is more natural to
define arithmetic theta lifting just as a cocycle on the generic fiber in order to consider the canonical
height.
57
In the following discussion, let m = 2n and r = n. Let π be an irreducible cuspidal automorphic
AF ), χ a character of E ×A×F \A×E such that π∞ is a discrete series representation
representation of Hn (
χ
χ
of weight (n − k2 , n + k2 ) and (π, χ) = −1. Then the (equal-rank) theta correspondence of πι (under
ωχ ) is the trivial representation of U(2n, 0)R for every archimedean place ι. Therefore,
a totally positive definite incoherent hermitian space over
AE .
V(π, χ) is
Now we fix an incoherent hermitian
V that is totally positive definite of rank 2n and let (ShK )K be the attached Shimura varieties.
We fix an embedding ι◦ : E ,→ C inducing ι : F ,→ C if F 6= Q. Then similarly we have the
class map cl : CH• (ShK )C → H2•
Bet (ShK,ι (C), C). By a theorem of S. Zucker [Zuc1982, Section 6]
space
◦
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;
C


IH• (Sh#
K, )
ShK is non-proper,
and let H•(2) (Sh) = limK H•(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.
f
Proposition 3.3.4. The class cl(Θfφ ) = 0 in H2n
(2) (Sh), i.e., if Sh is proper (resp. non-proper), Θφ is
∼
cohomologically trivial (resp. such that cl(Θfφ ) ∈ H2n
∂ (Sh )).
Proof. If Sh is non-proper, we can assume that n > 1. By our definition of the arithmetic theta lifting,
for φ = φ∞ φf with fixed φ∞ , cl(Θfφ ) defines an element in
n
Vfin
) ⊗ πfin , H2n
(2) (Sh)
HomHn (AF,fin ) S(
,
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
Zω∼ (h)φ (g) = Th∗ Zφ∼ (g) for all h ∈ H (ι) (AF,fin ) where Th is the Hecke operator of h, we see that cl(Θfφ )
where Hn (
χ
in fact defines an element
n
Vfin
) ⊗ πfin , H2n
(2) (Sh)
n
∨
Vfin
), πfin
⊗ H2n
(2) (Sh) ,
HΘ,φ∞ ∈ HomHn (AF,fin )×H (ι) (AF,fin ) S(
= HomHn (AF,fin )×H (ι) (AF,fin ) S(
n
AF,fin) × H (ι)(AF,fin) acts on S(Vfin
) through the Weil representation ωχ and H (ι) (AF,fin )
(ι)
acts on H2•
(AF,fin )-representation, we
(2) (Sh) through Hecke operators and on πfin trivially. As an H
where Hn (
58
have the following well-known decomposition (cf. e.g., [BW2000, Chapter XIV])
H2n
(2) (Sh) =
M
(ι)
mdisc (σ)H2n (Lie H∞
, KH (ι) ; σ∞ ) ⊗ σfin ,
∞
σ
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
(ι)
mdisc (σ)H2n (Lie H∞
, KH (ι) ; σ∞ ) 6= 0
∞
∨
∨
is the theta correspondence θ(πfin
) of πfin
.
We define a character χ̃ of E ×,1 \
x=
e
eτ
for some e ∈
A×,1
in the following way.
E
For every x ∈
A×E and define χ̃(x) = χ(e) which is well-defined since χ|A
×
F
AE×,1, we can write
= 1.
(ι)
For all finite place v such that v - 2 and ψv , χv , πv unramified, we have Hv
∼
= Hn,v . Let
Σ ∈ AP(HAF )disc be the A-packet containing σ and Π ∈ AP(Hn,AF )disc containing π. Then by
(ι)
Corollary A.3.6, we have that for v as above, JL(Σ)v = JL(Σv ) = Σv and Πv ⊗ χ̃v contain the same
unramified representation. Therefore, JL(Σ) and Π ⊗ χ̃ coincide. In particular,
JL(Σ∞ ) = JL(Σ)∞ = Π∞ ⊗ χ̃∞
which implies that Σ∞ is a discrete series L-packet (cf. [Ada]). This contradicts our assumption since
(ι)
for every discrete series representation σ∞ , H• (Lie H∞ , KH (ι) ; σ∞ ) 6= 0 happens only in the middle
∞
dimension which is 2n − 1, not 2n (cf. [BW2000, Chapter II, Theorem 5.4]). Thus HΘ,φ∞ = 0 and we
prove the proposition.
The proposition says that Θfφ is automatically cohomologically trivial at least in the proper case.
We would like to propose the following conjecture.
∼
Conjecture 3.3.5. When Sh is non-proper, cl(Θfφ ) ∈ H2n
∂ (Sh ) is 0 for every cusp form f ∈ π and
φ as above.
When n = 1, this follows from Corollary 3.4.2 by just computing the degree of the generating
series which is the linear combination of an Eisenstein series and (possibly) an automorphic character
AF ).
(i.e., 1-dimensional automorphic representation) of H1 (
Therefore, cl(Θfφ ) is zero since f is
cuspidal. For the general case, we believe that the same phenomenon will happen.
59
3.3.2
Arithmetic inner product formula: the general conjecture
Let us further assume Conjecture 3.3.5 and the existence of Beilinson–Bloch height paring [Beı̆1987,
Blo1984] on smooth proper schemes over number fields. Then Θfφ is cohomologically trivial and we
let
∨
hΘfφ , Θfφ∨ iK
BB
be the Beilinson–Bloch height paring on ShK (resp. Sh∼
K ) if it is proper (resp. non-proper) for
sufficiently small K. Let Vol(K) be the volume defined in Definition 4.3.3. Then
∨
∨
hΘfφ , Θfφ∨ iBB := Vol(K)hΘfφ , Θfφ∨ iK
BB
is a well-defined number that is independent of K.
If
V V(π, χ), then hΘfφ, Θfφ
∨
∨
iBB = 0 for every f , f ∨ and φ, φ∨ since otherwise, it defines a
nonzero functional
∨
Vfin, χfin), πfin
χfin πfin ) ,
γ(f, f ∨ , φ, φ∨ ) ∈ HomHn (AF,fin )×Hn (AF,fin ) (R(
which contradicts the fact that the latter space is zero. This will imply that, assuming the conjecture
that the Beilinson–Bloch height pairing is non-degenerate, every arithmetic theta lifting Θfφ = 0.
If
V ∼= V(π), then we have the following main conjecture.
Conjecture 3.3.6 (Arithmetic inner product formula). Let π be an irreducible cuspidal automorphic
AF ), χ a character of E ×A×F \A×E , such that π∞ is a discrete series representation of weight (n − k2 , n + k2 ), (π, χ) = −1 and V ∼
= V(π, χ). Then for every f ∈ π, f ∨ ∈ π ∨ and
every φ, φ∨ ∈ S(Vn )U that are decomposable, we have
representation of Hn (
χ
χ
∞
∨
L0 ( 1 , π, χ) Y ∗
hΘfφ , Θfφ∨ iBB = Q2n 2
Z (0, χv , fv , fv∨ , φv ⊗ φ∨
v ),
i
L(i,
)
i=1
v
E/F
where in the last product, almost all factors are 1.
∨
We would like to remark that when n = 1, the height paring hΘfφ , Θfφ∨ iBB is just the classical
Néron–Tate height paring which will be recalled in 3.4.1, hence is defined unconditionally.
60
3.4
3.4.1
Arithmetic kernel functions
Néron–Tate height pairing on curves
we will review the general theory of the Néron–Tate height paring on curves over number fields and
some related facts.
Let E be a number field, not necessarily CM; M a connected smooth projective curve over E,
not necessarily geometrically connected. Let CH1 (M )0C be the group of cohomologically trivial cycles
which is the kernel of the following map
2
deg : CH1 (M )C → Hét
(ME ac ,
Z`(1))Γ
E
⊗Z `
C ∼= C
for a fixed rational prime number `. Let M be a regular model of M , i.e., a regular scheme, flat and
projective over Spec OE 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
C
C
function, i.e., Green (0,0)-form of logarithmic type [Sou1992, II.2], for the divisor Zι◦ ( ) on Mι◦ ( )
for each embedding ι◦ : E ,→
C.
1
bC (M) the group of arithmetic
We denote by Z
C-divisors.
For a
nonzero rational function f on M, we define the associated principal arithmetic divisor to be
c ) = div(f ), − log |fι◦ ,C |2 .
div(f
b1 (M) divided by the
The quotient of Z
C
C-subspace generated by principal arithmetic divisors is the
1
1
dC (M). Inside CH
dC (M), there is a subspace CH1fin (M)C , which
arithmetic Chow group, denoted by CH
C-generated by (Z, 0) with Z supported on special fibres.
orthogonal complement under the C-bilinear pairing
is
1
1
1
dC (M) be the
Let CH1fin (M)⊥
C ⊂ CH
dC (M) × CH
dC (M) →
h−, −iGS : CH
C.
C
An arithmetic divisor (Z, gι◦ ) is flat if we have the following equality in the space D1,1 (Mι◦ ( )) 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.
1
1
1
dC (M). Now we introduce the subgroup CH
dC (M)0 of CH
dC (M) consisting
Flatness is well-defined in CH
61
of elements (represented by) (Z, gι◦ ) satisfying:
1. (Z, gι◦ ) is inside CH1fin (M)⊥
C.
2. ZE is inside CH1 (M )0C .
3. (Z, gι ) is flat.
Therefore, we have a natural map
1
d (M)0 → CH1 (M )0
pM : CH
C
C
(3.9)
(Z, gι ) 7→ ZE
which is surjective. Now we can define the Néron–Tate height paring as
h−, −iNT : CH1 (M )0C × CH1 (M )0C →
C
(3.10)
(Z1 , Z2 ) 7→ h(Z1 , g1,ι◦ ), (Z2 , g2,ι◦ )iGS
where (Zi , gi,ι◦ ) (i = 1, 2) is any preimage of Zi under pM . It is easy to see that this is independent
of the choices of preimages and also the regular model M.
In practice, the cycles we are interested in are not automatically cohomologically trivial. We need
to make some modifications with respect to some auxiliary data. This is quite easy if we are working
c
over a curve. Let Pic(M
) be the abelian group of isomorphism classes of hermitian line bundles on
M . Recall that a hermitian line bundle is the data L = (L , k • kι◦ ), where L ∈ Pic(M ) and k • kι◦ is
a (smooth) hermitian metric on the holomorphic line bundle Lι◦ ,C . We assume that deg c1 (L ) 6= 0.
For every Z ∈ CH1 (M ), the divisor
0
ZL
=Z−
deg Z
c1 (L )
deg c1 (L )
is in CH1 (M )0C .
Now we define the modified height pairing with respect to L to be
0
0
hZ1 , Z2 iL := hZ1,L
, Z2,L
iNT
for every Zi ∈ CH1 (M )C (i = 1, 2). In particular, we need to choose suitable Green function on Zi
when computing via (3.10). We say that the Green function gι◦ of Z is L -admissible if the following
62
equalities of (1, 1)-currents hold
deg Z
ddc [gι◦ ] + δZι◦ (C) =
[c1 (Lι◦ ,C , k • kι◦ )];
deg c1 (L )
Z
gι◦ · c1 (Lι◦ ,C , k • kι◦ ) = 0,
C
Mι◦ ( )
C
where c1 (Lι◦ ,C , k • kι◦ ) ∈ A1,1 (Mι◦ ( )) is the Chern form associated to the hermitian holomorphic
line bundle (Lι◦ ,C , k • kι◦ ) which is a (1, 1)-form.
3.4.2
Degree of generating series
V is a totally
positive definite hermitian space over AE of rank 2 and we write H 0 = H1 . Let H = ResA /A U(V) be
the associated unitary group over A. For every (sufficiently small; Assumption 3.1.1) open compact
subgroup K of H(Afin ), there is a Shimura curve ShK (H), smooth over the reflex field E. For every
embedding ι◦ : E ,→ C over ι ∈ Σ∞ , we have the following ι◦ -adic uniformization
We adopt the construction in 3.1.1 in the case where m = 2 and r = 1. In particular,
F
H
Q
◦
(ι)
∼
ShK ( )an
H
(
)\
D(ι ) ×
◦
=
ι
The underlying real symmetric domain D(ι
map h(ι) :
◦
)
H(Afin)/K
.
R
is identified with the H ι ( )-conjugacy class of the Hodge
S → HR(ι) ∼= U(1, 1)R × U(2, 0)Rd−1 given by


 1
h(ι) (z) = 
z/z



 , 12 , . . . , 12  .
H
The Shimura curves ShK ( ) are non-proper if and only if F =
Q and Σ(V) = {∞}. In this case, we
can compactify them by add cusps. We denote by MK the compactified (resp. original) Shimura curve
H
if ShK ( ) is non-proper (resp. proper), and M the projective system (MK )K with respect to the
K
projections πK
: MK 0 → MK (3.1). On each MK , we have the Hodge bundle LK ∈ Pic(MK )Q which
0
K
is ample. They are compatible under pullbacks of πK
, hence define an element L ∈ Pic(M )Q :=
0
limK Pic(MK )Q .
−→
63
Adapting the definition of (compactified) generating series in 3.2.2, we have
Zφ (g) =
X
x∈K\
Zφ∼ (g) =
V
ωχ (g)φ(T (x), x)Z(x)K ,
fin



Zφ (g)
H
if Sh( )K is proper;


Zφ (g) + W0 ( 1 , g, φ)c1 (LK∨ )
2
if not,
V
A
respectively, as vector-valued series in CH1 (MK )C for φ ∈ S( )U∞ K and g ∈ H 0 ( F ), where
Q
W0 (s, g, φ) = v W0 (s, gv , φv ) that is holomorphic at s = 21 . It is easy to see that Zφ (g) and Zφ∼ (g)
K
are compatible under pullbacks of πK
, hence define series in CH1 (M )C := limK CH1 (MK )C . Readers
−→
may view the modification in the non-proper case as an analogy of the classical Eisenstein series G2 (τ )
0
(which is not a modular form!). It becomes modular if we add a term −π/ Im τ at the price of being
non-holomorphic (cf. e.g., [DS2005, Page 18]).
We apply the construction in 3.4.1 to the curve MK . The cycles whose heights we want to compute
are the generating series Zφ∼ (g) which are not necessarily cohomologically trivial. We use the dual
Hodge bundle L ∨ = (LK∨ )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 Hι0 -invariant metric
kvkι◦ =
(ι)
for v ∈ Vι
1
(v, v)ι
2
c
and the hermitian form (−, −)ι of V (ι) at ι. We denote by L = (L K )K ∈ Pic(M
)
V
the corresponding metrized line bundle. Since L is ample, deg c1 (LK ) 6= 0. For φ ∈ S( )U∞ K and
AF ), we define the arithmetic theta series to be
g ∈ H 0(
Θφ (g) = Zφ∼ (g) −
deg Zφ∼ (g)
c1 (LK∨0 )
deg c1 (LK∨0 )
on every curve MK 0 with K 0 ⊂ K. The ratio
D(g, φ) :=
deg Zφ∼ (g)
deg c1 (LK∨0 )
is independent of the choice of K 0 .
V
We compute the degree function D(g, φ). From φ ∈ S( )U∞ K that is decomposable, we can form
64
an Eisenstein series
1
X
E(s, g, φ) =
(ωχ (γg)φ) (0)λP 0 (γg)s− 2
γ∈P 0 (F )\H 0 (F )
AF ),
on H 0 (
which is absolutely convergent if <(s) >
1
2
and has a meromorphic continuation to
the entire complex plane. We take Tamagawa measures (with respect to ψ) dh on
H(A), dh̄ on
der
A×,1
(A) and dhx on H(A)x , which is the stabilizer of x ∈ V in H(A).
E = H/H
For every v ∈ Σ, let b ∈ Fv× such that Ωb := {x ∈ Vv | T (x) = b} =
6 ∅. Then the local Whittaker
integral Wb (s, e, φv ) has a holomorphic continuation to the entire complex plane and Wb ( 12 , e, φv ) is
not identically zero. Therefore, on the one hand, we have an Nv -intertwining map
Vv ) → CN ,φ
S(
v
b
1
φv →
7 Wb ( , e, φv ).
2
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
1
Wb ( , e, φv ) = γVv
2
Z
φv (x)dµv,b (x)
(3.11)
Ωb
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 =
1
2
and
D(g, φ) = E(s, g, φ) |s= 21 .
Proof. We can assume that φ is decomposable. For b ∈ F × , let
Db (g, φ) =
1
deg c1 (LK∨0 )
X
V
(ωχ (g)φ) (b, x) deg Z(x)K 0
x∈K 0 \ fin
T (x)=b
be the b-th Fourier coefficient of D(g, φ). We first compute the degree of Z(x)K 0 when T (x) = b
is totally positive. Without lost of generality, let us assume that x is contained in the image of
some (rational) nearby hermitian space V (ι) ,→
Vfin and K 0 is sufficiently small.
The isomorphism
65
(ι)
det : Hx → E ×,1 induces a surjective map
×,1
H Afin)x/(K 0 ∩ H(Afin)x) → E ×,1\Afin,E
/ det K 0 .
Hx(ι) \ (
Therefore,
deg Z(x)
K0
Vol(det K 0 , dh̄fin )
det K 0
=
.
= 0
K ∩ ( fin )x
Vol(K 0 ∩ ( fin )x , dhf,x )
HA
HA
When b 6= 0 that is not totally positive, deg Z(x)K 0 = 0 by definition. Therefore, on the one hand,
Db (g, φ) =
1
deg c1 (LK∨0 )
X
V
Vol(det K 0 )
Vol(K 0 ∩ ( fin )x )
HA
(ωχ (g)φ) (b, x)
x∈K 0 \ fin
T (x)=b
=
ωχ (g∞ )φ∞ (b) Vol(det K 0 )
deg c1 (LK∨0 ) Vol(K 0 )
=
Z
ωχ (g∞ )φ∞ (b) Vol(det K 0 ) Y
(ωχ (gv )φv ) (x)dµv,b (x)
deg c1 (LK∨0 ) Vol(K 0 )
Ωb
Z
V
x∈ fin
T (x)=b
(ωχ (g)φ) (x)dµb (x)
v∈Σfin
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= 12 = 0; otherwise,
Y
1
1
Eb (s, g, φ)|s= 12 = Wb ( , g, φ) =
Wb ( , gv , φv )
2
2
v∈Σ
Z
(3.11) Y
=
γVv
ωχ (gv )φv (x)dµv,b (x)
Ωb
v∈Σ
= − Vol(Ω∞ ) (ωχ (g∞ )φ∞ ) (b)
Y Z
v∈Σfin
(ωχ (gv )φv ) (x)dµv,b (x),
Ωb
where Vol(Ω∞ ) = Vol(Ω∞,b ) for every b that is totally positive. Let
D=
Vol(det K 0 )
.
Vol(Ω∞ ) deg c1 (LK 0 ) Vol(K 0 )
Now we compute the constant term
1
D0 (g, φ) = ωχ (g)φ(0) + W0 ( , g, φ),
2
66
on the one hand. On the other hand, the constant term of E( 21 , g, φ) is
1
1
E0 ( , g, φ) = ωχ (g)φ(0) + W0 ( , g, φ).
2
2
H
V
Here the intertwining term W0 ( 21 , g, φ) is nonzero only if ShK ( ) is not proper, i.e., |Σ( )| = 1.
There are two cases:
H
• If ShK ( ) is proper, then we can apply Theorem 3.1.6 to see that D(g, φ) is already an automorphic form. Comparing the ratio of the constant term and non-constant terms, we find that
D = 1.
H
• If ShK ( ) is not proper, we calculate the degree of the Hodge bundle as the classical way on
modular curves and find that D = 1.
Therefore, the proposition follows.
We let
1
E(g, φ) = E(s, g, φ)|s= 21 − W0 ( , g, φ).
2
Then
Θφ (g) = Zφ (g) − E(g, φ)c1 (LK∨ ).
V
If |Σ( )| > 1, W0 ( 21 , g, φ) = 0; otherwise, it equals C · χ̃ ◦ det where C is a constant and χ̃ is the
descent of χ to
AE×,1, as in the proof of Proposition 3.3.4. In all cases, E(g, φ) is a linear combination
of an Eisenstein series and an automorphic character.
Proposition 3.4.1 implies the following corollary on the modularity of the generating series in the
compactified case.
Corollary 3.4.2. Let ` be a linear functional of CH1 (M )C . Then `(Zφ∼ )(g) and hence `(Θφ )(g) are
absolutely convergent and are automorphic forms of H 0 .
H(Afin). We only need to prove that the identity
Zφ∼ (γg) = Zφ∼ (g) in Pic(MK )C for every γ ∈ H 0 (Q). By Theorem 3.1.6 and the fact that the Hodge
bundle is supported on the cusps, Zφ∼ (γg) = Zφ∼ (g) in CH1 (ShK (H))C . So their difference must be
Proof. Assume that φ is invariant under K ⊆
supported on the set of cusps. By a theorem of Manin–Drinfeld (cf. [Man1972, Dri1973]) which posits
that every two cusps are same in CH1 (MK )C , we have an exact sequence
C → CH1(MK )C → CH1(ShK (H))C → 0.
67
Therefore, we only need to prove that deg Zφ∼ (γg) = deg Zφ∼ (g), which is true by the Proposition
3.4.1.
In particular, Definition 3.3.2 specializes to the following unconditional definition.
AF ) = H1(AF ).
Definition 3.4.3. Let π be an irreducible cuspidal automorphic representation of H 0 (
V
For every cusp form f ∈ π and φ ∈ S( )U∞ , we define the arithmetic theta lifting of f to be the
following integral
Θfφ
Z
=
A
H 0 (F )\H 0 (
F)
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 =
P
V
φi,1 ⊗ φi,2 with φi,α ∈ S( )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 0)
X
0
hΘφi,1 (g1 ), Θφi,2 (g2 )iK
NT ,
where the superscript K 0 means that we are taking the Néron–Tate height paring on the curve MK 0 for
some K 0 ⊂ K, of which the definition is independent. By Corollary 3.4.2,
E(g1, g2; Φ) is in A(H 0 ×H 0).
Now let us work over MK and choose a regular model MK of it. We choose an arithmetic line bundle
∨
∨
ω
bK extending L K . In particular, the metrics on ω
bK 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
MK and write
b φ (g) := p−1 (Θφ (g)) = ([Zφ (g)]Zar , gι◦ ) + (Vφ (g), 0) − E(g, φ)b
Θ
ωK ,
MK
where gι◦ is an L K -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
Zbφ (g) = ([Zφ (g)]Zar , gι◦ ) + (Vφ (g), 0).
68
V
Then we have for φα ∈ S( )U∞ K (α = 1, 2),
E(g1, g2; φ1 ⊗ φ2)
= Vol(K)hΘφ1 (g1 ), Θφ2 (g2 )iK
NT
b φ (g1 ), Θ
b φ (g2 )iGS
= − Vol(K)hΘ
1
2
bφ (g1 ) − E(g1 , φ1 )b
bφ (g2 ) − E(g2 , φ2 )b
= − Vol(K)hZ
ωK , Z
ωK iGS
1
2
bφ (g1 ), Zbφ (g2 )iGS + E(g1 , φ1 ) Vol(K)hb
b φ (g2 )iGS
= − Vol(K)hZ
ωK , Θ
1
2
2
b φ (g1 ), ω
+ E(g2 , φ2 ) Vol(K)hΘ
bK iGS + E(g1 , φ1 )E(g2 , φ2 ) Vol(K)hb
ωK , ω
bK iGS ,
1
(3.12)
where the Gillet–Soulé parings are taken on the model MK . By Corollary 3.4.2,
b φ (g)iGS
A(g, φ) := Vol(K)hb
ωK , Θ
is an automorphic form of H 0 , which may depend on K and also the model MK , since we do not
require any canonicality of p−1
ωK , ω
bK iGS . Then
MK . Let C = Vol(K)hb
(3.12) = − Vol(K)hZbφ1 (g1 ), Zbφ2 (g2 )iGS
+ E(g1 , φ1 )A(g2 , φ2 ) + A(g1 , φ1 )E(g2 , φ2 ) + CE(g1 , φ1 )E(g2 , φ2 ).
(3.13)
Vv2 )reg for some v ∈ Σfin. Then
Zφ (g1 ) and Zφ (g2 ) will not intersect on the generic fiber if gα ∈ Pv0 H 0 (AvF ) (α = 1, 2). Then
We assume that φ1 and φ2 are decomposable, and φ1,v ⊗ φ2,v ∈ S(
1
2
bφ (g2 )iGS =
hZbφ1 (g1 ), Z
2
X
bφ (g2 )iv◦ ,
hZbφ1 (g1 ), Z
2
(3.14)
v ◦ ∈Σ◦
C
where the intersection h−, −iv◦ is taken on the local model MK;p◦ := MK ×OE OEp◦ (resp. MK,ι◦ ( ))
if v ◦ = p◦ is finite (resp. if v ◦ = ι◦ is infinite). Combining (3.13) and (3.14), we have for such φα and
gα (α = 1, 2),
E(g1, g2; φ1 ⊗ φ2) = − Vol(K)
X
bφ (g2 )iv◦
hZbφ1 (g1 ), Z
2
v ◦ ∈Σ◦
+ E(g1 , φ1 )A(g2 , φ2 ) + A(g1 , φ1 )E(g2 , φ2 ) + CE(g1 , φ1 )E(g2 , φ2 ).
(3.15)
69
Chapter 4
Comparison at infinite places
We compare local terms of analytic and arithmetic kernel functions at an archimedean place. Section
4.1 is dedicated to the computation on the analytic side. We calculate certain Whittaker integral and
its derivative, following the method of G. Shimura. In 4.2, we introduce a local height function on the
hermitian domain in terms of the Kudla–Millson form, and prove an important invariance property
of such height. The actual comparison of the derivative of Whittaker integrals and the local height
function is accomplished in 4.3.
70
4.1
Archimedean Whittaker integrals
In this section, we calculate the Whittaker integral WT (s, g, Φ) and its derivative (at s = 0) at an
archimedean place. In particular, we fix archimedean places ι : F →
C and ι0 : E → C over ι that
will be suppressed from notations. As in 2.1.2, we identify H 0 = Hι0 (resp. H 00 = Hι00 ) with U(n, n)R
(resp. U(2n, 2n)R ). We have the parabolic subgroup P = Pι of H 00 = U(2n, 2n)R as in Notation 2.2.1.
Moreover, we have the hermitian space V =
Vι of rank 2n over C, which is the standard positive
definite 2n-dimensional complex hermitian space, and Φ0 ∈ S(V 2n ) the Gaussian. We have also a
character χ = χι :
4.1.1
C× → C× that is trivial on R×. Therefore χ(z) = z2`/(zz)` for some integer `.
Elementary reduction steps
We study the integral
WT (s, g, Φ0 ) =
Z
C
ϕΦ0 ,s (wn(u)g)ψT (n(u))−1 du
(4.1)
Her2n ( )
C
for T ∈ Her2n ( ) and Re s > n, where w = wn and du is the selfdual measure with respect to the
additive character ψ(t) = exp(2πit). We have the formula
ωχ ([k1 , k2 ])Φ0 = (det k1 )n+` (det k2 )−n+` Φ0
for [k1 , k2 ] ∈ K (cf. 2.1.2). Write g = n(b)m(a)[k1 , k2 ] under the Iwasawa decomposition. Then
Z
(4.1) =
ωχ (wn(u)n(b)m(a)[k1 , k2 ]) Φ0 (0)λP (wn(u)n(b)m(a)[k1 , k2 ])s ψ(− tr T u)du
C
Her2n ( )
= ψ(tr T b)(det k1 )n+` (det k2 )−n+`
Z
ωχ (wn(u)m(a))Φ0 (0)λP (wn(u)m(a))s ψ(− tr T u)du.
C
Her2n ( )
Since
wn(u)m(a) = wm(a)n(a−1 u t a−1 ) = m(t a−1 )wn(a−1 u t a−1 ),
(4.2)
71
changing variable du = | det a|2n
C d(a−1 u t a−1 ), we have
n+`
−n+`
(4.2) = ψ(tr T b)| det a|n−s
C χ(det a)(det k1 ) (det k2 )
Z
ωχ (wn(u))Φ0 (0)λP (wn(u))s ψ(− tr t aT au)du
C
Her2n ( )
n+`
−n+`
= ψ(tr T b)| det a|n−s
Wt aT a (s, e, Φ0 ).
C χ(det a)(det k1 ) (det k2 )
(4.3)
Therefore, we need to only study WT (s, e, Φ0 ). In what follows, we will not restrict ourselves to the
case of even dimension. In other words, V will be the standard positive definite complex hermitian
C
space of dimension m > 0. For T ∈ Herm ( ), the Whittaker integral WT (s, e, Φ0 ) is absolutely
convergent for Re s >
m
2.
C
Lemma 4.1.1. For u ∈ Herm ( ),
ωχ (wn(u))Φ0 (0) = γV det(1m − iu)m ,
where γV is the Weil constant.
Proof. By definition,
γV−1 ωχ (wn(u))Φ0 (0) =
t
Z
V
ωχ (n(u))Φ0 (x)dx =
m
Write u = k diag[u1 , . . . , um ] k with uj ∈
Z
V
ψ(tr uT (x))Φ0 (x)dx.
(4.4)
m
R (j = 1, . . . , m) and k ∈ U(m)R. Then
t
ψ(tr k diag[u1 , . . . , um ] kT (x)kk −1 )Φ0 (x)dx
(4.4) =
ZV
Z
m
=
ψ(tr diag[u1 , . . . , um ]T (xk)) exp(−2π tr T (x))dx.
(4.5)
Vm
Changing variable x 7→ xk and since tr T (x) = tr T (xk), we have
Z
exp(2πi tr diag[u1 , . . . , um ]T (x) − 2π tr T (x))dx
(4.5) =
=
Vm
m Z
Y
j=1
Identifying V with
exp(2πiuj T (xj ) − 2πT (xj ))dxj .
(4.6)
V
Cm such that (−, −) coincides with the standard hermitian form on Cm, then the
72
selfdual measure dxj on V is simply the usual Lebesgue measure dx on
(4.6) =
=
=
m Z
Y
R
exp(−π(1 − iuj )kxk2 )dx
2m
j=1
m Z ∞
Y
j=1
m
Y
Cm ∼= R2m. Therefore,
2
2m
exp(−π(1 − iuj )t )dt
−∞
(1 − iuj )−m = det(1m − iu)−m .
j=1
Therefore, the lemma follows.
C
Lemma 4.1.2. For u ∈ Herm ( ), λP (wn(u)) = det(1m + u2 )−1 .
Proof. We have the following identities




 i1m  
wn(u) 
=
1m
−1m

1m   1m




u   i1m  
1m


=
.
1m
1m
−i1m − u
Then,
1m (−i1m − u)−1 = −u(1m + u2 )−1 + i(1m + u2 )−1 .
Therefore, λP (wn(u)) = det(1m + u2 )−1 that is a positive real number.
Combining Lemmas 4.1.1 and 4.1.2, we have that for Re s >
γV−1 WT (s, e, Φ0 ) =
Z
C
m
2,
ψ(− tr T u) det(1m + iu)−s det(1m − iu)−s−m du.
Herm ( )
We proceed as in [Shi1982, Case II]. We introduce some new notations that may be different from
those in [Shi1982]. Set
C
C
Her+
m ( ) = {x ∈ Herm ( ) | x > 0};
C
C
h0m = {x + iy | x ∈ Her+
m (C), y ∈ Herm (C)}.
hm = {x + iy | x ∈ Herm ( ), y ∈ Her+
m ( )};
The following lemma is proved in [Shi1982, Section 1].
Lemma 4.1.3 (Siegel). We have
73
1. For z ∈ h0m and Re s > m − 1, we have
Z
C
Her+
m( )
exp(− tr zx)(det x)s−m dx = Γm (s)(det z)−s ,
C
where dx is induced from the selfdual measure on Herm ( ), and
Γm (s) = (2π)
m(m−1)
2
m−1
Y
Γ(s − j).
j=0
C
C
2. For x ∈ Herm ( ), b ∈ Her+
m ( ) and Re s > 2m − 1, we have
Z
Γm (s)
C
exp(2πi tr ux) det(b + 2πiu)−s du =
Herm ( )



exp(− tr xb)(det x)s−m
x ∈ Her+
m ( );


0
if not.
C
By Lemma 4.1.3 (1), for Re s > m − 1,
γV−1 WT (s, e, Φ0 )
Z
=
CZ
ψ(− tr T u)
Herm ( )
=
1
Γm (s)
C
1
Γm (s)
Z
C
Her+
m( )
exp(− tr(1m + iu)x)(det x)s−m det(1m − iu)−s−m dxdu
exp(− tr x)(det x)s−m
Her+
m( )
Z
C
exp(−i tr(x + 2πT )u) det(1m − iu)−s−m dudx.
Herm ( )
(4.7)
u
Applying Lemma 4.1.3 (2) to (1m , x + 2π, s + m), and changing variable u 7→ − 2π
, we have
1
(4.7) =
Γm (s)
2
Z
s−m
exp(− tr x)(det x)
x>0,x+2πT >0
(2π)m
exp(− tr(x + 2πT )) det(x + 2πT )s dx.
Γm (s + m)
(4.8)
In [Shi1982, (1.26)], the author introduced the function
Z
η(g, h; α, β) =
x>−h, x>h
exp(− tr gx) det(x + h)α−m det(x − h)β−m dx
74
C
C
for g ∈ Her+
m ( ), h ∈ Herm ( ), and Re α 0, Re β 0. Changing variable x 7→
2
(4.8) =
(2π)m π 2ms
Γm (s)Γm (s + m)
Z
x
π
+ T , we have
exp(− tr 2πx) det(x + T )s det(x − T )s−m dx
x>−T,x>T
2
(2π)m π 2ms
=
η(2π1m , T ; s + m, s).
Γm (s)Γm (s + m)
(4.9)
In what follows, we assume that T is nonsingular with sign T = (p, q) for p + q = m. Write
t
T = k diag[t1 , . . . , tp , −tp+1 , . . . , −tm ] k
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) = | det T |α+β−m η(a∗ ga, εp,q ; α, β);
(4.10)
η(g, εp,q ; α, β) = 2m(α+β−m) exp(− tr g)ζp,q (2g; α, β).
(4.11)
We recall the definition of ζp,q (g; α, β) introduced in [Shi1982, (4.16)]. Let


 1p
εp = 
0q

;

 0p
ε0q = 

1q

.
C
Then for g ∈ Her+
m ( ), and Re α 0, Re β 0,
Z
ζp,q (g; α, β) =
exp(− tr gx) det(x + εp )α−m det(x + ε0q )β−m dx,
Xp,q
where
C
Xp,q := {x ∈ Herm ( ) | x + εp > 0, x + ε0q > 0}
C
C
with the measure induced from the selfdual one on Herm ( ). In particular, Xm,0 = Her+
m ( ).
75
4.1.2
Analytic continuation
Following [Shi1982, (4.17)], we set
ωp,q (g; α, β) = Γq (α − p)−1 Γp (β − q)−1 (det+ εp,q g)β−q/2 (det− εp,q g)α−p/2 ζp,q (g; α, β),
(4.12)
C
where for a nonsingular element h ∈ Herm ( ), det+ h (resp. det− h) is the absolute value of the
product of all positive (resp. negative) (real) eigenvalues of h if they exist; 1 otherwise. It is proved in
[Shi1982, Section 4] that ωp,q (g; α, β) has a holomorphic continuation in (α, β) to the whole
C2, and
satisfies the following functional equation
ωp,q (g; m − β, m − α) = ωp,q (g; α, β).
Lemma 4.1.4. If p = m and q = 0, then ωm,0 (g; m, β) = ωm,0 (g; α, 0) = 1.
Proof. The integral
Z
ζm,0 (g; m, β) =
C
exp(− tr gx)(det x)β−m dx
Her+
m( )
is absolutely convergent for Re β > 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.
C
Proposition 4.1.5. Suppose T ∈ Herm ( ) is nonsingular with sign T = (p, q). Then
1. ords=0 WT (s, e, Φ0 ) ≥ q; and
2. If T is positive definite, i.e., p = m and q = 0, then
2
(2π)m
WT (0, e, Φ ) = γV
exp(−2π tr T ).
Γm (m)
0
Proof.
1. Combining (4.9), (4.10), (4.11) and (4.12), we have
γV−1 WT (s, e, Φ0 ) =
2
Γq (m + s − p)Γp (s − q)
(2π)m +2ms | det T |2s exp(−2π tr t aa)
Γm (s)Γm (s + m)
(det+ 4πT )q/2−s (det− 4πT )p/2−m−s ωp,q (4π t aa; m + s, s).
All terms except the Gamma factors, are holomorphic for all s ∈
C. Since
m(m−1)
Γq (m + s − p)Γp (s − q)
(2π)−pq− 2
=
,
Γm (s)Γm (s + m)
Γ(s) · · · Γ(s − q + 1) × Γ(s + m) · · · Γ(s + m − p + 1)
(4.13)
76
we have
ords=0 WT (s, e, Φ0 ) ≥ −ords=0 Γ(s) · · · Γ(s − q + 1) = q.
2. If T is positive definite, then tr t aa = tr T . By (4.13) and Lemma 4.1.4, we have
2
2
γV−1 WT (0, e, Φ0 ) =
4.1.3
(2π)m
(2π)m
exp(−2π tr T )ωm,0 (4π t aa; m, 0) =
exp(−2π tr T ).
Γm (m)
Γm (m)
First-order derivatives
By Proposition 4.1.5 (1), the T -th coefficient will not contribute to the analytic kernel function
E 0 (0, g, Φ) if sign T = (p, q) with q ≥ 2. Therefore, we focus on the case where q = 1, and study
the functions ζm−1,1 (g; α, β) and ωm−1,1 (g; α, β)1 .
We assume that

 a
g=

b

,
C
a ∈ Her+
m−1 ( ), b ∈
R>0.
We write elements in Xm−1,1 in the following form

 x

t
z

z 
,
y
C
x ∈ Herm−1 ( ), 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 t z, y + 1 > t zx−1 z}
= {(x, y, z) | x + 1m−1 > 0, y + 1 > 0, x > z(y + 1)−1 t z, y > t z(x + 1m−1 )−1 z}.
We have
α−m

 x + 1m−1
exp(− tr ax − by) det 
t
Xm−1,1
z
Z
ζm−1,1 (g; α, β) =
 x
det 
t
z
z 

z
β−m

z
y+1


dxdydz,
(4.14)
C
where we apply the selfdual measure dx on Herm−1 ( ), the Lebesgue measure dy on
C
R, and the
measure dz that is 2m−1 times the Lebesgue measure on Matm−1,1 ( ). We make change of variables
1 Two
letters g appearing here stand for different meanings.
77
as in [Shi1982, Page 289] as follows. Put
f = (x + 1m−1 )−1/2 z(y + 1)−1/2 .
t
Then 1m−1 − f > 0. Put
t
t
r = (1 − f f )1/2 ;
s = (1m−1 − f f )1/2 ;
w = s−1 f = f r−1 ;
u = x − w t w;
v = y − t ww.
C R>0×Matm−1,1(C),
Then the map (x, y, z) 7→ (u, v, w) maps Xm−1,1 bijectively onto Y = Her+
m−1 ( )×
and the Jacobian
∂(x, y, z)
= det(1m−1 + x)(1 + y)m−1 (1 + t ww)−m
∂(u, v, w)
C
for the measure ∂(u, v, w) on Y induced from that on Herm−1 ( ) ×
R × Matm−1,1(C) as an open
subset. Since


 x + 1m−1 z
det 
t
z
y

z
 x
det 
t
z y+1

t
t
−1
 = det(u + 1m−1 + w w)v det(1m−1 w w) ;


t
−1
t
 = (v + 1 + ww)(det u) det(1m−1 w w) ,
we obtain, on the one hand, that
Z
exp − tr(au + aw t w) − (bv + b t ww) det(1m−1 + w t w)m−α−β
(4.14) =
Y
det(u + 1m−1 + w t w)α−m+1 (det u)β−m (v + 1 + t ww)β−1 v α−m dudvdw
Z
=
C
exp(− tr aw t w − b t ww)ζ1,0 (b(1 + t ww); β, α − m + 1)
Matm−1,1 ( )
Z
C)
Her+
m−1 (
exp(− tr au) det(u + 1m−1 + w t w)α−m−1 (det u)β−m dudw.
(4.15)
By (4.10), (4.11) and (4.12), we have, on the other hand, that
2
γV−1 WT (s, e, Φ0 )
(2π)m +2ms | det T |2s
=
exp(−2π tr t aa)ζm−1,1 (4π t aa; m + s, s).
Γm (s)Γm (s + m)
t
Assume that T = k diag[a1 , . . . , am−1 , −b] k with a1 , . . . , am−1 , b ∈
R>0
(4.16)
and k ∈ U(m)R . Then
78
t
aa = diag[a1 , . . . , am−1 , b]. By (4.16) and (4.13),
γV−1 WT0 (0, e, Φ0 )
2
(2π)m
exp (−2π(a1 + · · · + am−1 + b)) ζm−1,1 (4π diag[a1 , . . . , am−1 , b]; m, s). (4.17)
= lim
s→0 sΓm (s)Γm (m)
Plugging (4.15) with (α, β) = (m, s),
2
2m−1 (2π)m
(4.17) = lim
exp (−2π(a1 + · · · + am−1 + b))
s→0 sΓm (s)Γm (m)
Z
exp (−4π[(a1 + am )w1 w1 + · · · + (am−1 + am )wm−1 wm−1 ]) ζ1,0 (4πb(1 + t ww); 0, 1)
m−1
C
Z
exp (−4π tr diag[a1 , . . . , am−1 ]u) det(u + 1m−1 + w t w)(det u)s−m dudw1 · · · dwm−1 .
C
Her+
m−1 ( )
(4.18)
It is easy to see that
ζ1,0 (4πb(1 + t ww); 0, 1) = − exp 4πb(1 + t ww) Ei −4πb(1 + t ww) ,
(4.19)
where Ei is the exponential integral
∞
Z
Ei(z) = −
1
exp(zt)
dt.
t
C
We evaluate the inside integral, i.e., the one over Her+
m−1 ( ). Temporarily let g0 = 4π diag[a1 , . . . , am−1 ],
and consider the integral
Z
C
Her+
m−1 ( )
exp(− tr ug0 ) det(u + 1m−1 + w t w)(det u)s−m du.
Define a differential operator
∆ = det
∂
∂gjk
m−1
.
j,k=1
Then
∆ exp(− tr ug) = (−1)m−1 (det u) exp(− tr ug).
(4.20)
79
Therefore,
(4.20) = exp tr(1m−1 + w t w)g0
Z
exp − tr(u + 1m−1 + w t w)g0 det(u + 1m−1 + w t w)(det x)s−m du
C
Her+
m−1 ( )
= (−1)m−1 exp tr(1m−1 + w t w)g0
Z
∆ |g=g0 exp − tr(u + 1m−1 + w t w)g (det x)s−m du.
C
(4.21)
Her+
m−1 ( )
We exchange ∆ and the integration by analytic continuation. Then
(4.21) = (−1)m−1 exp tr(1m−1 + w t w)g0
Z
exp − tr(u + 1m−1 + w t w)g (det x)s−m du
∆ |g=g0
C
Her+
m−1 ( )
= (−1)m−1 exp tr(1m−1 + w t w)g0 ∆ |g=g0 exp − tr(1m−1 + w t w)g ζm−1 (g; m − 1, s − 1)
= (−1)m−1 exp tr(1m−1 + w t w)g0 ∆ |g=g0 exp − tr(1m−1 + w t w)g (det g)1−s Γm−1 (s − 1)
= (−1)m−1 Γm−1 (s − 1) exp tr(1m−1 + w t w)g0 ∆ |g=g0 exp − tr(1m−1 + w t w)g (det g)1−s .
(4.22)
Plugging (4.19) and (4.22) into (4.18), we obtain
2
(4.18) = lim
s→0
Z
Γm−1 (s − 1)(−2)m−1 (2π)m
exp(−2π tr T )
sΓm (s)Γm (m)
C
exp (−4π(a1 w1 w1 + · · · + am−1 wm−1 wm−1 ))
m−1
exp tr(1m−1 + w t w)g0 ∆ |g=g0 exp − tr(1m−1 + w t w)g (det g)1−s
(− Ei) −4πb(1 + t ww) dw1 · · · dwm−1
Z
2
(2π)m (−2)m−1
=
exp(−2π
tr
T
)
exp (−4π(a1 w1 w1 + · · · + am−1 wm−1 wm−1 ))
Γm (m)(2π)m−1
Cm−1
exp tr(1m−1 + w t w)g0 ∆ |g=g0 exp − tr(1m−1 + w t w)g (det g)
(− Ei) −4πb(1 + t ww) dw1 · · · dwm−1 .
(4.23)
To compare with the local height later, we make a change of variables. Let
Dm−1 = {z = (z1 , . . . , zm−1 ) ∈
Cm−1 | zz := z1z1 + · · · + zm−1zm−1 < 1}
80
be the open unit disc in
Cm−1. Then the map
wj =
is a homeomorphism from
zj
(1 − zz)
1/2
,
j = 1, . . . , m − 1
Cm−1 to Dm−1 as real manifolds.
(4.24)
To calculate the Jacobian, let wj =
uj + vj i and zj = xj + yj i be the corresponding real and imaginary parts. Then
xj xk
∂uj
=
,
k 6= j;
3/2
∂xk
(1 − zz)
x2j
1
∂uj
+
;
=
3/2
1/2
∂xj
(1 − zz)
(1 − zz)
∂uj
xj yk
=
;
3/2
∂yk
(1 − zz)
∂vj
yj yk
,
k 6= j;
=
3/2
∂yk
(1 − zz)
yj2
∂vj
1
=
+
;
3/2
1/2
∂yj
(1 − zz)
(1 − zz)
∂vj
yj xk
.
=
3/2
∂xk
(1 − zz)
t
In Lemma 4.1.6 below, we let n = 2m + 2, = 1 − zz and c = (c1 , . . . , c2m−2 ) with cj = xj ,
cm+1−j = yj for j = 1, . . . , m − 1. Then
∂(u1 , v1 , . . . , um−1 , vm−1 )
∂(x1 , y1 , . . . , xm−1 , ym−1 )
∂(u1 , . . . , um−1 ; v1 , . . . , vm−1 )
=
∂(x1 , . . . , xm−1 ; y1 , . . . , ym−1 )
= (1 − zz)
−3(m−1)
det (1 − zz) 12m−2 + c t c
= (1 − zz)
−3(m−1)
(1 − zz)
= (1 − zz)
−m
2m−3
2
1 − zz + x21 + · · · + x2m−1 + y12 + · · · + ym−1
.
(4.25)
C
Lemma 4.1.6. Let c ∈ Matn×1 ( ). Then
1. det 1n + c t c = 1 + t cc;
2. For > 0, det 1n + c t c = n−1 + t cc .
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 t c = 1 + s t cc for all c ∈ . Since they
R
81
are both polynomials in s, we need only to prove for s < 0. We have

 1n

√

− −sc   1n
 √
1
−s t c
√



t

1n
−sc  
  1n + sc c

 √
=
,
− −s t c 1
1
1
and




1n
  1n
 √
√ t
− −s c 1
−s t c
√

−sc   1n

1
√


− −sc   1n
=
1

1 + s t cc

.
Therefore, det 1n + sc t c = 1 + s t cc.
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
Ω,
(−2i)m−1
where
Ω=
m−1
^
(dzj ∧ dzj ) .
(4.26)
j=1
Here, we view dzj as a (1, 0)-form, not the Lebesgue measure. By (4.25), we have
Z
2
(2π)m
−m
(4.23) =
exp (−2π tr T )
exp (−4π(a1 w1 w1 + · · · + am−1 wm−1 wm−1 )) (1 − zz)
Γm (m)(2πi)m−1
Dm−1
exp tr 1m−1 + w t w g0 ∆ |g=g0 exp − tr 1m−1 + w t w g (det g)
(− Ei) −4πb 1 + t ww 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 t w g (det g)
= exp − tr 1m−1 + w t w g
X
(−4π)t (m − 1 − t)!(as1 · · · ast ) (1 + ws1 ws1 + · · · + wst wst ) ,
1≤s1 <···<st ≤m−1
82
where the sum is taken over all subsets of {1, . . . , m − 1}.
Proof. Let
ujk = − (1 + wj wk ) ;
m−1
g = (gjk )j,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. g J,K ) the (square)
matrix obtained by keeping (resp. discarding) the rows indexed in J and the columns indexed in
K. Therefore, g J,K = gJ,K , where J (resp. K) is the complement set {1, . . . , m − 1} − J (resp.
{1, . . . , m − 1} − K). Let Sm−1 be the group of (m − 1)-permutations. For σ ∈ Sm−1 and a subset
J = {j1 < · · · < jt } ⊂ {1, . . . , m−1}, let J (σ) ∈ {±1} be a factor that depends only on J and σ. This
factor comes from the combinatorics in taking successive partial derivatives. In later calculation, we
only need to know its value in the case where σ maps J to itself. Then, if we let σJ be the restriction
of σ to J, we have J (σ) = (−1)|σJ | .
We compute that
∂
(exp (tr ug) |g|) = uσ(1),1 exp (tr ug) |g| + {1} (σ) exp(tr ug) g {1},{σ(1)} ;
∂g1,σ(1)
∂
∂
(exp(tr ug)|g|) = uσ(2),2 uσ(1),1 exp(tr ug)|g| + {2} (σ)uσ(1),1 exp(tr ug) g {2},{σ(2)} ∂g2,σ(2) ∂g1,σ(1)
+ {1} (σ)uσ(2),2 exp(tr ug) g {1},{σ(1)} + {1,2} (σ) exp(tr ug) g {1,2},{σ(1),σ(2)} .
By induction, we have
∂
=
∂gm−1,σ(m−1)
X
···
∂
∂g1,σ(1)
(exp(tr ug)|g|)
{j1 ,...,jt } (σ)uσ(sm−1−t ),sm−1−t · · · uσ(s1 ),s1 exp(tr ug) g {j1 ,...,jt },{σ(j1 ),...,σ(jt )} ,
1≤j1 <···<jt ≤m−1
where {s1 < · · · < sm−1−t } is the complement of {j1 , . . . , jt }. Summing over σ, we have
∆ |g=g0 (exp(tr ug)|g|) = exp(tr ug0 )
X
σ∈Sm−1
(−1)|σ|
X
1≤j1 <···<jt ≤m−1
{j ,...,jt },{σ(j1 ),...,σ(jt )} {j1 ,...,jt } (σ)uσ(sm−1−t ),sm−1−t · · · uσ(s1 ),s1 g0 1
.
83
Changing the order of summation, since g0 is diagonal, we have
∆ |g=g0 (exp(tr ug)|g|)
X
X
= exp(tr ug0 )
(−1)|σ| (−1)|σJ | uσ(sm−1−t ),sm−1−t · · · uσ(s1 ),s1 g0J,J J={j1 <···<jt } σ(J)=J
X
= exp(tr ug0 )
X
0
t! g0J,J (−1)|σ | uσ(sm−1−t ),sm−1−t · · · uσ(s1 ),s1
J={j1 <···<jt }
X
= exp(tr ug0 )
σ 0 :J→J
t! g0J,J uJ,J J={j1 <···<jt }
X
= exp(tr ug0 )
(m − 1 − t)! |(g0 )J 0 ,J 0 | |uJ 0 ,J 0 | .
J 0 ={s1 <···<st }
The lemma follows by Lemma 4.1.6 (1).
In conclusion, combining (4.27), we obtain the following proposition.
t
Proposition 4.1.8. For T = k diag[a1 , . . . , am−1 , −b] k of signature (m − 1, 1) as above, we have
WT0 (0, e, Φ0 )
= γV
Z
2
(2π)m
exp(−2π
tr
T
)
exp (−4π(a1 w1 w1 + · · · + am−1 wm−1 wm−1 ))
Γm (m)(2πi)m−1
Dm−1
X
(−4π)t (m − 1 − t)!(as1 · · · ast ) (1 + ws1 ws1 + · · · + wst wst )
1≤s1 <···<st ≤m−1
−m
(− Ei) (−4πb(1 + w∗ w)) (1 − zz)
Ω,
where wj are functions in z as in (4.24), and Ω (4.26) is the volume form in z.
4.2
Archimedean local height
In this section, we introduce a notion of height on the symmetric domain that will eventually contribute
to the local height pairing at an archimedean place. We also prove some properties of such height. A
basic reference for archimedean Green currents and height paring is [Sou1992, Chapter II]. We keep
the notations in 4.1. We fix an integer m ≥ 2.
84
4.2.1
Green currents
Let V 0 '
Cm be the complex hermitian space with the form
0
0
(z 0 , z) = z10 z1 + · · · + zm−1
zm−1 − zm
zm ;
0
z = (z1 , . . . , zm ), z 0 = (z10 , . . . , zm
)∈
Cm .
In particular, the signature of V 0 is (m−1, 1). The symmetric hermitian domain D of U(V 0 ), introduced
in 3.1.1, can be identified with the (m − 1)-dimensional open complex unit disc Dm−1 via the map
z = [z1 : · · · : zm ] ∈ D 7→
zm−1
z1
,...,
zm
zm
∈ Dm−1 .
In what follows, we will not distinguished between D and Dm−1 .
Given any x ∈ V 0r (1 ≤ r ≤ m − 1) with nonsingular moment matrix T (x), let Dx be the subspace
of Dm−1 consisting of lines perpendicular to all components in x. Then Dx is nonempty if and
only if T (x) is positive definite. Suppose r = 1, for z ∈ Dm−1 , let x = xz + xz be the orthogonal
decomposition with respect to the line z, i.e., xz ∈ z and xz ⊥ z. Let R(x, z) = −(xz , xz ) that is
nonnegative since z is negative definite, and R(x, z) = 0 if and only if x = 0 or z ∈ Dx . Explicitly, let
x = (x1 , . . . , xm ) ∈ V 0 , z = (z1 , . . . , zm−1 ) ∈ Dm−1 . Then
R(x, z) =
(x1 z1 + · · · + xm−1 zm−1 − xm ) (x1 z1 + · · · + xm−1 zm−1 − xm )
,
1 − zz
where we recall that zz = z1 z1 + · · · + zm−1 zm−1 . We define
ξ(x, z) = − Ei(−2πR(x, z)).
For each nonzero element x ∈ V 0 , ξ(x, •) is a smooth function on Dm−1 − Dx , and has logarithmic
growth along Dx if not empty. Therefore, we can view it as a current [ξ(x)] = [ξ(x, •)] on Dm−1 .
U(V 0 )
We recall the Kudla–Millson form ϕ ∈ (S(V 0r ) ⊗ Ar,r (Dm−1 ))
(1 ≤ r ≤ m − 1) constructed in
[KM1986], and let
ω(x) = ω(x, •) = exp(2π tr T (x))ϕ(x, •).
We have the following proposition.
Proposition 4.2.1. Let x ∈ V 0 be a nonzero elements. Then we have
ddc [ξ(x)] + δDx = [ω(x)]
85
as currents on Dm−1 .
We will only give a proof for m = 2, and the proof for general m is similar but involves tedious
computations.
Proof. We start from showing that ddc ξ(x) = ω(x) holds away from Dx . Let x = (x1 , x2 ) and
z ∈ D1 − Dx . Sometimes we simply write R instead of R(x) for short. Then we have the formula
ddc ξ(x) =
1
2πi
exp(−2πR)
exp(−2πR)
R∂∂R
−
∂R
∧
∂R
−
2π
∂R
∧
∂R
.
R2
R
(4.28)
Computing each term, we have
R(x, z) =
∂R =
∂R =
∂∂R =
∂R ∧ ∂R =
(x1 z − x2 ) (x1 z − x2 )
;
1 − zz
x1 (x1 z − x2 ) (1 − zz) + (x1 z − x2 ) (x1 z − x2 ) z
2
dz;
2
dz;
(1 − zz)
x1 (x1 z − x2 ) (1 − zz) + (x1 z − x2 ) (x1 z − x2 ) z
(1 − zz)
!
x1 z (x1 z − x2 ) + (x1 z − x2 ) (2x1 z − x2 ) + 2Rzz
x1 x1
+
dz ∧ dz;
2
1 − zz
(1 − zz)
!
x1 x1 R x1 z (x1 z − x2 ) R + x1 z (x1 z − x2 ) R + R2 zz
+
dz ∧ dz.
2
1 − zz
(1 − zz)
Therefore,
R∂∂R − ∂R ∧ ∂R =
(x1 z − x2 ) (x1 z − x2 ) R
2
(1 − zz)
−
!
R2 zz
2
(1 − zz)
dz ∧ dz = R2
dz ∧ dz
2;
(1 − zz)
(4.29)
and
∂R ∧ ∂R = (x1 x1 (1 − zz) + x1 z (x1 z − x2 ) + x1 z (x1 z − x2 ) + Rzz)
= (x1 x1 + x1 zx2 + x1 zx2 + x1 x1 zz)
Rdz ∧ dz
2
(1 − zz)
Rdz ∧ dz
2
(1 − zz)
Rdz ∧ dz
= ((x, x) + (x1 z − x2 ) (x1 z − x2 ) + Rzz)
2
(1 − zz)
Rdz ∧ dz
= (R(x, z) + (x, x))
2.
(1 − zz)
(4.30)
Plugging (4.29) and (4.30), we have that
(4.28) = (1 − 2π(R(x, z) + (x, x))) exp(−2πR(x, z))
dz ∧ dz
2πi (1 − zz)
2
= ω(x, z).
86
The remaining discussion is same as in the proof of [Kud1997, Proposition 11.1], from Lemma 11.2
on Page 606. We will not repeat the detail.
The above proposition says that ξ(x) is a Green function of logarithmic type for Dx . Now we
consider x = (x1 , . . . , xr ) ∈ V 0r with nonsingular moment matrix T (x). Then using the star product
of Green currents, we have a Green current
Ξx = [ξ(x1 )] ∗ · · · ∗ [ξ(xr )]
for Dx . As currents of degree (r, r), we have
ddc ([ξ(x1 )] ∗ · · · ∗ [ξ(xr )]) + δDx = [ω(x1 ) ∧ · · · ∧ ω(xr )] = [ω(x)].
Definition 4.2.2 (Height functions (on D)). For x = (x1 , . . . , xm ) ∈ V 0m with nonsingular moment
matrix T (x), we define the height function (on D) to be
H(x)∞ = h1, Ξx i = h1, [ξ(x1 )] ∗ · · · ∗ [ξ(xm )]i.
Since ξ(hx, hz) = ξ(x, z) for h ∈ U(V 0 ), the height function satisfies H(hx)∞ = H(x)∞ , and thus
depends only on the (nonsingular) moment matrix T (x). Sometimes we simply write H(T )∞ for this
function.
The following proposition claims that H(T )∞ is in fact invariant under the conjugation action of
U(m)R .
Proposition 4.2.3. The height function H(T )∞ depends only on the eigenvalues of T . In other
t
words, for every k ∈ U(m)R , H(kT k)∞ = H(T )∞ .
Proof. We prove by induction on m. The case m = 2 in left to the next subsection. Suppose that m ≥ 3
and the proposition holds for m − 1. Since U(m)R is generated by diagonal matrices, permutation
matrices, and the matrices of form


0
 k

1

,
k 0 ∈ U(m − 1)R ,
We only need to prove that H((x0 k 0 , xm ))∞ = H((x0 , xm ))∞ , where x = (x0 , xm ) ∈ V 0m−1 ⊕ V 0 = V 0m
87
with T (x) = T . By definition,
H((x0 , xm ))∞ = h1, [ξ(x1 )] ∗ · · · ∗ [ξ(xm−1 )]|Dxm i +
= H(x0 )∞ +
Z
Z
ω(x1 ) ∧ · · · ∧ ω(xm−1 ) ∧ ξ(xm )
Dm−1
ω(x0 ) ∧ ξ(xm ).
Dm−1
By induction, H(x0 k 0 )∞ = H(x0 )∞ and by [KM1986, Theorem 3.2 (ii)], ω(x0 ) = ω(x0 k 0 ). Therefore,
H((x0 k 0 , xm ))∞ = H((x0 , xm ))∞ .
4.2.2
Invariance under U(2): an exercise in Calculus
We consider the case m = 2. Suppose


 d1
T =
m
with d1 , d2 ∈
m 

d2
R and m ∈ C. Choose a complex number with norm 1 such that 2m ∈ R. Then
 
 t


−1
  d1

m
m    
d2
−1



2
  d1
=
2 m
R
m 
 ∈ Sym2 ( ).
d2
We write elements of SO(2) in the form


 cos θ
kθ := 
− sin θ
sin θ 
,
cos θ
θ∈
R,
t
and write T [θ] = kθ T kθ = kθ T t kθ . Since ξ(x) = ξ(x) for every x ∈ V 0 , to finish the proof of
Proposition 4.2.3, we need only to prove the following proposition.
R
Proposition 4.2.4. For every T ∈ Sym2 ( ) with sign T = (1, 1), we have H(T [θ])∞ = H(T )∞ .
Suppose T = diag[a, −b] with a, b > 0,

 d1,θ
T [θ] = 
mθ

R
mθ 
 ∈ Sym2 ( ).
d2,θ
88
√
√
Let x0 = ( 2a, 0) ∈ V 0 , y0 = (0, 2b) ∈ V 0 . For θ ∈
R, let
xθ = x0 kθ = (x1,θ , x2,θ ) = cos θ · x0 − sin θ · y0 ;
yθ = y0 kθ = (y1,θ , y2,θ ) = sin θ · x0 + cos θ · y0 .
We have
dyθ
= xθ ;
dθ
dxθ
= −yθ ;
dθ
H(T [θ])∞ = H((xθ , yθ ))∞ .
Let
zx,θ =
x2,θ
;
x1,θ
zy,θ =
y2,θ
.
y1,θ
Then
Dxθ = [zx,θ , 1];
Dyθ = [zy,θ , 1]
if not empty. In what follows, we adopt the convention that if |z| ≥ 1, f (z) = 0 for any function f .
We record the following lemma that is [Kud1997, Lemma 11.4].
Lemma 4.2.5. We have that
Z
H((xθ , yθ ))∞ = ξ(xθ , zy,θ ) +
ξ(yθ )ω(xθ )
D1
Z
= ξ(yθ , zx,θ ) +
ξ(xθ )ω(yθ )
Z
= ξ(xθ , zy,θ ) + ξ(yθ , zx,θ ) −
dξ(xθ ) ∧ dc ξ(yθ ).
D1
D1
We consider the last integral above in general. Write x = (x1 , x2 ) ∈ V 0 , y = (y1 , y2 ) ∈ V 0 ,
R1 = R(x), and R2 = R(y). Define
Z
dξ(x) ∧ dc ξ(y)
Z
1
∂ + ∂ ξ(x) ∧ ∂ − ∂ ξ(y)
=−
4πi D1
Z
i
=−
∂ξ(x) ∧ ∂ξ(y) + ∂ξ(y) ∧ ∂ξ(x)
4π D1
Z
i
exp (−2π(R1 + R2 ))
=−
∂R1 ∧ ∂R2 + ∂R2 ∧ ∂R1 .
4π D1
R1 R2
I(T ) = I((x, y)) = −
D1
For z ∈ D1 , let
−1/2
x(z) = (1 − zz)
(z, 1) ∈ V 0 ;
M = (x, x(z))(y, x(z)).
89
We have the following lemma.
Lemma 4.2.6. Let 2m = (x, y). Then
∂R1 ∧ ∂R2 + ∂R2 ∧ ∂R1 = 2 R1 R2 + mM + mM
∂R1 ∧ ∂R2 − ∂R2 ∧ ∂R1 = 2 mM − mM
dz ∧ dz
2;
(1 − zz)
dz ∧ dz
(1 − zz)
2.
Proof. By definition,
R1 =
(x1 z − x2 ) (x1 z − x2 )
.
1 − zz
∂R1 =
(x1 z − x2 ) x1 + zR1
dz,
1 − zz
Therefore,
and similarly for ∂R2 , ∂R1 , and ∂R2 . We compute that
∂R1 ∧ ∂R2
= ((x1 z − x2 ) (y1 z − y2 ) x1 y1 + (y1 z − y2 ) y1 zR1 + (x1 z − x2 ) x1 zR2 + zzR1 R2 )
= ((x1 z − x2 ) (y1 z − y2 ) x1 y1 + (y1 z − y2 ) y2 R1 + (x1 z − x2 ) x1 zR2 + R1 R2 )
dz ∧ dz
2
(1 − zz)
dz ∧ dz
2
(1 − zz)
y2 (x1 z − x2 )
dz ∧ dz
= (x1 z − x2 ) (y1 z − y2 ) x1 y1 +
+ (x1 z − x2 ) x1 zR2 + R1 R2
2
1 − zz
(1 − zz)
x1 y1 − x2 y2 − x1 z (y1 z − y2 )
dz ∧ dz
= (x1 z − x2 ) (y1 z − y2 )
+ (x1 z − x2 ) x1 zR2 + R1 R2
2
1 − zz
(1 − zz)
dz ∧ dz
= (2mM + R1 R2 )
2.
(1 − zz)
The lemma follows from a similar calculation for ∂R2 ∧ ∂R1 .
We define a morphism
α:
R × D1 → Her2(C)det=0 = {h ∈ Her2(C) | det h = 0}
of 3-dimensional real analytic spaces by the formula

 R1
α(θ, z) = 
M
 
M   (xθ , x(z))(xθ , x(z))
=
R2
(xθ , x(z))(yθ , x(z))

(xθ , x(z))(yθ , x(z)) 
,
(yθ , x(z))(yθ , x(z))
90
and set αθ = α(θ, •). By an easy computation, we see that
dR1
=− M +M ;
dθ
dR2
= M + M;
dθ
dM
= R1 + R2 .
dθ
(4.31)
Therefore, R1 + R2 and M − M are independent of θ, which are the values at θ = 0, respectively. In
other words,
2azz + 2b
2(a + b)
= −2a +
;
1 − zz
1 − zz
√
2 ab (z − z)
M −M =
.
1 − zz
R1 + R2 =
By Lemma 4.2.6, and the fact that 2mθ = (xθ , yθ ) ∈
i
I(T [θ]) = −
2π
Z
D1
(4.32)
(4.33)
R, we have
dz ∧ dz
exp(−2π(R1 + R2 ))
R1 R2 + m M + M
2
R1 R2
(1 − zz)
= I 0 (T [θ]) + I 00 (T [θ]),
(4.34)
where
I 0 (T [θ]) = −
I 00 (T [θ]) = −
i
2π
Z
i
2π
Z
exp(−2π(R1 + R2 ))
D1
D1
dz ∧ dz
(1 − zz)
2;
dz ∧ dz
exp(−2π(R1 + R2 ))
m M +M
2.
R1 R2
(1 − zz)
By (4.32) and (4.33), the integral I 0 (T [θ]) is independent of θ. There we only need to consider the
C
second term I 00 (T [θ]). Define a differential form of degree two on (the smooth locus of) Her2 ( )det=0
as follows.
i exp(−2π(R1 + R2 )) M + M
dR1 ∧ dR2 ,
4π
R1 R2
M −M
which has singularities along the locus R1 R2 M − M = 0. We have the following lemma.
Ξ=−
Lemma 4.2.7. We have
1. For a fixed θ ∈
R,
αθ∗ (Ξ) = −
dz ∧ dz
i exp(−2π(R1 + R2 ))
m M +M
2;
2π
R1 R2
(1 − zz)
91
C
2. On Her2 ( )det=0 , we have
dΞ =
Proof.
i exp(−2π(R1 + R2 ))
2
d M − M ∧ dR1 ∧ dR2 .
π M −M
M +M
1. It follows from Lemma 4.2.6.
2. By the equality
M +M
2
− M −M
2
= 4R1 R2 ,
we have
d
M +M
4R1 R2
=−
2
.
d M −M M −M
M +M
M −M
Then it follows.
Let D1+ = {z ∈ D1 | Im z ≥ 0}. Since αθ∗ (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 00 (T [θ]) − I 00 (T [θ0 ])
Z
Z
= ξ(xθ1 , zy,θ1 ) + ξ(yθ1 , zx,θ1 ) − ξ(xθ0 , zy,θ0 ) − ξ(yθ0 , zx,θ0 ) +
αθ∗1 (Ξ) −
αθ∗0 (Ξ)
D1
D1
Z
Z
= ξ(xθ1 , zy,θ1 ) + ξ(yθ1 , zx,θ1 ) − ξ(xθ0 , zy,θ0 ) − ξ(yθ0 , zx,θ0 ) + 2
αθ∗0 (Ξ).
αθ∗1 (Ξ) − 2
(4.35)
D1+
D1+
We see that the form αθ∗ (Ξ) has (possible) singularities when R1 R2 = 0, i.e., at the (possible) points
zx,θ , zy,θ . An easy calculation shows that
zx,θ
x2,θ
= − tan θ ·
=
x1,θ
r
b
∈
a
R;
zy,θ
y2,θ
=
= cot θ ·
y1,θ
r
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
Z
D1+
αθ∗0 (Ξ) −
Z
D1+
αθ∗1 (Ξ).
For any > 0 small enough, let
• B1, be the (oriented) path {z = r exp(i) | r ∈ [0, 1)} from r = 0 to r = 1;
92
• B2, the path {z = r exp(i(π − )) | r ∈ [0, 1)} from r = 1 to r = 0; and
• D ⊂ D1+ the area containing points on or above the lines B1, and B2, .
By our assumption, αθ∗ (Ξ) is nonsingular on D for every θ ∈ [θ0 , θ1 ]. By Stokes’ Theorem and the
fact that exp(−2π(R1 + R2 )) decays rapidly as |z| goes to 1, we have
Z
α∗ (dΞ) =
[θ0 ,θ1 ]×D
Z
D
αθ∗1 (Ξ) −
Z
D
αθ∗0 (Ξ) +
Z
α∗ (Ξ).
(4.36)
[θ0 ,θ1 ]×(B1, ∪B2, )
Lemma 4.2.8. We have
Z
α∗ (dΞ) = 0.
[θ0 ,θ1 ]×D
Proof. By (4.31), (4.32) and (4.33), we have
dR1 = ∂R1 + ∂R1 − M + M dθ;
dR2 = ∂R2 + ∂R2 + M + M dθ;
√
z−z
z−z
+∂
.
d M − M = 2 ab ∂
1 − zz
1 − zz
Therefore,
α∗ d M − M ∧ dR1 ∧ dR2
√
z−z
z−z
= 2 ab M + M ∂
∧ ∂(R1 + R2 ) − ∂(R1 + R2 ) ∧ ∂
1 − zz
1 − zz
√
z−z
1
1
z−z
∧∂
−∂
∧∂
= 4 ab(a + b) M + M ∂
.
1 − zz
1 − zz
1 − zz
1 − zz
Then by Lemma 4.2.7 (2), we have
√
4i ab(a + b) exp(−2π(R1 + R2 ))
z−z
1
1
z−z
α (dΞ) =
∂
∧
∂
−
∂
∧
∂
2
π
1 − zz
1 − zz
1 − zz
1 − zz
M −M
√
4i ab(a + b) exp(−2π(R1 + R2 )) z + z
=
dz ∧ dz.
2
π
(1 − zz)3
M −M
∗
Therefore, z 7→ −z stabilizes the domain [θ0 , θ1 ] × D , and maps α∗ (dΞ) /dz ∧ dz to its negative, the
integral is zero.
93
By the above lemma and (4.36), we have
Z
D1+
αθ∗0 (Ξ) −
Z
αθ∗1 (Ξ) = lim
→0
D1+
Z
D
αθ∗0 (Ξ) − lim
→0
Z
D
αθ∗1 (Ξ) = lim
→0
Z
α∗ (Ξ). (4.37)
[θ0 ,θ1 ]×(B1, ∪B2, )
A simple computation shows that on [θ0 , θ1 ] × (B2, ∪ B1, ), we have
2
r
−i(a + b) exp(−2π(R1 + R2 )) M + M
dr ∧ dθ
α (Ξ) =
π
R1 R2
M − M (1 − r2 )2
−i(a + b) exp(−2π(R1 + R2 )) M − M
r
=
dr ∧ dθ
π
R1 R2
(1 − r2 )2
−4i(a + b) exp(−2π(R1 + R2 ))
r
+
dr ∧ dθ.
π
(1 − r2 )2
M −M
∗
Since the integrations of the second term on two paths cancel each other, we have
θ1
Z
exp(−2π(R1 + R2 )) M − M
−i(a + b)
r
lim
dr
→0
π
R
R
(1
−
r2 )2
1 2
θ0
B1, ∪B2,
√
Z θ1
Z
exp(−2π(R1 + R2 ))
4 ab(a + b)
r2
dθ
=
lim sin dr.
→0
π
R1 R2
(1 − r2 )3
B1, ∪B2,
θ0
Z
(4.37) =
dθ
(4.38)
To proceed, we need the following lemma.
Lemma 4.2.9. Let f (r) be a smooth function on [0, 1) that is rapidly decreasing as r → 1. Then for
any c1 , c2 , d1 , d2 > 0,
Z
lim
1
sin 2 r 2 + c2 − 2c c r cos ) (d2 r 2 + d2 + 2d d r cos ) f (r)dr
(c
1 2
1 2
0
1
2
1
2
πc1
f cc12
c1 > c2 ;
d +c d )2
→0+
=




c2 (c1
2
2 1


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
Z
lim
→0+
0
1
c21 r2
+
c22
π
sin dr =
.
− 2c1 c2 r cos c1 c2
(4.39)
94
The integral of the left-hand side of (4.39) (for small > 0) equals
Z
1
1
sin 2
(c1 r − c2 cos ) + c22 (1 − cos )
−c2 cos Z c1 √
c2 1−cos sin 1
c1 r − c2 cos √
√
=
d
2
cos c1 c2 1 − cos − √1−cos
c2 1 − cos c1 r−c
√ 2 cos +1
0
c2
sin √
=
c1 c2 1 − cos 1−cos cos c1 r − c2 cos + arctan √
arctan √
.
c2 1 − cos 1 − cos Let → 0+ , the limit is π/c1 c2 .
Applying the above lemma, we have, on the one hand, that
Z
θ1
√
ab(a + b)
(4.38) =
θ0
exp (−2πR2 (zx,θ )) x1,θ x2,θ
exp (−2πR1 (zy,θ )) y1,θ y2,θ
+
2
R1 (zy,θ )
d2,θ
R2 (zx,θ )
d21,θ
!
dθ.
(4.40)
On the other hand, we have
d
dR1 (zy,θ )
d
4(a + b)r
=
(R1 (zy,θ ) + R2 (zy,θ )) =
| y2,θ
dθ
dθ
(1 − r2 )2 r= y1,θ dθ
√
x1,θ x2,θ
dR2 (zx,θ )
= 2 ab(a + b) 2
.
dθ
d1,θ
y2,θ
y1,θ
√
y1,θ y2,θ
= 2 ab(a + b) 2
;
d2,θ
Therefore,
1
(4.40) =
2
=
Z
R1 (zy,θ1 )
R1 (zy,θ0 )
exp (−2πR1 (zy,θ ))
dR1 (zy,θ ) +
R1 (zy,θ )
Z
R2 (zx,θ1 )
R2 (zx,θ0 )
exp (−2πR2 (zx,θ ))
dR2 (zx,θ )
R2 (zx,θ )
!
1
(ξ(xθ1 , zy,θ1 ) + ξ(yθ1 , zx,θ1 ) − ξ(xθ0 , zy,θ0 ) − ξ(yθ0 , zx,θ0 )) ,
2
which, by (4.35), implies that
H(T [θ1 ])∞ − H(T [θ0 ])∞ = 0
(4.41)
for 0 < θ0 ≤ θ1 < π/2. Same argument works for intervals (π/2, π), (π, 3π/2) and (3π/2, 2π), other
than (0, π/2). The constancy of H(T [θ])∞ for all θ ∈
This finishes the proof of Proposition 4.2.4.
R then follows from (4.41) and the continuity.
95
4.3
An archimedean local Siegel–Weil formula
In this section, we set up a relation between derivatives of Whittaker integrals and the height functions
defined in the previous section. Furthermore, we prove a local arithmetic analogue of the Siegel–Weil
formula at an archimedean place for arbitrary dimensions.
4.3.1
Comparison on the hermitian domain
We propose and prove the following theorem, which we call the archimedean local arithmetic Siegel–
Weil formula.
C
Theorem 4.3.1. Let T ∈ Herm ( ) such that sign T = (m − 1, 1). Then we have
2
WT0 (0, e, Φ0 ) = γV
(2π)m
exp(−2π tr T )H(T )∞ ,
Γm (m)
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 ∈
p
p
2aj , . . . ) ∈ m ∼
= V 0 with the j-th entry 2aj and all others zero for
>0 . We let xj = (. . . ,
√
j = 1, . . . , m − 1, and xm = (0, . . . , 0, 2b). Then H(T )∞ = H((x1 , . . . , xm ))∞ . Since (xm , xm ) < 0,
R
C
we have Dxm = ∅, and
Z
ω(x1 ) ∧ · · · ∧ ω(xm−1 ) ∧ ξ(xm ).
H(T )∞ =
Dm−1
Proof of Theorem 4.3.1. By Proposition 4.1.8, we need to prove that
m−1
Z
ω(x1 ) ∧ · · · ∧ ω(xm−1 ) ∧ ξ(xm )
(2πi)
Dm−1
Z
exp (−4π (a1 w1 w1 + · · · + am−1 wm−1 wm−1 ))
=
Dm−1
X
(−4π)t (m − 1 − t)!(as1 · · · ast ) (1 + ws1 ws1 + · · · + wst wst )
1≤s1 <···<st ≤m−1
(− Ei) −4πb 1 + t ww
−m
(1 − zz)
Ω.
(4.42)
96
By definition and (4.24), we have
2aj zj zj
j = 1, . . . , m − 1;
= 2aj wj wj ,
1 − zz
−2b
Rm (z) : = R(xm , z) =
= −2b 1 + t ww .
1 − zz
Rj (z) : = R(xj , z) =
Therefore, ξ(xm ) = − Ei −4πb 1 + t ww . 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 = 2aj zj zj ,
∂ (1 − zz) Rj + (1 − zz) ∂Rj = 2aj zj dzj ;
∂Rj =
(4.43)
2aj zj dzj + Rj ∂ (zz)
.
1 − zz
(4.44)
Similarly,
∂Rj =
2aj zj dzj + 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 = 2aj dzj dzj ,
which implies that
Rj =
2aj (1 − zz) dzj dzj + 2aj zj dzj ∂ (zz) + 2aj zj ∂ (zz) dzj + 2Rj ∂ (zz) ∂ (zz) + Rj (1 − zz) ∂∂ (zz)
2
(1 − zz)
.
(4.46)
Taking wedge of (4.44) and (4.45), we have
∂Rj ∧ ∂Rj =
4a2j zj zj dzj dzj + 2aj Rj zj dzj ∂ (zz) + 2aj Rj zj ∂ (zz) dzj + Rj2 ∂ (zz) ∂ (zz)
(1 − zz)
2
.
(4.47)
Combining (4.46) and (4.47), we have
∂ (zz) ∂ (zz) ∂∂ (zz)
1
Rj ∂∂Rj − ∂Rj ∧ ∂Rj =
2 + 1 − zz .
Rj2
(1 − zz)
(4.48)
97
For simplicity, we make some substitutions. Let
ω = ∂ (zz) ∂ (zz) + (1 − zz) ∂∂ (zz) ;
ωj = (1 − zz) zj dzj dzj + zj dzj ∂ (zz) + zj ∂ (zz) dzj + wj wj ∂ (zz) ∂ (zz) ,
j = 1, · · · , m − 1.
Then we have
2
2πiω(xj ) = −∂∂ξ(xj ) = exp (−4πaj wj wj ) (ω − 4πaj ωj ) (1 − zz) .
Therefore, to prove (4.42), we only need to prove the following equality of (m − 1, m − 1)-forms on
Dm−1 ,
m−1
^
X
(ω − 4πaj ωj ) =
(−4π)t (m − 1 − t)!(as1 · · · ast ) (1 + ws1 ws1 + · · · + wst wst ) (1 − zz)
m−2
Ω,
s1 <···<st
j=1
which follows from the claim that for every subset {s1 < · · · < st } ⊂ {1, . . . , m − 1}, we have
ωs1 ∧ · · · ∧ ωst ∧ ω m−1−t = (m − 1 − t)! (1 + ws1 ws1 + · · · + wst wst )(1 − zz)
m−2
Ω.
This will be proved in the next lemma where, without lost of generality, we assume that sj = j. The
theorem then follows.
Lemma 4.3.2. Let wj , Ω, ω and ωj be as above. For any integer 0 ≤ t ≤ m−1, we have the following
equality of (m − 1, m − 1)-forms on Dm−1

t
^


ωj 

^
ω m−1−t = (m − 1 − t)! 1 +
j=1
t
X

m−2
wj wj  (1 − zz)
Ω.
j=1
The proof will occupy the next subsection.
4.3.2
Proof of Lemma 4.3.2
For j = 1, . . . , m − 1, we let
σj = zj dzj ∂ (zz) ;
σj0 = zj ∂ (zz) dzj ;
δj = (1 − zz) dzj dzj .
98
Then
m−1
X
k=1
σk =
m−1
X
σk0 ;
ω=
k=1
m−1
X
ωj = δj + σj + σj0 + wj wj
(σk + δk ) ;
m−1
X
σk ;
k=1
k=1
and
σj0 ∧ σj0 = 0;
σj ∧ σj = 0;
δj ∧ δj = 0.
Introduce the following (m − 1) × (m − 1) matrix

z1 z1
z2 z1
···
zm−1 z1
z1 z2
..
.
z2 z2
..
.
···
..
.
zm−1 z2
..
.
z1 zm−1
z2 zm−1
···
zm−1 zm−1




Z=








.



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, Z{j},{k} = zj zk
and Z∅,∅ = 1.
Consider three subsets I, J, K ⊂ {1, . . . , m − 1} with |I| + |J| + |K| = m − 1. Write
σI =
^
σi ;
σJ0 =
i∈I
^
j∈J
σj0 ;
δK =
^
δk .
k∈K
We have the following equality
σI σJ0 δK := σI ∧ σJ0 ∧ δK =




|K|
Ω
Z
Z
I,J,K
I,J∪K
I∪K,J (1 − zz)


0
(I ∪ J) ∩ K = ∅;
(I ∪ J) ∩ K 6= ∅.
Here, I,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,
σI σJ0 δK =


m−3


zi zi zj zj (1 − zz)
Ω
i 6= j, I = {i}, J = {j}, K = I ∪ J;







−zi zj zj zi (1 − zz)m−3 Ω i =
6 j, I = J = {i}, K = I ∪ {j};

m−2


zi zi (1 − zz)
Ω







(1 − zz)m−1 Ω
I ∪ J = {i}, K = {i};
I = J = ∅, K = {1, . . . , m − 1}.
99
For a subset P of {1, . . . , m}, we set wP =

t
^


ωj 
^
ω m−1−t =
j=1
t
^
Q
p∈P
Q
p∈P
wp . Then
!
^
m−1
X
wp and wP =
δj + σj + σj0 + wj wj
j=1
m−1
X
σk
k=1
=
L
`
`
M
N
`
m−1
X
0
δL σM σN
wP wP
X
Q⊂1,...,m−1
|Q|≤m−1−t
X
=
L
`
M
`
N
`
m−1
X
δk
k=1
!|P | 
σk

(m − 1 − t)!
(m − 1 − t − |Q|)!
X

!m−1−t−|Q|

δQ 

σk
k=1
TL,M,N,P,Q ,
(4.49)
P ={1,...,t} Q⊂1,...,m−1
|Q|≤m−1−t
where
TL,M,N,P,Q
!m−1−t
k=1
P ={1,...,t}




σk +
k=1

X
m−1
X
m−1
X
(m − 1 − t)!
0
δL∪Q σM σN
wP wP
=
(m − 1 − t − |Q|)!
!|P |+m−1−t−|Q|
σk
.
k=1
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 corresponding terms is
X
TL,M,N,P,Q = (m − 1 − t)!
t
X
m−3
zm zm zn zn (1 − zz)
Ω.
(4.50)
m,n=1
m6=n
Case I-1b: M ∪ N = {m} for 1 ≤ m ≤ t. Then the sum of corresponding terms is
X
TL,M,N,P,Q = 2(m − 1 − t)!
t
X
m−2
zm zm (1 − zz)
Ω.
(4.51)
m=1
Case I-1c: M = N = ∅. Then the corresponding term is
m−1
TL,M,N,P,Q = T{1,...,t},∅,∅,∅,{t+1,...,m−1} = (m − 1 − t)! (1 − zz)
Ω.
(4.52)
Case I-2: |P | = 1. Then M = N = ∅. Suppose P = {p} for 1 ≤ p ≤ t. Then Q =
{p, t+1, . . . , m−1}−{q} for some q inside {p, t+1, . . . , m−1}. The sum of the corresponding
100
terms is
X
TL,M,N,P,Q = (m − 1 − t)!
t
X
wp wp
zp zp +
m−1
X
!
m−2
(1 − zz)
zq zq
Ω.
(4.53)
q=t+1
p=1
Case II: |Q| = m − 2 − t. Then M = N = P = ∅ and |Q| = {t + 1, . . . , m − 1} − {q} for some q
inside {t + 1, . . . , m − 1}. The sum of the corresponding terms is
X
TL,M,N,P,Q = (m − 1 − t)!
m−1
X
zq zq (1 − zz)
m−2
Ω.
(4.54)
q=t+1
Taking the sum from (4.50) to (4.54), we have
m−2
Ω
(4.49) = (m − 1 − t)! (1 − zz)


! Ptm,n=1 z z z z
t
t
m−1
m−1
m
n
n
m
X
X
X
X

m6=n
wp wp zp zp +
zm zm +
zq zq +
+2
zq zq + (1 − zz)

1
−
zz
p=1
m=1
q=t+1
q=t+1
m−2
Ω
= (m − 1 − t)! (1 − zz)


Pt
Pt
Pm−1
z
z
z
z
+
z
z
z
+
z
z
z
m,n=1
t
m
m
n
n
p
p
p
q
p
q
p=1
q=t+1
X


m6=n
zm zm +
1 +

1
−
zz
m=1
Pt
zm zm
= (m − 1 − t)! (1 − zz)
Ω m=1
1 − zz


t
X
m−2
wj wj  (1 − zz)
= (m − 1 − t)! 1 +
Ω.
m−2
j=1
Lemma 4.3.2 is proved.
4.3.3
Comparison on Shimura varieties
We use previous results to compute the archimedean local height paring on the unitary Shimura
varieties with respect to suitable Green currents. We recall some notations from 3.1.1 and 3.1.2. Let
V over AE
of rank 2n, and H = ResA /A U(V). For (sufficiently small) open compact subgroup K of H(Afin ),
Q
we have the Shimura variety ShK := ShK (H). Suppose that K = p∈Σ Kp is decomposable. Let
n K
φα = φ0∞ ⊗ φα,fin (α = 1, 2) be decomposable Schwartz functions with φα,fin ∈ S(Vfin
) . Suppose that
φ1,p ⊗ φ2,p ∈ S(Vp2n )reg for some finite place p of F . Then the generating series Zφ (g1 ) and Zφ (g2 )
do not meet on ShK providing gα ∈ Pp0 H 0 (ApF ).
n ≥ 1 be an integer. We have a totally positive definition incoherent hermitian space
F
fin
1
2
101
Definition 4.3.3 (Volume of open compact). For every finite place p of F , we define a measure dh
H(Afin), which depends only on the additive character ψfin, as follows. By [Ral1987, Lemma 4.2],
there is a unique Haar measure d0 hp on U(Vp ) such that for every nonsingular matrix T ∈ Her2n (Ep )
such that OT is nonempty, and Φ ∈ S(Vp2n ),
on
WT (0, e, Φ) = γVp b2n,p (0)−1
Z
V
U(
0
Φ(h−1
v xT )d hv ,
p)
where xT is any element in OT , and b2n,p is defined in (2.6). By Lemma 2.3.6, for almost all p, the
volume of Kv with respect to d0 hp is 1. We define
dh =
where b2n =
Q
v∈Σ b2n,v
1
Y
2b2n (0)
d0 hp ,
p∈Σfin
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 ι0 ∈ {ι◦ , ι• }. Let V = V (ι) , H =
ResF/Q U(V ), and D = D(ι ) . Assume that there exists a finite place p of F such that φp (0) = 0.
0
Let Her+
n (E) be the subset of Hern (E) consisting of totally positive definite matrices. For every
ApF ), we have
n
such that T (x) = T . Then for g ∈ Pp0 H 0 (
T ∈ Her+
n (E), we choose an element xT ∈ V
the generating series
Zφ (g) =
X
X
T ∈Her+
n (E) h∈HxT (
A
fin )\H(
A
(ωχ (g)φ) (T, h−1 xT )Z(h−1 xT )K ,
fin )/K
where HxT ⊂ H is the stabilizer of xT . Recall that under the uniformization
Q
Afin)/K,
an
(ShK )ι0 ∼
= H( )\D × H(
the special cycle Z(h−1 xT )K is represented by the points (z, h0 h), where z ⊥ VxT and h0 fixes all
elements in VxT . In other words, if we identify D with D2n−1 , then z is in DxT . Choose a set of
Q
Afin)/K such that h1 is the identity element.
representatives h1 , . . . , hl of the double coset H( )\H(
0
Let [Zφ (g)ι0 ]an
h1 be the restriction of (the ι -analytification of) Zφ (g) to the neutral component. Then
it is the image of
X
(ωχ (g)φ) (T (x), x)Dx
x∈V n ,T (x)∈Her+
n (E)
Q
under the projection map D → (H( ) ∩ K)\D. Write gι = n(b)m(a)[k1 , k2 ] under the Iwasawa
102
decomposition as in 4.1.1. Then
X
Ξφ (g)ι0 ,h1 =
x∈V
(ωχ (g)φ) (T (x), x)Ξxa
(4.55)
n ,T (x)∈Her+ (E)
n
Q
descends to a current on (H( )∩K)\D, which is a Green current for [Zφ (g)ι0 ]an
h1 . By Hecke translation
under hi (i = 2, . . . , l), we obtain the a Green current Ξφ (g)ι0 for Zφ (g)ι0 . The following is our main
theorem of this chapter.
Theorem 4.3.4. Let ι, ι0 be as above. Let φα = φ0∞ ⊗ φα,fin (α = 1, 2) be decomposable Schwartz
n K
Vfin
) .
functions with φα,fin ∈ S(
Then for gα ∈ Pp0 H 0 (
Vp2n)reg for some finite place p of F .
Suppose that φ1,p ⊗ φ2,p ∈ S(
ApF ),
Eι (0, ı(g1 , g2∨ ), φ1 ⊗ φ2 ) = −2 Vol(K)h(Zφ1 (g1 ), Ξφ1 (g1 )ι0 ) , (Zφ2 (g2 ), Ξφ2 (g2 )ι0 )iShK ,
where Eι is defined as (2.14); and the right-hand side is the local height pairing on ShK at the place
ι0 .
Proof. It is clear that we can assume that gα,ι = m(aα ) with aα ∈ GLn (Eι0 ) for α = 1, 2. Then we
have
h(Zφ1 (g1 ), Ξφ1 (g1 )ι0 ) , (Zφ2 (g2 ), Ξφ2 (g2 )ι0 )iShK
l Z
X
=
Ξωχ (hi )φ1 (g1 )ι0 ,h1 ∗ Ξωχ (hi )φ2 (g2 )ι0 ,h1
Q
(H( )∩K)\D
i=1
=
X
Q

(ωχ (g1 )φ1 ) (T (x1 ), h−1
i x1 )Ξx1 a1

(H( )∩K)\D
i=1


l Z
X
x1 ∈V n ,T (x1 )∈Her+
n (E)

∗

X

(ωχ (g2 )φ2 ) (T (x2 ), h−1
i x2 )Ξx2 a2
x2 ∈V n ,T (x2 )∈Her+
n (E)
=
l
X
X
X
ωχ00 (ı(g1 , g2∨ )) (φ1 ⊗ φ2 ) (T ((x1 , x2 )), h−1
i (x1 , x2 ))
+
n
i=1 x1 ∈V n ,T (x1 )∈Her+
n (E) x2 ∈V ,T (x2 )∈Hern (E)
Z
D
Let
Ξx1 a1 ∗ Ξx2 a2
(4.56)


 a1
a=

.
a2
103
Then
(4.56) =
l
X
X
t
ωχ00 (ı(g1 , g2∨ )) (φ1 ⊗ φ2 ) (T (x), h−1
i x)H( aT (x)a)∞
i=1 x∈V 2n ,T (x)∈Her+ (E)
2n
=
X
Y
H(t aT a)∞
T
∨
ωχ00v ı(g1,v , g2,v
) Φ0v (T )
v∈Σ∞
Y
X
V
p∈Σfin hp ∈U(
∨
ωχ00p ı(g1,p , g2,p
) (φ1,p ⊗ φ2,p ) (h−1
p xT ),
(4.57)
p )/Kp
2n
where the sum is taken over all T ∈ Her+
. There
2n (E) that is the moment matrix of some xT ∈ V
are three cases.
Case I: v = ι. By (4.3) and Theorem 4.3.1 for t aT a, we have
−1 Γ2n (2n)
∨
∨
H(t aT a)∞ ωχ00ι ı(g1,ι , g2,ι
) Φ0ι (T ) = γV
W 0 (0, ı(g1,ι , g2,ι
), Φ0ι ).
ι
(2π)4n2 T
By (2.6), we have
−1
∨
∨
H(t aT a)∞ ωχ00ι ı(g1,ι , g2,ι
) Φ0ι (T ) = γV
b (0)WT0 (0, ı(g1,ι , g2,ι
), Φ0ι ).
ι 2n,ι
(4.58)
Case II: v ∈ Σ∞ and v 6= ι. By (4.3) and Proposition 4.1.5 (2), we have
−1
∨
∨
ωχ00v ı(g1,v , g2,v
) Φ0v (T ) = γV
b
(0)WT (0, ı(g1,v , g2,v
), Φ0v ).
v 2n,v
(4.59)
Case III: v ∈ Σfin . By Definition 4.3.3,
X
V
hp ∈U(
∨
ωχ00p ı(g1,p , g2,p
) (φ1,p ⊗ φ2,p ) (h−1
p xT )
p )/Kp
−1
= γV
b
(0)
p 2n,p
Z
!−1
d0 hp
∨
WT (0, ı(g1,p , g2,p
), φ1,p ⊗ φ2,p )
Kp
After plugging (4.58), (4.59) and (4.60) in (4.57), the theorem follows.
(4.60)
104
Chapter 5
Comparison at finite places: good
reduction
We compare local terms of analytic and arithmetic kernel functions at an unramified finite place. In
5.1, we introduce the integral models for the Shimura curves, and extend the special cycles and the
generating series to the models. At a good place, we need to calculate certain intersection multiplicity
on the smooth model. This is done in 5.2. The computation of the derivative of Whittaker integrals
at an unramified place and the comparison of the analytic and arithmetic side are the content of 5.3.
We fix some notations for this chapter, which may differ from the previous ones. Let F/
Qp be a
finite extension and E/F a quadratic extension of fields with Gal(E/F ) = {1, τ }. We fix a uniformizer
$ of F and let q be the cardinality of the residue field of F . Let V ± be the 2-dimensional E-hermitian
space with (V ± ) = ±1, which is unique up to isometry and H ± = U(V ± ). Let Λ± be a maximal
OE -lattice in V ± , on which the hermitian form takes values in OE . Let K0± be the stabilizer of Λ±
in H ± , which is a maximal compact subgroup. Recall that we have local groups H 0 = H1 , H 00 ∼
= H2 ,
P , etc.
Recall that we let ιi (i = 1, . . . , d) be all embeddings of F into
C and ι◦i , 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/Q
Gm,F and TF1
0
= ResF/Q F 0×,1 for any quadratic extension
105
F 0 /F , where F 0×,1 = ker (Nm : F 0× → F × ). If F is totally real, let F + be the set of all totally
positive elements.
• For every finite extension L/
Qp of local fields with ring of integers OL and maximal ideal q ⊂ OL,
s
0
let ULs be the subgroup of O×
L congruent to 1 modulo q . Denote by L the maximal unramified
extension of L. For s ≥ 0, let Ls be the finite extension of L0 corresponding to ULs through local
cs be the completion of Ls , whose ring of integers is denoted by
class field theory. Finally, let L
OLcs .
• We fix an algebraic closure of
Fp to be F.
106
5.1
Integral models
In the next four subsections, we will assume that F 6=
Q.
The rest case is slightly different and in
fact simpler, which will be summarized in the last subsection.
5.1.1
Change of Shimura data
Let p = p1 , p2 , . . . , pr (1 ≤ r ≤ d) be all places of F dividing p and p◦ one of E above p. We assume
that the embedding ι(p) ◦ ι◦1 : E ,→
Cp induces the place p◦. As before, we suppress ι◦1 and ι1 for the
nearby objects. We have the hermitian space V over E of dimension 2 whose signature is (1, 1) at
Q, and the Shimura curve MK = ShK (H, X) for a
sufficiently small open compact subgroup K ⊂ H(Afin ), which is a smooth projective curve defined
over ι◦1 (E). Recall that X is the conjugacy class of the Hodge map h : S → HR defined by
ι1 and (2, 0) elsewhere, the unitary group H over

 x
z = x + iy 7→ 
−y
R
−1

y 

 × z, 1, . . . , 1 ∈ H( ) ⊂ GL2 ( ) ×R×
x
R
R
C× × H× ×R C×d−1 ,
×
C×)d through (ι◦1 , . . . , ι◦d). We denote by ν : H → TE1 the determinant
where we identify TE ( ) with (
map. Then we have the 0-dimensional Shimura variety LK = Shν(K) (TE1 , ν(X)), and a smooth
morphism (also denoted by) ν : MK → LK of ι◦1 (E)-schemes such that the fiber of each geometric
point is connected.
Let us define a subgroup Kp,n of U(Vp ) for every integer n ≥ 0. Since Vp is (isometric to) either
V + or V − , we have the lattice Λ± (if p is split, we only have the positive one). We define Kp,n to be
the subgroup of K0± consisting of elements that have trivial action on Λ± /$n Λ± . Then Kp,0 = K0±
is a maximal compact subgroup. For K = Kp,n × K p , we write Mn,K p (resp. Ln,K p ) for MK (resp.
LK ).
Notation 5.1.1. For simplicity, we introduce the following notation
p
Hfin
= U(V ⊗F
ApF,fin).
p
Then K p is an open compact subgroup of Hfin
.
Since F 6=
Q, the Shimura datum (H, X) is not of PEL type.
We need to change Shimura data
in order to obtain the moduli interpretations and integral models. This is analogous to the case
considered in [Car1986, YZZa, Zha2001a, Zha2001b], and we refer the detailed proof of various facts
107
√
( λ) is split at p, and
Q
Q
to [Car1986]. We choose a negative number λ ∈
such that the extension
√
the CM extension F † = F ( λ)/F with Gal(F † /F ) = {1, τ † } is not isomorphic to E/F . We fix also a
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 λ0 (resp. −λ0 ). Since p
√
is split in Q( λ), pi for i = 1, . . . , r are all split in F † . We denote by p1i (resp. p2i ) the place above pi
square root λ0 of λ in
which sends
√
λ to λp (resp. −λp ), and assume that ι(p) ◦ ι11 induces p11 .
By the Hasse principle, there is a unique up to isometry quaternion algebra B over F , such that
B, as an F -quadratic space (of dimension 4), is isometric to V as an F -quadratic space with the
quadratic form TrE/F (−, −), where (−, −) is the hermitian form on V . More precisely, when v is
R
R
finite, Bv = B ⊗F Fv is division if and only if v is nonsplit and Vv ∼
= V − ; and Bι1 ( ) ∼
= Mat2 ( ),
R
Bιi ( ) ∼
=
H for i > 1. We identify two quadratic spaces B and V
through a fixed isometry and hence
V has both left and right multiplication by B. We fix an embedding E ,→ B, through which the
action of E induced from the left multiplication of B coincides with the E-vector space structure of
V . Let G = ResF/Q B × with center T ∼
= TF and
ν†
G† = G ×T TF † −→ T × TF1 † ,
where
†
z ν † (g × z) = Nm g · zz τ , τ † .
z
Consider the subtorus T † =
Hodge map h† :
Gm,Q × TF1 , and let H † be the preimage of T † under ν †.
†
Define the
S → HR† ⊂ GR ×TR TF ,R by
†

 x
z = x + iy 7→ 
−y
−1
y 

x


× 1, 12 × z −1 , . . . , 12 × z −1  ,
R
R
and let X † be the H † ( )-conjugacy class of h† , where we identify TF † ( ) with (
(5.1)
C×)d
through
†
†
†
1
†
(ι11 , . . . , ι1d ). We then have the Shimura curve MK
† = ShK † (H , X ) that is defined over ι1 (F ) for an
Afin). Similarly, we have the smooth morphism
open compact subgroup K † of H † (
†
†
ν † : MK
† → LK † .
Moreover, h† (i) defines a complex structure on Vι1 , and hence Vι1 becomes a complex hermitian space
of dimension 2 that is isometric to its original complex hermitian space structure inherited from the
108
E-hermitian space V . In such a way, X † can be identified with the set of negative definite complex
lines in Vι1 . Therefore, X † is isomorphic to X as hermitian symmetric domains.
As in [Car1986, Section 2.2], H † is a group of symplectic similitude. In fact, let B † = B ⊗F F † ,
and b 7→ b be the involution of the second kind on B † , which is the tensor product of the canonical
Q-vector space V † of B†. Define
involution on B and the conjugation on F † . Consider the underlying
a symplectic form by
√
ψ † (v, w) = TrF † /Q
λ TrB † /F † (vw)
for v, w ∈ B † . Then H † can be identified with the group of B † -linear symplectic similitude of
Q
(V † , ψ † ) through the left action hv = v · h−1 . In particular, H † ( p ) can be identified with the group
Qr
Qr
×
×
×
†,p
†,p
of H † ( pfin ), we simply
p × i=1 Bpi . For every open compact subgroups Kp of
i=2 Bpi , and K
Q
write M †
0,Kp†,p ,K †,p
†
†
for MK
=
† , where K
Z×p × O×B
p
A
× Kp†,p × K †,p , and similarly for L†
0,Kp†,p ,K †,p
.
We let
MK;p◦ = MK ×E Ep◦ ;
LK;p◦ = LK ×E Ep◦ ;
†
†
†
MK
† ;p = MK † ×F Fp1
1
L†K † ;p
=
L†K †
×F Fp†1 ,
1
where Fp†1 is naturally a subfield of Ep◦ , which is identified with Fp . Since H and H † have the same
1
derived subgroup, which is also the derived subgroup of G, we have the follow result of Carayol.
p
Proposition 5.1.2 (Section 4 of [Car1986]). Let K p ⊂ Hfin
(Notation 5.1.1) be an open compact
subgroup that is decomposable and sufficiently small. Then there is an open compact subgroup Kp†,p ×
Qr
†,◦
◦
K †,p ⊂ i=2 Bp×i × H † ( pfin ), such that the geometric neutral components M0,K
p ;p◦ and M
†,p
†,p
A
are defined and isomorphic over
5.1.2
0,Kp ,K
;p
Ep0◦ .
Moduli interpretations and integral models: minimal level
From the Hodge map h† (5.1), we have a Hodge filtration
0 ⊂ Fil0 (VC† ) = (VC† )0,−1 ⊂ VC† .
We define
t† (b) = tr(b; VC† / Fil0 (VC† )) ∈ ι11 (F † )
†
for b ∈ B † . For sufficiently small open compact subgroup K † , the curve MK
† represents the following
functor in the category of locally noetherian ι11 (F † )-schemes (cf. [Kot1992]): for every such scheme
109
†
S, MK
† (S) is the set of equivalence classes of quadruples (A, θ, i, η̄), where
• A is an abelian scheme over S of dimension 4d;
• θ : A → A∨ is a polarization;
• i : B † ,→ End0S (A) is a monomorphism of
Q-algebras, such that tr(i(b); LieS (A)) = t†(b) and
θ ◦ i(b) = i(b)∨ ◦ θ for all b ∈ B † ;
• η̄ is a K-level structure, that is, for chosen geometric point s on each connected component of
S, η̄ is a π1 (S, s)-invariant K † -orbit of B † ⊗
Hét
1 (As ,
Afin-linear symplectic similitude η : V † ⊗ Afin →
Afin), where the pairing on the latter space is the θ-Weil pairing.
Two quadruples (A, θ, i, η̄) and (A0 , θ0 , i0 , η̄ 0 ) are equivalent if there is an isogeny A → A0 that sends θ
to a
Q×-multiple of θ0, i to i0, and η̄ to η̄0.
†
1
† ∧ ∼
Taking the base change by ι(p) , we obtain the functor MK
† ;p over the completion [ι1 (F )]ι(p) = Fp .
†
†
†
For every element (A, θ, i, η̄) in MK
† ;p (S), LieS (A) is a Bp = B ⊗Q
Bp† = B ⊗F (F † ⊗
Qp) decomposes as
Qp-module.
Since the algebra
Bp† = B11 ⊕ B21 ⊕ · · · ⊕ Br1 ⊕ B12 ⊕ B22 ⊕ · · · ⊕ Br2 ,
(5.2)
where Bij = B † ⊗F Fp†j is isomorphic to Bpi as an Fpi -algebra, the Bp† -module LieS (A) decomposes as
i
LieS (A) =
r
M
!
LieS (A)1i
r
M M
i=1
!
LieS (A)2i
,
i=1
and
Ap ∞ =
r
M
(Ap∞ )1i
!
r
M M
(Ap∞ )2i
i=1
!
,
i=1
for the p-divisible group Ap∞ of A. Since the involution b 7→ b on Bp† interchanges the factors Bi1 and
Bi2 , by computing the trace, we see that the condition tr(i(b); LieS (A)) = t† (b) is equivalent to the
following:
tr(b ∈ B12 ; LieS (A)21 ) = TrB12 /Fp (b);
LieS (A)2i = 0, i ≥ 2.
(5.3)
Fix a maximal order Λ2i = OBpi of Bi2 for each i = 1, . . . , r, and let Λ1i be the dual of Λ2i . Then
Λp =
r
M
i=1
!
Λ1i
r
M M
i=1
!
Λ2i
⊂
r
M
i=1
!
(Vp† )1i
r
M M
i=1
!
(Vp† )2i
= Vp† := V † ⊗
Qp
110
Zp-lattice in Vp† that is selfdual under ψ†. There is a unique maximal Z(p)-order O† ⊂ B† such that
O† = O† , and O†p = OB that acts on Λ2i , where O†p is the Bi2 -component of O† ⊗Z Zp ⊂ B † ⊗Q Qp =
is a
2
i
Bp†
2
i
pi
under the decomposition (5.2). Then the functor
(p)
M † †,p †,p
0,Kp ,K ;p
is isomorphic to the following one
in the category of locally noetherian Fp -schemes: for every such scheme S, M †
0,Kp†,p ,K †,p ;p
(S) is the
set of equivalence classes of quintuples (A, θ, i, η̄ p , η̄pp ) where
• A is an abelian scheme over S of dimension 4d;
• θ : A → A∨ is a prime-to-p polarization;
Z(p) such that (5.3) is satisfied, and θ ◦ i(b) = i(b)∨ ◦ θ for all b ∈ O†;
• i : O† ,→ EndS (A) ⊗
• η̄ p is a K †,p -level structure, that is, a π1 (S, s)-invariant K †,p -orbit of B † ⊗
similitude η p : V † ⊗
Apfin → Hét1(As, Apfin);
Apfin-linear symplectic
• η̄pp is a Kp†,p -level structure, that is, a π1 (S, s)-invariant Kp†,p -orbit of isomorphisms of O† -modules
Lr
Lr
2
ηpp : i=2 Λ2i → i=2 Hét
1 (As , p )i .
Z
Two quintuples (A, θ, i, η̄ p , η̄pp ) and (A0 , θ0 , i0 , (η̄ p )0 , (η̄pp )0 ) are equivalent if there is a prime-to-p isogeny
A → A0 that carries θ to a
Z×(p)-multiple of θ0, i to i0, η̄p to (η̄p)0, and η̄pp to (η̄pp)0.
We will extend the previous moduli functor to the category of locally noetherian schemes over
Spec OFp to construct an integral model of M †
0,Kp†,p ,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∞ )2i are Cartier dual to each other. We replace (5.3) by the following condition:
tr(b ∈ OBp ⊂ B12 ; LieS (A)21 ) = TrB12 /Fp (b) ∈ OFp ;
LieS (A)2i = 0, i ≥ 2.
(5.4)
This means that the p-divisible group (Ap∞ )2i is ind-étale for i ≥ 2. We let Tp A = limn A[pn ] that is a
←−
†
2
pro-scheme over S. It has an action by O ⊗Z(p) p . Then Tp A(S)i is isomorphic to Λ2i as O† -modules
Z
if S is connected and simply-connected.
We define a functor M†
0,Kp†,p ,K †,p
such scheme S, M†
0,Kp†,p ,K †,p
in the category of locally noetherian schemes over OFp : for every
(S) is the set of equivalence classes of quintuple (A, θ, i, η̄ p , η̄pp ) where
• (A, θ, i) is as in the last moduli problem but satisfies (5.4);
• η̄ p K †,p -level structure;
• η̄pp is a π1 (S, s)-invariant Kp†,p -orbit of isomorphisms of O† -modules ηpp :
Lr
i=2
Λ2i →
Lr
i=2
Tp A(s)2i .
111
Two quintuples (A, θ, i, η̄ p , η̄pp ) and (A0 , θ0 , i0 , (η̄ p )0 , (η̄pp )0 ) are equivalent if there exists a prime-to-p
isogeny A → A0 satisfying the same requirements in the last moduli problem. For sufficiently small
Kp†,p × K †,p , this moduli functor is represented by a regular scheme denoted by M†
0,Kp†,p ,K †,p
, which
is flat and projective over Spec OFp . By Proposition 5.1.2, we obtain a regular scheme M0,K p that
is flat and projective over Spec OEp◦ whose generic fiber is (isomorphic to) M0,K p ;p◦ . Here, we also
need to use the fact that M†
0,Kp†,p ,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,Kp†,p ,K †,p
×OFp OE 0◦
p
and M0,K p ×OEp◦ OE 0◦ are isomorphic.
p
We denote by (A, θ, i) (that is a part of the datum of) the universal object over M†
0,Kp†,p ,K †,p
. We
also denote by X† = (Ap∞ )21 → M† †,p †,p the universal p-divisible group with the action by OBp
0,Kp ,K
Qr
×
and another action by i=2 Bpi × H † ( pfin ) that is compatible with the one on the underlying scheme
A
M†
0,Kp†,p ,K †,p
p
. We have also a p-divisible group X → M0,K p with an action by Hfin
that is compatible
with the one on M0,K p .
Remark 5.1.3. In fact, when p | 2 and Bp is division, the condition (5.4) is not enough. One needs
to impose that (Ap∞ )21 is special (cf. [BC1991, Section II.2]) for geometric points of characteristic p.
Let us consider the case where (Vp ) = 1, that is, Vp is isometric to V + ; Bp is isomorphic to
Mat2 (Fp ); or U(Vp ) is quasi-split. Before we proceed, we introduce some notations. Let R be a
(commutative) ring (with a unit) and M a (left) R-module (or p-divisible group according to the
context). Let m > 0 be an integer. We denote by M ] = M m (arranged in a column) as a left
Matm (R)-module in the natural way. Conversely, for any left Matm (R)-module N , we denote by
N [ = eN the (left) R-module, where e = diag[1, 0, . . . , 0] ∈ Matm (R), and the action is given by
r.(en) = (e × diag[r, . . . , r]).n for r ∈ R and n ∈ N . It is easy to see that the pair of adjoint
functors (−] , −[ ) induce an equivalence between the category of left R-modules and the category of
left Matm (R)-modules.
We identify Λ21 = OBp with Mat2 (OFp ), and hence O×
Bp with GL2 (OFp ), respectively. By the above
discussion, we can replace the first part of (5.4) by the following one:
tr(b ∈ OFp ; LieS (A)2,[
1 ) = b,
in the moduli problem M†
0,Kp†,p ,K †,p
(5.5)
.
Consider a geometric point s : Spec
F → M†0,K
†,p
†,p
p ,K
d
of characteristic p and let O
(s) be the
completion of the henselization of the local ring at s. By the theorem of Serre–Tate, it is the universal
112
deformation ring of (As , θs , is ) that is the isomorphic to the one of (As,p∞ , θs , is ). This is in turn
2,[
isomorphic to the deformation ring of the p-divisible group X†,[
s = (As,p∞ )1 , which is an OFp -module
d
of dimension 1 and height 2. Therefore, O
(s) is isomorphic to OF
c0 [[t]]. We have the following result
p
of Carayol [Car1986, Section 6].
Proposition 5.1.4. The scheme M†
0,Kp†,p ,K †,p
(resp. M0,K p ) is smooth and projective over OFp (resp.
OEp◦ ).
For a geometric point s of characteristic p of M†
0,Kp†,p ,K †,p
(resp. M0,K p ), 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,Kp†,p ,K †,p
M†
0,Kp†,p ,K †,p
5.1.3
]ss (resp. [M0,K p ]ss ) the supersingular locus of the scheme
(resp. M0,K p ).
Basic abelian scheme
In order to obtain the moduli interpretation of special cycles, we will construct construct a special
abelian scheme, which we name the basic abelian scheme. We fix an imaginary element µ in E, that
is, an element µ 6= 0 such that µτ = −µ. Since we are interested only in the place p, we identify the
following commutative diagram
ι◦1 (E)
O

/ ι(p) (ι◦ (E))
1
f
ι11 (F † ) ;
3S
/ ι(p) (ι1 (F † ))
8 1

=
? - 
ι1 (F ) / ι(p) (ι1 (F ))
with
EO 
/ Ep◦
†
F
?
`
0P
/ Fp
>

=
? / 
F
where the closure is taken inside
/ Fp ,
Cp .
Let E † = E ⊗F F † be a CM field of degree 4d that is a subalgebra of B † extending the fixed
embedding E ,→ B. The involution on B † induces e 7→ e that fixes the maximal totally real subfield
113
contained in E † . The maps
ι◦i ⊗ ιji : E ⊗F F † →
C ⊗ R C → C;
ι•i ⊗ ιji : E ⊗F F † →
for i = 1, . . . , d and j = 1, 2 provide 4d different embeddings of E † into
C ⊗R C → C
C, where C ⊗R C → C is the
usual multiplication. We choose a CM type
Φ = {ι◦1 ⊗ ι11 , ι◦1 ⊗ ι21 ; ι◦i ⊗ ι1i , ι•i ⊗ ι1i | i = 2, . . . , d}
of E † . Then Φ determines a Hodge map h‡ :
consisting of elements e such that ee ∈
S → TR‡ , where T ‡ is the subtorus of ResE /Q Gm,E
†
Gm,Q. We have the Shimura varieties
‡
MK
‡
†
= ShK ‡ (T ‡ , {h‡ })
that parameterize abelian varieties over E †,Φ with Complex Multiplication by E † of type Φ. It is
finite and projective over Spec E †,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
√
λ TrE † /F † (vw)
for v, w ∈ E † . Then T ‡ can be identified with the group of E † -linear symplectic similitude of (V ‡ , ψ ‡ ),
Qr
×
‡
and T ‡ ( p ) can be identified with ×
p ×
i=1 Epi . The Hodge map h induces a filtration 0 ⊂
Q
‡
Q
‡
Fil0 (VC ) ⊂ VC such that
t‡ (e) = tr(e; VC‡ / Fil0 (VC‡ )) =
X
ι(e)
ι∈Φ
for e ∈ E † . Since we have identified E (resp. F † ) with its embedding through ι◦1 (resp. ι11 ), we can
identify E † with its embedding through ι◦1 ⊗ ι11 , that is, with ι◦1 (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) × (x0 + y 0 λ0 ) be an element in E † with x, y, x0 , y 0 ∈ F . Then
t‡ (e) = (x + yµ)(2x0 ) +
d
X
2ιi (x)(ιi (x0 ) + ιi (y 0 )λ0 )
i=2
= 2 TrF/Q (xx0 ) + 2 TrF/Q (xy 0 )λ0 + 2yx0 µ − 2xy 0 λ0 .
Therefore, E †,Φ = E † .
114
As before, the algebra Ep† = E † ⊗Q
r
M
Ep† =
!
Ei1
Qp decomposes as
r
M M
i=1
!
Ei2
∼
=
i=1
r
M
!
Epi
r
M M
i=1
!
,
Epi
i=1
and also for its modules. Let π11 be the projection of Ep† to the first factor E11 . The additive map
t‡ extends to a map t‡p : Ep† → Ep† . From the calculation in the above lemma, we find that for
(eji ) = (e11 , . . . , e1r ; e21 , . . . , e2r ) ∈ Ep† ,
π11 ◦ t‡p ((eji )) =
r
X
TrEpi /Qp (e1i ) + e11 + e21 − TrEp1 /Fp1 (e11 ).
(5.6)
i=1
Z(p)-order in E † such that O‡ = O‡ and O‡p = OE is
is the projection of O‡ ⊗Z Zp to the Ei2 component. For any abelian
Let O‡ = E † ∩ O† be the unique maximal
the ring of integers, where O‡p2
i
2
i
pi
(p)
variety A over an Ep◦ -scheme S, which is equipped with an action by O‡ , LieS (A) is an Ep† -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 ⊂ E12 ; LieS (A)21 ) = ep◦ ∈ Ep◦ ;
LieS (A)2i = 0, i ≥ 2.
(5.7)
Let
K‡ =
Z×p ×
r
Y
‡,p
O×
Ep × K
i
i=1
‡
Afin), and we denote M00,K
be an open compact subgroup of T ‡ (
‡,p
‡
‡
= MK
‡ . Let M00,K ‡,p ;p◦ be its
base change under ι(p) ◦ (ι◦1 ⊗ ι11 ) : E † → Ep◦ . We fix a sufficiently small compact subgroup K ‡,p of
T ‡(
‡
Apfin), M00,K
‡,p ;p◦
represents the following functor in the category of locally noetherian schemes
‡
over Ep◦ : for every such scheme S, M00,K
‡,p ;p◦ (S) is the set of equivalence classes of quadruples
(A, ϑ, j, η̄ p ), 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‡;
• η̄ p is a K ‡,p -level structure, that is, a π1 (S, s)-invariant K ‡,p -orbit of E † ⊗
similitude η p : V ‡ ⊗
Apfin → Hét1(As, Apfin).
Apfin-linear symplectic
The notion of being equivalent is similarly defined as before. Moreover, we can extend this modular
115
functor to the category of locally noetherian schemes over OEp◦ . We omit the detailed definition.
One can similarly prove that the extended moduli functor, denoted by M‡00,K ‡,p , is connected, and
finite, projective, smooth over Spec OEp◦ . Therefore, it is isomorphic to Spec OE \ for some finite
unramified extension of local fields E \ /Ep◦ . We fix an embedding ι\ : E \ ,→ Ep0◦ . Let (E, ϑ, j) be the
universal object over M‡00,K ‡,p ×OE\ ,ι\ OE 0◦ ∼
= Spec OEp0◦ , and denote Y = (Ep∞ )21 . Fix a geometric
p
point s : OE 0◦ ,→
p
Bet
C of characteristic 0 and an O‡-generator x of HBet
is the
1 (Es , Z(p) ), where H•
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 Spec OE 0◦ is simply
A
p
fin -sheaf
p
A
p
fin )
connected, x extends to a unique section x of the lisse
over Spec OE 0◦ , and
p
L
r
p
2
determines canonically an element xp of i=2 (Tp E)i . For any scheme S over Spec OE 0◦ , we denote
p
Hét
1 (E,
p
by
xpS
and
xpp,S
for the corresponding base change respectively.
F
Let (E, ϑE , jE ) be the special fiber of (E, ϑ, j), where E is an abelian variety over Spec . Let
Y = (Ep∞ )21 and iY : OEp◦ → End(Y) be the induced OEp◦ -action. Therefore, Y is the special fiber
F
of Y, which is an OFp -module over Spec .
5.1.4
The nearby space
In this and the next subsections, we assume that p is nonsplit in E, and fix an
supersingular locus of the common neutral component of
M† †,p †,p
0,Kp ,K
F-point s in the
and M0,K p , which corresponds
to a quintuple (A, θA , iA , η̄ p , η̄pp ). Let
V̆ † = 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 (x̆, y̆) to
−1
∨
(x̆, y̆)0 := jE
◦ ϑ−1
E ◦ y̆ ◦ θA ◦ x̆
induces a E † -hermitian form on V̆ † . If we let (A0 , θA0 , iA0 ) be the isogeny class of (A, θA , iA ), then
B̆ † = End(A0 , iA0 ) is a quaternion algebra over F † . Moreover, the underlying F † -quadratic spaces
of V̆ † and B̆ † are isometric. The hermitian form (−, −)0 induces a
Q-symplectic form on V̆ †.
If we
let H̆ † be the corresponding group of symplectic similitude, then Aut(A0 , θA0 , iA0 ) can be identified
1 Here
(A, iA ) really means (A, iA | O‡ ), and we apply the similar convention in what follows.
116
Q
Q
†
with H̆ † ( ). We fix an E-subspace V̆ of V̆ † of dimension 2 that is stable under the action of H̆der
( )
and such that the restricted hermitian form (−, −)0 |V̆ takes value in E. Then V̆ † = V̆ ⊗E E † , and V̆v
is isometric to Vv exactly for v away from {ι1 , p}. Let H̆ = ResF/Q U(V̆ ). We fix an isometry
γ p = (γpp , γ p ) : V̆ ⊗F
ApF,fin → V ⊗F ApF,fin
such that
%∗ (xp ) ∈ η̄ p (γ p (%))
%∗ (xpp ) ∈ η̄pp (γpp (%));
p
p
for every element % ∈ V̆ . In particular, γ p identifies H̆fin
with Hfin
.
5.1.5
Integral special subschemes: minimal level
For every admissible x ∈ V ⊗
Afin,E , we have the subscheme Z(x)K on MK that is a special cycle
introduced in 3.1.2. For K = Kp,0 K p , let us consider the curve M0,K p ;p◦ , which is the base change
MK ×E Ep◦ , and its subscheme Z(x)0,K p ;p◦ , which is the corresponding base change of Z(x)K .
Afin), such that
Consider x† ∈ V † and h† ∈ H † (
• the (Vp† )21 -component of h†,−1 x† is inside Λ21 ; and
Afin,E (inside V † ⊗ Afin) contains a K-orbit that has totally positive definite
×
†,p
†,p
Here, K † = Z×
and K = Kp,0 K p for K p as in Proposition
p × OB × Kp × K
• K † h†,−1 x† ∩ V ⊗
norm in E.
p
5.1.2.
We define a functor Z† (x† , h† )00,K †,p ,K †,p in the category of locally noetherian schemes over OE 0◦ as
p
p
follows: for every such scheme S, Z† (x† , h† )00,K †,p ,K †,p (S) is the set of equivalence classes of sextuples
p
(A, θ, i, η̄ p , η̄pp , %A ) where
• (A, θ, i, η̄ p , η̄pp ) is an element of M†
0,Kp†,p ,K †,p
(S);
• %A : E ×Spec OE0 S → A is a quasi-homomorphism that satisfies the following conditions:
p◦
1. For any e ∈ O‡ , the following diagram commutes:
E ×Spec OE0 S
%A
/A
%A
/ A;
p◦
j(e)
iA (e)
E ×Spec OE0 S
p◦
117
2. %A induces a homomorphism from Y ×Spec OE0 S to (Ap∞ )21 ;
p◦
3. For the geometric point s defining the K †,p -level structure, the map ρAs ,∗ : Hét
1 (Es ,
Hét
1 (As ,
Apfin) sends xps into η̄p(h†,−1x†);
4. The map ρAs ,∗ :
Lr
2
i=2 (Tp Es )i
→
Lr
2
i=2 (Tp As )i
Apfin) →
⊗OFp Fpi sends xpp,s into η̄pp (h†,−1 x† ).
i
The equivalence relations are defined in the similar way. The evident morphism
Z† (x† , h† )00,K †,p ,K †,p → M†
0,Kp†,p ,K †,p
p
×OEp◦ OE 0◦
p
is finite and its image is a 1-dimensional closed subscheme that is stable under the action of the
Galois group Gal(OE 0◦ /OEp◦ ). Therefore, Z† (x† , h† )00,K †,p ,K †,p defines a 1-dimensional closed subp
p
scheme Z (x , h )0,Kp†,p ,K †,p of M†
†
†
†
0,Kp†,p ,K
, which only depends on the orbit K † h†,−1 x† . Moreover,
†,p
†,◦
◦
by definition, the intersection Z (x† , h† )0,Kp†,p ,K †,p ∩ M0,K
p ;p◦ inside M
†
0,Kp†,p ,K †,p
coincides with the
special cycle Z(x)◦0,K p ;p◦ if x ∈ K † h†,−1 x† has totally positive definite norm in E. Conversely, for
every admissible x ∈ V ⊗
AF,fin with xp ∈ Λp, we have a 1-dimensional closed subscheme Z(x)0,K
M0,K p obtained by the Hecke translation of Z† (x† , h† )0,Kp†,p ,K †,p ∩ M†,◦
0,Kp†,p ,K †,p
p
of
, whose generic fiber is
Z(x)0,K p ;p◦ . It only depends on the orbits Kp,0 K p x.
From now on, we assume that (p is nonsplit in E) and (Vp ) = 1. Following [Car1986, 11], we have
the following isomorphisms of sets:
F
[M† †,p †,p ]ss (
0,Kp ,K
F
Q
)∼
= H̆ † ( )\
Z×
r
Y
Bp×i /Kp†,p
†
× H̆ (
A
p
†,p
fin )/K
!
;
i=2
Q
p
[M0,K p ]ss ( ) ∼
= H̆( )\H̆fin /K p ,
such that in both cases, the neutral double coset corresponds to the point s. Moreover, we have the
following lemma.
Lemma 5.1.6. The special fiber [Z† (x† , h† )0,Kp†,p ,K †,p ]sp (resp. [Z(x)0,K p ]sp ) locates in the supersingular locus [M†
0,Kp†,p ,K †,p
]ss (resp. [M0,K p ]ss ).
Proof. We only need to prove for [Z† (x† , h† )0,Kp†,p ,K †,p ]sp . Let s = (A, θ, i, η̄ p , η̄pp , %A ) be an
F-point
of Z† (x† , h† )0,Kp†,p ,K †,p . We have a nontrivial homomorphism between OFp -modules %A,∗ : Y →
⊕2
⊕2
(Ap∞ )21 = X†s ∼
. Therefore, there is at least one projection (X†,[
→ X†,[
= (X†,[
s )
s )
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 =
118
Q
We briefly explain the constructions in the above four subsections in the case F =
Q.
Therefore,
Q is a imaginary quadratic extension. Let ι◦, ι• be two different embeddings of E into C such that
ι(p) ◦ ι◦1 : E ,→ Cp induces the place p◦ . We identify E with a subfield of C via the embedding ι◦ . If
E/
p is split in E, we denote p• the other place of E above p. We have the hermitian space V over E of
dimension 2 and signature (1, 1), the unitary group H over
Q. For a sufficiently small open compact
Afin), the Shimura curve ShK (H, X) is a smooth and quasi-projective curve defined
subgroup K ⊂ H(
over ι◦ (E), and is proper if and only if V is anisotropic.
By the Hasse principle, there is a unique up to isometry quaternion algebra B over
B, as an
Q-quadratic space (of dimension 4), is isometric to V
Q, such that
as an F -quadratic space with the
quadratic form TrE/Q (−, −), where (−, −) is the hermitian form on V . We identify two quadratic
spaces B and V through a fixed isometry and hence V has both left and right multiplication by B.
We fix an embedding E ,→ B, through which the action of E induced from the left multiplication of
B coincides with the E-vector space structure of V . We let H † = B × , which is different from the
F 6= Q. Then similarly, we have the Shimura curve ShK (H †, X †) defined over Q. We can view
as a symplectic space over Q and H † the group of E-linear symplectic similitude.
case
V
†
Let
Sh(H)n,K p ;p◦ = ShKp,n K p (H, X) ×E Ep◦ ;
†
†
Sh(H † )n,K †,p ;p = ShKp,n
†
K †,p (H , X ) ×Q
Qp.
We have the following proposition that is parallel to Proposition 5.1.2.
p
Proposition 5.1.7. Let K p ⊂ Hfin
be an open compact subgroup that is decomposable and sufficiently
Apfin), such that the geometric neutral
small. Then there is an open compact subgroup K †,p ⊂ H † (
components Sh(H)◦0,K p ;p◦ and Sh(H † )◦0,K †,p ;p are defined and isomorphic over Ep0◦ .
We are going to define a functor M†0,K †,p in the category of locally noetherian schemes over
Zp .
There are two case: the anisotropic case, i.e., B is division, and the isotropic case, i.e., B is isomorphic
to the matrix algebra. In the anisotropic case, for every
Zp-scheme S, define M†0,K
†,p
(S) to be the set
of equivalence classes of quadruples (A, θ, i, η̄ p ) where
• A is an abelian surface over S;
• θ : A → A∨ is a prime-to-p polarization;
• ι : OB ,→ EndS (A) of a monomorphism of rings such that det(ι(b); LieS (A)) = Nm b;
119
• η̄ p is a K †,p -level structure, that is, a π1 (S, s)-invariant K †,p -orbit of B ⊗
similitude η p : V ⊗
Apfin → Hét1(As, Apfin).
In the isotropic case, for every
Zp-scheme S, define M†0,K
†,p
Apfin-linear symplectic
(S) to be the set of equivalence classes of
pairs (A[ , η̄ [,p ) where A[ is a generalized elliptic curve over S, and η̄ [,p is a K †,p -level structure (cf. [KM1985]). 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,K
p
that
is smooth and projective over Spec OEp◦ whose generic fiber is (the Baily–Borel compactification of)
Sh(H)0,K p ;p◦ . By construction, the neutral components of M†0,K †,p ×Zp OE 0◦ and M0,K p ×OEp◦ OE 0◦
p
p
[ ]
p
[,p ]
[
are isomorphic. In the isotropic case, we will denote A = A , η̄ = η̄
when A is an elliptic
Z
curve. Moreover, we have the canonical polarization θ : A → A∨ and the OB = Mat2 ( )-action
i : OB → EndS (A).
When F =
Q, in the basic unitary datum (E, ϑ, j; x), E is an elliptic curve over Spec OE
0
p◦
with the
principal polarization ϑ and the OE -action j : OE → EndOE0 (E). Let Y = (Ep∞ ). Let (E, ϑE , jE ) be
p◦
F
the special fiber of (E, ϑ, j), where E is an elliptic curve over Spec . 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,K p of M0,K p , whose generic fiber is
Z(x)0,K p ;p◦ . It only depends on the orbits Kp,0 K p x. In the isotropic case, Z(x)0,K p is disjoint from
the set of cusps. If we further assume that (Vp ) = 1, then we have the following isomorphism of sets:
F
Q
p
[M0,K p ]ss ( ) ∼
= H̆( )\H̆fin /K p ,
such that the neutral double coset corresponds to the point s. Moreover, we have that the special
fiber [Z(x)0,K p ]sp locates in the supersingular locus [M0,K p ]ss .
5.2
Local intersection numbers
In this section, we study the formal scheme N and its special formal subschemes Z (x̆). Therefore, p
will be a finite place of F that is nonsplit in E and such that (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
†,p
†,p
p ,K
and M0,K p ,
which corresponds to a quintuple (A, θA , iA , η̄ p , η̄pp ). Let X = (Ap∞ )21 and iX : Mat2 (OFp ) → End(X)
120
be the induced Mat2 (OFp )-action. Then X[ is a formal OFp -module of dimension 1 and height 2.
We define a functor N in the category of schemes over Spec OFp0 where $ is locally nilpotent: for
every such scheme S, N (S) is the set of equivalence classes of pairs (G, ρG ) where
• G is an OFp -module over S of dimension 1 and height 2;
• ρG : X[ ×Spec F Ssp → G ×S Ssp is a quasi-isogeny of height 0 (which is in fact an isomorphism).
F
Here, Ssp = S ×Spec O d0 Spec .
Fp
Two pairs (G, ρG ) and (G0 , ρG0 ) are equivalent if there is an isomorphism G0 → G sending ρG to
ρG0 . Then N is represented by the formal scheme N , which is isomorphic to Spf RFp ,2 , where
RFp ,2 = OFc0 [[t]]. Let N 0 = N ×O d0 OEd
0 .
Fp
p
p◦
By the theorem of Serre–Tate, the formal completion of M†
ically isomorphic to N (resp. N 0 ). Therefore, if we denote
(resp. M0,K p ) at s is canon-
0,Kp†,p ,K †,p
by [M† †,p †,p ]∧
ss
0,Kp ,K
(resp. [M0,K p ]∧
ss ) the
formal completion along the supersingular locus, then we have the following p-adic uniformization:
[M† †,p †,p ]∧
ss
0,Kp ,K
×OFp OFc0
p
Q
∼
= H̆ † ( )\ N ×
Z×
r
Y
Bp×i /Kp†,p
†
A
× H̆ (
!
p
†,p
fin )/K
;
i=2
Q
p
∼
[M0,K p ]∧
= H̆( )\N 0 × H̆fin /K p .
d
0
ss ×OEp◦ OE
p◦
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 V̆ † and a E-subvector space V̆ . 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 V̆p ∼
= V − . For every
x̆ ∈ Mor ((Y, jY ), (X, iX ))reg := Mor ((Y, jY ), (X, iX )) − {0},
we define a subfunctor Z (x̆) of N as follows: for every scheme S in the previously mentioned
category, Z (x̆)(S) is the set of equivalence classes of (G, ρG ) ∈ N (S) such that the following composed
homomorphism
F
ρ]G
x̆
Y ×Spec O d0 Spec
×Spec F Ssp = Y ×Spec F Ssp −
→ X ×Spec F Ssp −−→
G] ×S Ssp
Fp
121
extends to a homomorphism Y ×Spec O d0 S → G] . Then Z (x̆) is represented by a closed formal
Fp
subscheme Z (x̆) of N . In fact, one can show that it is a relative divisor of N by the same argument
in [KR2011, Proposition 3.5]. We will use the same notation for the base change of Z (x̆) in N 0 .
Let φ = φ0∞ ⊗ (⊗v∈Σfin φv ) such that
ApF,fin))K
• φpfin := ⊗v∈Σfin −{p} φv is in S(V (
p
;
• φp = φ0p is the characteristic function of the a the self-dual lattice Λ+ of Vp ;
• φ(0) = 0.
AF ) such that gp ∈ n(bp)K0p for some unipotent element n(bp) ∈ N 0(Fp). Consider
Let g = (gv ) ∈ H 0 (
the generating series
X
Zφ (g) =
A
x∈K\V (
(ωχ (g)φ) (x)Z(x)K
F,fin )−{0}
X
=
A
x∈K\V (
ψp (bp T (x)) φ0p ⊗ (ωχ (g p )φp ) (x)Z(x)Kp,0 K p .
F,fin )−{0}
Define
X
Zφ (g)0,K p =
A
x∈K\V (
ψp (bp T (x)) φ0p ⊗ (ωχ (g p )φp ) (x)Z(x)0,K p ,
F,fin )−{0}
whose generic fiber is the base change of Zφ (g) to Ep◦ . Moreover, its completion at the
[Zφ (g)0,K p ]∧
s =
X
ψp (bp T (x̆)) φ̆0p ⊗ (ωχ (g p )φp ) (x̆)Z (x̆).
F-point s is
(5.8)
x̆∈V̆
Here, φ̆0p is the characteristic function of Mor ((Y, jY ), (X, iX )), and in particular, the notation Z (x̆)
makes sense since φ(0) = 0.
5.2.3
A formula for local intersection multiplicity
In this and the next subsections, we will assume that p is not divided by 2 and is inert in E. For every
2
pair (x̆1 , x̆2 ) ∈ Mor ((Y, jY ), (X, iX ))reg that are linearly independent, the formal divisors Z (x̆1 ) and
Z (x̆2 ) intersects properly at the unique closed point of N . We would like to calculate the intersection
number Z (x̆1 ) · Z (x̆2 ). Assuming that (y̆1 , y̆2 ) = (x̆1 , x̆2 )g for some g ∈ GL2 (OEp◦ ) such that (y̆1 , y̆2 )
122
has the moment matrix


 $
T ((y̆1 , y̆2 )) = 
a
$
b


with a nonnegative even and b positive odd. If we denote by Def(X[ , (x̆1 , x̆2 )) the subring of Def(X[ ) =
OFc0 [[t]] where (x̆1 , x̆2 ) deforms, then
p
Z (x̆1 ) · Z (x̆2 ) = lengthO d0 Def(X[ , (x̆1 , x̆2 )) = lengthO d0 Def(X[ , (y̆1 , y̆2 )) = Z (y̆1 ) · Z (y̆2 ).
Fp
Fp
Therefore, we only need to study the intersection number Z (y̆1 ) · Z (y̆2 ).
Let Yτ be the unique (up to isomorphism) formal OFp -module of dimension 1 and height 2 with
an OEp◦ action jYτ , such that
• As OFp -modules, there is an isomorphism Y ∼
= Yτ (and we fix such an isomorphism); and
jY
τ
• jYτ is given by the composition OEp◦ −
→ OEp◦ −−→ 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−1
X ◦ y̆α =



incα ◦ Πa
α=1


incα ◦ Πb
α = 2,
where
• incα (α = 1, 2) denotes the inclusion of Y into the α-th component of Y × Yτ ∼
= Y × Y; and
• Π is a fixed uniformizer of the division algebra End(Y).
In what follows, we will identify X with Y × Yτ via the isomorphism ρX .
For an integer s ≥ 0, let Fs be a quasi-canonical lifting of level s, which is an OFp -module over
Spf OFcs , unique up to the Galois action (cf. [Gro1986]). Therefore, it defines a morphism Spf OFcs →
p
p
N that is a closed immersion. Let Zs be the divisor of N defined by the image, which is independent
of Fs we choose. We have the following proposition generalizing [KR2011, Proposition 8.1] from
to Fp .
Qp
123
Proposition 5.2.1. As divisors on N , we have
Z (y̆1 ) =
a
X
Zs ;
Z (y̆2 ) =
s=0
even
b
X
Zs .
s=1
odd
Proof. The original proof of [KR2011, Proposition 8.1] works again for one direction. Namely,
a
X
b
X
Zs ≤ Z (y̆1 );
s=0
even
Zs ≤ Z (y̆2 ).
s=1
odd
To prove the other direction, we need only to prove that the intersection multiplicities of both sides
F
in two cases with the special fiber Nsp = Spf [[t]] are the same. For the left-hand side, we have
a
X
Zs · Nsp =
s=0
even
b
X
a
X
s
[O×
Fp : UFp ] =
s=0
even
Zs · Nsp =
s=1
odd
q a+1 − 1
;
q−1
b
X
q b+1 − 1
s
,
[O×
:
U
]
=
Fp
Fp
q−1
s=1
odd
where q is the cardinality of the residue field of Fp . Then the assertion follows from the following
proposition that generalizes [KR2011, Proposition 8.2].
Proposition 5.2.2. For y̆ ∈ HomOEp◦ (Y, Y × Yτ ), the intersection multiplicity
Z (y̆) · Nsp =
q v+1 − 1
,
q−1
where v ≥ 0 is the valuation of (y̆, y̆)0 , i.e., (y̆, y̆)0 ∈ $v O×
Fp .
We keep the assumptions and notations in the above subsection. The results in [ARG2007] cited
in the proof of [KR2011, Proposition 8.4] also work for Fp , not just
we have
Z (y̆1 ) · Zs =
Qp. Therefor, for 0 < s ≤ b odd,


a+1

 q −1
a < s;
q−1

s

s
 q −1 + 1 (a + 1 − s)[O×
Fp : UFp ] a ≥ s.
q−1
2
Summing over s, we get the following local arithmetic Siegel-Weil formula at a good finite place.
2
Theorem 5.2.3. For every pair (x̆1 , x̆2 ) ∈ Mor ((Y, jY ), (X, iX ))reg that are linearly independent,
the intersection multiplicity Z (x̆1 ) · Z (x̆2 ) depends only on the GL2 (OEp◦ )-equivalence class of the
124
moment matrix T = T ((x̆1 , x̆2 )). Moreover, if

 $
T ∼

a
$b
0 ≤ a < b,


then we have
a
1X l
Hp (T ) := Z (x̆1 ) · Z (x̆2 ) =
q (a + b + 1 − 2l),
2
l=0
where q is the cardinality of the residue field of Fp .
5.2.4
Proof of Proposition 5.2.2
We generalize the proof of [KR2011, Proposition 8.2] to the case Fp 6=
Qp, still by using the theory of
windows and displays of p-divisible groups developed by T. Zink in [Zin2001, Zin2002]. In the proof,
we simply write F = Fp , E = Ep◦ . Moreover, we let e and f be the ramification index and the
extension degree of residue fields of F/
F
Qp, respectively.
In particular, q = pf . Let R =
F[[t]] and
A = W [[t]], where W = W ( ) is the Witt ring. We extend the Frobenius automorphism σ on W to
A by letting σ(t) = tp . For any s ≥ 1, we let Rs = R/ts and As = A/ts . Then A (resp. As ) is a
frame of R (resp. Rs ). The category of formal p-divisible groups over R is equivalent to the category
of pairs (M, α) where
• M is a free A-module of finite rank; and
• α : M → M (σ) := Aσ ⊗A M is an A-linear injective homomorphism such that coker α is a free
R-module.
The baby case would be the one where f = 1. Consider the p-divisible group Y over
F of dimension
1 and (absolute) height 2e with action by OE . It corresponds to the pair (N, β), where
N = N0 ⊕ N1 = OFc0 n0 ⊕ OFc0 n1
is the
Z/2-graded free OFc
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 OFc0 OF -linearly.
Similarly as in the proof of [KR2011, Proposition 8.2], the p-divisible group X = Y × Yτ over
F
corresponds to (M, α) described there, and its universal deformation is (M, αt ). The only difference
is that we should replace p by $. The rest of the proof follows in the same way.
125
Now we treat the general case and hence assume that f ≥ 2. Consider the p-divisible group Y
over
F. It corresponds to the pair (N, β), where N is a Z/2-graded free OF ⊗Z
p
Since
OF ⊗Zp W =
f −1
M
j=0
(σ j )
O c0 :=
F
f −1
M
W -module of rank 2.
OF ⊗W (k),σj W,
j=0
where k is the residue field of F , we can write




f −1
f −1
M
M
M
j
j
(σ )
(σ )

N =
O c0 e0,j 
O c0 e1,j  ,
j=0
F
j=0
F
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 (N τ , β τ ), which we write as




f −1
f −1
M
M
M
j
j
(σ )
(σ )

O c0 eτ1,j  ,
Nτ = 
O c0 eτ0,j 
j=0
F
j=0
F
and
• β τ (eτi,j ) = eτi,j+1 for i = 0, 1, 0 6 j < f − 1;
• β τ (eτ1,f −1 ) = eτ0,0 ;
• β τ (eτ0,f −1 ) = $eτ1,0 .
We extend (N, β) (resp. (N τ , β τ )) 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) ⊕ (N τ , β τ ). Under the
basis
{e0,0 , eτ1,0 , . . . , e0,f −1 , eτ1,f −1 ; e1,0 , eτ0,0 , . . . , e1,f −1 , eτ0,f −1 },
126
the matrix of α is


$






 1




α=










..


$ 








.










.
1
1
..
.
1
Let (M, αt ) corresponds to the universal deformation of (X, iX ) over
F[[t]].
basis,


 1










αt = 










t
−t
1
1
..
.
1
1
..
.
1











 · α.










Explicitly, we have
αt (ei,j ) = ei,j+1
i = 0, 1 and j = 0, . . . , f − 2;
αt (e0,f −1 ) = e1,0 − teτ1,0 ;
αt (e1,f −1 ) = $e0,0 ;
αt (eτi,j ) = eτi,j+1
i = 0, 1 and j = 0, . . . , f − 2;
αt (eτ1,f −1 ) = eτ0,0 + te0,0 ;
αt (eτ0,f −1 ) = $eτ1,0 .
Then under the same
127
If we denote by σ k (α) : M (σ
k
)
→ M (σ
k+1
)
the induced homomorphism for k > 0. Then formally, we
have
σ k (α)−1 (ei,j ) = ei,j−1
i = 1, 2 and j = 1, . . . , f − 1;
1
e1,f −1 ;
$
k
tp τ
k
−1
σ (α) (e1,0 ) = e0,f −1 +
e
;
$ 0,f −1
σ k (α)−1 (e0,0 ) =
σ k (α)−1 (eτi,j ) = eτi,j−1
i = 1, 2 and j = 1, . . . , f − 1;
1 τ
e
;
$ 0,f −1
k
tp
k
−1 τ
τ
e1,f −1 .
σ (α) (e0,0 ) = e1,f −1 −
$
σ k (α)−1 (eτ1,0 ) =
Now let y̆ correspond to the graded A1 -linear homomorphism γ : N ⊗A A1 → M . Then the length
Z (y̆) · Nsp of the deformation space of γ is the maximal number a such that there exists a diagram
β
N
/ N (σ)
γ̃
M
αt
γ̃ (σ)
/ M (σ) ,
which commutes modulo ta , and γ̃ lifts γ.
Case i: v = 2r is even. We may assume that γ = $r inc1 that is represented by the following
4f × 2f matrix










X(0) = 








$r
0
···
0
0
···
0
$r
···
0
..
.
0
..
.
···
..
.
0
0
···
0
0
···
0



0 


0 



0 .

.. 
. 



$r 

0
If r = 0, in order to lift γ to γ̃ mod tp , we search for a 4f × 2f matrix X(1) with entries in Ap
such that X(1) ≡ X(0) in A1 and satisfies
αt ◦ X(1) = σ(X(1)) ◦ β.
128
But σ(X(1)) = σ(X(0)) = X(0). Therefore, we need to find the largest integer a ≤ p such that
αt−1 ◦ X(0) ◦ β has integral entries
mod ta . Since the entry at the place (e0,f −1 , eτ0,f −1 ) is
t
$,
the
largest a is just 1. It follows that when v = r = 0, the proposition holds.
2r
If r > 0, we first show that we can lift γ to γ̃ mod tq . By induction, we introduce X(k) for
k ≥ 1, i.e., the one satisfying X(k + 1) ≡ X(k) in Apk , and
αt ◦ X(k + 1) = σ(X(k + 1)) ◦ β.
Since σ(X(k + 1)) = σ(X(k)), we formally have
X(k + 1) = αt−1 ◦ σ(X(k)) ◦ β.
We need to show that
X(2rf ) = αt−1 ◦ σ(αt )−1 ◦ · · · ◦ σ 2rf −1 (αt )−1 ◦ X(0) ◦ β 2rf
has integral entries. Let xi,j;i0 ,j 0 (resp. xτi,j;i0 ,j 0 ) be the entry of X(2rf ) mod $ at the place (ei,j , ei0 ,j 0 )
(resp. (ei,j , eτi0 ,j 0 )). Then among all these terms, the only nonzero terms are
xτ0,j;0,j = (−1)r−1 tp
x1,j;1,j = (−1)r tp
f −1−j
f −1−j
(q 2r−2 +q 2r−3 +···+1)
(q 2r−1 +q 2r−2 +···+1)
j = 0, . . . , f − 1;
j = 0, . . . , f − 1,
2r
2r
which implies that we can lift γ to γ̃ mod tq . Next, we consider the lift of γ to γ̃ mod tpq .
Therefore, we consider the matrix
X(2rf + 1) = αt−1 ◦ 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
f −1
2r−1
(−1)r q2r +q2r−1 +···+1
t
+q 2r−2 +···+1)
(−1)r tp·p (q
=
t
.
$
$
It turns out that the length Z (y̆) · Nsp is exactly
q 2r+1 −1
q−1
=
q v+1 −1
q−1 .
Case ii: v = 2r + 1 is odd. We may assume that γ = $s inc2 ◦ Π, where Π is the endomorphism
of Y determined by Π(e0,j ) = e0,j and Π(e1,j ) = $e0,j for j = 0, . . . , f − 1. Then γ is represented by
129
the following 4f × 2f matrix

















 0

 s
 $

 .
 ..



 0

0
···
0
···
..
.
0
..
.
···
0
···
$s
0
···
0
$s+1
..
.
···
..
.
0
..
.
0
···
0
0
···
$s+1















.













Similarly, we introduce the matrix Y (k) for k ≥ 0. We first show that γ can be lifted to γ̃
mod tq
2s+1
, i.e., the matrix
Y ((2s + 1)f ) = αt−1 ◦ σ(αt )−1 ◦ · · · ◦ σ (2s+1)f −1 (αt )−1 ◦ Y (0) ◦ β (2s+1)f
τ
has integral entries. Let yi,j;i0 ,j 0 (resp. yi,j;i
0 ,j 0 ) be the entry of Y ((2s + 1)f ) mod $ at the place
(ei,j , ei0 ,j 0 ) (resp. (ei,j , eτi0 ,j 0 )). Then among all these terms, the only nonzero terms are
f −1−j
τ
y0,j;0,j
= (−1)s tp
y1,j;1,j = (−1)s+1 tp
(q 2s−1 +q 2s−2 +···+1)
f −1−j
(q 2s +q 2s−1 +···+1)
which implies that we can lift γ to γ̃ mod tq
2s+1
j = 0, . . . , f − 1;
j = 0, . . . , f − 1,
. Next we consider the lift of γ to γ̃ mod tpq
2s+1
Therefore, we consider the matrix
Y ((2s + 1)f + 1) = αt−1 ◦ 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
f −1
2s
2s−1
t
(−1)s+1 q2s+1 +q2s +···+1
+···+1)
(−1)s+1 tp·p (q +q
=
t
.
$
$
.
130
It turns out that the length Z (y̆) · Nsp is exactly
q
2s+2
−1
q−1
=
q
v+1
−1
q−1 .
Therefore, the proposition is
proved.
5.3
5.3.1
Comparison
Non-archimedean Whittaker integrals
We calculate certain Whittaker integrals WT (s, g, Φ) and their derivatives (at s = 0) at a nonarchimedean place when T is of rank 2.
Assume that E/F is unramified and p > 2. We fix a selfdual OE -lattice Λ+ in V + and let
φ0+ ∈ S(V + ) (resp. Φ0+ ∈ S((V + )2 )) be the characteristic function of Λ+ (resp. (Λ+ )2 ). Let ψ
be the unramified character of F . For T ∈ Her2 (E) that is nonsingular, we consider the Whittaker
integral
WT (s, g, Φ0+ ) =
Z
ϕΦ0+ ,s (wn(u)g)ψT (n(u))−1 du
(5.9)
Her2 (E)
for Re s > 1, where du is the selfdual measure with respect to ψ. Write g = n(b)m(a)k under the
Iwasawa decomposition of H 00 . Then
Z
(5.9) =
ω100 (wn(u)n(b)m(a)k)Φ0+ (0)λP (wn(u)n(b)m(a)k)s ψ(− tr T u)du
Her2 (E)
Z
λP (wn(u)m(a))s ψ(− tr T u)du
= ψ(tr T b)
Her2 (E)
0+
= ψ(tr T b)| det a|1−s
).
E Wt aτ T a (s, e, Φ
Therefore, we only need to consider the integral WT (s, e, Φ0+ ).
If T is not in Her2 (OE ), then
WT (s, e, Φ0+ ) is identically 0. For T ∈ Her2 (OE ), it is well-known (e.g., [Kud1997, Appendix]) that
for an integer r > 1, WT (r, e, Φ0+ ) = γV + αF (12+r , T ), where γV + is the Weil constant and αF is the
classical representation density (for hermitian matrices). By [Hir1999], we see that for r ≥ 0,
αF (12+r , T ) = PF (12 , T ; (−q)−r )
for a polynomial PF (12 , T ; X) ∈
Q[X]. By analytic continuation, we see that
WT (s, e, Φ0+ ) = γV + PF (12 , T ; (−q)−s ).
131
If ord(det T ) 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
WT0 (0, e, Φ0+ ) = −γV + log q ·
d
PF (12 , T ; X) |X=1 .
dX
Moreover, we have the following results.
Proposition 5.3.1 (Hironaka, [Hir1999]). Suppose that T is GL2 (OE )-equivalent to diag[$a , $b ] with
0 ≤ a < b. Then
PF (12 , T ; X) = (1 + q
−1
X)(1 − q
−2
X)
a
X
l
(qX)
l=0
a+b−2l
X
!
k
(−X)
.
k=0
Corollary 5.3.2. If a + b is odd, then
a
WT0 (0, e, Φ0+ ) = γV + b2 (0)−1 log q ·
1X l
q (a + b − 2l + 1).
2
l=0
5.3.2
Comparison on Shimura curves
bφ (g1 ), Zbφ (g2 )ip◦ at a finite place p◦ of E
In this subsection, we calculate the local height paring hZ
1
2
that is good. Recall that we have a (compactified) Shimura curve MK constructed from the hermitian
space
V, and assume that K is sufficiently small and decomposable. We also assume that φα (α = 1, 2)
are decomposable and K = Kp K 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;
• (
Vp) = 1;
• φα,p = φ0p (α = 1, 2) are the characteristic function of a selfdual lattice Λp = Λ+ ⊂
Vp;
Vp) stabilizing Λp, i.e., Kp is a hyperspecial maximal compact
• Kp = Kp,0 is the subgroup of U(
subgroup;
• χ and ψ are both unramified at p.
We say a finite place p◦ of E is good if it is not lying over some place in S. Assume that φα (0) = 0.
Consider the generating series Zφα (gα ) for α = 1, 2. Write gα,p = n(bα,p )m(aα,p )kα,p in the Iwasawa
×
−1
decomposition, and choose some element eα ∈ E × such that e−1
α aα,p ∈ OEp . Let g̃α = m(eα )gα , and
132
we have g̃α,p = n(b̃α,p )kα,p in the Iwasawa decomposition. Then Zφα (gα ) = Zφα (g̃α ). As in 5.2.2, we
have the series Zφα (g̃α )0,K p for α = 1, 2. We let Zφα (gα ) = Zφα (g̃α )0,K p . The following is the main
theorem of this chapter.
Theorem 5.3.3. Suppose that φ1,v ⊗φ2,v ∈ S(
Vv2 )reg for at least one place v ∈ S, and gα ∈ Pv0 H 0(AvF )
for α = 1, 2. Let p be a finite place that is not in S, MK;p◦ = M0,K p the smooth local model introduced
in 5.1.2, and Zφα (gα ) the series introduced above. Then we have
1. If p is nonsplit in E, then
Ep (0, ı(g1 , g2∨ ), φ1 ⊗ φ2 ) = − Vol(K)hZbφ1 (g1 ), Zbφ2 (g2 )ip◦ ,
where
• by definition,
bφ (g1 ), Zbφ (g2 )ip◦ = log q 2 (Zφ (g1 ) · Zφ (g2 ));
hZ
1
2
1
2
• Ep (0, ı(g1 , g2∨ ), φ1 ⊗ φ2 ) is defined in (2.14); and
• Vol(K) is the defined in Definition 4.3.3.
2. if p is split in E, then
Zφ1 (g1 ) · Zφ2 (g2 ) = 0.
Combining this with Theorem 4.3.4, we have the following corollary.
Corollary 5.3.4. Assume that
• φα = φ0∞ φα,fin (α = 1, 2) are decomposable as above;
VS2)reg , i.e., φ1,v ⊗ φ2,v ∈ S(Vv2 )reg for every v ∈ S;
• φ1,S ⊗ φ2,S is in S(
Vv2 )reg,d
• φ1,v ⊗ φ2,v ∈ S(
v
for every v ∈ S that is nonsplit and some dv ≥ dψv (cf. 2.4.2);
ASF ) (α = 1, 2);
• gα ∈ eS H 0 (
• the local model MK;p◦ is M0,K p for all finite places p◦ above p that are not in S.
Then
E 0 (0, ı(g1 , g2∨ ), φ1 ⊗ φ2 ) = − Vol(K)
X
v ◦ |v
v6∈S
bφ (g2 )iv◦ ,
hZbφ1 (g1 ), Z
2
133
where the Green functions used in archimedean places are those defined in (4.55), which are not the
admissible Green functions in the sense in 3.4.1.
1. By Lemma 5.1.6, the special fiber of Zφα (gα ) locates in the supersin-
Proof of Theorem 5.3.3.
gular locus [M0,K p ]ss . If we denote by [Zφα (gα )]∧
sp (α = 1, 2) the completion along the special
fiber, then
∧
∧
∧
Zφ1 (g1 ) · Zφ2 (g2 ) = [Zφ1 (g1 )]∧
sp · [Zφ2 (g2 )]sp = [Zφ1 (g̃1 )0,K p ]sp · [Zφ2 (g̃2 )0,K p ]sp .
F
(5.10)
Q
p
Let h̆i (i = 1, . . . , l) be a set of representatives of the double coset [M0,K p ]ss ( ) ∼
= H̆( )\H̆fin /K p .
Then
(5.10) =
l
X
∧
[Zωχ (h̆i )φ1 (g̃1 )0,K p ]∧
s · [Zωχ (h̆i )φ2 (g̃2 )0,K p ]s ,
(5.11)
i=1
where we recall that ωχ (h̆i )φα (x) = φα (h̆−1
i x). By (5.8),
(5.11) =
l
X


X

i=1
ψp (b̃1,p T (x̆1 )) φ̆0p ⊗ ωχ (g̃1p , h̆i )φp1
(x̆1 )Z (x̆1 )
x̆1 ∈V̆

X
·

ψp (b̃2,p T (x̆2 )) φ̆0p ⊗ ωχ (g̃2p , h̆i )φp2 (x̆2 )Z (x̆2 )
x̆2 ∈V̆
X
=
ψp (tr b̃p T ((x̆1 , x̆2 )))
(x̆1 ,x̆2 )∈(V̆ ∩Mor((Y,jY ),(X,iX )))
l
X
2
ωχ00 ı(g̃1p , g̃2p,∨ ) φp1 ⊗ φp2 (h̆−1
i (x̆1 , x̆2 ))Z (x̆1 ) · Z (x̆2 ),
(5.12)
i=1
where


 b̃1,p
b̃p = 
0
0
b̃2,p

.
Let x̆T be a representative of the pairs (x̆1 , x̆2 ) such that T ((x̆1 , x̆2 )) = T . Then
(5.12) =
X
T ∈Her2 (Ep◦ )∩GL2 (OEp◦ )
ψp (tr b̃p T )
X
ωχ00 ı(g̃1p , g̃2p,∨ )
φp1 ⊗ φp2
(h̆−1 x̆T )Hp (T ).
p
h̆∈H̆fin
/K p
By Theorem 5.2.3, Corollary 5.3.2, and following the same steps in the proof of Theorem 4.3.4,
134
we obtain that
− Vol(K) log q 2 (Zφ1 (g1 ) · Zφ2 (g2 )) = Ep (0, ı(g̃1 , g̃2∨ ), φ1 ⊗ φ2 ).
Let


 e1
e=

.
e2
(5.13)
By definition,
(5.13) =
X
V
∨
WT0 (0, ı(g̃1,p , g̃2,p
), φ1,p ⊗ φ2,p )
=
V
=
V
Y
∨
We0 τ T e (0, ı(g1,p , g2,p
), φ1,p ⊗ φ2,p )
∨
Weτ T e (0, ı(g1,v , g2,v
), φ1,v ⊗ φ2,v )
v6=p
Diff(T, )={p}
X
∨
WT (0, ı(g̃1,v , g̃2,v
), φ1,v ⊗ φ2,v )
v6=p
Diff(T, )={p}
X
Y
∨
WT0 (0, ı(g1,p , g2,p
), φ1,p ⊗ φ2,p )
Diff(T, )={p}
Y
∨
WT (0, ı(g1,v , g2,v
), φ1,v ⊗ φ2,v )
v6=p
= Ep (0, ı(g1 , g2∨ ), φ1 ⊗ φ2 ).
Therefore, (1) is proved.
2. It will be proved in a more general context in Lemma 6.1.1.
135
Chapter 6
Comparison at finite places: bad
reduction
bφ (g2 )ip◦ on certain model MK at every
In this chapter, we discuss the local height paring hZbφ1 (g1 ), Z
2
bad place p ∈ S. We will assume that
• φα = φ0∞ φα,fin (α = 1, 2) are decomposable;
• φα,S ∈ S(
VS)reg (α = 1, 2);
• φ1,S ⊗ φ2,S is in S(
VS2)reg ;
ASF ) for α = 1, 2.
and gα ∈ eS H 0 (
In 6.1, we discuss the contribution of the local height paring at a finite place p in S that is split in
E. In 6.2, we discuss the contribution of the local height paring at a finite place p in S that is nonsplit
in E and such that (Vp ) = 1. In 6.3, we discuss the contribution of the local height paring at a finite
place p in S that is nonsplit in E and such that (Vp ) = −1.
136
6.1
Split case
In this section, we discuss the contribution of the local height paring at a finite place p in S that is
split in E. Let p◦ be a place in Σ◦fin lying over p.
6.1.1
Integral models and ordinary reduction
Afin) with K p sufficiently small and Kp = Kp,n
Let K = Kp K p be an open compact subgroup of H(
for n ≥ 0. Therefore, MK = Mn,K p . In 5.1.2, we construct a smooth integral model M0,K p for
M0,K p ;p◦ on which there is a p-divisible group X. Then X[ → M0,K p is an OFp -module of dimension
1 and height 2. Recall that a Drinfeld $n -structure for an OFp -module X of dimension 1 and height
2 over an OFp -scheme S is an OFp -homomorphism
αn : (OFp /$n OFp )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,K p =
M0,K p (n) be the universal scheme over M0,K p of the Drinfeld $n -structure (cf. [HT2001, Lemma
0
II.2.1]). Then Mn,K p is regular and finite over M0,K p , whose generic fiber is Mn,K p . Let Mn,K
p =
Mn,K p ×Fp Fpn be the base change, and M0n,K p the normalization of Mn,K p ×OFp OFpn that is regular.
We denote by [M0n,K p ]ord the ordinary locus, which is an open subscheme of the special fiber [M0n,K p ]sp
and also the smooth locus.
The set of connected components of [M0n,K p ]sp corresponds canonically to the set of geometric
AE×,1/ν(K).
connected components of Mn,K p , hence to E ×,1 \
The set of irreducible components on
each connected component of [M0n,K p ]sp , that is, the Igusa curves, corresponds to the set
P(Vp)/Kp,n.
P(Vp) is the set of all rank 1 Ep ∼= Fp ⊕ Fp-submodules in Vp, where U(Vp) acts from right by
l.h = h−1 l for l ∈ P(Vp ) and h ∈ U(Vp ). Together, the set of irreducible components of [M0n,K ]sp can
Here,
p
be identified with
Ign,K p =
P(Vp)/Kp,n ×
AE×,1/ν(K)
E ×,1 \
.
We consider special cycles. We keep the same notations for the base change of special cycles Z(x)K
0
and the generating series Zφα (gα ) on Mn,K
p . We let Z(x)K (resp. Zφα (gα )) be the Zariski closure of
Z(x)K (resp. Zφα (gα )) in M0n,K p . Since p is split in E, the special fiber [Zφα (gα )]sp is contained in
the ordinary locus [M0n,K p ]ord . Let
P(V )+ be the set of totally positive definite E-lines in V .
Then
137
0
the set of geometric special points of Mn,K p (also of Mn,K p and Mn,K
p ) can be identified with
Q P
Afin)/K =
a
SpK = H( )\ (V )+ × H(
Q P
Q
Afin)/K,
Hl ( )\H(
l∈H( )\ (V )+
F
and the set [M0n,K p ]ord ( ) can be identified with
Q
a
Q P
p
Hl ( )\ (Nl \ U(Vp )/Kp,n ) × Hfin
/K p ,
l∈H( )\ (V )
where Nl ⊂ U(Vp ) is the unipotent radical of the parabolic subgroup stabilizing l. The reduction map
F
SpK → [M0n,K p ]ord ( ) → Ign,K p
(6.1)
is given by
p
(l, h) 7→ (l, hp , hp ) 7→ (h−1
p l, ν(hp h ))
(cf. [Zha2001b, 5.4] for a discussion).
6.1.2
Coherence for intersection numbers
bφ = Zφ (gα ) + Vφ (gα )
We compute the local height paring on the integral model M0n,K p . Write Z
α
α
α
for some divisor Vφα (gα ) supported on the special fiber as in 3.4.3. We have
bφ (g2 )ip◦
(log q)−1 hZbφ1 (g1 ), Z
2
= (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.
Vv2 )reg
Lemma 6.1.1. Under the weaker hypotheses that φ1,v ⊗ φ2,v ∈ S(
= S(Vv2 )reg and gα ∈
AvF ) (α = 1, 2) for some finite place v other than p, Zφ (g1) and Zφ (g2) do not intersect.
Pv0 H 0 (
1
2
Proof. It follows immediately from the first map in (6.1).
Second, we define a function ν(•, φ2 , g2 ) on Vp − {0} in the following way. For any nonzero x ∈ Vp ,
let lx be the line spanned by x that is an element in
P(Vp). Set ν(x, φ2, g2) to be the coefficient of the
138
geometric irreducible component represented by (lx , 1) in Ign,K p in Vφ2 (g2 ). It is a locally constant
function and
ν(•, φ1,p ; φ2 , g2 ) =
Vol(det K)
φ1,p ⊗ ν(•, φ2 , g2 )
Vol(K)
extends to a function in S(Vp ) such that ν(0, φ1,p ; φ2 , g2 ) = 0 since φ1,p (0) = 0. Then the intersection
number
Zφ1 (g1 ) · Vφ2 (g2 ) =
X
A
x∈K\V ⊗
=
(ωχ (g1 )φ1 ) (x)Z(x)K · Vφ2 (g2 )
(6.3)
fin
X
A
x∈K\V ⊗ fin
T (x)∈F +
Vol(K)
Vol(K ∩ H(
Afin)x)
ν(•, φ1,p ; φ2 , g2 ) ⊗ ωχ (g1p )φp1
(x)
(6.4)
ApF ). If we let
since g1 ∈ ep H 0 (
X
E(s, g, ν(•, φ1,p ; φ2 , g2 ) ⊗ φp1 ) =
ωχ (γg) ν(•, φ1,p ; φ2 , g2 ) ⊗ φp1
1
(0)λP 0 (γg)s− 2
γ∈P 0 (F )\H 0 (F )
be an Eisenstein series that is holomorphic at s = 12 . Then we have
1
(6.3) = E(s, g1 , ν(•, φ1,p ; φ2 , g2 ) ⊗ φp1 ) |s= 12 −W0 ( , g1 , ν(•, φ1,p ; φ2 , g2 ) ⊗ φp1 )
2
by the standard Siegel–Weil argument, which is similar to Proposition 3.4.1. For simplicity, we write
1
E(p◦ ) (φ1 , g1 ; φ2 , g2 ) = log q E(s, g1 , ν(•, φ1,p ; φ2 , g2 ) ⊗ φp1 ) |s= 12 −W0 ( , g1 , ν(•, φ1,p ; φ2 , g2 ) ⊗ φp1 ) .
2
Finally, we let
A(p◦ ) (g1 , φ1 ) = log q (Vφ1 (g1 ) · ωK ) .
Then we have the following proposition.
Proposition 6.1.2. Suppose that φα,S ∈ S(
eS H 0 (
ASF ) for α = 1, 2. Then
VS)reg (α = 1, 2), φ1,S ⊗ φ2,S is in S(VS2)reg , and gα ∈
bφ (g2 )ip◦ = E(p◦ ) (φ1 , g1 ; φ2 , g2 ) + A(p◦ ) (g1 , φ1 )E(g2 , φ2 ).
hZbφ1 (g1 ), Z
2
139
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 is
nonsplit in E and such that (Vp ) = 1. Let p◦ be the only place in Σ◦fin lying over p. We identify Vp
with Mat2 (Fp ) and the lattice Λ+ with Mat2 (OFp ).
6.2.1
Integral models and supersingular reduction
Afin) with K p sufficiently small and Kp = Kp,n for
Let K = Kp K p be an open compact subgroup of H(
n large such that Fpn contains Ep◦ . Therefore, MK = Mn,K p . In 5.1.2, we construct a smooth integral
model M0,K p for M0,K p ;p◦ on which there is a p-divisible group X. Let Mn,K p be the normalization
0
of M0,K p in Mn,K p , which is regular and finite over M0,K p . Consider the base change Mn,K
p =
Mn,K p ×Ep◦ Fpn . Let M0n,K p be the normalization of Mn,K p ×OEp◦ OFpn that is a regular mode of
0
Mn,K
p . We have the following description of the supersingular locus
F
Q
×,1
p
[M0n,K p ]ss ( ) ∼
= H̆( )\ Ep◦ /ν(Kp,n ) × H̆fin /K p ,
Q
where H̆( ) 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 Spec RFp ,2,n = (Spec RFp ,2 )(n) is the universal scheme of the Drinfeld $n structure for Xuniv,[ that is in fact defined over Spec RFp ,2 . Let Spec Rn0 be the normalization of
n . The set of connected components of Spec R0 is
Spec RFp ,2 ⊗O d0 OFcn in Spec RFp ,2,n ⊗O d0 Fc
p
n
Fp
p
Fp
†
0
0
parameterized by coset O×
Fp /ν(Kp,n ). Let Nn be the neutral connected component of Spf Rn , which
0
n
is Galois over Nη0 = Spf RFp ,2 ⊗O d0 Fc
is finite over N 0 ×O d
OFcn . Its generic fiber Nn,η
p with the
0
E ◦
p
p
Fp
Galois group
∧2
K p,n := ker GL(Λ+,[ /$n Λ+,[ ) −−→ GL(OFp /$n OFp ) ,
which fits into the following exact sequence
1
/ K p,n
/ Kp,0 /Kp,n
ν
/ E ×,1
p◦ /ν(Kp,n )
/1.
140
We define the universal p-divisible group X0n over Nn0 by the following Cartesian diagram
X0n
/ Nn0
Xuniv ×O d0 OFcn
/ N ×O O cn .
d
F
0
Fp
p
p
Fp
Moreover, we have the universal Drinfeld $n -structure
0
n
0
αn,η
: Λ+,[ /$n Λ+,[ → X0[
n,η [$ ](Nn,η )
0
for X0[
n,η . In particular, we have the following p-adic uniformization for the completion of Mn,K p along
the supersingular locus
Q
×,1
p
0
p
∼
.
[M0n,K p ]∧
ss = H̆( )\ Nn × Ep◦ /ν(Kp,n ) × H̆fin /K
In 5.1.5, we have defined a 1-dimensional closed subscheme Z(x0 )0,K p of M0,K p whose generic fiber
is Z(x0 )0,K p ;p◦ , for a Kp,0 K p -orbit Kp,0 K p x0 in V (
AF,fin) that is admissible.
Define Z0 (x0 )n,K p by
the following Cartesian diagram
Z0 (x0 )n,K p
/ M0
n,K p
Mn,K p ×OEp◦ OFpn
Z(x0 )0,K p ×OEp◦ OFpn
/ M0,K p ×O
OFpn .
Ep◦
It is easy to see that the Kp,n K p -orbits inside Kp,0 K p x0 is parameterized by the finite group K p,n . For
every such orbit Kp,n K p x, we define Z(x)n,K p to be the union of irreducible components of Z0 (x0 )n,K p
0
whose generic fiber contributes to the special divisor Z(x)n,K p ×Ep◦ Fpn on the generic fiber Mn,K
p.
In 5.2.2, we have defined Z (x̆) that is a formal divisor of N , for x̆ ∈ Mor ((Y, jY ), (X, iX ))reg .
141
Define Z 0 (x̆)n by the following Cartesian diagram
Z 0 (x̆)n
/ Nn0
Z (x̆) ×O d0 OFcn
/ N ×O O cn .
d
F
0
p
Fp
Fp
p
b the completion of Y along the special fiber, we have a universal homomorphism
If we denote by Y
b ×Spf O Z 0 (x̆)n → X0 | Z 0 (x̆)n
[x̆] : Y
n
d
◦
F
p
of p-divisible groups over the formal scheme Z 0 (x̆)n . Pick up any geometric point z of characteristic
0 on Z 0 (x̆)n and choose a similitude X0n,z ∼
= Λ+ of OFp -symplectic modules that induces the universal
†
0
at the point s. Then [x̆]∗ (xp ) defines a Kp,n
-orbit in Λ+ ⊂ Vp such that
Drinfeld $n -structure αn,η
Ht(x̆) = Ht(x) for every x in the orbit. Here,
• xp is induced from x in the fixed unitary data;
• Ht(x̆) is the integer h (≥ 0) such that
1
2
TrEp◦ /Fp (x̆, x̆)0 ∈ $h O×
Fp ;
• Ht(x) is the largest integer h (≥ 0) such that det x ∈ $h O×
Fp .
Therefore, we have the following decomposition
[
Z 0 (x̆)n =
Z (x̆, x)n
†
x∈Kp,n
\Λ+
Ht(x)=Ht(x̆)
into (finitely many) formal divisors of Nn0 .
Let φ = φ0∞ ⊗ (⊗v∈Σfin φv ) such that
• φpfin := ⊗v∈Σfin −{p} φv is in S(V (
ApF,fin))K
p
;
†
• φp is in S(Vp )reg ∩ S(Vp )Kp,n , and its support is contained in Λ+ .
In particular, φ(0) = 0. Consider the generating series
Zφ (ep g p ) =
X
x∈Kp,n K p \V (
A
(φp ⊗ (ωχ (g p )φp )) (x)Z(x)Kp,n K p .
F,fin )−{0}
142
Define
X
Zφ (ep g p )n,K p =
x∈Kp,n K p \V (
A
(φp ⊗ (ωχ (g p )φp )) (x)Z(x)n,K p ,
F,fin )−{0}
whose generic fiber is the base change of Zφ (ep g p ) to Fpn . Moreover, its completion at the closed point
sn of Nn0 is
[Zφ (ep g p )n,K p ]∧
sn =
X
X
(ωχ (g p )φp ) (x̆)φp (x)Z (x̆, x)n .
(6.5)
+
x̆∈V̆ x∈Kp,n \Λ
(x,x)=(x̆,x̆)0
6.2.2
Coherence for intersection numbers
Define the subset
+
Redqs
n = {(x1 , x2 ; x̆1 , x̆2 )} ⊂ Λ
2
2
× Mor ((Y, jY ), (X, iX ))reg
by the conditions that
• Ht(xα ) = Ht(x̆α ) for α = 1, 2;
†
†
• Kp,n
x1 and Kp,n
x2 are linearly independent.
Then by definition, for (x1 , x2 ; x̆1 , x̆2 ) ∈ Redqs
n , the formal divisors Z (x̆1 , x1 )n and Z (x̆2 , x2 )n will
intersect properly. We let
m(x1 , x2 ; x̆1 , x̆2 ) = Z (x̆1 , x1 )n · Z (x̆2 , x2 )n
be the intersection multiplicity, which is a well-defined continuous function on Redqs
n . The following
lemma is straightforward.
†
Lemma 6.2.1. Suppose that for α = 1, 2, φα,p ∈ S(Vp )reg ∩ S(Vp )Kp,n whose support is contained in
Λ+ and such that φ1,p ⊗ φ2,p is in S(Vp2 )reg . Then the following function
µ(x̆1 , x̆2 ; φ1,p ⊗ φ2,p ) =
X
X
+
(φ1,p ⊗ φ2,p ) (x1 , x2 )m(x1 , x2 ; x̆1 , x̆2 )
+
x1 ∈Kp,n \Λ
x2 ∈Kp,n \Λ
(x1 ,x1 )=(x̆1 ,x̆1 )0 (x2 ,x2 )=(x̆2 ,x̆2 )0
2
that is a priori defined on Mor ((Y, jY ), (X, iX ))reg is a Schwartz function in S(V̆p2 ) via extension by
zero.
143
Let hp,j ∈ Kp,0 (j = 1, . . . , m) be a set of representatives of the coset K p,n \Kp,0 /Kp,n , and define
µ̆(x̆1 , x̆2 ; φ1,p ⊗ φ2,p ) =
m
X
µ x̆1 , x̆2 ; ωχ00 (hp,j ) (φ1,p ⊗ φ2,p ) ,
j=1
whose value does not depend on the representatives we choose.
VS)reg (α = 1, 2), φ1,S ⊗ φ2,S is
We return to our original assumption for this chapter that φα,S ∈ S(
VS2)reg , and gα ∈ eSH 0(ASF ) for α = 1, 2. Choose some element eα ∈ E × such that ωχ(m(eα))φα,p
in S(
is supported on Λ+ for α = 1, 2. We have
Zφα (gα ) =
X
xα ∈K\
=
(ωχ (gα )φα ) (xα )Z(xα )K
fin
X
xα ∈K\
=
V
V
(ωχ (gα )φα ) (xα eα )Z(xα eα )K
fin
X
xα ∈K\
V
(ωχ (gα )φα ) (xα eα )Z(xα )K .
fin
†
Therefore, we can add one more assumption that φα,p is in S(Vp )reg ∩ S(Vp )Kp,n , and its support is
contained in Λ+ . We let Zφα (gα ) = Zφ (ep gαp )n,K p and denote by [Zφ (gα )]∧
sp its completion along the
special fiber that is contained in the supersingular locus [M0n,K p ]ss . We have a similar decomposition
as (6.2).
First, we consider
p
p
∧
∧
∧
Zφ1 (g1 ) · Zφ2 (g2 ) = [Zφ1 (g1 )]∧
sp · [Zφ2 (g2 )]sp = [Zφ1 (ep g1 )n,K p ]sp · [Zφ2 (ep g2 )n,K p ]sp .
(6.6)
Q
p
p
Let h̆pi ∈ H̆fin
(i = 1, . . . , l) be a set of representatives of the double coset H̆( )\H̆fin
/K p .
Then
(6.6) =
l X
m
X
p
∧
[Zωχ (hp,j h̆p )φ1 (ep g1p )n,K p ]∧
sn · [Zωχ (hp,j h̆p )φ2 (ep g2 )n,K p ]sn .
i
i=1 j=1
i
(6.7)
144
By (6.5),


(6.7) =
l X
m  X
X



i=1 j=1
x̆1 ∈V̆


−1
x̆
)φ
(h
x
)Z
(x̆
,
x
)
ωχ (g1p )φp1 (h̆p,−1
1 1,p p,j 1
1
1 n
i

X
x1 ∈Kp,n \Λ+
(x1 ,x1 )=(x̆1 ,x̆1 )0


X

·

x2 ∈Kp,n \Λ+
(x2 ,x2 )=(x̆2 ,x̆2 )0
x̆2 ∈V̆
=
l
X


−1
ωχ (g2p )φp2 (h̆p,−1
x̆
)φ
(h
x
)Z
(x̆
,
x
)

2
2,p
2
2
2
n
i
p,j

X
X
ωχ00 ı(g1p , g2p,∨ )
φp1 ⊗ φp2
(x̆1 , x̆2 ))µ̆(x̆1 , x̆2 ; φ1,p ⊗ φ2,p ).
(h̆p,−1
i
(6.8)
i=1 (x̆1 ,x̆2 )∈V̆ 2
We define
Φhor =
l X
ωχ00 (h̆pi ) φp1 ⊗ φp2
⊗ µ̆(•; φ1,p ⊗ φ2,p )
i=1
AF )2). Then
that is a function in S(V̆ (
X
(6.8) =
ωχ00 ı(g1p , g2p,∨ ) Φhor ((x̆1 , x̆2 )) .
(x̆1 ,x̆2 )∈V̆ 2
We define the following theta series
hor
θ(p
◦ ) (•; φ1 , φ2 ) = log q
X
ωχ00 (•)Φhor ((x̆1 , x̆2 )) ,
(x̆1 ,x̆2 )∈V̆ 2
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
hor
∨
log q (Zφ1 (g1 ) · Zφ2 (g2 )) = θ(p
◦ ) (ı(g1 , g2 ); φ1 , φ2 ).
Second, we consider
p
∧
Zφ1 (g1 ) · Vφ2 (g2 ) = [Zφ1 (g1 )]∧
sp · Vφ2 (g2 ) = [Zφ1 (ep g1 )n,K p ]sp · Vφ2 (g2 ).
(6.9)
145
p
Let h̆pi ∈ H̆fin
(i = 1, . . . , l) and hp,j ∈ Kp,0 (j = 1, . . . , m) be as above. Then
(6.9) =
l X
m
X
[Zωχ (hp,j h̆p )φ1 (ep g1p )n,K p ]∧
sn · Vωχ (hp,j h̆p )φ2 (g2 )
i
i
i=1 j=1
=
l X
m X
X
X
ωχ (g1p )φp1 (h̆p,−1
x̆1 )φ1,p (h−1
i
p,j x1 )Z (x̆1 , x1 )n · Vωχ (hp,j h̆p )φ2 (g2 ).
i
i=1 j=1 x̆1 ∈V̆
x1 ∈Kp,n \Λ+
(x1 ,x1 )=(x̆1 ,x̆1 )0
(6.10)
The function
ν(x̆1 ; φ1,p , φ2 , g2 ) =
m
X
j=1
X
φ1,p (h−1
p,j x1 )Z (x̆1 , x1 )n · Vωχ (hp,j )φ2 (g2 ),
x1 ∈Kp,n \Λ+
(x1 ,x1 )=(x̆1 ,x̆1 )0
which is originally defined for x̆1 ∈ Mor ((Y, jY ), (X, iX ))reg , can be extended by zero to a function
in S(V̆p ). Then
(6.10) =
l
X
X
ωχ (g1p )φp1 (h̆p,−1
x̆1 )ν(x̆1 ; φ1,p , φ2 , g2 ).
i
i=1 x̆1 ∈V̆
We define
φver =
l X
ωχ (h̆pi )φp1 ⊗ ν(•; φ1,p , φ2 , g2 )
i=1
AF )). Then
that is a function in S(V̆ (
(6.11) =
X
ωχ g1p φver (x̆1 ).
x̆1 ∈V̆
We define the following theta series
ver
θ(p
◦ ) (•; φ1 , φ2 , g2 ) = log q
X
(ωχ (•)φver ) (x̆1 ).
x̆1 ∈V̆
Then in summary, we have the following lemma.
Lemma 6.2.3. Under the previous assumption, we have
ver
log q (Zφ1 (g1 ) · Vφ2 (g2 )) = θ(p
◦ ) (g1 ; φ1 , φ2 , g2 )
that is a theta series for g1 ∈ ep H 0 (
ApF ).
(6.11)
146
Finally, we let
A(p◦ ) (g1 , φ1 ) = log q (Vφ1 (g1 ) · ωK ) .
Then in summary, we have the following proposition.
VS2)reg and gα ∈ eSH 0(ASF ) (α = 1, 2),
Proposition 6.2.4. For φ1,S ⊗ φ2,S ∈ S(
bφ (g2 )ip◦ = θhor◦ (ı(g1 , g ∨ ); φ1 , φ2 ) + θver◦ (g1 , φ1 , φ2 , g2 ) + A(p◦ ) (g1 , φ1 )E(g2 , φ2 ).
hZbφ1 (g1 ), Z
2
2
(p )
(p )
6.3
Nonsplit case
In this section, we discuss the contribution of the local height paring at a finite place p in S that is
nonsplit in E and such that (Vp ) = −1. Let p◦ be the only place in Σ◦fin lying over p. We identify Vp
with Bp , the unique division quaternion algebra over Fp , and the lattice Λ− with OBp .
6.3.1
Integral models and Čerednik–Drinfeld uniformization: minimal level
Let K = Kp K p be an open compact subgroup of H(
Afin) with K p 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,K p
defined over Spec OEp◦ , whose generic fiber is M0,K p ;p◦ . We fix an
component of M†
0,Kp†,p ,K †,p
F-point s on the common neutral
and M0,K p , which corresponds to a quintuple (A, θA , iA , η̄ p , η̄pp ). Repeat
the process of 5.1.4, we obtain the E-hermitian space V̆ with a fixed isometry γ p = (γpp , γ p ) : V̆ ⊗F
ApF,fin → V ⊗F ApF,fin, and the reductive group H̆ = ResF/Q U(V̆ ).
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 Spec
F (cf. [Dri1976, BC1991, KR2000]).
In particular X is of
dimension 2 and height 4. Then Aut(X, iX ) acts on Mor ((Y, jY ), (X, iX )) and can be identified with
GL2 (OFp ). For every integer h, we define a functor Ωh in the category of schemes over Spec OFp0
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 ×Spec F Ssp → X ×Spec F Ssp 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 OFc0 . Let
p
147
Ω = Ω01 and Ω0 = Ω ×O d0 OEd
0 .
p◦
Fp
If we denote by
[M† †,p †,p ]∧
sp
0,Kp ,K
(resp. [M0,K p ]∧
sp ) the formal completion along the special fiber,
then we have the following Čerednik–Drinfeld uniformization (cf. [Dri1976, BC1991, RZ1996]):
[M† †,p †,p ]∧
sp
0,Kp ,K
×OFp OFc0
Q
∼
= H̆ † ( )\
p
a
h∈
Q
Z
h
Ω ×
r
Y
Bp×i /Kp†,p
†
A
× H̆ (
p
†,p
fin )/K
!
;
i=2
p
∼
[M0,K p ]∧
= H̆( )\Ω0 × H̆fin /K p .
d
0
sp ×OEp◦ OE
p◦
In fact, the above isomorphisms underly the corresponding uniformization for the universal p-divisible
groups. For example, let Xuniv → Ω and X0 → Ω0 be the universal special formal OBp -modules,
respectively. Then we have the following commutative diagram:
Q
/ H̆( )\X0 × H̆ p /K p
fin
∼
[X]∧
d
0
sp ×OEp◦ OE
p◦
[M0,K p ]∧
×
OEp◦ OE
d
0
sp
/ H̆( )\Ω0 × H̆ p /K p .
fin
Q
∼
p◦
For every nonzero element x̆ ∈ Mor ((Y, jY ), (X, iX )), we define a subfunctor Z (x̆) of Ω0 as
follows: for every scheme S in the previously mentioned category, Z (x̆)(S) is the set of equivalence
classes of (X, ρX ) ∈ Ω0 (S) such that the following composed homomorphism
Y ×Spec O d0 Spec
Fp
F
×Spec F Ssp = Y ×Spec F Ssp −
→ X ×Spec F Ssp −−→ X ×S Ssp
x̆
ρX
extends to a homomorphism Y ×Spec O d0 S → X. Then Z (x̆) is represented by a closed formal
Fp
subscheme Z (x̆) of Ω . We denote by Z (x̆)hor the horizontal part of the associated divisor of Z (x̆),
0
which is empty if (x̆, x̆) = 0.
6.3.2
Integral models and Čerednik–Drinfeld uniformization: higher level
Afin) with K p sufficiently small and Kp = Kp,n
Let K = Kp K p be an open compact subgroup of H(
for n large such that Fpn contains Ep◦ . Therefore, MK = Mn,K p .
Let Ωn,η = Xuniv
[$n ] − Xuniv
[$n−1 ] be the étale covering over the generic fiber Ωη , viewed as
η
η
†
†
a rigid space, with the Galois group Kp,0
/Kp,n
= (OBp /$n OBp )× . Let Ωn be the normalization
n
of Ω ×O d0 OFcn in Ωn,η ×Fc0 Fc
p , whose set of connected components is parameterized by the coset
Fp
1 This
p
p
is not to be confused with the differential form Ω in Chapter 4
148
†
0
0
O×
OFcn .
Fp /ν(Kp,n ). Let Ωn be the neutral connected component of Ωn , which is finite over Ω ×O d
0
E ◦
p
The formal scheme
Ω0n
is not regular but has double points. We replace
Ω0n
p
by blowing up all its
double points and denote by the same notation. It is easy to see that
n
Mn,K p ×E Fc
p
Q
rig
×,1
p
∼
= H̆( )\ Ω0n,η × Ep◦ /ν(Kp,n ) × H̆fin /K p .
Q
The group H̆( ) acts on Ω0n by the universal property of normalization and blowing-up of double
points. The quotient
Q
p
p
H̆( )\ Ω0n × Ep×,1
◦ /ν(Kp,n ) × H̆fin /K
is regular, flat and projective over Spf OFcn (for sufficiently small K p ). By Grothendieck Existence
p
Theorem, we have a regular scheme Mn,K p that is flat and projective over Spec OFpn , and a morphism
pn : Mn,K p → M0,K p ×OEp◦ OFpn , such that the following diagram commutes:
Q
p
p
/ H̆( )\ Ω0n × E ×,1
/ν(K
)
×
H̆
/K
◦
p,n
p
fin
∼
[Mn,K p ]∧
sp
[pn ]∧
sp
[M0,K p ]∧
cn
sp ×OEp◦ OF
/ H̆( )\Ω0 × H̆ p /K p ×O
OFcn .
d
fin
0
Q
∼
p
E ◦
p
p
Define Z 0 (x̆)n by the following Cartesian diagram
/ Ω0n
Z 0 (x̆)n
Z (x̆)hor ×O d0 OFcn
Fp
/ Ω ×O
p
d
0
Fp
OFcn .
p
By the similar argument as in 6.2.1, we have the following decomposition
Z 0 (x̆)n =
[
Z (x̆, x)n
†
x∈Kp,n
\Λ−
Ht(x)=Ht(x̆)
into (finitely many) formal divisors of Ω0n .
In 5.1.5, we have defined a 1-dimensional closed subscheme Z(x0 )0,K p of M0,K p whose generic fiber
is Z(x0 )0,K p ;p◦ , for a Kp,0 K p -orbit Kp,0 K p x0 in V (
AF,fin) that is admissible.
Define Z0 (x0 )n,K p by
149
the following Cartesian diagram
Z0 (x0 )n,K p
/ Mn,K p
Z(x0 )hor
0,K p ×OEp◦ OFpn
/ M0,K p ×O
OFpn ,
Ep◦
p
where Z(x0 )hor
0,K p denotes the horizontal part of the associated divisor. For every orbit Kp,n K x inside
Kp,0 K p x0 , we define Z(x)n,K p to be the union of irreducible components of Z0 (x0 )n,K p whose generic
fiber contributes to the special divisor Z(x)n,K p ×Ep◦ Fpn on the generic fiber Mn,K p ×E Fpn .
Let φ = φ0∞ ⊗ (⊗v∈Σfin φv ) such that
• φpfin := ⊗v∈Σfin −{p} φv is in S(V (
ApF,fin))K
p
;
†
• φp is in S(Vp )reg ∩ S(Vp )Kp,n , and its support is contained in Λ− .
In particular, φ(0) = 0. Consider the generating series
X
Zφ (ep g p ) =
x∈Kp,n K p \V (
A
(φp ⊗ (ωχ (g p )φp )) (x)Z(x)Kp,n K p .
F,fin )−{0}
Define
X
Zφ (ep g p )n,K p =
A
x∈Kp,n K p \V (
(φp ⊗ (ωχ (g p )φp )) (x)Z(x)n,K p ,
F,fin )−{0}
whose generic fiber is the base change of Zφ (ep g p ) to Fpn . Moreover, its completion along the neutral
component is (the image of)
X
X
(ωχ (g p )φp ) (x̆)φp (x)Z (x̆, x)n .
(6.12)
+
x̆∈V̆ x∈Kp,n \Λ
(x,x)=(x̆,x̆)0
6.3.3
Coherence for intersection numbers
Let Mor ((Y, jY ), (X, iX ))reg ⊂ Mor ((Y, jY ), (X, iX )) be the subset consisting of x̆ with (x̆, x̆) 6= 0.
Define the subset
−
Redns
n = {(x1 , x2 ; x̆1 , x̆2 )} ⊂ Λ
by the conditions that
2
2
× Mor ((Y, jY ), (X, iX ))reg
150
• Ht(xα ) = Ht(x̆α ) for α = 1, 2;
†
†
• Kp,n
x1 and Kp,n
x2 are linearly independent.
Then by definition, for (x1 , x2 ; x̆1 , x̆2 ) ∈ Redns
n , the formal divisors Z (x̆1 , x1 )n and Z (x̆2 , x2 )n will
intersect properly. We let
m(x1 , x2 ; x̆1 , x̆2 ) = Z (x̆1 , x1 )n · Z (x̆2 , x2 )n
be the intersection multiplicity, which is a well-defined continuous function on Redns
n . The following
lemma is straightforward.
†
Lemma 6.3.1. Suppose that for α = 1, 2, φα,p ∈ S(Vp )reg ∩ S(Vp )Kp,n whose support is contained in
Λ− and such that φ1,p ⊗ φ2,p is in S(Vp2 )reg . Then the following function
µ(x̆1 , x̆2 ; φ1,p ⊗ φ2,p ) =
X
X
−
(φ1,p ⊗ φ2,p ) (x1 , x2 )m(x1 , x2 ; x̆1 , x̆2 )
−
x1 ∈Kp,n \Λ
x2 ∈Kp,n \Λ
(x1 ,x1 )=(x̆1 ,x̆1 )0 (x2 ,x2 )=(x̆2 ,x̆2 )0
2
that is a priori defined on Mor ((Y, jY ), (X, iX ))reg is a Schwartz function in S(V̆p2 ) via extension by
zero.
The remaining discussion follows similarly as in 6.2.2. We sketch the process. Let hp,j ∈ Kp,0
(j = 1, . . . , m) be similarly defined previously. Let
m
X
µ̆(x̆1 , x̆2 ; φ1,p ⊗ φ2,p ) =
µ x̆1 , x̆2 ; ωχ00 (hp,j ) (φ1,p ⊗ φ2,p ) ,
j=1
whose value does not depend on the representatives we choose.
VS)reg (α = 1, 2), φ1,S ⊗ φ2,S is
We return to our original assumption for this chapter that φα,S ∈ S(
in S(
VS2)reg , and gα ∈ eSH 0(ASF ) for α = 1, 2. Choose some element eα ∈ E × such that ωχ(m(eα))φα,p
is supported on Λ+ for α = 1, 2. We have
Zφα (gα ) =
X
xα ∈K\
V
(ωχ (gα )φα ) (xα eα )Z(xα )K .
fin
†
Therefore, we can add one more assumption that φα,p is in S(Vp )reg ∩ S(Vp )Kp,n , and its support is
contained in Λ− . We let Zφα (gα ) = Zφ (ep gαp )n,K p and denote by [Zφ (gα )]∧
sp its completion along the
special fiber. We have a similar decomposition as (6.2).
151
First, we consider
p
p
∧
∧
∧
Zφ1 (g1 ) · Zφ2 (g2 ) = [Zφ1 (g1 )]∧
sp · [Zφ2 (g2 )]sp = [Zφ1 (ep g1 )n,K p ]sp · [Zφ2 (ep g2 )n,K p ]sp .
(6.13)
Q
p
p
Let h̆pi ∈ H̆fin
(i = 1, . . . , l) be a set of representatives of the double coset H̆( )\H̆fin
/K p . We assume
that h̆p1 is the identity.
Then
l X
m
X
(6.13) =
p
∧
[Zωχ (hp,j h̆p )φ1 (ep g1p )n,K p ]∧
h1 · [Zωχ (hp,j h̆p )φ2 (ep g2 )n,K p ]h1 ,
(6.14)
i
i
i=1 j=1
where the subscript h1 means that we take completion along the component indexed by h1 . By (6.12),


m  X
l X
X

(6.14) =


i=1 j=1
x̆1 ∈V̆
X
x1 ∈Kp,n \Λ−
(x1 ,x1 )=(x̆1 ,x̆1 )0


ωχ (g1p )φp1 (h̆p,−1
x̆1 )φ1,p (h−1
i
p,j x1 )Z (x̆1 , x1 )n 



X

·

x2 ∈Kp,n \Λ−
(x2 ,x2 )=(x̆2 ,x̆2 )0
x̆2 ∈V̆
=
l
X


−1
ωχ (g2p )φp2 (h̆p,−1
x̆
)φ
(h
x
)Z
(x̆
,
x
)

2
2,p
2
2
2
n
i
p,j

X
X
ωχ00 ı(g1p , g2p,∨ ) φp1 ⊗ φp2 (h̆p,−1
(x̆1 , x̆2 ))µ̆(x̆1 , x̆2 ; φ1,p ⊗ φ2,p ).
i
(6.15)
i=1 (x̆1 ,x̆2 )∈V̆ 2
We define
Φhor =
l X
ωχ00 (h̆pi ) φp1 ⊗ φp2
⊗ µ̆(•; φ1,p ⊗ φ2,p )
i=1
that is a function in S(V̆ (
AF )2). Then
X
(6.15) =
ωχ00 ı(g1p , g2p,∨ ) Φhor ((x̆1 , x̆2 )) .
(x̆1 ,x̆2 )∈V̆ 2
We define the following theta series
hor
θ(p
◦ ) (•; φ1 , φ2 ) = log q
X
ωχ00 (•)Φhor ((x̆1 , x̆2 )) ,
(x̆1 ,x̆2 )∈V̆ 2
where q is the cardinality of the residue field of Ep◦ . Then in summary, we have the following lemma.
152
Lemma 6.3.2. Under the previous assumption, we have
hor
∨
log q (Zφ1 (g1 ) · Zφ2 (g2 )) = θ(p
◦ ) (ı(g1 , g2 ); φ1 , φ2 ).
Second, we consider
p
∧
Zφ1 (g1 ) · Vφ2 (g2 ) = [Zφ1 (g1 )]∧
sp · Vφ2 (g2 ) = [Zφ1 (ep g1 )n,K p ]sp · Vφ2 (g2 ).
(6.16)
p
Let h̆pi ∈ H̆fin
(i = 1, . . . , l) and hp,j ∈ Kp,0 (j = 1, . . . , m) be as above. Then
(6.16) =
l X
m
X
[Zωχ (hp,j h̆p )φ1 (ep g1p )n,K p ]∧
h1 · Vωχ (hp,j h̆p )φ2 (g2 )
i
i
i=1 j=1
=
m X
l X
X
X
ωχ (g1p )φp1 (h̆p,−1
x̆1 )φ1,p (h−1
i
p,j x1 )Z (x̆1 , x1 )n · Vωχ (hp,j h̆p )φ2 (g2 ).
i
i=1 j=1 x̆1 ∈V̆
x1 ∈Kp,n \Λ−
(x1 ,x1 )=(x̆1 ,x̆1 )0
(6.17)
The function
ν(x̆1 ; φ1,p , φ2 , g2 ) =
m
X
j=1
X
φ1,p (h−1
p,j x1 )Z (x̆1 , x1 )n · Vωχ (hp,j )φ2 (g2 ),
x1 ∈Kp,n \Λ−
(x1 ,x1 )=(x̆1 ,x̆1 )0
which is originally defined for x̆1 ∈ Mor ((Y, jY ), (X, iX ))reg , can be extended by zero to a function
in S(V̆p ). Then
(6.17) =
l
X
X
ωχ (g1p )φp1 (h̆p,−1
x̆1 )ν(x̆1 ; φ1,p , φ2 , g2 ).
i
i=1 x̆1 ∈V̆
We define
φver =
l X
ωχ (h̆pi )φp1 ⊗ ν(•; φ1,p , φ2 , g2 )
i=1
AF )). Then
that is a function in S(V̆ (
(6.18) =
X
x̆1 ∈V̆
ωχ g1p φver (x̆1 ).
(6.18)
153
We define the following theta series
ver
θ(p
◦ ) (•; φ1 , φ2 , g2 ) = log q
X
(ωχ (•)φver ) (x̆1 ).
x̆1 ∈V̆
Then in summary, we have the following lemma.
Lemma 6.3.3. Under the previous assumption, we have
ver
log q (Zφ1 (g1 ) · Vφ2 (g2 )) = θ(p
◦ ) (g1 ; φ1 , φ2 , g2 )
that is a theta series for g1 ∈ ep H 0 (
ApF ).
Finally, we let
A(p◦ ) (g1 , φ1 ) = log q (Vφ1 (g1 ) · ωK ) .
Then in summary, we have the following proposition.
VS2)reg and gα ∈ eSH 0(ASF ) (α = 1, 2),
Proposition 6.3.4. For φ1,S ⊗ φ2,S ∈ S(
bφ (g2 )ip◦ = θhor◦ (ı(g1 , g2∨ ); φ1 , φ2 ) + θver◦ (g1 , φ1 , φ2 , g2 ) + A(p◦ ) (g1 , φ1 )E(g2 , φ2 ).
hZbφ1 (g1 ), Z
2
(p )
(p )
154
Chapter 7
Arithmetic inner product formula:
the main theorem
We finish the proof of the main theorem, i.e., the arithmetic inner product formula for n = 1 in this
chapter. In 7.1, we introduce the general theory of holomorphic projection for the group U(1, 1)F , and
compute such projection for the analytic kernel function. In particular, this process will relate the
archimedean local terms to admissible height parings (at archimedean places) and leaves local terms
at finite places unchanged. Based on all these previous results, we come to the final stage of the proof
in 7.2.
155
7.1
Holomorphic projection
In this section, we define and compute the holomorphic projection of the analytic kernel function
E 0 (0, ı(g1 , g2∨ ), 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 0 = 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 0 ) ⊂ A(H 0 ) the subspace of cusp
forms. Let Ak0 (H 0 ) ⊂ A0 (H 0 ) be the subspace consisting of those cusp forms of weight (1 + k, 1 − k)
(cf. Definition 2.1.2). Let Z 0 be the center of H 0 , which is isomorphic to E ×,1 , as an F -torus. We
0
by letting ζιk (zι ) = zι2kι . Let A(H 0 , ζ k ) ⊂ A(H 0 ) be the subspace consisting
define a character ζ k of Z∞
of the forms that have archimedean central character ζ k . It is clear that Ak0 (H 0 ) ⊂ A(H 0 , ζ k ). Recall
that F + ⊂ F is the subset of totally positive elements. For each fixed t ∈ F + , the t-th Whittaker
functions (with respect to the standard additive character) of all members of Ak0 (H 0 ) are the same
one, namely, the function Wtk defined by
Wtk (n(b)m(a)[k1 , k2 ]) =
Y
1+kι 1−kι
exp (2πit (bι + iaι aι )) (aι aι ) k1,ι
k2,ι ,
ι∈Σ∞
×
, b = (bι ) ∈ F∞ , and [k1 , k2 ] = ([k1,ι , k2,ι ]) ∈ K0∞ .
where a = (aι ) ∈ E∞
We denote by Ak0 (H 0 × H 0 ) ⊂ A0 (H 0 × H 0 ) the subspace consisting of cusp forms f (•, •) such that
for every g ∈ H 0 (
AF ), both f (g, •) and f (•, g) are in Ak0(H 0).
The space A0 (H 0 × H 0 ) is a Hilbert
space with norm given by the Petersson inner product h−, −iH 0 ×H 0 .
Definition 7.1.1 (Holomorphic projection). We define a linear map
Pr : A(H 0 × H 0 , ζ k ) → Ak0 (H 0 × H 0 ),
which sends f to the unique element Pr(f ) in the later space satisfying hf , f 0 iH 0 ×H 0 = hPr(f ), f 0 iH 0 ×H 0
for every f 0 ∈ Ak0 (H 0 × H 0 ).
AF such that ψ∞0 (x) = ψ∞(tx) for t ∈ F∞× . We recall that ψ∞ is the
Let ψ 0 be a character of F \
standard additive character at archimedean places as assumed in the beginning of Chapter 3. For an
156
automorphic form f ∈ A(H 0 × H 0 , ζ k ), we define
fψ0 ,s (g1 , g2 ) = (4π)2d Wtk (g1,∞ )Wtk (g2,∞ )
ZZ
λP 0 (h1 )s λP 0 (h2 )s
[Z 0 (F∞ )N 0 (F∞ )\H 0 (F∞ )]2
fψ0 (h1 g1,fin , h2 g2,fin )Wtk (h1 )Wtk (h2 )dh1 dh2 .
Proposition 7.1.2. Suppose that f ∈ A(H 0 × H 0 , ζ k ) has the following asymptotic behavior: for some
> 0,
f (m(a1 )g1 , m(a2 )g2 ) = Og1 ,g2 |a1 a2 |1−
AE
as |a1 a2 |AE → ∞ for aα ∈
A×E (α = 1, 2). Then the holomorphic projection Pr(f ) has the ψ0-Whittaker
function
Pr(f )ψ0 (g1 , g2 ) = lim fψ0 ,s (g1 , g2 ).
s→0
AF ) such that ζα,∞ = ζ k. Consider two
α
(gα,fin ) (α = 1, 2) of H 0 (AF ) with central character
ψ 0 -Whittaker functions W α (gα ) = Wtk (gα,∞ )Wfin
α
ζα , such that Wfin
is compactly supported modulo Z 0 (AF,fin )N 0 (AF,fin ). We define the Poincaré series
Proof. Let ζα (α = 1, 2) be two automorphic characters of Z 0 (
to be
X
PW α (gα ) = lim+
s→0
W α (γgα )λP 0 (γ∞ gα,∞ )s .
γ∈Z 0 (F )N 0 (F )\H 0 (F )
For a function f 0 ∈ A(H 0 × H 0 , ζ k ), we let
fζ01 ,ζ2 (g1 , g2 ) =
ZZ
A
[Z 0 (F )\Z 0 (
2
F )]
f 0 (g1 z1 , g2 z2 )ζ1−1 (z1 )ζ2−1 (z2 )dz1 dz2 .
Assume that f has the asymptotic behavior as in the proposition. Then on one hand,
hf , PW 1 ⊗ PW 2 iH 0 ×H 0
ZZ
=
A )H (F )\H (A
ZZ
[Z 0 (
= lim+
s→0
F
0
[Z 0 (
A
0
fζ1 ,ζ2 (g1 , g2 )PW 1 (g1 )PW 2 (g2 )dg1 dg2
2
F )]
A
0
0
F )N (F )\H (
fζ1 ,ζ2 (g1 , g2 )W 1 (g1 )W 2 (g2 )λP 0 (g1,∞ )s λP 0 (g2,∞ )s dg1 dg2
2
F )]
ZZ
= lim+
s→0
[Z 0 (
A
F
A
)N 0 (
F
A
)\H 0 (
(fζ1 ,ζ2 )ψ (g1 , g2 )W 1 (g1 )W 2 (g2 )λP 0 (g1,∞ )s λP 0 (g2,∞ )s dg1 dg2 . (7.1)
F
)]2
157
On the other hand, we have
hPr(f ), PW 1 ⊗ PW 2 iH 0 ×H 0
ZZ
=
Wtk (g1 )Wtk (g2 )Wtk (g1 )Wtk (g2 )dg1 dg2
[Z 0 (F∞ )N 0 (F∞ )\H 0 (F∞ )]2
ZZ
AZ Z
[Z 0 (
= (4π)−2d
A
0
F,fin )N (
A
[Z 0 (
0
F,fin )\H (
F,fin
A
)N 0 (
A
F,fin
2
F,fin )]
)\H 0 (
1 (g
2
(Pr(f )ζ1 ,ζ2 )ψ0 (g1,fin , g2,fin )Wfin
1,fin )Wfin (g2,fin )dg1,fin dg2,fin
A
F,fin
)]2
1 (g
2
(Pr(f )ζ1 ,ζ2 )ψ0 (g1,fin , g2,fin )Wfin
1,fin )Wfin (g2,fin )dg1,fin dg2,fin .
(7.2)
Since, by definition, hPr(f ), PW 1 ⊗ PW 2 iH 0 ×H 0 = hf , PW 1 ⊗ PW 2 iH 0 ×H 0 , (7.1) and (7.2) are equal for
1
2
all (ζ1 , ζ2 ) and (Wfin
, Wfin
). Therefore, the proposition follows.
For general f ∈ A(H 0 × H 0 , ζ k ) 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 0 × H 0 , ζ k ), we define
f )ψ0 (g1 , g2 ) = consts=0 fψ0 ,s (g1 , g2 ),
Pr(f
where consts=a denotes the constant term at s = a (possibly after meromorphic continuation around
a). We define the quasi-holomorphic projection of f to be
f )(g1 , g2 ) =
Pr(f
X
f )ψ0 (g1 , g2 ),
Pr(f
ψ0
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 0 (F ) × N 0 (F ). In
f to such functions or even currents.
some middle steps of later calculation, we need to apply Pr
Then Proposition 7.1.2 amounts equivalently to say that if f satisfies the asymptotic behavior
f ) = Pr(f ).
there, we have Pr(f
158
7.1.2
Siegel–Fourier expansion of Eisenstein series on U(2, 2)
AF ) =
We study the (Siegel–)Fourier expansion of the Eisenstein series E(s, g, Φ) at g = e ∈ H 00 (
AF ) for Φ = φ1 ⊗ φ2 ∈ S(V2), where V is a hermitian space of rank 2 over AE . By definition
H2 (
X
E(s, g, φ1 ⊗ φ2 ) =
(ωχ (γg)(φ1 ⊗ φ2 )) (0)λP (γg)s ,
γ∈P (F )\H 00 (F )
which is absolutely convergent when Re s > 1. The following lemma is straightforward.
Lemma 7.1.4. The group H 00 (F ) is the disjoint union of the following cosets:
1.





P (F )w2 



1
b11
1
b21
1
b12



b22 

,



1

 b11
b=
b21

b12 
 ∈ Her2 (E);
b22
2.

1


 c

P (F )w2,1 



0
1
0
1
0



b22 

,
τ 
−c 

1
c ∈ E, b22 ∈ F ;
3. P (F ),
where we recall that



w2 = 

−12
12 
;
w2,1




=




1
1
−1


1 

.



In particular, we have
E(s, g, φ1 ⊗ φ2 ) = E 2 (s, g, φ1 ⊗ φ2 ) + E 0 (s, g, φ1 ⊗ φ2 ) +
X
c∈E
E 1,c (s, g, φ1 ⊗ φ2 ),
159
where
X
E 2 (s, g, φ1 ⊗ φ2 ) =
(ωχ (w2 n(b)g)(φ1 ⊗ φ2 )) (0)λP (w2 n(b)g)s ;
b∈Her2 (E)

 
 

 
X  

  0 0  
 1
E 1,c (s, g, φ1 ⊗ φ2 ) =
 g  (φ1 ⊗ φ2 ) (0)
 n 
ωχ w2,1 m 
0 b22
c 1
b22 ∈F
 s

 

 1
  0 0  

λP w2,1 m 
 g 
 n 
0 b22
c 1
E 0 (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
ETβ (s, e, φ1
Z
⊗ φ2 ) :=
N (F )\N (
A
E β (s, n(b0 ), φ1 ⊗ φ2 )ψ(tr T b0 )−1 dn
F)
for β = 2, 0, and 1, c for c ∈ E, respectively.
For β = 2, we have
ET2 (s, e, φ1 ⊗ φ2 ) =
Z
A
N(
(ωχ (w2 n(b0 ))(φ1 ⊗ φ2 )) (0)λP (w2 n(b0 ))s ψ(tr T b0 )−1 dn.
(7.3)
F)
For β = 0, we have
ET0 (s, e, φ1 ⊗ φ2 ) =



(φ1 ⊗ φ2 ) (0)
if T = 02 ;


0
otherwise.
(7.4)
Now we calculate for β = 1, c with c ∈ F . To simplify the formulae, we ignore the term λP (•)s in
the following calculation. We have
ET1,c (s, e, φ1 ⊗ φ2 )

Z
Z
=
b11 ∈F \

A
F
b12 ∈E\


 b11
ψ tr T 
bτ12
Z
A
E
A
b22 ∈
F


 
 1
ωχ w2,1 m 
c
−1
b12 
 db11 db12 db22 .
b22
 
  b11
 n 
1
bτ12


b12 

 (φ1 ⊗ φ2 ) (0)
b22
(7.5)
160
Change variables

 b11

bτ12


b12 
 1
 7→ 
c
b22

  b11

1
bτ12

b12   1

b22

cτ 
.
1
Then

Z
(7.5) =
b11 ∈F \
A
F
 
 
ψ tr 
Z
=
b11 ∈F \
A
F
 
 
ψ tr 








 b11 b12   1
 
 (φ1 ⊗ φ2 ) (0)
 m 
ωχ w2,1 n 
τ
b12 ∈E\AE b22 ∈AF
c 1
b12 b22
−1

 
1 −cτ   1
  b11 b12 
 db11 db12 db22

T 
bτ12 b22
−c 1
1
 

 


Z
Z
0
0
1
 

 


ωχ w2,1 n 
 m 
 (φ1 ⊗ φ2 ) (0)
b12 ∈E\AE b22 ∈AF
0 b22
c 1
−1

 
b
b
1 −cτ   1
  11 12 
 db11 db12 db22 .

T 
bτ12 b22
−c 1
1
Z
Z
Therefore, ET1,c (s, e, φ1 ⊗ φ2 ) = 0 unless

 1
T =

cτ   0

0
1

0  1

c
t

1


for some t ∈ F . In the later case,
ET1,c (s, e, φ1 ⊗ φ2 )


 
  
Z
Z


   0 0   1
−1
=
 (φ1 ⊗ φ2 ) (0, x)ψ(tb) dbdx
ωχ n 
 m 
b∈AF x∈V
c 1
0 b
  


Z
Z
1
  


=
ωχ m 
 (φ1 ⊗ φ2 ) (0, x)ψ ((q(x) − t)b) dbdx
b∈AF x∈V
c 1
Z
Z
=
φ1 (cx)φ2 (x)ψ ((q(x) − t)b) dbdx.
(7.6)
b∈
A
F
x∈
V
We remark that in the above calculation, all terms involving λP (•)s are ignored. In particular, when
c = 0,
1
ET1,0 (s, e, φ1 ⊗ φ2 ) = φ1 (0) × Wt (s + , e, φ2 ).
2
(7.7)
161
7.1.3
Holomorphic projection of analytic kernel functions
We apply 7.1.1 to the analytic kernel function E 0 (0, ı(g1 , g2∨ ), φ1 ⊗ φ2 )). Here, φα = φ0∞ ⊗ φα,fin is in
V
S( ) for a totally positive definite incoherent hermitian space over
which is ζ
kχ
2
AE of rank 2.
0
Let χ0 = χ | Z∞
,
. Then the analytic kernel function is in A(H 0 × H 0 , χ0 ). Unfortunately, it does not have
the asymptotic behavior in Proposition 7.1.2. To find its holomorphic projection, we introduce the
following function
1
1
F (s; g1 , g2 ; φ1 , φ2 ) = E(s + , g1 , φ1 )E(s + , g2 , φ2 ) ∈ A(H 0 × H 0 , χ◦ ),
2
2
which is holomorphic at s = 0.
Proposition 7.1.5. The difference
E 0 (0, ı(g1 , g2∨ ), φ1 ⊗ φ2 ) − F 0 (0; g1 , g2 ; φ1 , φ2 )
has the asymptotic behavior in Proposition 7.1.2.
Proof. Since the problem is symmetric in g1 and g2 , we prove the asymptotic behavior only in g1 .
×
Moreover, we can assume that a1 has 1 as its finite part. Let a = diag[a∞ , 1] with a∞ ∈ E∞
. We
consider the behavior of the difference
E 0 (0, m(a), φe1 ⊗ φe2 ) − F 0 (0; m(a∞ ), e; φe1 , φe2 )
(7.8)
as |a∞ | → ∞, where φeα = ωχ (gα )φα for α = 1, 2. We apply the calculation in 7.1.2 to φe1 ⊗ φe2 . We
see that the following terms
d
|s=0 ET2 (s, m(a), φe1 ⊗ φe2 ),
ds
d
|s=0 ET1,c (s, m(a), φe1 ⊗ φe2 ),
ds
T ∈ Her2 (E);
c 6= 0, T ∈ Her2 (E)
are bounded by O(log |a|). Therefore, we only need to consider the difference
d
|s=0
ds
ET0 (s, m(a), φe1
⊗ φe2 ) +
X
t∈F
1,0
Ediag[0,t]
(s, m(a), φe1
!
e
e
e
⊗ φ2 ) − F (s; m(a∞ ), e; φ1 , φ2 ) .
(7.9)
162
By (7.4) and (7.7), we have on the one hand,
ET0 (s, m(a), φe1 ⊗ φe2 ) +
X
1,0
Ediag[0,t]
(s, m(a), φe1 ⊗ φe2 )
t∈F
= ωχ (a∞ )φe1 (0)|a∞ |s × φe2 (0) +
X
t∈F
= ωχ (a∞ )φe1
1
ωχ (a∞ )φe1 (0)|a∞ |s × Wt (s + , e, φe2 )
2
1
(0)|a∞ |s × E(s + , e, φe2 ).
2
On the other hand,
F (s; m(a∞ ), e; φe1 , φe2 ) =
X
1
ωχ (a∞ )φe1 (0)|a∞ |s +
Wt (s + , m(a∞ ), φe1 )
2
t∈F
!
1
× E(s + , e, φe2 ).
2
Therefore,
(7.9) =
X
t∈F
X
1
1
1
1
Wt ( , m(a∞ ), φe1 ) × E 0 ( , e, φe2 ),
Wt0 ( , m(a∞ ), φe1 ) × E( , e, φe2 ) +
2
2
2
2
t∈F
which is bounded by O(log |a|). The proposition then follows.
By the above proposition, we have
Pr (E 0 (0, ı(g1 , g2∨ ), φ1 ⊗ φ2 ))
= Pr (E 0 (0, ı(g1 , g2∨ ), φ1 ⊗ φ2 ) − F 0 (0; g1 , g2 ; φ1 , φ2 )) + Pr (F 0 (0; g1 , g2 ; φ1 , φ2 ))
f (E 0 (0, ı(g1 , g2∨ ), φ1 ⊗ φ2 ) − F 0 (0; g1 , g2 ; φ1 , φ2 )) + Pr (F 0 (0; g1 , g2 ; φ1 , φ2 )) .
= Pr
(7.10)
Since
1
1
1
1
F 0 (0; g1 , g2 ; φ1 , φ2 ) = E 0 ( , g1 , φ1 )E( , g2 , φ2 ) + E( , g1 , φ1 )E 0 ( , g2 , φ2 ),
2
2
2
2
its holomorphic projection vanishes. Therefore,
f (E 0 (0, ı(g1 , g2∨ ), φ1 ⊗ φ2 ) − F 0 (0; g1 , g2 ; φ1 , φ2 ))
(7.10) = Pr
f (E 0 (0, ı(g1 , g2∨ ), φ1 ⊗ φ2 )) − Pr
f (F 0 (0; g1 , g2 ; φ1 , φ2 ))
= Pr
f (E 0 (0, ı(g1 , g2∨ ), φ1 ⊗ φ2 )) − Pr
f E 0 ( 1 , g1 , φ1 )E( 1 , g2 , φ2 ) − Pr
f E( 1 , g1 , φ1 )E 0 ( 1 , g2 , φ2 ) .
= Pr
2
2
2
2
163
But for {α, α0 } = {1, 2},
f E 0 ( 1 , gα , φα )E( 1 , gα0 , φα0 ) = Pr
f E( 1 , gα0 , φα0 )
f E 0 ( 1 , gα , φα ) Pr
Pr
2
2
2
2
1
1
f E 0 ( , gα , φα ) E∗ ( , gα0 , φα0 ),
= Pr
2
2
where
X
1
1
1
E∗ ( , gα , φα ) =
Wt ( , gα , φα ) = E( , gα , φα ) − (ωχ (gα )φα ) (0).
2
2
2
t∈F
In summary, we have the following proposition.
Proposition 7.1.6. Assume that there is a finite place p such that φα,p (0) = 0 for α = 1, 2. Then
for gα ∈ Pp0 H 0 (
ApF ),
Pr (E 0 (0, ı(g1 , g2∨ ), φ1 ⊗ φ2 ))
f (E 0 (0, ı(g1 , g ∨ ), φ1 ⊗ φ2 )) − Pr
f E 0 ( 1 , g1 , φ1 ) E( 1 , g2 , φ2 ) − E( 1 , g1 , φ1 )Pr
f E 0 ( 1 , g2 , φ2 ) .
= Pr
2
2
2
2
2
7.1.4
Quasi-holomorphic projection of analytic kernel functions
ASF ) under the following assumptions:
f (E 0 (0, ı(g1 , g ∨ ), φ1 ⊗ φ2 )) for gα ∈ eS H 0 (
We calculate Pr
2
• φα = φ0∞ φα,fin (α = 1, 2) are decomposable;
• φ1,S ⊗ φ2,S is in S(
• φ1,v ⊗ φ2,v ∈ S(
VS2)reg ;
Vv2 )reg,d
v
for every v ∈ S that is nonsplit and some dv ≥ dψv (cf. 2.4.2).
Then we recall the formula (2.22)
E 0 (0, ı(g1 , g2∨ ), φ1 ⊗ φ2 ) =
X
Ev0 (0, ı(g1 , g2∨ ), φ1 ⊗ φ2 ).
v6∈S
f to the above summation, all terms except Eι0 (0, ı(g1 , g ∨ ), φ1 ⊗ φ2 )
If we apply the formula defining Pr
2
f (Eι0 (0, ı(g1 , g ∨ ), φ1 ⊗ φ2 )). By
for ι ∈ Σ∞ will not change. Therefore, we fix a ι ∈ Σ∞ and consider Pr
2
Theorem 4.3.4, it amounts to consider
f (−2 Vol(K)h(Zφ (g1 ), Ξφ (g1 )ι0 ) , (Zφ (g2 ), Ξφ (g2 )ι0 )iSh ) .
Pr
1
1
2
2
K
164
For simplicity, we will omit the term −2 Vol(K) in the following computation. Recall (4.56),
h(Zφ1 (g1 ), Ξφ1 (g1 )ι0 ) , (Zφ2 (g2 ), Ξφ2 (g2 )ι0 )iShK


l Z
X
X


=
(ωχ (g1 )φ1 ) (T (x1 ), h−1
i x1 )Ξx1 a(g1 )
i=1
Q
(H( )∩K)\D
x1 ∈V n ,T (x1 )∈Her+
n (E)


X
∗
,
(ωχ (g2 )φ2 ) (T (x2 ), h−1
i x2 )Ξx2 a(g2 )
x2 ∈V n ,T (x2 )∈Her+
n (E)
where a(gα ) is denoted as aα before. Since it is symmetric in g1 and g2 , we only need to compute


X
f
Pr
x∈V
(ωχ (•)φ) (T (x), x)Ξxa(•)  .
(7.11)
n ,T (x)∈Her+ (E)
n
By Definition 7.1.3,


X
(7.11) =
X
f
Pr
t∈F +
(ωχ (•)φ) (t, x)Ξxa(•) 
T (x)=t
.
ψt
For each term,


f
Pr
X
(ωχ (•)φ) (t, x)Ξxa(•) 
T (x)=t
(g)
ψt
kχ
2
X Z
= consts→0 (4πt)Wt (gι )
T (x)=t
λP 0 (h)s (ωχ (g ι h)φ) (t, x)Ξxa(h) dh.
(7.12)
Zι0 Nι0 \Hι0
Taking substitution y = aa, we have
kχ
X
(7.12) = consts→0 (4πt)Wt 2 (gι )
(ωχ (g ι )φι ) (t, x)
Z
= consts→0 (4πt)
Z
(ωχ (g)φ) (t, x)
T (x)=t
∞
Ξx√y y s exp(−4πty)dy.
0
Let
δx (z) =
Ξx√y y s exp(−4πty)dy
0
T (x)=t
X
∞
R(x, z)
(xz , xz )
=−
.
2t
(x, x)
(7.13)
165
Then
∞
Z
∞
exp(−4πtyuδx (z))
du y s exp(−4πty)dy
u
1
0
T (x)=t
Z
Z
∞
∞
X
exp(−4πyuδx (z)) s
= consts→0 (4πt)
(ωχ (g)φ) (t, x)t−1−s
y exp(−4πty)dudy
u
1
0
T (x)=t
Z ∞ Z ∞
X
1
−1−s
exp (−4πy(1 + uδx (z))) du
= consts→0 (4πt)
(ωχ (g)φ) (t, x)t
u
0
1
T (x)=t
Z
X
Γ(1 + s) ∞
du
= consts→0 (4πt)
(ωχ (g)φ) (t, x)
1+s
(4πt)
u(1 + uδx (z))1+s
1
T (x)=t
Z ∞
X
du
= consts→1
.
(7.14)
(ωχ (g)φ) (t, x)
u(1
+
uδx (z))s
1
X
(7.13) = consts→0 (4πt)
Z
(ωχ (g)φ) (t, x)
T (x)=t
Following [GZ1986], we introduce the Legendre function of the second type as follows
Z
Qs−1 (t) =
∞
t+
p
−s
du,
t2 − 1 cosh u
t > 1, s > 0.
0
Then the admissible Green function attach to the divisor
Ξadm
(g)t = consts→1 2
φ
X
P
T (x)=t
(ωχ (g)φ) (t, x)Zx is
(ωχ (g)φ) (t, x)Qs−1 (1 + 2δx (z)).
T (x)=t
By [GZ1986], we have
Z
1
∞
du
= 2Qs−1 (1 + 2c) + O(c−s−1 ),
u(1 + uc)s
c → +∞.
(7.15)
Combining (7.14), (7.15), Corollary 5.3.4 and Proposition 7.1.6, we have the following proposition.
Proposition 7.1.7. Assume the three assumptions on φα at the beginning of the subsection. Then
the following identity
Pr (E 0 (0, ı(g1 , g2∨ ), φ1 ⊗ φ2 ))
= − Vol(K)
X
f E 0 ( 1 , g2 , φ2 )
bφ (g1 ), Zbφ (g2 )iv◦ − Pr
f E 0 ( 1 , g1 , φ1 ) E( 1 , g2 , φ2 ) − E( 1 , g1 , φ1 )Pr
hZ
1
2
2
2
2
2
v ◦ |v,v6∈S
ASF ). In particular, the archimedean local height paring is computed via admissible
holds for gα ∈ eS H 0 (
Green functions.
166
7.2
Proof of the main theorem
We prove the arithmetic inner product formula for n = 1.
7.2.1
Difference of kernel functions
We assume that
• φα = φ0∞ φα,fin (α = 1, 2) are decomposable;
• φα,S ∈ S(
VS)reg (α = 1, 2);
• φ1,S ⊗ φ2,S is in S(
• φ1,v ⊗ φ2,v ∈ S(
VS2)reg ;
Vv2 )reg,d
v
for every v ∈ S that is nonsplit and some dv ≥ dψv (cf. 2.4.2).
Let
E(g1 , g2 ; φ1 ⊗ φ2 ) = Pr (E 0 (0, ı(g1 , g2∨ ), φ1 ⊗ φ2 )) −
E(g1, g2; φ1 ⊗ φ2),
which is a function in A(H 0 × H 0 , χ0 ). By (3.15), Propositions 6.1.2, 6.2.4, 6.3.4 and 7.1.7, we have
that for gα ∈ eS H 0 (
ASF ) (α = 1, 2), E(g1, g2; φ1 ⊗ φ2) is the sum of the following terms:
EI (g1 , g2 ; φ1 ⊗ φ2 ) = −E(g1 , φ1 )A(g2 , φ2 ) − A(g1 , φ1 )E(g2 , φ2 ) − CE(g1 , φ1 )E(g2 , φ2 );
X
EII (g1 , g2 ; φ1 ⊗ φ2 ) =
A(p◦ ) (g1 , φ1 )E(g2 , φ2 );
p◦ |p
p∈S
X
EIII (g1 , g2 ; φ1 ⊗ φ2 ) =
E(p◦ ) (g1 , φ1 ; g2 , φ2 );
p◦ |p split
p∈S
X
EIV (g1 , g2 ; φ1 ⊗ φ2 ) =
hor
∨
ver
θ(p
◦ ) (ı(g1 , g2 ); φ1 , φ2 ) + θ(p◦ ) (g1 , φ1 ; g2 , φ2 );
◦
p |p non-split
p∈S
1
1
1
0 1
EV (g1 , g2 ; φ1 , φ2 ) = −Pr E ( , g1 , φ1 ) E( , g2 , φ2 ) − E( , g1 , φ1 )Pr E ( , g2 , φ2 ) .
2
2
2
2
0
Let π be an irreducible cuspidal automorphic representation of H 0 such that π∞ is a discrete series of
weight (1 −
kχ
2 ,1
+
kχ
2 ),
and (π, χ) = −1. Then for every f ∈ π and f ∨ ∈ π ∨ , the integral
ZZ
A
[eS H 0 (
S )]2
F
f (g1 )f ∨ (g2∨ )χ−1 (det g2 )E♥ (g1 , g2 ; φ1 ⊗ φ2 ) = 0,
for ♥ = I, II, III, IV, V. This is due to the facts that each term inside E♥ involves only Eisenstein
ASF ) for at least one α, and that eSH 0(ASF )
series, automorphic characters or theta series in gα ∈ eS H 0 (
167
AF ).
is a dense subset of H 0 (F )\H 0 (
7.2.2
The final step
From the above subsection, we obtain the following identity.
ZZ
[H 0 (F )\H 0 (
A
f (g1 )f ∨ (g2 )χ−1 (det g2 )E 0 (0, ı(g1 , g2 ), φ ⊗ φ∨ )
2
F )]
ZZ
=
A
[H 0 (F )\H 0 (
2
F )]
E
f (g1 )f ∨ (g2 )χ−1 (det g2 ) (g1 , g2∨ ; φ ⊗ φ∨ )
ZZ
= Vol(K)
A
[H 0 (F )\H 0 (
=
F
)]2
f (g1 )f ∨ (g2 )hΘφ (g1 ), Θφ∨ (g2 )iK
NT
∨
hΘfφ , Θfφ∨ iNT ,
(7.16)
where (φ, φ∨ ) satisfy the assumptions in the previous subsection.
Theorem 7.2.1 (Arithmetic inner product formula). Let π be an irreducible cuspidal automorphic
AF ), χ a character of E ×A×F \A×E , such that π∞ is a discrete series representation of weight (1 − k2 , 1 + k2 ), (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 .
representation of H1 (
χ
χ
∞
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
∨
hΘfφ , Θfφ∨ iNT =
Y
L0 ( 12 , π, χ)
Z ∗ (0, χv , fv , fv∨ , φv ⊗ φ∨
v ),
LF (2)L(1, E/F ) 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∨ ) =
Y
Z ∗ (0, χv , fv , fv∨ , φv ⊗ φ∨
v)
v
in
N
v∈Σ
HomHv0 ×Hv0 (R(
Vv , χv ), πv∨ χv πv ), which is nonzero since V ∼= V(π, χ). By Proposition
2.4.10 and the assumption that π∞ is a discrete series representation of weight (1 −
kχ
kχ
2 , 1 + 2 ),
168
∨
we can choose local components fv , fv∨ 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∨ ) := hΘfφ , Θfφ∨ iNT = Vol(K)hΘfφ , Θfφ∨ iK
NT ,
V
N
∨
0
0
which is also in
v∈Σ HomHv ×Hv (R( v , χv ), πv χv πv ). By Proposition A.2.1, the space
N
∨
0
0
v∈Σ HomHv ×Hv (R( v , χv ), πv χv πv ) has dimension 1. In other words, γ/α is a constant.
V
Applying (7.16) to the data we choose previously, and by (2.13), we have
L0 ( 12 , π, χ)
γ
=
.
α
LF (2)L(1, E/F )
Therefore, (2) of the theorem follows.
2. The functional γ defined above is zero since
V is not isometric to V(π, χ).
and f ∨ = f , then Θfφ = 0 since the paring h−, −iNT is positive definite.
If we take φ∨ = φ
169
Appendix
There are three appendix. In A.1, we prove the theta dichotomy for unitary groups over nonarchimedean fields. For symplectic-orthogonal pairs, this is proved by S. Kudla and S. Rallis. In
A.2, we prove a multiplicity one result in the theory of local theta correspondence in a special case.
In A.3, we study the theta correspondence of unramified representations for unitary groups, following
the method of S. Rallis.
170
A.1
Theta dichotomy for unitary groups
In this section, we prove the theta dichotomy for unitary groups over non-archimedean fields. For
symplectic-orthogonal pairs, this is proved by Kudla–Rallis in [KR2005]. We will follow the same
line1 .
Let F be a non-archimedean local field whose characteristic is not 2, and a quadratic field extension
E/F with Gal(E/F ) = {1, τ }. Let OF (resp. OE ) be the ring of integers, $ (resp. $E ) a uniformizer
of F (resp. E), and q = |OF /$OF | (resp. qE = |OE /$E OE |).
Let n ≥ 1 be an integer. We denote by Vn± the (left) hermitian spaces over E of dimension n such
that (Vn± ) = ±1, and Hn± = U(Vn± ). Let (W, h−, −i) be a (right) skew-hermitian space over E of the
same dimension n, and H 0 = U(W ).
As in 2.2.1, we have the doubling space W ⊕ (−W ), and H 00 = U(W ⊕ (−W )), where (−W ) =
(W, −h−, −i). We fix a character χ : E × →
C× such that χ|F
×
= nE/F . Let In (s, χ) be the space
of degenerate principal series, which is an admissible representation of H 00 . We have In (0, χ) =
R(Vn+ , χ) ⊕ R(Vn− , χ). For any irreducible admissible representation π of H 0 ,
HomH 0 ×H 0 (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 0 ×H 0 (R(Vn+ , χ), π χπ ∨ );
HomH 0 ×H 0 (R(Vn− , χ), π χπ ∨ ),
which is nonzero.
Proof. From the discussion above, we know that there is at least one space that is nonzero. If they are
both nonzero, then HomH 0 (S((Vn+ )n ), π) 6= 0 and HomH 0 (S((Vn− )n ), π) 6= 0, where H 0 acts on both
spaces of Schwartz functions through the Weil representation ωχ .
Let g0 ∈ GLF (W ) be a τ -linear automorphism such that
hw1 g0 , w2 g0 i = hw1 , w2 i
1 During the preparation of this article, Gong and Grenié [GG2011] also prove the same results independently. Our
methods are similar and we refer to their article for more details
171
for all w1 , w2 ∈ W . By [MVW1987],
π ◦ Ad g0 ∼
= π∨ ,
and we have
ωχ ◦ Ad g0 , S((Vn− )n ) ∼
= ωχ−1 , S((−Vn− )n ) .
Therefore,
HomH 0 (ωχ , S((Vn+ )n )), π =
6 0;
HomH 0 (ωχ−1 , S((−Vn− )n )), π ∨ 6= 0.
Taking product and the paring between π and π ∨ , we have
HomH 0 (ω1 , S((Vn+ ⊕ (−Vn− ))n )), 1 6= 0,
−
. We have
where 1 is the trivial representation of H 0 . It is easy to see that Vn+ ⊕ (−Vn− ) ∼
= V2n
− n
HomH 0 (S((V2n
) ), 1) 6= 0. By Lemma A.1.2 below, we have
−
HomH 0 ×H 0 (Rn (V2n
, 1), 1) 6= 0.
−
Then by [KS1997, Theorem 1.2 (4)], Rn (V2n
, 1) is the unique irreducible submodule of In ( n2 , 1), which
is also the image of the intertwining map
n
n
−
M ∗ (− , 1) : In (− , 1) → Rn (V2n
, 1).
2
2
By [KS1997, Proposition 4.3 (2)], ker(M ∗ (− n2 , 1)) =
C · ϕ0, where ϕ0 ∈ In(− n2 , 1) is the constant
function with value 1. Therefore, there is a nonzero functional
n
ζ ∈ HomH 0 ×H 0 In (− , 1), 1
2
such that ζ(ϕ0 ) = 0. By Lemma A.1.3 below,
n
HomH 0 ×H 0 In (− , 1), 1 =
2
C · Z(− n2 , 1, 1, −)
and Z(− n2 , 1, 1, ϕ0 ) 6= 0, which is a contradiction. The proposition is proved.
172
Lemma A.1.2. For any hermitian space V over E of dimension m and χ such that χ|F × = m
E/F ,
the following are equivalent:
• HomH 0 ((ωχ , S(V n )), π) 6= 0;
• HomH 0 ×H 0 (Rn (V, χ), π χπ ∨ ) 6= 0.
Proof. Since HomH 0 ((ωχ , S(V n )), π) 6= 0 is equivalent to Θχ (π, V ) 6= 0, and Θχ (π, V ) admits an
irreducible quotient by [MVW1987, Chapitre 3, IV, Theorème 4 (2-a)], the lemma follows by [HKS1996, Proposition 3.1].
Before we proceed, let us recall some notations from [HKS1996, Section 4]. Let Y ⊂ W ⊕ (−W ) be
the graph of the identity map and PY ⊂ H 00 the parabolic subgroup that stabilizes Y . Let r0 be the
Witt index of W . Then PY \H 00 can be canonically identified with the set of isotropic n-dimensional
subspaces of W ⊕ (−W ). The H 0 × H 0 -orbit of Z in PY \H 00 is uniquely determined by
d = dim(Z ∩ W ) = dim(Z ∩ (−W )).
Therefore,
H 00 =
a
Ωd :=
0≤d≤r0
a
PY δd (H 0 × H 0 )
0≤d≤r0
for some representative δd ∈ H 00 . The (topological) closure
Ωr =
a
Ωr 0 .
r 0 ≥r
Let
In (s, χ) = In(r0 ) (s, χ) ⊃ In(r0 −1) (s, χ) ⊃ · · · ⊃ In(0) (s, χ)
be the filtration given by support, and
0
s+ r2
s+ r
×H 0
(r)
(r−1)
0
Q(r)
(s, χ) ∼
χ| • |E 2 ⊗ S(Hn−2r
) ,
= IndH
n = In (s, χ) ⊃ In
Pr ×Pr χ| • |E
where the induction is normalized. Here Pr is the parabolic subgroup of H 0 with the Levi quotient
0
0
Mr isomorphic to GLr (E) × Hn−2r
, and the maximal unipotent subgroup Nr , where Hn−2r
is the
unitary group of some skew-hermitian space of dimension n − 2r.
173
For every section ϕs ∈ In (s, χ) and matrix coefficient φ of π ∨ , we define the zeta integral
Z
Z(s, χ, φ, ϕ) =
φ(g)ϕs ((g, 1))dg.
H0
If the section is standard, it has a meromorphic continuation to the entire complex. If for some s0 ∈
C
at which every such continuation is holomorphic, then the zeta integral defines a nonzero element
Z(s0 , χ, −, −) ∈ HomH 0 ×H 0 (In (s0 , χ), π χπ ∨ ).
Moreover, if HomH 0 ×H 0 (Qr (s0 , χ), πχπ ∨ ) = 0 for every r ≥ 1, then all zeta integrals are holomorphic
at s0 , and
HomH 0 ×H 0 (In (s0 , χ), π χπ ∨ ) =
C · Z(s0, χ, −, −).
For the trivial representation, we have the following lemma.
Lemma A.1.3. For χ = 1 and s0 = − n2 , HomH 0 ×H 0 (Qr (s0 , 1), 1) = 0 for every r ≥ 1. In particular,
Z(s0 , 1, 1, ϕ0 ) 6= 0.
Proof. For r ≥ 1 and every irreducible admissible representation π of H 0 ,
HomH 0 ×H (Qr (s, 1), π π ∨ )
0
−s− r2
−s− r
×H 0
0
= HomH 0 ×H 0 π ∨ π, IndH
| • |E 2 ⊗ C ∞ (Hn−2r
))
Pr ×Pr (| • |E
−s+ n −r
−s+ n −r
0
) .
= HomMr ×Mr (π ∨ )Nr πNr , | • |E 2 | • |E 2 ⊗ C ∞ (Hn−2r
In particular, when π = 1, we have
HomH 0 ×H (Qr (s, 1), 1)
−s+ n −r
−s+ n −r
0
= HomMr ×Mr 1, | • |E 2 | • |E 2 ⊗ C ∞ (Hn−2r
)
−s+ n −r
−s+ n −r
= HomGLr (E)×GLr (E) 1, | • |E 2 | • |E 2
.
Therefore, if s0 = − n2 , then HomH 0 ×H 0 (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 ϕ0s |K = 1, where K is the maximal compact
Qn−1
subgroup of H 00 . Let bn (s) = i=0 L(2s + n − i, iE/F ). Then we have three cases:
174
1. If n = 2m (m ≥ 1) and r0 = m,
m−1
Y
1
i−s −1
Z(s, 1, 1, ϕ ) =
1 − qE
.
bn (s) i=−m
0
2. If n = 2m (m ≥ 1) and r0 = m − 1,
Z(s, 1, 1, ϕ0 ) =
Y
m−1
1
−s
i−s −1
1 − qE
1 − qE
.
bn (s)
i=−m
3. If n = 2m + 1 (m ≥ 0) and r0 = m,
m Y
1
i−s− 1 −1
.
1 − qE 2
bn (s) i=−m
Z(s, 1, 1, ϕ0 ) =
Proof. For (1) and (3), if E/F is unramified, they are [GPSR1987, Part A, Proposition 6.2] (for n
even) and [Li1992, Theorem 3.1] (for n odd). The calculation for the integral
Z
H0
ϕ0s ((g, 1))dg
in these papers works also for E/F ramified, since the only thing we need to check is the ratio cw (χ)
of the intertwining operator (cf. [Li1992, Page 189]). The formula for such ratio holds more generally,
in particular, when E/F is ramified, by [Cas1980, Theorem 3.1].
For (2), one can prove similarly as in [Li1992, Section 3]. In fact, (1) is enough for our use.
A.2
Uniqueness of local invariant functionals
In this section, we fix a place v ∈ Σ and suppress it from notations. We prove that the space
HomH 0 ×H 0 (I2 (0, χ), π ∨ χπ) is of dimension 1, following [HKS1996]. Here, χ is a character of E × . If
χ is trivial on F × , i.e., χ−1 = χτ , then we define a character πχ of H 0 as follows. For every element
g ∈ H 0 , det g is in E ×,1 . In particular, det g = eg /eτg for some eg ∈ E × by Hilbert’s Theorem 90. Set
πχ (g) = χ(eg ) that is independent of eg we choose.
Proposition A.2.1. Let χ be a character of E × , and π an irreducible admissible representation of
H 0 that is not πχ−1 if χ−1 = χτ . Then we have
dimC HomH 0 ×H 0 (I2 (0, χ), π ∨ χπ) = 1,
175
and L(s, π, χ) is holomorphic at s = 21 .
In fact, the proposition holds for arbitrary representation π. Since we do not need such generality
in this article, we put this condition for simplicity. In what follows, we also assume that v is finite for
simplicity. The case for archimedean places is similar, and actually will not be used in the article.
Recall from 2.2.1, we have, as a special case, the following decomposition of double cosets H 00 =
`
Ω0 Ω1 , where Ω0 = P γ0 ı(H 0 × H 0 ) is open and Ω = P ı(H 0 × H 0 ) is closed. Therefore, we have a
(0)
subspace I2 (0, χ) ⊂ I2 (0, χ), where
(0)
I2 (0, χ) = {ϕ ∈ I2 (0, χ) | Supp ϕ ⊂ Ω0 }
that is invariant under the action of H 0 × H 0 by right translation via ı. As H 0 × H 0 -representations,
we denote
(0)
(0)
(1)
Q2 (0, χ) = I2 (0, χ);
(0)
Q2 (0, χ) = I2 (0, χ)/I2 (0, χ).
We have a linear isomorphism
(0)
Q2 (0, χ) → S(H 0 )(1 ⊗ χ)
ϕ 7→ Ψ(g) = ϕ (γ0 ı(g, 12 )) ,
where S(H) denotes the space the Schwartz functions on H 0 that is viewed as a representation of
H 0 × H 0 . Since
ϕ (γ0 ı(g, 12 )ı(g1 , g2 )) = ϕ γ0 ı(g2 , g2 )ı(g2−1 gg1 , 12 ) = χ(det g2 )ϕ γ0 ı(g2−1 gg1 , 12 ) ,
the above isomorphism is H 0 × H 0 -equivariant. There is a unique, up to constant, H 0 × H 0 -invariant
functional on S(H 0 ) ⊗ (π π ∨ ) given by
Ψ ⊗ (f ⊗ f ∨ ) 7→
Z
hπ(g)f, f ∨ iΨ(g)dg.
H0
Since
C
HomH 0 ×H 0 (S(H 0 ) ⊗ (π π ∨ ) , ) = HomH 0 ×H 0 (S(H 0 ), π ∨ π)
(0)
= HomH 0 ×H 0 (S(H 0 ) ⊗ (1 χ) , π ∨ χπ) = HomH 0 ×H 0 Q2 (0, χ), π ∨ χπ ,
176
we have
(0)
dimC HomH 0 ×H 0 Q2 (0, χ), π ∨ χπ = 0.
(1)
For Q2 (0, χ), we have the following lemma.
Lemma A.2.2. If χ−1 6= χτ , or χ−1 = χτ but π 6= πχ−1 , then
(1)
HomH 0 ×H 0 Q2 (0, χ), π ∨ χπ = 0.
Proof. It is easy to see that the following map
1
1
(1)
Q2 (0, χ) → I1 ( , χ) I1 ( , χ)
2
2
ϕ̄ 7→ ((g1 , g2 ) 7→ ϕ̄ (ı(g1 , g2 )))
is H 0 × H 0 -equivariant isomorphism. Therefore,
(1)
HomH 0 ×H 0 Q2 (0, χ), π ∨ χπ
1
1
= HomH 0 ×H 0 I1 ( , χ) I1 ( , χ), π ∨ χπ
2
2
1
1
= HomH 0 ×H 0 π χ−1 π ∨ , I1 (− , χ−1 ) I1 (− , χ−1 ) .
2
2
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 0 -submodule properly
contained in I1 (− 21 , χ−1 ) is isomorphic to πχ−1 . Therefore, the lemma follows by our assumption.
Proof of Proposition A.2.1. The normalized zeta integral (2.7) has already defined a nonzero element in HomH 0 ×H 0 (I2 (0, χ), π ∨ χπ). Therefore, the dimension is at least 1. If it is greater than
(0)
one, we can find a nonzero element in HomH 0 ×H 0 (I2 (0, χ), π ∨ χπ) whose restriction to I2 (0, χ)
(0)
is zero since dimC HomH 0 ×H 0 Q2 (0, χ), π ∨ χπ = 1. Then it defines a nonzero element in
(1)
HomH 0 ×H 0 Q2 (0, χ), π ∨ χπ that is 0 by the above lemma. Therefore,
dimC HomH 0 ×H 0 (I2 (0, χ), π ∨ χπ) = 1.
(0)
For the L-factor, the restriction of the normalized zeta integral to I2 (0, χ) is nonzero. Since the
(0)
original zeta integral is absolutely convergent at s = 0 if ϕ ∈ I2 (0, χ), L(s, π, χ) can not have a pole
at s = 12 , by realizing that b2 (s) is holomorphic and nonzero at s = 0.
177
A.3
Theta correspondence of unramified representations
In this section, we study the theta correspondence of unramified representations for unitary groups.
Let F be a p-adic local field with p 6= 2, E/F an unramified quadratic field extension with
Gal(E/F ) = {1, τ }. Let OF (resp. OE ) be the ring of integers of F (resp. E), $ a uniformizer of OF ,
and q the cardinality of OF /$OF . Let ψ be an unramified additive character of F , which determines
an unramified additive character of E by composing with 21 TrE/F . Let dx be the selfdual Haar
dx
measure of E with respect to ψ ◦ 12 TrE/F , and d× x = |x|
the Haar measure of E × , normalized
E
such that |$|E = q −2 . We will use slightly different notations from 2.1.1.
Let n, m ≥ 1 be two integers, and r = min{m, n}. Let (Wn , h−, −i) 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 

∗
}. Let Hn0 = U(Wn ) (resp. Hm = U(Vn )) be the corresponding
under a basis {f1 , . . . , fm ; f1∗ , . . . , fm
group of isometries, and
Kn0 = Hn0 ∩ GL(OE he1 , . . . , en ; e∗1 , . . . , e∗n i) (resp. Km = Hm ∩ GL(OE hf1 , . . . , fm ; f1∗ , . . . , fn∗ i))
be a hyperspecial maximal subgroup. We have a Weil representation ω = ωχ=1,ψ on the space S(Vmn )
of Schwartz functions, whose formulae are given in 2.1.1.
∗
∗
∗
∗
Let Wn,i
= spanE {e∗i+1 , . . . , e∗n } for 0 ≤ i ≤ n, and Vm,j
= spanE {fj+1
, . . . , fm
} for 0 ≤ j ≤ m.
∗
∗
, respectively as
Then we have filtration of the maximal isotropic subspaces Wn,0
and Vm,0
∗
∗
∗
Wn,0
⊃ Wn,1
⊃ · · · ⊃ Wn,n
= {0};
∗
∗
∗
Vm,0
⊃ Vm,1
⊃ · · · ⊃ Vm,m
= {0}.
0
Up to conjugacy, the maximal parabolic subgroups of Hn0 ×Hm are precisely those subgroups Pn,i
×Pm,j
∗
∗
that consists of elements (h0 , h) stabilizing the subspace Wn,i
⊗Vm,j
⊂ Wn ⊗Vm , for i = 0, 1, . . . , n and
0
0
0
j = 0, 1, . . . , m. Let Nn,i
× Nm,j be its unipotent radical. Then the Levi quotient Pn,i
× Pm,j /Nn,i
×
178
Nm,j is isomorphic to (GLn−i (E) × Hi0 ) × (GLm−j (E) × Hj ). For 0 ≤ t ≤ n, we define an algebraic
closed subsets Σt to be
Σt = {x = (x1 , . . . , xn ) ∈ Vmn | (xi , xj ) = 0 for t + 1 ≤ j ≤ n}.
We say that a function φ ∈ S(Vmn ) is spherical if it is invariant under the action of Kn0 × Km . The
same proof of [Ral1984, Proposition 2.2] implies the following lemma.
Lemma A.3.1. Let φ be a spherical function in S(Vmn ) such that for every h0 ∈ Hn0 , ω(h0 )φ vanishes
on the subset Σ0 . Then ω(h0 )φ vanishes identically.
∗
We identify Vmn with Mat2m,n (E) via the basis {f1 , . . . , fm ; f1∗ , . . . , fm
}.
Then the action of
GLn (E) × Hm is given by (A, h).X = hXA−1 . We have the following version of [Ral1984, Lemma 3.1],
(i)
Lemma A.3.2. Let Σ0
(i)
= {X ∈ Σ0 | rank X = i}. Then Σ0 , if nonempty, is an orbit under
(i)
(r)
GLn (E) × Hm ; and Σ0 is a disjoint union of orbits of the form Σ0 for i = 0, 1, . . . , r, in which Σ0
is the unique open one.
Let


 A

Bn0 = 



t
Aτ,−1
  1n




B 
|
A
is
lower
triangular
and
B
∈
Her
(E)
,

n


1n

whose Levi decomposition is Bn0 = Tn0 Un0 with
Tn0 = {diag[t1 , . . . , tn ; t1τ,−1 , . . . , tnτ,−1 ] | ti ∈ E × },
and Un0 begin the unipotent radical. Let
Bm,r


 A

= 



t
Aτ,−1
  1m



B 
 A1
|A=
1m


A2  
 ,

A3 
where A1 ∈ Matr,r (E) is lower triangular, A3 ∈ Matm−r,m−r (E) is upper triangular, A2 ∈ Matr,m−r (E),
0
and B is skew-hermitian. We have the Levi decomposition Bm,r = Tm Um,r with Tm = Tm
and Um,r
begin the unipotent radical. Then Bn0 × Bm,r is a minimal parabolic subgroup of Hn0 × Hm .
Review some facts about spherical representations of Hn0 × Hm . For ν = (ν1 , . . . , νn ) ∈
Cn, define
179
the space I 0 (ν) consisting of all locally constant functions ϕ : Hn0 →
0− 21
ϕ(h0 t0 u0 ) = δn
(t0 )
n
Y
C satisfying
ν
|t0i |Ei ϕ(h0 )
i=1
for all h0 ∈ Hn0 , t ∈ Tn0 , and u0 ∈ Un0 . Here,
δn0 =
n
Y
2i−1
|t0i |E
i=1
is the modulus function of Bn0 . These I 0 (ν) provide all spherical principal series of Hn0 . Let S(Hn0 //Kn0 )
be the spherical Hecke algebra of Hn0 . Then we have the Fourier–Satake isomorphism
FS : S(Hn0 //Kn0 ) →
C[X1, X1−1, . . . , Xn, Xn−1]W (H ),
0
n
such that for every f 0 ∈ S(Hn0 //Kn0 ),
FS(f 0 )(q 2ν1 , q −2ν1 , . . . , q 2νn , q −2νn ) = traceI 0 (ν) (f 0 ).
For µ = (µ1 , . . . µm ) ∈
C satisfying
Cm, define the space I(µ) consisting of all locally constant functions ϕ : Hm →
−1
2
(t)
ϕ(htu) = δm,r
m
Y
µ
|tj |Ej ϕ(h)
j=1
for all h ∈ Hm , t ∈ Tm , and u ∈ Um,r . Here,
δm,r (t) =
r
Y
2m−2r+2j−1
|tj |E
j=1
m
Y
2m−2j+1
|tj |E
j=r+1
is the modulus function of Bm,r . These I(µ) provide all spherical principal series of Hm . Let
S(Hm //Km ) be the spherical Hecke algebra of Hm . Then we have the Fourier–Satake isomorphism
FS : S(Hm //Km ) →
C[X1, X1−1, . . . , Xm, Xm−1]W (H
m)
,
such that for every f ∈ S(Hm //Km ),
FS(f )(q 2µ1 , q −2µ1 , . . . , q 2µm , q −2µm ) = traceI(µ) (f ).
Let Yr ⊂ GLr (E) be the group of lower-triangular matrices, which has the Levi decomposition
180
Yr = Ar Lr . Here, Ar = {diag[a1 , . . . , ar ]|ai ∈ E × } and Lr is the unipotent radical. We write elements
in Lr in the form (ljk )j>k . The group Yr has the following right invariant measure
dyr =
r
Y
2i−(r+1) ×
|ai |E
i=1
d ai
Y
d̃ljk ,
1≤k<j≤r
where d̃ljk is certain measure on Lr normalized as in [Ral1982, Page 490]. Let σ = (σ1 , . . . , σr ) ∈
Cr
such that Re σi 0. For every function φ ∈ S(Vmn ), the following integral

  yr
Zσ (φ) =
φ 
Yr
0
Z

r
0  Y
σ
|ai |E dyr

0
i=1
is absolutely convergent. Define a map Zσ sending φ ∈ S(Vmn ) to the function
(h0 , h) 7→ Zσ ω(h0−1 , h−1 )φ
0
-equivariant map from S(Vmn ) to S(Hn0 × Hm ).
on Hn0 × Hm . It is a nonzero Hn0 × Hm
Lemma A.3.3. For σ = (σ1 , . . . , σr ) ∈
Cr such that Re σi 0, the image of the above intertwining
map Zσ lies in I 0 (ν) ⊗ I(µ), where
3
1
1
3
;
ν = 2 + σ1 − m − , . . . , 2r + σr − m − , (r + 1) − m − , . . . , n − m −
2
2
2
2
3
3
1
1
µ = −2 − σ1 + m + , . . . , −2r + σr + m + , −(r + 1) + m + , . . . ,
.
2
2
2
2
Proof. We have


r
y
0
 r
 Y
Zσ (φ)(h0 t0 u0 , htu) =
ω u0−1 t0−1 h0−1 , u−1 t−1 h−1 φ 
|ai |σE dyr

Yr
0 0
i=1


Z
r
 yr 0  Y
=
ω t0−1 h0−1 , t−1 h−1 φ 
|ai |σE dyr

Yr
0 0
i=1
 


Z
r
y
0
−m
  r
 0−1  Y
=
|det t0 |E ω h0−1 , h−1 φ t 
|ai |σE dyr .
t 
Yr
0 0
i=1
Z
(A.1)
181
Changing variable yr 7→ yr00 = diag[t1 , . . . , tr ]yr , we have
r
Y
|ai |σE dyr =
i=1
r
Y
r−3i−σi +2
|ti |E
i=1
r
Y
|ti ai |σE dyr00 .
i=1
0−1
Changing variable yr 7→ yr0 = diag[t0−1
1 , . . . , tr ], we have
r
Y
|ai |σE dyr =
i=1
r
Y
i+σi −1
|ti |E
i=1
r
Y
0−1 σ 0
t ai dyr .
i
E
i=1
Therefore,
(A.1) =
r
Y
i+σi −m−1
|t0i |E
i=1
=
r
Y
n
Y
−m
|t0i |E
i=r+1
i+σi −m−1
|t0i |E
i=1
n
Y
r
Y

r−3j−σj +2
Z
|tj |E
Yr
j=1
−m
|t0i |E
i=r+1
r
Y
r−3j−σj +2
|tj |E

r
0  Y
|ai |σE dyr

0
i=1
  yr
ω h0−1 , h−1 φ 
0
(Zσ (φ)) (h0 , h).
j=1
The lemma follows immediately.
The above lemma implies the following facts. If m ≥ n = r, then there is a surjective homomorphism
Φm,n : S(Hm //Km ) → S(Hn0 //Kn0 )
satisfying that
Zσ ◦ (Φm,n (f ) − f ) = 0
for all f ∈ S(Hm //Km ) and Re σi 0. Using the Fourier–Satake isomorphism, the map Φm,n is given
by
C[X1, X1−1, . . . , Xm, Xm−1]W (H
m)
→
C[X1, X1−1, . . . , Xn, Xn−1]W (H ),
0
n
where
logq Xj 7→ logq Xj ,
j = 1, . . . , n;
logq Xj 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
Φ0n,m : S(Hn0 //Kn0 ) → S(Hm //Km )
182
satisfying that
Zσ ◦ (f 0 − Φ0n,m (f 0 )) = 0
for all f 0 ∈ S(Hn0 //Kn0 ) and Re σi 0. Using the Fourier–Satake isomorphism, the map Φ0n,m is given
by
C[X1, X1−1, . . . , Xn, Xn−1]W (H ) → C[X1, X1−1, . . . , Xm, Xm−1]W (H
0
n
m)
,
where
logq Xi 7→ logq Xi ,
i = 1, . . . , m;
logq Xi 7→ 2m − 2i + 1,
i = m + 1, . . . , n.
Lemma A.3.4. Suppose that φ ∈ S(Vmn ) is spherical, and Zσ (φ) = 0 for all σ ∈
Cr
such that
Re σi 0. Then ω(h0 )φ vanishes on Σ0 for all h0 ∈ Hn0 .
(r)
Proof. It suffices to show that ω(h0 )φ vanishes on Σ0
(r)
since it is dense open in Σ0 . Since Σ0
is
transitive under the action of GLn (E) × Hm , we need only to show that


 1r
(ω(h0 , h)φ) 
0
0 
 = 0
0
for all (h0 , h) ∈ GLn (E) × Hm . We can write h0 = b0 k 0 with b0 ∈ Bn0 , k 0 ∈ Kn0 , and h = bk with
0
b ∈ Bm,r
, k ∈ Km . Since φ is spherical by assumption, we have

  1r
(ω(h0 , h)φ) 
0


0 
 1r
0
 = (ω(b , b)φ) 
0
0


0 
 X
 = φ 
0
0

0 

0
with X ∈ Matr,r (E). Therefore, the lemma follows from [Ral1982, Lemma 5.2] for k = E.
Combining Lemmas A.3.1, A.3.3 and A.3.4, we have the following proposition.
Proposition A.3.5. The ideal
In,m = {f ∈ S(Hn0 //Kn0 ) ⊗ S(Hm //Km ) | ω(f ) = 0}
is generated by
{Φm,n (f ) − f | f ∈ S(Hm //Km )} (resp. {f 0 − Φ0n,m (f 0 ) | f 0 ∈ S(Hn0 //Kn0 )})
183
if m ≥ n (resp. m < n).
When E = F ⊕ F , the corresponding unitary group Hn0 (resp. Hm ) can be identified with GLn (F )
(resp. GLm (F )). The Weil representation ω, which realizes on the space S(Matm,n (F )), is simply given
by the formula (ω(g 0 , g)φ) (x) = φ(g −1 xg 0 ) for g 0 ∈ GLn (F ), g ∈ GLm (F ) and φ ∈ S(Matm,n (F ))
(cf. [Ral1982, Section 6]). Without lost of generality, we assume that n ≥ m. Then the ideal
Jn,m = {f ∈ S(GLn (F )// GLn (OF )) ⊗ S(GLm (F )// GLm (OF )) | ω(f ) = 0}
is generated by
{f − Ψn,m (f ) | f ∈ S(GLn (F )// GLn (OF ))}.
In terms of the Fourier–Satake isomorphism, the surjective homomorphism Ψn,m is given by
C[X1, X1−1, . . . , Xn, Xn−1]W (GL (F )) → C[X1, X1−1, . . . , Xm, Xm−1]W (GL
n
m (F ))
,
where
logq Xi 7→ − logq Xi +
logq Xi 7→ −i +
n+1
,
2
n−m
,
2
i = 1, . . . , m;
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 Hn0 and Hn are isomorphic.
1. Let π be an unramified irreducible admissible representation of Hn0 . Then the theta correspondence of π to Hn , with respect to the Weil representation ω, is nontrivial and isomorphic to
π.
2. Let π be an unramified irreducible admissible representation of GLn (F ), and χ an unramified
character of F × . Then the theta correspondence of π to GLn (F ), with respect to the Weil
representation ωχ defined by (ωχ (g 0 , g)φ) (x) = χ(det g 0 )φ(g −1 xg 0 ) for φ ∈ S(Matn,n (F )), is
nontrivial and isomorphic to π ∨ ⊗ χ.
184
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Index
L -admissible Green function, 61
height function on D, 86
0-dimensional Shimura variety, 106
height paring
Beilinson–Bloch, 59
admissible (E-subspace), 43
Néron–Tate, 61
analytic kernel function, 31
modified, 61
archimedean local arithmetic Siegel–Weil forhermitian space, 9
mula, 95
hermitian line bundle, 61
arithmetic Chow group, 60
hermitian space
arithmetic divisor, 60
nearby, 42
flat, 60
nondegenerate (skew-), 10
principal, 60
skew-, 9
arithmetic theta lifting, 56, 67
hermitian space (over adèles)
arithmetic theta series, 63
coherent, 16
basic unitary datum, 115
incoherent, 16
Beilinson–Bloch height paring, 59
nondegenerate, 16
holomorphic projection, 155
coherent hermitian space, 16
quasi-, 157
cohomologically trivial cycle, 60
compactified generating series, 54
incoherent hermitian space, 16
positive partial, 54
kernel function
compactified special cycle, 53
analytic, 31
degenerate principal series, 14
geometric, 67
flat arithmetic divisor, 60
local zeta integral, 23
generating series, 45
modified height pairing, 61
compactified, 54
moment matrix, 17
positive partial compactified, 54
Néron–Tate height paring, 61
geometric kernel function, 67
nearby hermitian space, 42
good section, 23
nearby quadratic space, 45
Green function, 60
L -admissible, 61
nondegenerate (skew-)hermitian space, 10
190
nondegenerate hermitian space (over adèles),
16
theta dichotomy, 25
theta function, 18
normalized zeta integral, 24
theta lifting, 21
of weight (a, b), 16
unramified section, 14
ordinary (point), 112
Weil representation, 17
positive partial compactified generating series,
Whittaker integral, 28
54
zeta integral
principal arithmetic divisor, 60
local, 23
quadratic space
nearby, 45
quasi-holomorphic projection, 157
regular test function, 32
section
good, 23
standard, 14
unramified, 14
Shimura variety
0-dimensional, 106
of orthogonal group, 45
of unitary group, 42
Siegel–Weil formula, 19
archimedean local arithmetic, 95
skew-hermitian space, 9
special cycle, 44
compactified, 53
standard section, 14
supersingular (point), 112
test function, 31
of discriminant d, 36
regular, 32
normalized, 24
191
Index of Notations
(−] , −[ ), 111
A0 (H 0 ), 155
Ak0 (H 0 × H 0 ), 155
Afin, A, 9
AK , AK,fin, K∞, 9
(E, ϑE , jE ), 115
E(g1, g2; Φ), 67
E(g, φ), 66
E(g, ϕs ), 14
E(p◦ ) (φ1 , g1 ; φ2 , g2 ), 138
(π, χ), 26
A(g, φ), 68
A(H), 29
A(H 0 , ζ k ), 155
Ak0 (H 0 ), 155
α(f, f ∨ , φ, φ∨ ), 27
(V ), 16
V
( ), 16
E(s, g, φ), 18
ET (s, g, Φ), 30
(E, ϑ, j; x), 115
am (s), 34
A(p◦ ) (φ1 , g1 ; φ2 , g2 ), 138
fψ0 ,s , 156
(A, θ, i), 111
γV , 17
β(f, f ∨ , φ, φ∨ ), 27
H̆, 116
bm (s), 24
H † , 107
CH1fin (M)C , 60
h† , 107
CH1 (M )0C , 60
Herr , 13
CH1 (M )C , 63
p
Hfin
, 106
1
dC (M), 60
CH
1
h(ι) , 42
dC (M)0 , 60
CH
Hp (T ), 124
CHr (Sh( ))C , 45
Hr , 13
CHr (Sh∼ )C , 54
H(x)∞ , H(T )∞ , 86
H
CHr (X), 10
ı, 20
consts=a , 157
ı0 , 20
dc , 60
ı, 17
D(g, φ), 63
ι(p) , 104
diag[a1 , . . . , ar ], 10
Ir (s, χ), 14
Diff(T,
V), 32
D(ι) , 42
ı̃χ,1 , 17
IV (g, φ), 18
192
K0± , 104
Mr , 13
kχ , 15
N , 21
Kp,n , 106
Nr , 13
Kr , 13
N , N 0 , 120
Λ± , 104
OT , 28
λPr , 18
Ω, Ω0 , 147
L†
, 108
†,p
0,Kp†,p ,K
L†K † ;p , 108
ωχ , 17
ωχ00 , 21
LK , 106
ω
bK , 67
LK , L , 43
LK;p◦ , 108
P , 21
LK∼ , L ∼ , 53
b 17
φ,
Ln,K p , 106
φ0∞ , 44
Ls , 105
ϕφ,s , 18
cs , 105
L
c
Pic(M
), 61
L(s, π, χ), 24
Pic(M )Q , 62
H
Pic(Sh( )), 43
M0,K p , 111
Pic(Sh∼ ), 53
[M0,K p ]ss , 112
[M0,K p ]∧
ss , 120
MK , 67
Mn,K p , 136, 139, 148
M0n,K p , 136, 139
0,Kp†,p ,K †,p
pM , 61
Pr , 13
Pr, 155
, 111
M † †,p †,p , 108
0,Kp ,K
[M† †,p †,p ]ss , 112
0,Kp ,K
†
MK † , 107
†
MK
† ;p ,
0
K
, 43
πK
$, 104
MK;p◦ , 68
M†
Pic(X), 11
108
MK , 62
f 157
Pr,
ψT , 28
RFp ,2 , 120
R(V, χ), 25
H
ShK (H), 42
Sh( ), 43
MK;p◦ , 108
Mn,K p , 106
Mor ((Y, jY ), (X, iX ))reg , 120
∼
Sh∼
K , Sh , 53
193
Σ, 13
V ± , 16
Σfin , 13
Vs , 16
Σ∞ , 13
V(π, χ), 26
Σ◦ , 13
Σ◦fin , 13
Σ◦∞ , 13
(−W ), 20
W 00 , 20
0
W 0 , W , 20
V
Σ( ), 16
Wr , 13
S, 42
WT (g, f ), 29
Std , 22
S(V 2n )d , 37
S(V
2n
)reg,d , 37
S(V r ), 17, 18
S(V r )reg , 32
VS2n)reg , 32
S(Vr )U , 44
S(Vr )U K , 44
S(
WT (g, ϕs ), 28
Wtk , 155
WT (s, g, Φ), 29
wr , 10
wr,d , 10
∞
(X, iX ), 120
∞
X, 111
TF1 0 , 104
TF , 104
Θfφ , 56
θφf , 21
X † , 107
X† , 111
Ξφ (g)ι0 , 102
Ξx , 86
θ(g, h; φ), 18
(Y, jY ), 115
b φ (g), 67
Θ
(Yτ , jYτ ), 122
hor
θ(p
◦ ) (•; φ1 , φ2 ), 144, 151
Y, 115
Θφ (g), 63
ver
θ(p
◦ ) (•; φ1 , φ2 , g2 ), 145, 153
Tp A, 110
T (x), 17
ULs , 105
Z† (x† , h† )0,Kp†,p ,K †,p , 117
Zφ (g)0,K p , 121
[Zφ (g)0,K p ]∧
s , 121
Z(x)0,K p , 117
Z(χ, f, f ∨ , ϕs ), 23
ζ k , 155
V̆ , 116
V (ι) , 42
Vol(K), 101
Z ∗ (χ, f, f ∨ , ϕs ), 24
Z ∗ (s, χ, f, f ∨ , Φ), 40
Z ∗ (s, χ, f, f ∨ , φ ⊗ φ∨ ), 26
194
(Z, gι◦ ), 60
b1C (M), 60
Z
bφ (g), 67
Z
Zφ (g), 45
Zφ∼ (g), 54
Zφ∼,+ (g), 54
Z (x̆), 121
Z(V1 )K , 44
Z(x)0,K p ;p◦ , 116
Z(x)K , 44
Z(x)∼
K , 53