ON MAASS LIFTS AND THE CENTRAL CRITICAL
VALUES OF TRIPLE PRODUCT L-FUNCTIONS
ATSUSHI ICHINO AND TAMOTSU IKEDA
To the memory of Prof. Tsuneo Arakawa
Abstract. We express certain period integrals of Maass lifts which
appear in the Gross-Prasad conjecture in terms of the central critical values of triple product L-functions in the imbalanced case.
Introduction
After Garrett [10] discovered an integral representation of triple
product L-functions, Harris and Kudla [16] determined the transcendental part of the central critical values of triple product L-functions.
More precisely, for i = 1, 2, 3, let fi ∈ Sκi (Γ0 (Ni ), χi ) be a primitive
form. We may assume that κ1 ≤ κ2 ≤ κ3 . We also assume that χ1 χ2 χ3
is trivial, so that w = κ1 + κ2 + κ3 − 3 is odd. Recall that the triple
product L-function L(s, f1 × f2 × f3 ) satisfies a functional equation
which replaces s with w + 1 − s. Set
(
hf1 , f1 ihf2 , f2 ihf3 , f3 i if κ1 + κ2 > κ3 ,
p(f1 , f2 , f3 ) =
hf3 , f3 i2
if κ1 + κ2 ≤ κ3 ,
where
Z
|fi (τ )|2 Im(τ )κi −2 dτ
hfi , fi i =
Γ0 (Ni )\H
is the Petersson norm of fi . Then the main theorems of Harris and
Kudla [16] say that
Y
L( w+1
, f1 × f2 × f3 )
2
Cp · C∞
= C2 ·
p(f1 , f2 , f3 )
p|N1 N2 N3
with some constants C ∈ Q(f1 , f2 , f3 ), Cp ∈ Q(f1 , f2 , f3 )× , and
(
π w+2 · Q× if κ1 + κ2 > κ3 ,
C∞ ∈
π 2κ3 · Q× if κ1 + κ2 ≤ κ3 .
Date: May 5, 2005.
2000 Mathematics Subject Classification. 11F67.
1
2
ATSUSHI ICHINO AND TAMOTSU IKEDA
Here Q(f1 , f2 , f3 ) is the field generated over Q by the Fourier coefficients of f1 , f2 , f3 . Moreover, Cp (resp. C∞ ) depends only on the
local components of π1 , π2 , π3 at p (resp. at ∞), where πi is the irreducible cuspidal automorphic representation of GL2 (AQ ) determined
by fi . We remark that in the case κ1 + κ2 > κ3 , the critical values have
also been studied by Garrett [10], Orloff [31], Satoh [35], Garrett and
Harris [11]. Moreover, Gross and Kudla [13], Böcherer and SchulzePillot [5] expressed certain height pairings as the algebraic part of the
central critical values in the case κ1 + κ2 > κ3 . By contrast, in the
case κ1 + κ2 ≤ κ3 , there are no results on the critical values except
[16] to our knowledge. In the present paper, we express certain period
integrals of Maass lifts which appear in the Gross-Prasad conjecture
[14], [15], as the algebraic part of the central critical values in the case
κ1 = κ2 = κ3 /2.
Now
√ we give a more precise description of our result. Let K =
Q( −D) be an imaginary quadratic field with discriminant −D < 0,
O the ring of integers of K, wK the number of roots of unity in K,
and χ the primitive Dirichlet character associated to K/Q. Let QD
denote the set of all primes dividing D. For each q ∈ QD , let χq be the
√
quadratic character of Q×
q associated to Qq ( −D)/Qq by class field
theory.
Let κ be an even positive integer such that wK | κ. Let g ∈
Sκ−1 (Γ0 (D), χ) be a primitive form. Then g determines a cusp form
g ∗ (τ ) =
∞
X
ag∗ (n)q n ∈ Sκ−1 (Γ0 (D), χ)
n=1
√
such that ag∗ (n) = 0 if aD (n) = 0, where q = exp(2π −1τ ) and
Y
(1 + χq (−n)).
aD (n) =
q∈QD
q-n
Note that g ∗ is a certain sum of “twists” of g (see §1 for more de(2)
tails). Let G ∈ Sκ (ΓK ) be the hermitian Maass lift of g, where
(2)
ΓK = U(2, 2)(Q) ∩ SL4 (O). Then G has a Fourier expansion of the
form
X
√
G(Z) =
AG (H) exp(2π −1 tr(HZ))
H
for Z ∈ H2 . Here H2 is the hermitian upper half space of degree 2, H
runs over all positive definite semi-integral hermitian matrices of size
3
2, and
AG (H) =
X
µ
κ−1
d
∗
αG
d|ε(H)
where
∗
αG
D det(H)
d2
¶
,
is a function on Z≥0 such that
∗
(n)
ag∗ (n) = aD (n)αG
and ε(H) = max{n ∈ N | n−1 H is semi-integral}.
Let f ∈ S2κ−2 (SL2 (Z)) be a normalized Hecke eigenform and
h(τ ) =
∞
X
+
c(n)q n ∈ Sκ−1/2
(Γ0 (4))
n=1
a Hecke eigenform associated to f by the Shimura correspondence.
Note that c(n) = 0 unless −n ≡ 0, 1 mod 4. Let F ∈ Sκ (Sp2 (Z)) be
the Saito-Kurokawa lift of f . Then F has a Fourier expansion of the
form
X
√
F (Z) =
AF (B) exp(2π −1 tr(BZ))
B
for Z ∈ H2 . Here H2 is the Siegel upper half space of degree 2, B runs
over all positive definite half-integral symmetric matrices of size 2, and
µµ
¶¶
µ
¶
X
4nm − r2
n r/2
κ−1
AF
=
d c
.
r/2 m
d2
d|(n,r,m)
Recall that H2 ⊂ H2 . We consider the period integral hG|H2 , F i given
by
Z
hG|H2 , F i =
G(Z)F (Z) det(Im(Z))κ−3 dZ.
Sp2 (Z)\H2
Let Λ(s, g × g × f ) be the completed triple product L-function given
by
Λ(s, g×g×f ) = (2π)−4s+4κ−8 Γ(s)Γ(s−2κ+4)Γ(s−κ+2)2 L(s, g×g×f ).
Our main result is as follows.
Theorem 3.1.
2
Λ(2κ − 3, g × g × f )
4κ−6 −2κ+3
2 hG|H2 , F i
=
−2
D
c(D)
.
hf, f i2
hF, F i2
This paper is organized as follows. In §1 and §2, we review the theory
of hermitian Maass lifts and Saito-Kurokawa lifts, respectively. In §3,
4
ATSUSHI ICHINO AND TAMOTSU IKEDA
we state our main result. In §4, we compute restrictions of hermitian
Maass lifts to H2 and prove an identity for the seesaw
O(4, 2)
f 2 × SL
f2
O(2, 2)
SL
PPP
PPP
n
PPP nnnn
PPP nnnnn
nP
nP
nnn PPPPP
nnn PPPPP
n
n
n
n
P
n
n
P
nn
O(3, 2) × O(1)
.
O(2, 1) × O(1)
SL2
To compute the central critical values of triple product L-functions, we
use the seesaw
Sp3 UUUU
O(2, 2) × O(2, 2) × O(2, 2) .
UUUU
UUUU iiiiiii
iUiUiUUUU
iiii
UUUU
iiii
U
O(2, 2)
SL2 × SL2 × SL2
In §5, we give an explicit formula for theta lifts from GL2 to GO(2, 2).
In §6, we compute the local zeta integrals which arise in the integral
representation of triple product L-functions. Using these two seesaw
identities, we prove Theorem 3.1 in §7. Finally, in §8, we interpret our
result in terms of the Gross-Prasad conjecture.
The authors would like to thank Prof. Hiroshi Saito and Dr. Kaoru
Hiraga for useful discussions.
Notation
√
Let K = Q( −D) be an imaginary quadratic √
field with discriminant
]
−D < 0, O the ring of integers of K, O = ( −D)−1 O the inverse
different ideal of K/Q, wK the number of roots of unity in K, x 7→ x̄
the non-trivial Galois automorphism of K over Q, and χ the primitive
Dirichlet character associated to K/Q. Let QD denote the set of all
primes dividing D. For each q ∈ QD , put Dq = q ordq (D) . Let χq be the
primitive Dirichlet character mod Dq defined by
χq (n) = χ(n0 )
for n ∈ Z with q - n, where n0 is an integer such that
(
n mod Dq ,
n0 ≡
1 mod Dq−1 D.
Then
χ=
Y
q∈QD
χq .
5
For each Q ⊂ QD , set
χQ =
Y
χq
Y
and χ0Q =
q∈Q
χq .
q∈QD −Q
×
Let χ = ⊗v χv be the Hecke character of A×
determined by χ.
Q /Q
√
×
Then χv is the quadratic character of Qv associated to Qv ( −D)/Qv
by class field theory. If q ∈ QD , then χq (n) = χq (n) for n ∈ Z with
q - n. One should not confuse χq with χq .
A hermitian matrix H = (Hij ) is called semi-integral if Hii ∈ Z and
Hij ∈ O] for all i, j. The unitary group U(n, n) and the symplectic
group Spn are defined by
¯ µ
½
¶
µ
¶¾
¯
0 −1n t
0 −1n
¯
U(n, n)(R) = g ∈ GL2n (R ⊗ K) ¯ g
ḡ =
,
1n
0
1n
0
¯ µ
µ
¶
¶¾
½
¯
0
−1
0
−1
n
n
t
g=
Spn (R) = g ∈ GL2n (R) ¯¯ g
,
1n
0
1n
0
for any Q-algebra R, respectively. The hermitian upper half space Hn
and the Siegel upper half space Hn are defined by
¯
¾
½
¯ 1
t
¯
Hn = Z ∈ Mn (C) ¯ √ (Z − Z̄) > 0 ,
2 −1
¯t
©
ª
Hn = Z ∈ Mn (C) ¯ Z = Z, Im(Z) > 0 ,
(n)
respectively. Set ΓK = U(n, n)(Q) ∩ SL2n (O). When n ≥ 2, the space
(n)
Mκ (ΓK ) of hermitian modular forms of degree n and weight κ consists
of all holomorphic functions G on Hn which satisfy
G((AZ + B)(CZ + D)−1 ) = det(CZ + D)κ G(Z)
for all
µ
A B
C D
¶
(n)
∈ ΓK
and Z ∈ Hn .
1. Hermitian Maass lifts
In this section, we review the theory of hermitian Maass lifts [25],
[39], [26].
Let κ be an even positive integer such that wK | κ. Krieg [26]
∗
introduced the space Mκ−1
(Γ0 (D), χ) of all modular forms
∗
g (τ ) =
∞
X
n=0
ag∗ (n)q n ∈ Mκ−1 (Γ0 (D), χ)
6
ATSUSHI ICHINO AND TAMOTSU IKEDA
such that ag∗ (n) = 0 if aD (n) = 0, where
Y
(1 + χq (−n)).
aD (n) =
q∈QD
Set
∗
∗
Sκ−1
(Γ0 (D), χ) = Sκ−1 (Γ0 (D), χ) ∩ Mκ−1
(Γ0 (D), χ).
Let
g(τ ) =
∞
X
ag (n)q n ∈ Mκ−1 (Γ0 (D), χ)
n=0
be a primitive form. By Theorem 4.6.16 of [30], there is a primitive
form
∞
X
gQ (τ ) =
agQ (n)q n ∈ Mκ−1 (Γ0 (D), χ)
n=0
such that, for each prime p,
agQ (p) =
(
χQ (p)ag (p)
if p ∈
/ Q,
χ0Q (p)ag (p)
if p ∈ Q.
Put
g∗ =
(1.1)
X
χQ (−1)gQ .
Q⊂QD
∗
Then g ∗ ∈ Mκ−1
(Γ0 (D), χ).
Lemma 1.1. Let g ∈ Sκ−1 (Γ0 (D), χ) be a primitive form. Then the
Fourier coefficients of g ∗ are purely imaginary.
Proof. For each Q ⊂ QD , the Fourier coefficients of gQ − g(QD −Q) are
purely imaginary. This yields the lemma.
¤
Let
G(Z) =
X
√
(2)
AG (H) exp(2π −1 tr(HZ)) ∈ Mκ (ΓK )
H
be a hermitian modular form of degree 2 and weight κ. Here H runs
over all positive semi-definite semi-integral hermitian matrices of size
2. We say that G satisfies the Maass relation if there exists a function
∗
αG
on Z≥0 such that
¶
µ
X
D det(H)
κ−1 ∗
AG (H) =
,
d αG
d2
d|ε(H)
7
(2)
where ε(H) = max{n ∈ N | n−1 H is semi-integral}. Let MκMaass (ΓK )
denote the space of hermitian modular forms of degree 2 and weight κ
(2)
which satisfy the Maass relation. For G ∈ MκMaass (ΓK ), put
Ω(G)(τ ) =
∞
X
∗
aD (n)αG
(n)q n .
n=0
Then Ω(G) ∈
∗
Mκ−1
(Γ0 (D), χ),
and the linear map
(2)
∗
(Γ0 (D), χ)
Ω : MκMaass (ΓK ) −→ Mκ−1
is an isomorphism. Note that we slightly modified Krieg’s definition of
Ω by a scalar.
2. Saito-Kurokawa lifts
In this section, we review the theory of Saito-Kurokawa lifts [28],
[29], [1], [41].
Let κ be an even positive integer. Kohnen [22] introduced the space
+
Mκ−1/2
(Γ0 (4)) of all modular forms
h(τ ) =
∞
X
c(n)q n ∈ Mκ−1/2 (Γ0 (4))
n=0
such that c(n) = 0 unless −n ≡ 0, 1 mod 4. Set
+
+
Sκ−1/2
(Γ0 (4)) = Sκ−1/2 (Γ0 (4)) ∩ Mκ−1/2
(Γ0 (4)).
Let
F (Z) =
X
√
AF (B) exp(2π −1 tr(BZ)) ∈ Mκ (Sp2 (Z))
B
be a Siegel modular form of degree 2 and weight κ. Here B runs over
all positive semi-definite half-integral symmetric matrices of size 2. We
say that F satisfies the Maass relation if there exists a function βF∗ on
Z≥0 such that βF∗ (n) = 0 unless −n ≡ 0, 1 mod 4 and such that
µµ
¶¶
µ
¶
X
4nm − r2
n r/2
κ−1 ∗
AF
=
d βF
.
r/2 m
d2
d|(n,r,m)
Let MκMaass (Sp2 (Z)) denote the space of Siegel modular forms of degree
2 and weight κ which satisfy the Maass relation. For F ∈ MκMaass (Sp2 (Z)),
put
∞
X
SK
Ω (F )(τ ) =
βF∗ (n)q n .
n=0
8
ATSUSHI ICHINO AND TAMOTSU IKEDA
+
Then ΩSK (F ) ∈ Mκ−1/2
(Γ0 (4)), and the linear map
+
ΩSK : MκMaass (Sp2 (Z)) −→ Mκ−1/2
(Γ0 (4))
is an isomorphism.
3. Statement of the main theorem
√
Recall that K = Q( −D) is an imaginary quadratic field with discriminant −D < 0, wK is the number of roots of unity in K, and χ is
the primitive Dirichlet character associated to K/Q. Let κ be an even
positive integer such that wK | κ. Let
∞
X
g(τ ) =
ag (n)q n ∈ Sκ−1 (Γ0 (D), χ)
n=1
be a primitive form. For each prime p ∈
/ QD , the Satake parameter
{αp , χ(p)αp−1 } of g at p is defined by
1 − ag (p)X + χ(p)pκ−2 X 2 = (1 − p(κ−2)/2 αp X)(1 − p(κ−2)/2 χ(p)αp−1 X).
For each q ∈ QD , put αq = q −(κ−2)/2 ag (q). Let G = Ω−1 (g ∗ ) ∈
(2)
∗
(Γ0 (D), χ)
SκMaass (ΓK ) be the hermitian Maass lift of g, where g ∗ ∈ Sκ−1
is given by (1.1). Recall that H2 ⊂ H2 . Let G|H2 denote the restriction
of G to H2 . Then G|H2 ∈ SκMaass (Sp2 (Z)) by [26].
Let
∞
X
af (n)q n ∈ S2κ−2 (SL2 (Z))
f (τ ) =
n=1
be a normalized Hecke eigenform. For each prime p, the Satake parameter {βp , βp−1 } of f at p is defined by
1 − af (p)X + p2κ−3 X 2 = (1 − p(2κ−3)/2 βp X)(1 − p(2κ−3)/2 βp−1 X).
Let
h(τ ) =
∞
X
+
c(n)q n ∈ Sκ−1/2
(Γ0 (4))
n=1
be a Hecke eigenform associated to f by the Shimura correspondence
[37], [22]. Note that h is unique up to scalars. Let F = (ΩSK )−1 (h) ∈
SκMaass (Sp2 (Z)) be the Saito-Kurokawa lift of f . The Petersson norms
of f and F are defined by
Z
hf, f i =
|f (τ )|2 Im(τ )2κ−4 dτ,
SL (Z)\H
Z 2
|F (Z)|2 det(Im(Z))κ−3 dZ,
hF, F i =
Sp2 (Z)\H2
9
respectively.
Put w = 4κ−7. We define the triple product L-function L(s, g×g×f )
by an Euler product
Y
L(s, g × g × f ) =
Lv (s, g × g × f )
v<∞
for Re(s) À 0. Here, for v = p ∈
/ QD ,
´
³
w
Lp s + , g × g × f = det(18 − Ap ⊗ Ap ⊗ Bp · p−s )−1
2
with
µ
¶
µ
¶
αp
0
βp 0
Ap =
, Bp =
,
0 χ(p)αp−1
0 βp−1
and for v = q ∈ QD ,
³
´
w
Lq s + , g × g × f
2
¤−1
£
= (1 − αq2 βq q −s )(1 − αq2 βq−1 q −s )(1 − αq−2 βq q −s )(1 − αq−2 βq−1 q −s )
.
Let Λ(s, g × g × f ) be the completed triple product L-function given
by
Λ(s, g×g×f ) = (2π)−4s+4κ−8 Γ(s)Γ(s−2κ+4)Γ(s−κ+2)2 L(s, g×g×f ).
By [32], [19], Λ(s, g × g × f ) has a holomorphic continuation to the
whole s-plane and satisfies a functional equation which replaces s with
w + 1 − s.
Our main result is as follows.
Theorem 3.1.
2
Λ(2κ − 3, g × g × f )
4κ−6 −2κ+3
2 hG|H2 , F i
=
−2
D
c(D)
.
hf, f i2
hF, F i2
Remark 3.2. We may
√ assume that c(n) ∈ Q(f ) for all n ∈ N. In particular, hG|H2 , F i ∈ −1R by Lemma 1.1. Since the Fourier coefficients
of G (resp. F ) are in Q(g) (resp. Q(f )), we have
hG|H2 , F i
∈ Q(g, f ).
hF, F i
Remark 3.3. Fix a normalized Hecke eigenform f ∈ S2κ−2 (SL2 (Z)) and
assume that c(D) =
6 0. By Theorem 1 of [7], the restriction map
(2)
SκMaass (ΓK ) −→ SκMaass (Sp2 (Z))
G 7−→ G|H2
10
ATSUSHI ICHINO AND TAMOTSU IKEDA
is surjective. Hence there exists a primitive form g ∈ Sκ−1 (Γ0 (D), χ)
such that
Λ(2κ − 3, g × g × f ) 6= 0.
Remark 3.4. Using the Maass space defined by Sugano [39], one might
remove the assumption that wK | κ. See also [21].
Example 3.5. We discuss the case D = 7, κ = 10. Let g ∈ S9 (Γ0 (7), χ)
be the primitive form with
1
27 2 56557
2508
a3 −
a −
a−
,
108290
15470
54145
245
22 3
359 2 268563
13716
ag (5) = −
a +
a −
a+
,
10829
3094
10829
49
p
√
√
where a = 8 + 2 46 + 2 −2148 − 213 46. Here we have used Stein’s
database [38]. Let f ∈ S18 (SL2 (Z)) be the normalized Hecke eigenform,
+
and h ∈ S19/2
(Γ0 (4)) the Hecke eigenform associated to f with c(3) = 1,
c(7) = −16. Then
q
√
hG|H2 , F i
∗
= βG|H (3) = 2(ag (3) + ag (5)) = 24 −6088 + 442 46.
2
hF, F i
ag (3) = −
Using Dokchitser’s computer program [8], we obtain
hf, f i = 0.0000045947361976392466101158732480223961 . . . ,
Λ(17, g × g × f ) = 0.0000007104357360884738902046072703848041 . . . .
Hence
Λ(17, g × g × f )
= 33651.438624311611653588297096717239 . . .
hf, f i2
hG|H2 , F i2
= −234 · 7−17 · c(7)2 ·
.
hF, F i2
4. Restrictions of hermitian Maass lifts to H2
In this section, we compute restrictions of hermitian Maass lifts to
H2 and prove the following seesaw identity.
(2)
Proposition 4.1. Let G = Ω−1 (g ∗ ) ∈ SκMaass (ΓK ) be the hermitian
Maass lift of a primitive form g ∈ Sκ−1 (Γ0 (D), χ). Then
(4.1)
+
2
S−D
(ΩSK (G|H2 )) = ag (D)2 TrD
1 (g ).
11
Here
+
+
S−D
: Sκ−1/2
(Γ0 (4)) −→ S2κ−2 (SL2 (Z))
∞
X
n
c(n)q 7−→
∞ X
X
n=1
κ−2
χ(d)d
n=1 d|n
µ
n2
c D 2
d
¶
qn
is the linear map defined by Kohnen [22], and
TrD
1 : S2κ−2 (Γ0 (D)) −→ S2κ−2 (SL2 (Z))
X
f 7−→
f |γ
γ∈Γ0 (D)\ SL2 (Z)
is the trace operator.
The rest of this section is devoted to the proof of Proposition 4.1.
First, we compute the right-hand side of (4.1). For each Q ⊂ QD , let
gQ ∈ Sκ−1 (Γ0 (D), χ) be the primitive form as in §1. For convenience,
we write aQ (n) = agQ (n) for n ∈ N and put aQ (n) = 0 for n ∈ Q − N.
Put


if D is odd or 2 ∈ Q,
DQ
Y
DQ =
Dq , D̃Q = 2DQ if D is even, D2 = 4, and 2 ∈
/ Q,


q∈Q
4DQ if D is even, D2 = 8, and 2 ∈
/ Q,
−1
−1
0
0
0
is a square-free integer. For
= D̃Q
D. Then D̃Q
= DQ
D, and D̃Q
DQ
m ∈ N and f ∈ S2κ−2 (Γ0 (D)), we define Um (f ) by
µ
¶ X
∞
1 X
τ +l
Um (f )(τ ) =
f
=
af (mn)q n .
m l mod m
m
n=1
Lemma 4.2.
2
ag (D)2 TrD
1 (g ) =
X
0 2
2
).
χQ (−1)ag (D̃Q
) UD̃Q (gQ
Q⊂QD
Proof. For each q ∈ QD , we choose γq ∈ SL2 (Z) such that
!
Ã

0
−1


mod Dq2 ,

 1 0
!
γq ≡ Ã

1
0



mod Dq−2 D2 .
 0 1
For each Q = {q1 , . . . , qr } ⊂ QD , put
µ
¶ µ −1 ¶
µ
¶
DQ 0
Dqr 0
Dq1 0
∈ SL2 (Z).
γQ = γq1
. . . γ qr
0 1
0 1
0 1
12
ATSUSHI ICHINO AND TAMOTSU IKEDA
We define a subset RQ of SL2 (Z)

0

if
RQ
0
00
RQ = RQ ∪ RQ
if

R0 ∪ R00 ∪ R000 if
Q
Q
Q
where
by
D is odd or 2 ∈ Q,
D is even, D2 = 4, and 2 ∈
/ Q,
D is even, D2 = 8, and 2 ∈
/ Q,
¶¯
¾
1 l ¯¯
= γQ
l mod DQ ,
0 1 ¯
½ µ
¶µ
¶¯
¾
1
0
1 l ¯¯
00
RQ = γQ −1 0
l mod DQ ,
2 DQ 1
0 1 ¯
½ µ
¶µ
¶¯
¾
1
0
1 l + l0 DQ ¯¯
000
0
RQ = γQ −1 0
¯ l mod DQ , l mod 2 .
4 DQ 1
0
1
½
µ
R0Q
Then
[
RQ
Q⊂QD
is a set of representatives for Γ0 (D)\ SL2 (Z). Thus, it suffices to show
that
X
0 2
2
ag (D)2
g 2 |γ = χQ (−1)ag (D̃Q
) UD̃Q (gQ
).
γ∈RQ
We only consider the case when D is even, D2 = 8, and 2 ∈
/ Q. The
other cases are similar. By Corollary 4.6.18 of [30],
µ
¶
q
Dq 0
g|γq
= χq (−1) χq (−1)Dq(κ−2)/2 ag (Dq )−1 g{q}
0 1
for q ∈ QD . Thus,
2
µ
g |γQ =
κ−2
2
χQ (−1)DQ
ag (DQ )−2 gQ
|
¶
−1
DQ
0
.
0 1
P
κ−2
0 2
)
Hence ag (D)2 γ∈RQ g 2 |γ is equal to the product of χQ (−1)DQ
ag (DQ
and
¶
µ
¶ µ −1 ¶ µ
¶
µ −1 ¶ µ
X ·
DQ 0
DQ 0
1 l
1 l
1
0
2
2
+ gQ | −1
gQ |
0 1
0 1
2 D 1
0 1
0 1
l mod DQ
µ
¶µ
¶ µ −1 ¶ µ
¶¸
X
DQ 0
1
0
1 l0
1 l
2
+
gQ | −1
.
4 D 1
0 1
0 1
0 1
0
l mod 2
For m ∈ Z, put
µ
gQ,m
¶µ
¶
4−1 0
1 m
= gQ |
.
0 1
0 1
13
It is easy to check that
µ
1
gQ,m | −1
2 D
¶µ
µ
1
0
1
gQ,m | −1
4 D 1
0
¶
0
= χ(1 − 2−1 Dm)gQ,m ,
1
¶
l0
= χ(1 + 4−1 Dm)gQ,−m+l0 ,
1
and
X
gQ = 4(κ−3)/2 aQ (4)−1
gQ,m .
m mod 4
By a direct calculation,
µ
¶
µ
¶µ
¶
X
1
0
1
0
1 l0
2
2
2
gQ + gQ | −1
+
gQ | −1
2 D 1
4 D 1
0 1
0
l mod 2
X
2
= 4κ−2 aQ (4)−2
gQ,m
−2
m mod 4
2
U4 (gQ ).
= ag (4)
P
Hence ag (D)2 γ∈RQ g 2 |γ is equal to
−κ+2
0 2
χQ (−1)DQ
ag (DQ
) ag (4)−2
X
µ
2
U4 (gQ
)|
l mod DQ
−1
= χQ (−1)ag (4
¶µ
¶
−1
DQ
0
1 l
0 1
0 1
0 2
2
DQ
) U4DQ (gQ
).
¤
Next, we compute the left-hand side of (4.1). For each Q ⊂ QD , set
¯
(
Ã
!)
¯
[
¯
−1
−1
O] (Q) = t ∈ O] ¯ NK/Q (t) ∈ D̃(Q
Z−
D̃(Q
Z
.
D −Q)
D −Q1 )
¯
Q1 )Q
For i = 0, 1 and n ∈ N, set
]
Oi,n
= {t ∈ O] | trK/Q (t) = i, NK/Q (t) ≤ n}
]
]
and Oi,n
(Q) = Oi,n
∩ O] (Q). Note that
[
]
]
Oi,n
=
Oi,n
(Q).
Q⊂QD
Put
bi,n (Q) =
X
]
t∈Oi,n
−1
NK/Q (t)∈D̃Q
Z
aQ (D(n − NK/Q (t))).
14
ATSUSHI ICHINO AND TAMOTSU IKEDA
∗
Lemma 4.3. Let βG|
be the function on Z≥0 attached to G|H2 as in
H2
§2. Then
X
∗
βG|
(4n
−
i)
=
χQ (−1)bi,n (Q)
H
2
Q⊂QD
for i = 0, 1 and n ∈ N.
Proof. By [26, p. 680],
∗
βG|
(4n − i) =
H
X
2
∗
(D(n − NK/Q (t))),
αG
]
t∈Oi,n
∗
where αG
is the function on Z≥0 attached to G as in §1. If t ∈ O] (Q),
∗
then αG (D(n − NK/Q (t))) is equal to
X
X
2−](QD −Q) χ(Q1 ∪Q2 ) (−1)a(Q1 ∪Q2 ) (D(n − NK/Q (t)))
Q1 ⊂Q Q2 ⊂QD −Q
=
X
χ(Q1 ∪(QD −Q)) (−1)a(Q1 ∪(QD −Q)) (D(n − NK/Q (t)))
Q1 ⊂Q
=
X
χQ1 (−1)aQ1 (D(n − NK/Q (t))).
Q1 ⊃QD −Q
∗
(4n − i) is equal to
Hence βG|
H2
X
X X
χQ1 (−1)aQ1 (D(n − NK/Q (t)))
Q⊂QD t∈O] (Q) Q1 ⊃QD −Q
i,n
=
X
X
X
χQ1 (−1)aQ1 (D(n − NK/Q (t)))
Q1 ⊂QD Q⊃QD −Q1 t∈O] (Q)
i,n
=
X
X
Q1 ⊂QD
]
t∈Oi,n
χQ1 (−1)aQ1 (D(n − NK/Q (t))).
−1
NK/Q (t)∈D̃Q
Z
1
¤
Now we prove Proposition 4.1. Note that
X
0
0
m2 ).
b0,n (Q) =
)aQ (D̃Q n − D̃Q
aQ (D̃Q
m∈Z
0 m2 ≤D̃ n
D̃Q
Q
Also, b1,n (Q) is equal to
X
m∈Z
0 m2 ≤D̃ (n−1/4)
D̃Q
Q
µ
0
aQ (D̃Q
)aQ
µ
D̃Q
1
n−
4
¶
¶
−
0
D̃Q
m2
15
if D2 = 8 or 2 ∈ Q, and is equal to
Ã
µ
¶
µ
¶2 !
X
1
1
0
0
)aQ D̃Q n −
m+
aQ (D̃Q
− D̃Q
4
2
m∈Z
0 (m+1/2)2 ≤D̃ (n−1/4)
D̃Q
Q
otherwise. For n = Dl2 with l ∈ N,
!
Ã
0
X
D̃Q n − D̃Q
m2
=
aQ
4
m∈Z
0 m2 ≤D̃ n
D̃Q
Q
hence
∗
βG|
(n) =
H
2
X
X
X
0
aQ (D̃Q
)aQ (m1 m2 ),
m1 ,m2 ∈Z≥0
m1 +m2 =D̃Q l
0
)aQ (m1 m2 )
χQ (−1)aQ (D̃Q
Q⊂QD m1 ,m2 ∈Z≥0
m1 +m2 =D̃Q l
+
by Lemma 4.3. As in the proof of Proposition 3 of [24], S−D
(ΩSK (G|H2 ))(τ )
is equal to
µ
¶
∞ X
X
n2
κ−2 ∗
χ(d)d βG|H D 2 q n
2
d
n=1
d|n
=
∞ X
X
X
n=1 d|n Q⊂QD
=
∞
X X
X
0 2
χ(d)dκ−2 χQ (−1)aQ (D̃Q
) aQ (m1 m2 )q n
m1 ,m2 ∈Z≥0
m1 +m2 =D̃Q n/d
X
0 2
χQ (−1)ag (D̃Q
)
Q⊂QD n=1 m1 ,m2 ∈Z≥0
m1 +m2 =D̃Q n
=
∞
X X
X
X
χ(d)dκ−2 aQ
d|(m1 ,m2 )
³m m ´
1 2
qn
d2
0 2
χQ (−1)ag (D̃Q
) aQ (m1 )aQ (m2 )q n
Q⊂QD n=1 m1 ,m2 ∈Z≥0
m1 +m2 =D̃Q n
=
X
0 2
2
χQ (−1)ag (D̃Q
) UD̃Q (gQ
)(τ ).
Q⊂QD
Therefore Proposition 4.1 follows from Lemma 4.2.
5. Theta lifts from GL2 to GO(2, 2)
In this section, we give an explicit formula for theta lifts from GL2
to GO(2, 2).
16
ATSUSHI ICHINO AND TAMOTSU IKEDA
5.1. Weil representations for similitudes. Let k be a number field
and A = Ak the adele ring of k. Fix a non-trivial additive character
ψ of A/k. Let GSpn denote the symplectic similitude group and ν :
GSpn → Gm the scale map. Let V = M2 (k) be the quadratic space
with bilinear form (x, y) = tr(xy ι ). Here ι is the main involution on V ,
that is,
µ
¶
µ
¶
x4 −x2
x1 x2
ι
x =
for x =
∈V
−x3 x1
x3 x4
Let H = GO(V ) denote the orthogonal similitude group and ν : H →
Gm the scale map. Set H̃ = (GL2 × GL2 ) o hti and H̃ 0 = GL2 × GL2 ,
where t is the involution on GL2 × GL2 defined by
t(h1 , h2 ) = ((hι2 )−1 , (hι1 )−1 )
for h1 , h2 ∈ GL2 . Recall that there is an exact sequence
ρ
1 −→ Gm −→ H̃ −→ H −→ 1,
ι
where ρ(h1 , h2 )x = h1 xh−1
2 , ρ(t)x = x for h1 , h2 ∈ GL2 and x ∈ V .
Let ω be the Weil representation of Spn (A) × O(V )(A) with respect
to ψ on the Schwartz space S(V n (A)). Set
R = {(g, h) ∈ GSpn ×H | ν(g) = ν(h)}.
Following [17, §5.1], we extend the Weil representation to a representation ω of R(A). Then
µ µ
¶ ¶
1n
0
ω(g, h)ϕ = ω g
, 1 L(h)ϕ
0 ν(g)−1 1n
for (g, h) ∈ R(A) and ϕ ∈ S(V n (A)), where
L(h)ϕ(x) = |ν(h)|−n ϕ(h−1 x)
for x ∈ V n (A). Let R̃ be the pullback of R to GSpn ×H̃. By abuse of
notation, we write ω for the pullback of ω to R̃(A).
When n = 1, we also use a different model of the Weil representation.
Let ϕ̂ ∈ S(V (A)) be the partial Fourier transform of ϕ ∈ S(V (A)) given
by
µµ
¶¶ Z
µµ
¶¶
x1 x2
x1 x02
ϕ
ψ(x2 x04 − x4 x02 ) dx02 dx04 .
ϕ̂
=
0
x3 x4
x
x
3
4
A2
Here dx02 , dx04 are the self-dual measures on A with respect to ψ. We
define a representation ω̂ of R(A) (and of R̃(A)) on S(V (A)) by
\
ω̂(g, h)ϕ̂ = ω(g,
h)ϕ.
Note that ω̂(g, 1)ϕ̂(x) = ϕ̂(xg) for g ∈ SL2 (A).
17
5.2. Fourier coefficients of theta lifts. Let n = 1. The theta function, defined for (g, h) ∈ R(A) and ϕ ∈ S(V (A)) by
X
Θ(g, h; ϕ) =
ω(g, h)ϕ(x),
x∈V (k)
is left R(k)-invariant. Let φ be a cusp form on GL2 (A). For h ∈ H(A),
choose g 0 ∈ GL2 (A) such that ν(g 0 ) = ν(h), and put
Z
θ(φ, ϕ)(h) =
φ(gg 0 )Θ(gg 0 , h; ϕ) dg.
SL2 (k)\ SL2 (A)
Here dg is the Tamagawa measure on SL2 (A). This integral does not depend on the choice of g 0 , and the theta lift θ(φ, ϕ) is left H(k)-invariant.
By abuse of notation, we write θ(φ, ϕ) for the pullback of θ(φ, ϕ) to
H̃(A). For an irreducible cuspidal automorphic representation π of
GL2 (A), set
θ(π) = {θ(φ, ϕ) | φ ∈ π, ϕ ∈ S(V (A))}.
By [36], [16, §7],
θ(π)|H̃ 0 (A) = π £ π ∨
(5.1)
as spaces of functions on H̃ 0 (A). Here π ∨ is the contragredient representation of π.
Define the Whittaker function Wφ of φ by
µµ
¶ ¶
Z
1 b
Wφ (g) =
φ
g ψ(b) db
0 1
k\A
for g ∈ GL2 (A). Here db is the self-dual measure on A with respect to
ψ. Similarly, define the Whittaker function Wθ(φ,ϕ) of θ(φ, ϕ) by
µµµ
¶ µ
¶¶ ¶
Z
1 b1
1 b2
Wθ(φ,ϕ) (h) =
θ(φ, ϕ)
,
h ψ(b1 )ψ(b2 ) db1 db2
0 1
0 1
(k\A)2
for h ∈ H̃(A).
Lemma 5.1.
Z
Wφ (gg 0 )ω̂(g 0 , h)ϕ̂(g) dg.
Wθ(φ,ϕ) (h) =
SL2 (A)
18
ATSUSHI ICHINO AND TAMOTSU IKEDA
Proof. By the Poisson summation formula,
¶¶ ¶
µµ µ
Z
1 b
θ(φ, ϕ)
1,
h ψ(b) db
0 1
k\A
Z Z
X
ω̂(gg 0 , h)ϕ̂(x)ψ(b(det(x) − 1)) dg db
=
φ(gg 0 )
k\A
Z
SL2 (k)\ SL2 (A)
φ(gg 0 )
=
SL2 (k)\ SL2 (A)
Z
x∈V (k)
X
ω̂(gg 0 , h)ϕ̂(x) dg
x∈SL2 (k)
φ(gg 0 )ω̂(g 0 , h)ϕ̂(g) dg.
=
SL2 (A)
Hence Wθ(φ,ϕ) (h) is equal to
µ µµ
¶ ¶ ¶
Z Z
1 b
0
0
φ(gg )ω̂ g ,
, 1 h ϕ̂(g)ψ(b) dg db
0 1
k\A SL2 (A)
¶ ¶
µµ
Z Z
1 −b
0
0
g ψ(b) dg db
φ(gg )ω̂(g , h)ϕ̂
=
0 1
k\A SL2 (A)
µµ
¶ ¶
Z Z
1 b
=
φ
gg 0 ω̂(g 0 , h)ϕ̂(g)ψ(b) dg db
0
1
k\A SL2 (A)
Z
Wφ (gg 0 )ω̂(g 0 , h)ϕ̂(g) dg.
=
SL2 (A)
¤
5.3. An explicit formula for theta lifts. Let k = Q. Let ψ =
⊗v ψv be
√ the standard additive character of A/Q, so that ψ∞ (x) =
exp(2π −1x) for x ∈ R. Let N be a positive integer. We temporarily
let χ denote an arbitrary primitive Dirichlet character mod N and
χ = ⊗v χv the Hecke character of A× /Q× determined by χ.
Let
∞
X
f (τ ) =
af (n)q n ∈ Sl (Γ0 (N ), χ)
n=1
be a primitive form. Then f determines a cusp form f on GL2 (A) by
the formula
√
√
f (g) = det(g∞ )l/2 j(g∞ , −1)−l f (g∞ ( −1))χ(k)−1
for g = γg∞ k ∈ GL2 (A) with γ ∈ GL2 (Q), g∞ ∈ GL+
2 (R), and k ∈
K0 (N ; Ẑ). Here
¯
½µ
¶
¾
¯
a b
¯
K0 (N ; Ẑ) =
∈ GL2 (Ẑ) ¯ c ≡ 0 mod N Ẑ
c d
19
and
µµ
¶¶
a b
χ
= χ(d) for
c d
By definition, f satisfies
(5.2)
(5.3)
µ
a b
c d
¶
∈ K0 (N ; Ẑ).
f (gk) = χ(k)−1 f (g),
√
f (gkθ ) = exp( −1lθ)f (g),
for all g ∈ GL2 (A), k ∈ K0 (N ; Ẑ), and
µ
¶
cos θ sin θ
kθ =
∈ SO(2).
− sin θ cos θ
When N = 1,
(5.4)
f (g) = f (gJ )
for all g ∈ GL2 (A), where
J =
µ
¶
−1 0
∈ GL2 (R).
0 1
Let π = ⊗v πv be the irreducible cuspidal automorphic representation
of GL2 (A) generated by f . Then πp is a principal series representation
GL (Q )
| |−sp ) of GL2 (Qp ) for each prime p, where B is the
IndB(Q2 p ) p (| |sp £ χ−1
p
standard Borel subgroup of GL2 and sp ∈ C. Also, π∞ is the (limit of)
discrete series representation of GL2 (R) of weight l. In the space of π,
the conditions (5.2), (5.3) characterize the cusp form f up to scalars.
Let Wf = ⊗v Wv be the Whittaker function of f . We may assume that
Wp (1) = 1 for all primes p. Then
√
µµ
¶ ¶ ( l/2
a exp(−2πa + −1lθ) if a > 0,
a 0
W∞
k =
0 1 θ
0
if a < 0.
We define ϕf = ⊗v ϕv ∈ S(V (A)) as follows.
• For each prime p - N , ϕp is the characteristic function of M2 (Zp ).
• For each prime q | N , the partial Fourier transform ϕ̂q of ϕq is
given by
(
χq (x4 ) if x1 , x2 ∈ Zq , x3 ∈ N Zq , x4 ∈ Z×
q ,
ϕ̂q (x) =
0
otherwise.
√
√
• ϕ∞ (x) = (x1 + −1x2 + −1x3 − x4 )l exp(−π tr(xt x)).
Set K = SO(2) SL2 (Ẑ) and K0 (N ) = K ∩ SO(2)K0 (N ; Ẑ). Recall that
µµ
¶µ
¶ ¶
Z
Z
1 x
a 0
φ(g) dg =
φ
k |a|−2 dx d× a dk
0 1
0 a−1
SL2 (A)
A×A× ×K
20
ATSUSHI ICHINO AND TAMOTSU IKEDA
for φ ∈ L1 (SL2 (A)). Here dg is the Tamagawa measure on SL2 (A), dx
is the Tamagawa measure on A, d× a is the Tamagawa measures on A× ,
and dk is the Haar measure on K such that vol(K) = πζ(2)−1 .
Proposition 5.2. As functions on H̃ 0 (A),
θ(f , ϕf ) = 2l vol(K0 (N ))(f ⊗ f χ(det)).
Proof. By (5.1), θ(f , ϕf ) ∈ π £ π ∨ . A routine calculation shows that
µµ
ω
ω(g, h)ϕp = ϕp ,
¶
¶
det(h1 h−1
)
0
2
, (h1 , h2 ) ϕq = χq (h1 h2 )−1 χq (det(h2 ))ϕq ,
0
1
√
ω(kθ , (kθ1 , kθ2 ))ϕ∞ = exp( −1l(−θ + θ1 + θ2 ))ϕ∞ ,
for (g, h) ∈ R(Zp ), h1 , h2 ∈ K0 (N ; Zq ), and kθ , kθ1 , kθ2 ∈ SO(2), where
p (resp. q) is a prime such that p - N (resp. q | N ). Hence there exists
a constant C such that
θ(f , ϕf ) = C(f ⊗ f χ(det)).
By Lemma 5.1,
C = Wf (1)−2 Wθ(f ,ϕf ) (1) = exp(4π) vol(K0 (N ))C∞ ,
where C∞ is equal to
µµ
¶µ
¶¶
µµ
¶µ
¶¶
Z
1 x
a 0
1 x
a 0
W∞
ϕ̂∞
|a|−2 dx d× a
−1
−1
0
1
0
a
0
1
0
a
×
R×R
Z ∞Z ∞
√
=2
exp(2π −1x)al exp(−2πa2 )
0
−∞
√
× (a − −1a−1 x + a−1 )l exp(−π(a2 + a−2 x2 + a−2 )) · a−2 dx d× a
= 2l exp(−4π).
This completes the proof.
¤
6. Local zeta integrals
In this section, following Gross and Kudla [13], we compute the local
zeta integrals of Garrett [10], Piatetski-Shapiro and Rallis [32].
6.1. Preliminaries. Let
¯
¾
½µ
¶
¯
A
∗
×
¯
P =
∈ GSp3 (Qv ) ¯ A ∈ GL3 (Qv ), ν ∈ Qv
0 ν t A−1
21
be the Siegel parabolic subgroup of GSp3 (Qv ). Let Z be the center of
GSp3 (Qv ). Set K = GSp3 (Zp ) if v = p, and
½µ
¶¯
¾
√
A B ¯¯
K=
A + −1B ∈ U(3)
−B A ¯
if v = ∞. Let
G = {(g1 , g2 , g3 ) ∈ GL2 (Qv )3 | det(g1 ) = det(g2 ) = det(g3 )}.
We regard G as a subgroup of GSp3 (Qv ) via the embedding


a1 0 0 b 1 0 0
 0 a2 0 0 b 2 0 
µµ
¶ µ
¶ µ
¶¶


a1 b 1
a2 b 2
a3 b 3
 0 0 a3 0 0 b 3 
,
,
7−→ 
.
c1 d1
c2 d2
c3 d3
 c1 0 0 d1 0 0 
0 c 0 0 d 0
2
2
0 0 c3 0 0 d3
Set
¶ µ
¶ µ
¶¶ ¯
¾
¯
a1 0
a2 0
a3 0
×
¯ ai ∈ Qv ,
T = t(a) =
,
,
¯
0 a−1
0 a−1
0 a−1
1
2
3
¾
½µµ
¶ µ
¶ µ
¶¶ ¯
¯
1 x1
1 x2
1 x3
¯
U0 =
,
,
¯ xi ∈ Qv , x1 + x2 + x3 = 0 ,
0 1
0 1
0 1
½
µ
µ
¶¶ ¯
¾
¯
1 x
¯
Ũ = u(x) = 1, 1,
¯ x ∈ Qv ,
0 1
½
µµ
and KG = G ∩ K.
GSp (Q ) s/2
For s ∈ C, let I(s) = IndP 3 v (δP ) denote the degenerate principal series representation of GSp3 (Qv ). Here δP is the modulus character
of P . Let f (s) be a holomorphic section of I(s). Note that
µµ
¶ ¶
A
∗
(s)
f
g = | det(A)|2s+2 |ν|−3s−3 f (s) (g).
0 ν t A−1
For i = 1, 2, 3, let πi be an irreducible admissible generic representation
of GL2 (Qv ) with central character ωi . We assume that ω1 ω2 ω3 is trivial.
Let Wi be a Whittaker function of πi . Define a function W = W1 ⊗
W2 ⊗ W3 on G by
W (g) = W1 (g1 )W2 (g2 )W3 (g3 )
for g = (g1 , g2 , g3 ) ∈ G. Then the local zeta integral is given by
Z
(s)
Z(f , W ) =
f (s) (δg)W (g) dg,
ZU0 \G
22
ATSUSHI ICHINO AND TAMOTSU IKEDA
where

1
0

0
δ=
1
0
0
1
1
0
1
0
0
1
0
1
1
0
0
−1
−1
−1
0
−1
−1
0
1
0
0
1
0

0
0

1
 ∈ Sp3 (Z).
0
0
1
As in §5.1, let V = M2 (Qv ) be the quadratic space with bilinear
form (x, y) = tr(xy ι ). Let Φ ∈ S(V 3 ). For g ∈ GSp3 (Qv ), choose
h ∈ GO(V )(Qv ) such that ν(h) = ν(g), and put
(0)
fΦ (g) = ω(g, h)Φ(0).
(0)
Then fΦ (g) does not depend on the choice of h, and defines an element
(0)
(s)
of I(0). We extend fΦ to a holomorphic section fΦ of I(s) so that
(s)
the restriction of fΦ to K does not depend on s. Using a Bruhat
decomposition of δ, we obtain the following.
Lemma 6.1.
Z
(0)
fΦ (δg)
=
ω(g, h)Φ(y, y, y) dy.
V
6.2. The non-archimedean case. Let v = q ∈ QD√
. Let ψ be the
additive character of Qq given by ψ(x) = exp(−2π −1x) for x ∈
Z[q −1 ]. Let χ = χq be the quadratic character of Q×
q associated to
√
Qq ( −D)/Qq by class field theory. Put d = ordq (D). Note that
(
1
if q 6= 2,
d=
2 or 3 if q = 2.
GL (Q )
0
GL (Q )
0
00
00
Let π = IndB(Q2 q ) q (| |s £ χ| |−s ) and σ = IndB(Q2 q ) q (| |s £ | |−s ),
0
where B is the standard Borel subgroup of GL2 . Put α = q −s and
00
β = q −s . Let Wπ be the Whittaker function of π with respect to ψ
such that Wπ (1) = 1,
Wπ (gk) = χ(k)Wπ (g)
for all g ∈ GL2 (Qq ) and k ∈ K0 (D; Zq ). Here
¯
½µ
¶
¾
¯
a b
K0 (D; Zq ) =
∈ GL2 (Zq ) ¯¯ c ≡ 0 mod DZq
c d
and
µµ
χ
a b
c d
¶¶
µ
= χ(d) for
a b
c d
¶
∈ K0 (D; Zq ).
23
Similarly, let Wσ be the Whittaker function of σ with respect to ψ such
that Wσ (1) = 1,
Wσ (gk) = Wσ (g)
for all g ∈ GL2 (Qq ) and k ∈ GL2 (Zq ). Note that Wπ and Wσ are
uniquely determined. Define ϕχ ∈ S(V ) so that
µµ
¶¶ (
χ(y4 ) if y1 , y2 ∈ Zq , y3 ∈ DZq , y4 ∈ Z×
y1 y2
q ,
ϕ̂χ
=
y3 y4
0
otherwise,
where ϕ̂χ ∈ S(V ) is the partial Fourier transform of ϕχ given by
µµ
¶¶ Z
µµ
¶¶
y1 y2
y1 y20
ϕ̂χ
=
ϕχ
ψ(y2 y40 − y4 y20 ) dy20 dy40 .
0
y3 y4
y
y
2
3
4
Q
q
Let ϕ0 ∈ S(V ) be the characteristic function of M2 (Zq ).
Proposition 6.2. Set W = Wπ ⊗ Wπ ⊗ Wσ and Φ = ϕχ ⊗ ϕχ ⊗ ϕ0 ∈
S(V 3 ). Then
µ
¶
1
(0)
−3d
−1 −2
−1 2 −4d
Z(fΦ , W ) = χ(−1)q (1+q ) (1−q ) α vol(KG )L
,π × π × σ .
2
Here
L(s, π × π × σ)
¤−1
£
= (1 − α2 βq −s )(1 − α2 β −1 q −s )(1 − α−2 βq −s )(1 − α−2 β −1 q −s )
.
First, we compute the function W . It is well-known that

µµ n ¶¶  −n/2 β n+1 − β −n−1
q
if n ≥ 0,
q 0
Wσ
=
β − β −1
0 1

0
otherwise.
Set
µ
¶
µ
¶
µ
¶
1 0
1 0
0 1
w=
, k1 = d−1
, k2 = d−2
.
−1 0
q
1
q
1
p
√
Lemma 6.3. Put ε = χ(−1) and ζ8 = exp(π −1/4).
(i)
µµ n ¶¶ ( −n/2 n
q
α if n ≥ 0,
q 0
Wπ
=
0 1
0
otherwise,
(
µµ n ¶ ¶
χ(q n+d )ε−1 q −(n+d)/2 α−n−2d if n ≥ −d,
q 0
Wπ
w =
0 1
0
otherwise.
24
ATSUSHI ICHINO AND TAMOTSU IKEDA
(ii) When q = 2,
µµ n ¶ ¶ ( 1/2 −1
q α
q 0
Wπ
k =
0 1 1
0
if n = −1,
otherwise.
(iii) When q = 2 and d = 3, for u ∈ 1 + 2Z,
µµ n
¶ ¶ (
χ(u)εζ8−u q 1/2 α−2
q u 0
Wπ
k2 =
0 1
0
if n = −2,
otherwise.
Proof. Define φ ∈ π so that supp(φ) ∩ GL2 (Zq ) = K0 (D; Zq ) and
φ(k) = χ(k)
for k ∈ K0 (D; Zq ). Let Wφ be the Whittaker function of π defined by
µ µ
¶ ¶
Z
1 x
Wφ (g) =
φ w
g ψ(x) dx
0 1
Qq
for g ∈ GL2 (Qq ). A standard calculation shows that
Wπ = χ(q d )εq d/2 α−2d · Wφ
and proves the formula for Wπ .
¤
(s)
Next, we compute the section fΦ . For each n ∈ Z, let φn denote
the characteristic function of q n Zq . We define φχ ∈ S(Qq ) by
(
χ(x) if x ∈ Z×
q ,
φχ (x) =
0
otherwise.
Let φ̂χ ∈ S(Qq ) be the Fourier transform of φχ given by
Z
φ̂χ (x) =
φχ (x0 )ψ(xx0 ) dx0 ,
Qq
where dx0 is the Haar measure on Qq such that vol(Zq ) = 1. Similarly,
let ξχ ∈ S(Qq ) be the Fourier transform of
φχ · (the characteristic function of 1 + q 2 Zq ).
It is easy to check that
¶¶
¶ ¶ µµ
µµ
¶µ n
y1 y2
1 x
q
0
0
,1 ϕ
ω
y3 y4
0 1
0 q −n
= q −2n φ−n (y1 )φ−n (y2 )φ−n (y3 )φ−n (y4 )ψ(x(y1 y4 − y2 y3 )).
Also, a routine calculation shows the following.
Lemma 6.4.
25
(i)
µµ
ω
¶ ¶ µµ
¶¶
qn 0
y1 y2
, 1 ϕχ
0 q −n
y3 y4
= q −2n φ−n (y1 )φ̂χ (q n y2 )φ−n+d (y3 )φ−n (y4 ).
(ii) For b ∈ Zq ,
µµ n
¶µ
¶
¶ µµ
¶¶
q
0
1 b
y1 y2
ω
w, 1 ϕχ
0 q −n
0 1
y3 y4
= q −2n−d φ−n (y1 )φ−n−d (y2 )φχ (q n y3 )φ−n (y4 )ψ(q 2n b(y1 y4 − y2 y3 )).
(iii) When q = 2,
µµ n
¶
¶ µµ
¶¶
q
0
y1 y2
ω
k , 1 ϕχ
0 q −n 1
y3 y4
= q −2n φ−n (y1 )φ−n (y4 )φχ (−q n−d+1 y3 )φ̂χ (−q 2n−d+1 y2 y3 ).
(iv) When q = 2 and d = 3, for e ∈ {±1},
µµ n
¶
¶ µµ
¶¶
q
0
y1 y2
e
ω
k , 1 ϕχ
0 q −n 2
y3 y4
= q −2n φ−n (y1 )φ−n (y4 )φχ (−q n−d+2 ey3 )ξχ (−q 2n−d+2 ey2 y3 ).
Set
µµ
t(n1 , n2 , n3 ) =
¶ µ n
¶ µ n
¶¶
q n1
0
q 2
0
q 3
0
,
,
.
0 q −n1
0 q −n2
0 q −n3
Put K0 = {1} and
Kd =
½µ
¶ ¯
¾
¯
1 b
¯
w ¯ b ∈ Zq /DZq .
0 1
When q = 2, put K1 = {k1 }. When q = 2 and d = 3, put K2 =
{k2 , k2−1 }. It is easy to check that
(s)
(0)
fΦ (δg) = (q n1 +n2 +n3 max(q −2n1 , q −2n2 , q −2n3 , |x|))−2s fΦ (δg)
for g = u(x)t(n1 , n2 , n3 )k with x ∈ Qq , n1 , n2 , n3 ∈ Z, k ∈ Ki ×Kj ×{1}.
Lemma 6.5. Let g = u(x)t(n1 , n2 , n3 )k with x ∈ Qq , n1 , n2 , n3 ∈ Z,
and k ∈ Ki × Kj × {1}. If i 6= j, then
(0)
fΦ (δg) = 0.
Proof. If i 6= j, then ω(g, 1)Φ(y, y, y) = 0 for all y ∈ V . This yields the
lemma.
¤
Lemma 6.6. Let g = u(x)t(n1 , n2 , n3 )k with x ∈ Qq , n1 , n2 , n3 ∈ Z,
and k = (h1 , h2 , 1) ∈ Ki × Ki × {1}.
26
ATSUSHI ICHINO AND TAMOTSU IKEDA
(0)
(i) For i = 0 and h1 = h2 = 1, fΦ (δg) is equal to
(
χ(−1)q −2n3 −d (1 − q −1 ) if n1 = n2 ≤ n3 − d, x ∈ q 2n1 Zq ,
0
otherwise.
(ii) For i = d and
h1 =
µ
¶
1 b1
w,
0 1
µ
b2 =
¶
1 b2
w
0 1
(0)
with b1 , b2 ∈ Zq , fΦ (δg) is equal to
 −n −n −2d
q 1 3 (1 − q −1 ) if n1 = n2 ≤ n3 < n1 + d





and q 2n1 b1 + q 2n1 b2 + x ∈ q n1 +n3 Zq ,

q −2n3 −d (1 − q −1 )
if n1 = n2 ≤ n3 − d



and q 2n1 b1 + q 2n1 b2 + x ∈ q 2n1 +d Zq ,



0
otherwise.
(iii) When q = 2, for i = 1 and

χ(−1)q −2n3 −d (1 − q −1 )






−χ(−1)q −2n3 −d (1 − q −1 )






0
(0)
h1 = h2 = k1 , fΦ (δg) is equal to
if n1 = n2 ≤ n3 − d
and x ∈ q 2n1 +1 Zq ,
if n1 = n2 ≤ n3 − d
and x ∈ q 2n1 + q 2n1 +1 Zq ,
otherwise.
(iv) When q = 2 and d = 3, for i = 2 and h1 = k2e1 , h2 = k2e2 with
(0)
e1 , e2 ∈ {±1}, fΦ (δg) = 0 unless n1 = n2 ≤ n3 − d, in which
(0)
case, fΦ (δg) is equal to
 −2n −d
q 3 (1 − q −1 )
if e1 = e2 , x ∈ −q 2n1 e1 + q 2n1 +2 Zq ,




−2n −d
−1

if e1 = e2 , x ∈ q 2n1 e1 + q 2n1 +2 Zq ,
−q 3 (1 − q )
χ(−1)q −2n3 −d (1 − q −1 )
if e1 6= e2 , x ∈ q 2n1 +2 Zq ,



−χ(−1)q −2n3 −d (1 − q −1 ) if e1 6= e2 , x ∈ q 2n1 +1 + q 2n1 +2 Zq ,



0
otherwise.
Proof. We only prove (ii). The other cases are similar. Put m =
min(n1 , n2 , n3 ), m0 = min(n1 +d, n2 +d, n3 ), and x0 = q 2n1 b1 +q 2n2 b2 +x.
Since
ω(g, 1)Φ(y, y, y) = q −2n1 −2n2 −2n3 −2d φ−m (y1 )φ−m0 (y2 )φ−m (y4 )
× φχ (q n1 y3 )φχ (q n2 y3 )φ−n3 (y3 )ψ(x0 (y1 y4 − y2 y3 )),
27
(0)
(0)
fΦ (δg) = 0 unless n1 = n2 ≤ n3 , in which case, fΦ (δg) is equal to
Z
0
q −3m+m −2n3 −2d φ−m (y4 )φχ (q m y3 )2 φ−n3 (y3 )φm (x0 y4 )φm0 (x0 y3 ) dy3 dy4 .
Q2q
This integral is equal to
Z
Z
−3m+m0 −2n3 −2d
q
y3 ∈q −m Z×
q
0
dy4 dy3 = q −m+m −2n3 −2d (1 − q −1 )
y4 ∈q −m Zq
0
if x0 ∈ q m+m Zq , and vanishes otherwise.
¤
(s)
Now we compute the local zeta integral Z(fΦ , W ). Set
¶ µ
¶ µ
¶¶
µµ
q 0
q 0
q 0
,
,
.
q=
0 1
0 1
0 1
(s)
(s)
Note that fΦ (δqg) = q −s−1 fΦ (δg). Since Ũ T KG ∪ q Ũ T KG is a fun(s)
damental domain of ZU0 \G, Z(fΦ , W ) is equal to
Z
(s)
fΦ (δu(x)t(a)k)W (u(x)t(a)k)|a|−2 dx d× a dk
× 3
Qq ×(Qq ) ×KG
Z
(s)
fΦ (δqu(x)t(a)k)W (qu(x)t(a)k)q 2 |a|−2 dx d× a dk
+
3
Qq ×(Q×
q ) ×KG
as in [13, §4]. Here dx is the Haar measure on Qq such that vol(Zq ) =
×
×
1, d× ai is the Haar measure on Q×
q such that vol(Zq ) = 1, d a =
d× a1 d× a2 d× a3 , and |a| = |a1 a2 a3 |. Set
0
KG
= G ∩ (K0 (D; Zq ) × K0 (D; Zq ) × GL2 (Zq )).
(s)
0
Then the function g 7→ fΦ (δg)W (g) on G is right KG
-invariant. Since
[
Ki × Kj × {1}
0≤i,j≤d
0
, we have
is a set of representatives for KG /KG
(s)
Z(fΦ , W )
=
0
vol(KG
)
d X
d X
1
X
i=0 j=0 l=0
(l)
Zij (s),
28
ATSUSHI ICHINO AND TAMOTSU IKEDA
where
(0)
Zij (s)
X
=
X
k∈Ki ×Kj ×{1} n1 ,n2 ,n3 ∈Z
Z
Qq
(s)
(1)
Zij (s) =
× fΦ (δu(x)t(n1 , n2 , n3 )k)W (u(x)t(n1 , n2 , n3 )k)q 2n1 +2n2 +2n3 dx,
X
X Z
k∈Ki ×Kj ×{1} n1 ,n2 ,n3 ∈Z
Qq
(s)
× fΦ (δqu(x)t(n1 , n2 , n3 )k)W (qu(x)t(n1 , n2 , n3 )k)q 2n1 +2n2 +2n3 +2 dx.
(l)
If i 6= j, then Zij (s) = 0 by Lemma 6.5.
Lemma 6.7.
(i)
(0)
Z00 (s) = χ(−1)q −2ds−2d (1 − q −1 )(β − β −1 )−1
£
× β 2d+1 (1 − β 2 q −2s−1 )−1 (1 − α4 β 2 q −2s−1 )−1
¤
− β −2d−1 (1 − β −2 q −2s−1 )−1 (1 − α4 β −2 q −2s−1 )−1 ,
(1)
Z00 (s) = χ(−1)q −2ds−s−2d−1/2 (1 − q −1 )α2 (β − β −1 )−1
£
× β 2d+2 (1 − β 2 q −2s−1 )−1 (1 − α4 β 2 q −2s−1 )−1
¤
− β −2d−2 (1 − β −2 q −2s−1 )−1 (1 − α4 β −2 q −2s−1 )−1 .
(ii)
(
(0)
Y0 (s)
if d = 1,
(0)
Zdd (s) =
(0)
(0)
Y0 (s) + Y1 (s) if d = 2 or 3,
(
(1)
(1)
Y0 (s) + Y1 (s)
if d = 1 or 2,
(1)
Zdd (s) =
(1)
(1)
(1)
Y0 (s) + Y1 (s) + Y2 (s) if d = 3,
29
where
(0)
Y0 (s) = χ(−1)q −d (1 − q −1 )α−4d (β − β −1 )−1
£
× β(1 − β 2 q −2s−1 )−1 (1 − α−4 β 2 q −2s−1 )−1
¤
− β −1 (1 − β −2 q −2s−1 )−1 (1 − α−4 β −2 q −2s−1 )−1 ,
(0)
Y1 (s) = χ(−1)q −2s−d−1 (1 − q −1 )α−4d+4 (β − β −1 )−1
£
¤
× β 3 (1 − β 2 q −2s−1 )−1 − β −3 (1 − β −2 q −2s−1 )−1 ,
(1)
Y0 (s) = χ(−1)q −s−d−1/2 (1 − q −1 )α−4d−2 (β − β −1 )−1
£
× β 2 (1 − β 2 q −2s−1 )−1 (1 − α−4 β 2 q −2s−1 )−1
¤
− β −2 (1 − β −2 q −2s−1 )−1 (1 − α−4 β −2 q −2s−1 )−1 ,
(1)
Y1 (s) = χ(−1)q −s−d−1/2 (1 − q −1 )α−4d+2 (β − β −1 )−1
£
¤
× β 2 (1 − β 2 q −2s−1 )−1 − β −2 (1 − β −2 q −2s−1 )−1 ,
(1)
Y2 (s) = χ(−1)q −3s−d−3/2 (1 − q −1 )α−4d+6 (β − β −1 )−1
£
¤
× β 4 (1 − β 2 q −2s−1 )−1 − β −4 (1 − β −2 q −2s−1 )−1 .
(0)
(iii) When q = 2, Z11 (s) = 0 and
(1)
Z11 (s) = χ(−1)q −2ds+s−2d+1/2 (1 − q −1 )α−2 (β − β −1 )−1
£
¤
× β 2d (1 − β 2 q −2s−1 )−1 − β −2d (1 − β −2 q −2s−1 )−1 .
(1)
(iv) When q = 2 and d = 3, Z22 (s) = 0 and
(0)
Z22 (s) = χ(−1)q −2ds+2s−2d+1 (1 − q −1 )α−4 (β − β −1 )−1
£
¤
× β 2d−1 (1 − β 2 q −2s−1 )−1 − β −2d+1 (1 − β −2 q −2s−1 )−1 .
(0)
Proof. We only compute Zdd (s). The other cases are similar. Let
x ∈ Qq , n1 , n2 , n3 ∈ Z, and
¶
¶
¶ µ
µµ
1 b2
1 b1
w, 1
w,
k=
0 1
0 1
(0)
with b1 , b2 ∈ Zq . Then the integrand of Zdd (s) vanishes unless n1 =
n2 ≤ n3 . Put
Υ(m, n) = χ(−1)q −n(2s+1)−3d (1−q −1 )α−4m−4d (β−β −1 )−1 (β 2n+1 −β −2n−1 ).
For n1 = n2 = m and n3 = n, the integral
Z
(s)
fΦ (δu(x)t(n1 , n2 , n3 )k)W (u(x)t(n1 , n2 , n3 )k)q 2n1 +2n2 +2n3 dx
Qq
30
ATSUSHI ICHINO AND TAMOTSU IKEDA
is equal to Υ(m, n) if
(
0≤m≤n
if d = 1,
0 ≤ m ≤ n, or m = −1 and n ≥ 1 if d = 2 or 3,
and vanishes otherwise. It is easy to check that
∞ X
∞
∞
X
X
(0)
(0)
2d
2d
q
Υ(m, n) = Y0 (s), q
Υ(−1, n) = Y1 (s).
m=0 n=m
n=1
This proves the formula for
(0)
Zdd (s).
¤
By a direct calculation,
(s)
Z(fΦ , W )
−d
µ
−1
= χ(−1)q (1−q )(1−q
−2s−1
)α
−4d
0
)L
vol(KG
¶
1
s + ,π × π × σ .
2
This completes the proof of Proposition 6.2.
6.3. The archimedean case. Let v √
= ∞. Let ψ be the additive
character of R given by ψ(x) = exp(2π −1x) for x ∈ R.
For each l ∈ N, let σl denote the (limit of) discrete series representation of GL2 (R) of weight l. Let Wl be the Whittaker function of σl
with respect to ψ given by
√
¶¶ ( l/2
¶µ
µµ
a exp(−2πa + −1lθ) if a > 0,
cos θ sin θ
a 0
=
Wl
− sin θ cos θ
0 1
0
if a < 0.
Define ϕl ∈ S(V ) by
µµ
¶¶
√
√
y1 y2
ϕl
= (y1 + −1y2 + −1y3 −y4 )l exp(−π(y12 +y22 +y32 +y42 )).
y3 y4
Proposition 6.8. Set W = Wl ⊗ Wl ⊗ W2l and Φ = ϕl ⊗ ϕl ⊗
ω(1, (J , J ))ϕ2l ∈ S(V 3 ), where
µ
¶
−1 0
J =
.
0 1
Then
(0)
Z(fΦ , W ) = 2−4l+1 π −4l+2 Γ(l)2 Γ(2l − 1) vol(KG ).
(s)
First, we compute the section fΦ .
Lemma 6.9. Let r ∈ R×
+ and x ∈ R. For each l ∈ Z≥0 , put
Z
Jl =
((y1 + y4 )2 + (y2 + y3 )2 )l exp(−πr(y12 + y22 + y32 + y42 ))
4
R
√
× exp(−2π −1x(y1 y4 + y2 y3 )) dy1 dy2 dy3 dy4 .
31
Then
Jl = 2l π −l l!(r +
Proof. Put
Fl (z) =
√
−1x)−l−1 (r −
√
−1x)−1 .
Z
((y1 + y4 )2 + (y2 + y3 )2 )l exp(−πr(y12 + y22 + y32 + y42 ))
√
× exp(−2π −1x(y1 y4 − y2 y3 ))
√
× exp(2π −1(z1 y1 + z2 y2 + z3 y3 + z4 y4 )) dy1 dy2 dy3 dy4 .
R4
Then
√
F0 (z) = R−1 exp(−πrR−1 (z12 +z22 +z32 +z42 )+2π −1R−1 x(z1 z4 +z2 z3 ))
and Fl (z) = ∇l F0 (z), where R = r2 + x2 and
õ
¶2 µ
¶2 !
∂
∂
∂
1
∂
√
+
+
+
.
∇=
∂z1 ∂z4
∂z2 ∂z3
(2π −1)2
Hence
Jl = Fl (0) = 2l π −l l!(r +
√
−1x)−l R−1 .
¤
(s)
3
Lemma 6.10. For x ∈ R and a = (a1 , a2 , a3 ) ∈ (R×
+ ) , fΦ (δu(x)t(a))
is equal to
√
√
22l π −2l (2l)!(r + −1x)−s−2l−1 (r − −1x)−s−1 a2s+l+2
a2s+l+2
a32s+2l+2 ,
1
2
where r = a21 + a22 + a23 .
Proof. It is easy to check that
(s)
(0)
fΦ (δu(x)t(a)) = (a1 a2 a3 (r2 + x2 )−1/2 )2s fΦ (δu(x)t(a))
(0)
l+2 2l+2
and fΦ (δu(x)t(a)) = al+2
J2l . This yields the lemma.
1 a2 a3
¤
(s)
Now we compute the local zeta integral Z(fΦ , W ). Note that supp(W ) =
Z SL2 (R)3 . Since Ũ T KG is a fundamental domain of U0 \ SL2 (R)3 ,
(s)
Z(fΦ , W ) is equal to
Z
1
(s)
fΦ (δu(x)t(a)k)W (u(x)t(a)k)|a|−2 dx d× a dk
2 R×(R× )3 ×KG
as in [13, §6]. Here dx, dai are the Lebesgue measures on R, d× ai =
|ai |−1 dai , d× a = d× a1 d× a2 d× a3 , and |a| = |a1 a2 a3 |. Since the function
32
ATSUSHI ICHINO AND TAMOTSU IKEDA
(s)
(s)
g 7→ fΦ (δg)W (g) on G is right KG -invariant, vol(KG )−1 Z(fΦ , W ) is
equal to
22l+2 π −2l (2l)!
Z
√
√
×
(a21 + a22 + a23 + −1x)−s−2l−1 (a21 + a22 + a23 − −1x)−s−1
3
R×(R×
+)
2(s+l) 2(s+l) 2(s+2l)
a2
a3
−4s−4l+1 −s−4l+2
× a1
=2
π
exp(−2π(a21 + a22 + a23 −
√
−1x)) dx d× a
(2l)!Γ(s + 2l + 1)−1 Γ(2s + 2l)−1
× Γ(s + l)2 Γ(s + 2l − 1)Γ(s + 2l)
by Lemma 2.6 of [20]. This completes the proof of Proposition 6.8.
7. Proof of Theorem 3.1
Recall that g ∈ Sκ−1 (Γ0 (D), χ) is a primitive form and f ∈ S2κ−2 (SL2 (Z))
is a normalized Hecke eigenform. As in §5.3, let g (resp. f ) be the
cusp form on GL2 (A) determined by g (resp. f ), π (resp. σ) the irreducible cuspidal automorphic representation of GL2 (A) generated by
g (resp. f ), and Wg (resp. Wf ) the Whittaker function of g (resp. f ).
Let V = M2 (Q) be the quadratic space as in §5.1. We define ϕg , ϕf ∈
S(V (A)) as in §5.3. Set W = Wg ⊗ Wg ⊗ Wf and
Φ = ϕg ⊗ ϕg ⊗ ω(1, (J , J ))ϕf ∈ S(V 3 (A)),
where
µ
J =
Put S = {∞} ∪ QD and
ZS (s) = vol(SO(2))−3
¶
−1 0
∈ GL2 (R).
0 1
Y
vol(SL2 (Zq ))−3 ·
q∈QD
Y
(s)
Z(fΦv , Wv ),
v∈S
(s)
where Z(fΦv , Wv ) is the local zeta integral as in §6.1. Let
G = {(g1 , g2 , g3 ) ∈ (GL2 )3 | det(g1 ) = det(g2 ) = det(g3 )},
H = {(h1 , h2 , h3 ) ∈ GO(V )3 | ν(h1 ) = ν(h2 ) = ν(h3 )}.
We regard F = g ⊗ g ⊗ f as a cusp form on G(A) and define the theta
lift θ(F, Φ) to H(A) as in §5.2. Put
Z
I(θ(F, Φ)) =
θ(F, Φ)(h, h, h) dh,
ZH̃ 0 (A)H̃ 0 (Q)\H̃ 0 (A)
where ZH̃ 0 is the center of H̃ 0 = GL2 × GL2 and dh is the Haar measure
on H̃ 0 (A) such that vol(ZH̃ 0 (A)H̃ 0 (Q)\H̃ 0 (A)) = 1.
33
By Main Identity 9.1 of [16],
3
(7.1)
S
−2
µ
vol(K) ZS (0)ζ (2) L
S
1
,π × π × σ
2
¶
= 2I(θ(F, Φ)),
where K = SO(2) SL2 (Ẑ) and vol(K) = πζ(2)−1 . By Propositions 6.2
and 6.8,
µ
¶
1
S
−2 S
ZS (0)ζ (2) L
,π × π × σ
2
= 2−4κ+5 π −4κ+6 Γ(κ − 1)2 Γ(2κ − 3)
Y
×
χq (−1)Dq2κ−7 (1 + q −1 )−2 (1 − q −1 )2 ag (Dq )−4
q∈QD
× ζ S (2)−2 L(2κ − 3, g × g × f )
Y
= −2π 2 D2κ−7 ag (D)−4
(1 + q −1 )−4
q∈QD
−2
× ζ(2) Λ(2κ − 3, g × g × f ).
By Proposition 5.2 and (5.4),
θ(F, Φ) = θ(g, ϕg ) ⊗ θ(g, ϕg ) ⊗ θ(f , ω(1, (J , J ))ϕf )
= 24κ−4 vol(K0 (D))2 vol(K)
× (g ⊗ gχ(det)) ⊗ (g ⊗ gχ(det)) ⊗ (f̄ ⊗ f̄ ).
Thus,
2
2
I(θ(F, Φ)) = 24κ−6 vol(K0 (D))4 vol(K)hTrD
1 (g ), f i .
Therefore
2
2
Λ(2κ − 3, g × g × f ) = −24κ−6 D−2κ+3 ag (D)4 hTrD
1 (g ), f i
+
= −24κ−6 D−2κ+3 hS−D
(ΩSK (G|H2 )), f i2
by Proposition 4.1 and (7.1). If c(D) = 0, then Λ(2κ−3, g ×g ×f ) = 0.
+
If c(D) 6= 0, then f = c(D)−1 S−D
(h) and
+
hS−D
(ΩSK (G|H2 )), f i
hΩSK (G|H2 ), hi
hG|H2 , F i
= c(D)
= c(D)
.
hf, f i
hh, hi
hF, F i
This completes the proof of Theorem 3.1.
8. The Gross-Prasad conjecture
In this section, we interpret our result in terms of the Gross-Prasad
conjecture [14], [15].
34
ATSUSHI ICHINO AND TAMOTSU IKEDA
Let H1 = SO(n + 1) and H0 = SO(n) be special orthogonal groups
over a number field k with embedding ι : H0 ,→ H1 . Let π1 ' ⊗v π1,v
and π0 ' ⊗v π0,v be irreducible cuspidal automorphic representations
of H1 (Ak ) and H0 (Ak ), respectively. We assume that
HomH0 (kv ) (π1,v , π0,v ) 6= 0
for all places v of k. Gross and Prasad conjectured that, when π1 and
π0 are tempered, the period integral
Z
hF1 |H0 , F0 i =
F1 (ι(h0 ))F0 (h0 ) dh0
H0 (k)\H0 (Ak )
does not vanish for some F1 ∈ π1 and some F0 ∈ π0 if and only if
µ
¶
1
L
, π1 × π0 6= 0.
2
To relate our result to the Gross-Prasad conjecture, we must
• remove the assumption that π1 and π0 are tempered,
• formulate an identity which relates the period integral to special
values of automorphic L-functions.
Following Ginzburg, Piatetski-Shapiro, and Rallis [12], we put
Pπ1 ,π0 (s) =
L(s +
L(s, π1 × π0 )
,
1
, π1 , Ad)L(s + 21 , π0 , Ad)
2
where Ad is the adjoint representation of L Hi on the Lie algebra of
L
Hi . Then the identity
µ ¶
|hF1 |H0 , F0 i|2
1
(8.1)
= Pπ1 ,π0
hF1 , F1 ihF0 , F0 i
2
would hold up to an elementary constant. This conjectural identity is
compatible with the results of Waldspurger [40] for n = 2, Harris and
Kudla [16], [18] for n = 3, Böcherer, Furusawa, and Schulze-Pillot [4]
for n = 4.
Now we discuss the case n = 5. We retain the notation of §3. Note
that H1 = SO(4, 2) ∼ SU(2, 2) and H0 = SO(3, 2) ∼ Sp2 . We may
assume that g ∗ 6= 0. Let π1 (resp. π0 ) be the irreducible cuspidal
automorphic representation of H1 (AQ ) (resp. H0 (AQ )) determined by
G (resp. F ). Then π1 and π0 are non-tempered. It is easy to check
that
L(s, π1 ) = L(s, Sym2 (π))ζ(s + 1)ζ(s)ζ(s − 1),
¶ µ
¶
µ
1
1
L(s, π0 ) = L(s, σ)ζ s +
ζ s−
,
2
2
35
and
µ ¶
L( 12 , Sym2 (π) × σ)L( 23 , σ)
1
Pπ1 ,π0
=
.
2
L(2, Sym2 (π))L(1, π, Ad)L(1, σ, Ad)
Here π (resp. σ) is the irreducible cuspidal automorphic representation
of GL2 (AQ ) determined by g (resp. f ). On the other hand, if c(D) 6= 0,
then
µ
¶
hf, f ihh, hi|hG|H2 , F i|2
1
2
∼L
, Sym (π) × σ
hF, F i2
2
by Theorem 3.1 and the Kohnen-Zagier formula [24]. According to [21,
§15], it is expected that
hG, Gi
∼ L(2, Sym2 (π))ζ(2).
hg ∗ , g ∗ i
Indeed, Raghavan and Sengupta [34] proved it for D = 4. By the result
of Kohnen and Skoruppa [23],
µ
¶
3
hF, F i
∼L
, σ ζ(2).
hh, hi
2
It is well-known that hg ∗ , g ∗ i ∼ L(1, π, Ad) and hf, f i ∼ L(1, σ, Ad).
Therefore (8.1) is compatible with Theorem 3.1.
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Department of Mathematics, Graduate School of Science, Osaka
City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan
E-mail address: [email protected]
Graduate school of mathematics, Kyoto University, Kitashirakawa,
Kyoto 606-8502, Japan
E-mail address: [email protected]