Special Value Formulae of
Rankin-Selberg L-Functions
A Dissertation
in
Mathematics
Presented to the Faculties of the University of Pennsylvania in
Partial Fulfillment of the Requirements for the Degree of Doctor
of Philosophy
2005
Supervisor of Dissertation
Graduate Group Chairperson
ii
Acknowledgement
I would like to express my indebtness to Shou-Wu Zhang for his wonderful
idea of using the trick of Eisenstein series as well as his fundamental papers [21]
and [22] from which I learned his new language to compute Fourier coefficients.
I also would like to thank my advisor Ted Chinburg for his constant support
and encouragement. At last but not the least, I wish to express my heartiest
thanks to Ye Tian who told me this interesting topic and constantly encourage
me to work out the problem and shared his idea with me always. In fact this
is the a part of a joint project with Ye Tian aiming to establishing a p-adic
analogue of the special value formula of a Hida family of Hilbert newforms.
ABSTRACT
Haiping Yuan
Advisor: Ted Chinburg
In this paper, we prove a special value formula of level N of Rankin-Selberg
L-function associated to a Hilbert modular form of higher weight and a ring
class character of an totally imaginary quadratic extension of a totally real
field. The formula relates the special value of the Rankin-Selberg L-functions
at s =
1
2
to the value of certain test form at some CM-point on a 0-dimensional
Shimura Variety associated to a quaterion algebra. The formula generalizes the
formula proved by Shou-Wu Zhang which is a vast generalization of classical
Gross-Zagier formula. The proof is based on a formula (level N D) of Hui
Xue combined with a technique of Eisenstein Series to compute the universal
constants which arise in the comparison of both formulae of level N and N D.
Contents
1. Introduction
1
2. Hilbert modular forms and automorphic representations
8
2.1. Hilbert modular forms
8
2.2. Classification of local admissible representations
13
2.3. Newform theory
15
2.4. Weil representation and Jacquet-Langlands correspondence
27
2.5. Rankin-Selberg L-functions
35
2.6. Unitary similitudes
39
3. Special value formula of level N D
44
3.1. Kernel function and quasi-newform
45
3.2. Geometric pairing and local Gross-Zagier formula
48
4. Special value formula of level N
56
4.1. Universal constants
58
4.2. Determination of universal constants
66
5. Appendix. Continuous spectrum of L2 (GL2 (F )\GL2 (A), ω)
79
References
85
v
1
1. Introduction
Let F be a totally real number field, and K/F a totally imaginary quadratic
extension of relative discriminant d. Fix a Hilbert cusp newform φ of GL2 over
F of level N which generates a cuspidal automorphic representation π(φ) of
GL2 (AF ), and a ring class character χ of K, i.e., a finite order Hecke character
×
of K which is trivial on A×
F and K . We denote by c(χ) the conductor of χ.
To φ and χ, one associates a Rankin-Selberg L-function L(s, φ, χ) (following
Jacquet), which has analytic continuation and satisfies the functional equation
under s → 1 − s. It is well known that the central critical value L(1/2, φ, χ) is
related to the “height” of certain CM divisor on some 0-dimensional Shimura
Variety. In [8], Gross proves a formula expressing the special value in terms
of the height of certain CM divisor on a 0-dimensional Shimura variety in the
special case that F = Q, and N and D are primes. A far reaching generalization is proposed by Gross ( [7], [9]). There has been breakthrough recently
on this conjecture. Shou-Wu Zhang( [21] and [22]) proves a special (but still
quite general) case of the conjecture. More precisely, let φ be a Hilbert cusp
newform of weight (2, · · · , 2), of level N , and of the trivial character. Assuming that N , c(χ), and d are coprime to each other, he shows that the special
value L(1/2, φ, χ) can be expressed as the height of certain CM divisor of 0dimensional Shimura variety, namely, an (torus) integral of certain form on the
torus given by K × . His results generalize Gross’ earlier results and give an refinement of Waldspurger’s results [19] concerning the equivalence between the
non-vanishing of L(1/2, φ, χ) and the non-vanishing of the torus integral. We
2
generalize Zhang’s results to higher weight (2k, · · · , 2k), based on Hui Xue’s
formula of level N D [20], thus partially proves Gross’ conjecture.
Here are a little more details. Let φ be a Hilbert cusp newform over F
of level N , of weight (2k, · · · , 2k), k ≥ 1, and of trivial central character, see
chapter 2 for the definition. The representation of GL2 (AF ) spanned by φ is
×
denoted by π. Fix a ring class character χ of A×
with conductor c(χ),
K /K
× ×
×
i.e., χ : A×
K /K AF → C is a homomorphism. We denote by ω the quadratic
character associated to K/F . The Rankin-Selberg L-function L(s, φ, χ) has
analytic continuation and satisfies the following functional equation
L(s, φ, χ) = (−1)#Σ NF/Q (N D)1−2s L(1 − s, φ, χ),
where D := c(χ)2 c(ω) and Σ is the following set of places of F
Σ = { v|v|∞,
The point s =
1
2
L(s, φ, χ) at s =
or v - ∞ and ωv (N ) = −1 } .
is the central critical point. We consider the special value of
1
2
under the assumption that #Σ is even and that N, c(χ), c(ε)
are coprime to each other. Then there exists a unique quaternion algebra B
over F ramified exactly at the places in Σ. Let G be the algebraic group given
by B × /F × . Associated to φ and χ, one can define two forms:
(1) the quasi-newform φ# ,
the unique form of level N D in the space π satisfying the relation
(φ# , φa ) = ν ∗ (a)(φ# , φ# ),
(a|D)
3
where
ν ∗ (a) =



 ν(a), if a|c(ε);

 0,

otherwise.
−1
0 
 a
and φa := ρ 
 φ;
0 1
e
(2) a test form φ,
an automorphic form on MU such that for each finite place v not dividing
N D, φe is the eigenform for Hecke operators Tv with the same eigenvalue
as φ, where MU = G(F )\G(Af )/U with U an order of B of reduced
discriminant N .
Zhang [21] (and Hui Xue [20] for higher weight case) proves the following special
value formula for L( 12 , φ, χ) in terms of the quasi-newform φ# :
(1.1)
C
1
# (1)L( , φ, χ) = p
||φ# ||2U0 (N D) · |(φχ , η)|2 ,
φc
2
NF/Q (c(ω))
k−1 [(k−1)!]2
where C = ( 2·4
(2k−1)!
# (1) the first Fourier coefficient of the quasi)g , φc
newform φ# , φχ a toric newform on MU suitably normalized, and ||φ# ||2U0 (N D)
is computed as L2 -norm with respect to the Haar measure dg which is the
product of the standard measure on N (AF )A(AF ), and the measure on the
standard maximal compact group with
vol(SO(F∞ )U0 (N D)) = 1.
In [8], Gross conjectures that the special value may be expressed in terms
of the test form φe in viewing of earlier work of Waldspurger [19] and taking
advantage of φe being of level N . Meanwhile various applications ( [2], [18])
4
suggest that it would be more natural to have a formula expressing the special
value in terms of φ and χ. Inspired by those motivations, Zhang proves the
conjecture of Gross using the technique of continuous spectrum to deduce the
formula of level N from his above formula in the case of k = 1. Our approach
basically follows Zhang’s. Notice that a test form φe is a form in the space
of π 0 = ⊗v πv0 , the representation of G(A) corresponding to π via JacquetLanglands correspondence. In the special case that k = 1, πv0 is trivial for
v|∞ so that one can ignore the archimedean part. This is treated by Zhang.
In general, it is of finite dimension, since πv0 is irreducible representation of
G(Fv ) ∼
= SO3 (R), which is compact. In fact, the dimension is 2k − 1. The
formula can be stated as follows.
Theorem. Assume that #Σ is even, then
(1.2)
Here Pχ :=
1
C
e Pχ )|2 ,
L( , φ, χ) = p
||φ||2U0 (N ) |(φ,
2
NF/Q (D)
X
χ−1 (σ)[Pcσ ] with Pc a CM point in G(F )\G(Af )/U.
σ∈Gal(Hc /K)
To explain the idea of the proof, one rewrites the formula (1.1) in the following
way:
# (1)
e Pχ )|2 1
||φ||2U0 (N ) |(φ,
C
φc
e Pχ )|2 .
L( , φ, χ) = p
||φ||2U0 (N ) |(φ,
2
#
2
NF/Q c(χ) ||φ ||U0 (N D) |(φχ , η)|
2
N(D)
We define the archimedean components of the test form φe to be same as those
of toric newform φχ . Thus the norm of the φv at archimedean place is same
as that of toric newform. Therefore to deduce the formula (1.2) from formula
5
(1.1), it suffices to prove that the product
# (1)
e Pχ )|2
||φ||2U0 (N ) |(φ,
φc
= 1.
NF/Q c(χ) ||φ# ||2U0 (N D) |(φχ , η)|2
(1.3)
Now the key idea is that instead of fixing the Hilbert cusp newform φ, one
views φ as a form varying in the space L2 (GL2 (F )\GL2 (AF )), a space which
decomposes into a discrete part where the form φ lies in and a continuous part
whose elements are continuous sums of Eisenstein series. A crucial step is that
the left hand side of the equation (1.3) is actually a universal function of local
parameters in the sense that it is independent of the form φ. More precisely, for
a finite place v|D, there exists a rational function Qv (t) ∈ C(t), which depends
1
1
only on χv , Qv (0) = 1, and Qv (t) is regular for |t| ≤ |$v | 2 + |$v |− 2 such that
# (1)
Y
e Pχ )|2
||φ||2U0 (N ) |(φ,
φc
Qv (λv ),
=
C(χ)
NF/Q c(χ) ||φ# ||2U0 (N D) |(φχ , η)|2
v|D
where C(χ) is a constant depending only on χ, and λv is the parameter in
Lv (s, φ) = (1 − λv |$v |s + |$v |2s )−1 . The hard part of the above is to show the
ratio
e χ )|2
|(φ,P
|(φχ ,η)|2
has the similar property which requires to carefully analyze both
test form and toric newform. Thus replace the form φ by a form E in the
continuous spectrum L2cont (GL2 (F )\GL2 (AF )). The entire proof of the formula
(1.1) can be carried over to L(s, E, χ) to obtain a similar special value formula
for L(s, E, χ) through which the same constants occur. Now choosing the form
E appropriately, one reduces it to computing the periods of Eisenstein series, a
feasible computation. Finally it turns out that both C(χ) and Qv (t) are equal
to 1 for v|D.
6
For the purpose of self-contained paper, in chapter 2, we collect the materials
needed for our proof. We briefly describe the relationship between Hilbert modular forms and automorphic representations, adelic newform theory, Weil representation and Jacquet-Langlands correspondence, Rankin-Selberg L-functions
(following Jacquet). In addition, we add a proof of a theorem concerning the
isomorphism of the group GL2 × T /∆Gm (F ) with GU (F ), the group of F rational points of a group of unitary similitude, which is a beginning part of
programme proposed by Gross [9] refining Waldspurger’s general results. We
wish to come to this topic in future. Chapter 3 is a brief review of proof of
Xue’s formula of level N D and meanwhile we explain various term occurring in
the formula in order to carry out the computation in chapter 4. We prove the
final formula using the technique of Eisenstein series. In order not to interrupt
the continuity of the proof, an appendix of the spectral decomposition of the
L2 -space L2 (GL2 (F )\GL2 (A), ω) is added.
Notations.
We fix a totally real field F of degree g, and a totally imaginary quadratic
field extension K of F . Let $v denote the uniformizer of Ov , the integer ring
of the completion of F at v, and A := AF be the ring of adeles of F .
Let ψ be a fixed additive character of F \A. Write ψ = ⊗v ψv , and δv ∈ Fv×
denotes the conductor of ψv , i.e., if v is finite, then δv−1 Ov is the maximal
fractional ideal of Fv such that ψv |δv−1 Ov ≡ 1, if v is infinite, ψv (x) = e2πδv x . Set
7
δ=
Q
δv ∈ A× . Then one sees that
|δ|−1 = dF .
Unless specifically mentioned, we usually normalize a Haar measure on A
such that vol(F \A) = 1. The measure is decomposed as
dx = ⊗v dxv
such that dxv is self-dual with respect to ψv , and the multiplicative measure
dx× = ⊗v dx×
v has the property that
dx×
v =
dxv
xv
if v|∞, and
vol(Ov× ) = 1 if v - ∞.
For the torus T given by K × /F × , a Haar measure is chosen so that
vol(T (Fv )) = 1
if v|∞. If B is a quaternion algebra over F which is ramified at all infinite
places. We fix a Haar measure
dg = ⊗v dgv
such that vol(G(Fv )) = 1 if v|∞, and vol(U ) = 1 for some open compact
subgroup U ⊂ G(Af ) depending on the subgroup U . For the group PGL2 ,
we choose the standard Haar measure dg = ⊗v dgv , i.e., vol(PGL2 (Ov )) = 1 if
v - ∞ and vol(SO2 (R)) = 1 if v|∞.
8
2. Hilbert modular forms and automorphic
representations
In this section,we shall discuss the basic relationship between Hilbert modular
forms and automorphic representations. We shall also collect some further
results which are needed later.
2.1. Hilbert modular forms. We fix some notations and give the definition
of Hilbert modular form. Then we discuss the representations generated by
Hilbert modular forms.
Let F be a totally real field of degree d, I the set of embedding F ,→ R. The
A-points of GL2 is denoted by GL2 (A). It’s easily checked that one has
GL2 (A) = GL2 (Af ) × GL2 (F∞ ),
where F∞ = F ⊗Q R ∼
= RI . Fix k = (ki ) ∈ ZI such that each component ki ≥ 2
and such that all components have the same parity. Set t = (1, · · · , 1), and
√
√
z0 = ( −1, · · · , −1) ∈ HI , where H stands for the Poincaré upper half plane.
Let m = k − 2t and we choose v ∈ ZI suchthat wv
≥ 0, wv = 0 for some v and
 a b 
m + 2w = µt for some µ ∈ Z≥0 For g = 
 ∈ GL2 (F∞ ) and z ∈ HI ,
c d
one defines
jk,w (g, z) = (ad − bc)t−w−k (cz + d)k .
Let U be an open compact subgroup of GL2 (Af ). Following Hida [12] and
Tayler [17], one gives
9
Definition 2.1.1 A Hilbert modular form of weight (k, w) with respect to the
group U is a function f : GL2 (A) → C, satisfying the following two conditions:
(i) (f |k u)(x) := jk,w (u∞ , z0 )−1 f (αxu) = f (x) for α ∈ GL2 (F ), and u ∈ U ·(R× ·
SO2 (R))I ;
(ii) For all x ∈ GL2 (Af ), the function fx : HI → C defined by
u∞ z0 7→ f (xu∞ )jk,w (u∞ , z0 )
for u∞ ∈ GL2 (F∞ ) is holomorphic. If F = Q, one has to assume that the
function fx is holomorphic at cusps for each x ∈ GL2 (Af ). In addition, if
Z

 
 1 a  
f 
 g  da = 0
F \A
0 1
for all g ∈ GL2 (A), then f is called a cuspidal Hilbert modular form.
Remark 2.1.2: Using the formula in condition (i), one easily verifies that the
function fx in condition (ii) is well defined.
We denote by Mk,w (U )(Sk,w (U )) the space of (cuspidal) Hilbert modular
forms of weight (k, w) with respect to group U .
In particular, the Hilbert modular form defined above is an automorphic
form. Recall that an automorphic form is a function φ : GL2 (A) → C with the
following properties:
• φ(zγg) = ω(z)φ(g) for z ∈ Z(A), γ ∈ GL2 (F ), where Z is the center of GL2 ;
• φ is invariant under the right action of some open compact subgroup of
GL2 (Af );
10
• For each v|∞, φ is smooth in gv ∈ GL2 (Fv ) and SO
2 (Fv )-finite, i.e., φ(gr(θ))

 cos θ sin θ 
form a finite dimensional vector space, where r(θ) = 
;
− sin θ cos θ
• For each v|∞, φ is gl2 (Fv )-finite, i.e., for X ∈ gl2 (Fv ) ∼
= Z := center of the
universal enveloping algebra of gl2 (Fv ), Xφ form a finite dimensional vector
space, where
Xφ(g) :=
d
φ(g exp(tX)) |t=0
dt
• φ has moderate growth, i.e., for any compact subset Ω there exist positive
numbers C, t, such that
¯ 
 ¯
¯
¯
¯
¯
¯  a 0  ¯
−1 t
¯φ 
 g ¯ < C(|a| + |a| ) , ∀g ∈ Ω.
¯
¯
0 1
¯
¯
Let A(GL2 (F )\GL2 (A), ω) denote the space of automorphic forms with central
character ω.
In addition, if
Z

 
 1 a  
φ 
 g  = 0, ∀g ∈ GL2 (A),
F \A
0 1
then one says that φ is cuspidal. And we denote by A0 (GL2 (F )\GL2 (A), ω) the
space of cuspidal automorphic forms.
The group GL2 (Af ) acts on A(GL2 (F )\GL2 (A), ω) via right translation, i.e.,
for g ∈ GL2 (Af ), and φ ∈ A(GL2 (F )\GL2 (A), ω),
π(g)φ(x) := φ(xg).
11
Unfortunately the group GL2 (F∞ ) doesn’t act on the space A(GL2 (F )\GL2 (A), ω),
since π(g)φ may not be SO2 (Fv )-finite, for g ∈ GL2 (F∞ ). But we do have actions of both SO2 (Fv ) and gl2 (Fv ). For X ∈ gl2 (Fv ), define
Xφ(g) :=
d
φ(g exp(tX)) |t=0
dt
One can show that if φ is SO2 (Fv )-finite, then Xφ is an automorphic form and
is SO2 (Fv )-finite. Moreover one requires that both actions are compatible in the
following sense: for g ∈ SO2 (Fv ), X ∈ gl2 (Fv ) and φ ∈ A(GL2 (F )\GL2 (A), ω),
π(g)π(X)π(g −1 )φ = π(Ad(g)X)φ,
where Ad : GL2 (Fv ) → Aut(gl2 (Fv )) is the adjoint representation of GL2 (Fv ).
A vector space V equipped with such representations of SO2 (Fv ) and gl2 (Fv ) is
called a (SO2 (Fv ),gl2 (Fv ))-module.
Definition 2.1.3 A representation π of GL2 (A) or more precisely a representation of GL2 (Af ) and a commuting (SO2 (Fv ),gl2 (Fv ))-module is called an automorphic representation if π is isomorphic to a subquotient of A(GL2 (F )\GL2 (A), ω)
(a quotient of submodule of A(GL2 (F )\GL2 (A), ω)).
For the purpose of the paper, we are particularly interested in the following
two open compact subgroups U0 (N ) and U1 (N ) of GL2 (Af ). Let N be an ideal
of F , define




¯


 a b
¯


b
¯
U0 (N ) = 
 ∈ GL2 (OF )¯c ≡ 0 (mod N ) ,



 c d
12



¯
 a b
¯


U1 (N ) = 
 ∈ U0 (N )¯¯d ≡ 1

 c d




(mod N ) ,



 cos θ sin θ 
For each v|∞, let r(θ) = 
 . one easily checks that the Hilbert
− sin θ cos θ
modular form f satisfies the following property:
f (gr(θ)) = f (g)ekv θi ,
which inspires the following
Definition 2.1.4 An automorphic form φ is said to have weight k = (kv ), if
for each v|∞,
φ(gr(θ)) = φ(g)ekv θi .
Similar to classical Hilbert modular form, one says that φ is of level N , if
φ(gu) = φ(g), ∀u ∈ U1 (N ).
As already mentioned in the Introduction, we are primarily interested in
Hilbert modular forms with trivial central character of level N and weight
k. Having defined Hilbert modular forms and automorphic representation, we
now briefly review the classification of local admissible representations. See
Gelbart [5] for more details.
13
2.2. Classification of local admissible representations. Let (π, V ) be an
automorphic representation of GL2 (A). It’s well known that for each place v
of F , there exists an irreducible admissible representation πv of GL2 (Fv ) such
that π is isomorphic to the restricted tensor product of πv :
π∼
= ⊗ v πv .
Thus it boils down to the local irreducible admissible representations of GL2 (Fv ).
One has complete classification of all irreducible admissible infinite-dimensional
representations of GL2 (Fv ). See [5] and [14] for details
I. If v is nonarchimedean place of F .
(1) Principal series. These are the representations induced from a quasicharacter of the Borel subgroup determined by two quasi-characters of F × . Let
µ1 , µ2 : F × → C× , be two quasi-characters, Define the space of locally constant
functions on GL2 (Fv )
2
B(µ1 , µ2 ) = IndGL
B (µ1 µ2 )



 

¯

¯ ¯1


¯  a b  
¯d¯2
¯
¯
¯
= f : GL2 (F ) → C¯f 
 g  = µ1 (a)µ2 (d) ¯ ¯ f (g)


a


0 d
The group GL2 (Fv ) acts on B(µ1 , µ2 ) via right translation and the resulting
representation is called a principal series if it is irreducible and is denoted
π(µ1 , µ2 ). Using the Iwasawa decomposition
GL2 (Fv ) = B(Fv )Kv ,
14




¯

 a b


 ¯¯
×
where B(Fv ) = 
 ¯a, d ∈ Fv , b ∈ Fv and Kv = GL2 (Ov ), one sees


 0 d

easily that such a representation is admissible.
(2) Special representation. If the above representation is not irreducible,
then one must have µ(x) := µ1 (x)µ2 (x) = |x|±1 . If µ(x) = |x|−1 , then π(µ1 , µ2 )
contains a one-dimensional invariant subspace and the representation induced
on the quotient space is irreducible. If µ(x) = |x|, then π(µ1 , µ2 ) contains
an irreducibly invariant subspace of codimension one. In both cases, the irreducible subquotients of π(µ1 , µ2 ) are called special representation and denoted
σ(µ1 , µ2 ).
(3) Supercuspidal representation. If an irreducible admissible representation
is neither principal nor special, then it is called supercuspidal.
II. if v is archimedean.
(1) principal series. Similar to the non-archimedean case, one still defines the
induced representation of a character of Borel subgroup to GL2 (Fv ) ∼
= GL2 (R).
Let µ1 , µ2 → C× , be two characters, Kv = O2 (Fv ). Define the space of Kv -finite
functions on GL2 (Fv ):


 

1
¯


¯
¯


¯d¯2
¯  a b  
¯
¯
=
µ
(a)µ
(d)
g
f
(g)
B(µ1 , µ2 ) = f : GL2 (R) → C¯¯f 
.
 
1
2
¯a¯




0 d
The Lie algebra gl2 (R) acts on B(µ1 , µ2 ) by
X · φ(g) =
d
φ(g exp(tx))|t=0 ,
dt
15
where X ∈ gl2 (R). The compact group Kv acts via the right translation, thus
ε
k−1
produces a (gl2 (R), Kv )-module. If µ(x) := µ1 µ−1
, where
2 (x) 6= sgn(x) |x|
ε = 0 or 1, and k is an integer of the same parity as ε, then the representation
is irreducible and denoted by π(µ1 , µ2 ).
(2) Discrete series. If µx = sgn(x)ε |x|k−1 , then B(µ1 , µ2 ) contains an unique
nonzero subspace V0 which is either finite dimensional or infinite dimensional
depending that if k > 1 or k < 1. In this case, one denotes by σ(µ1 , µ2 ) and
calls it discrete series.
The following strong multiplicity one theorem is extremely useful in applications.
Theorem 2.2.1 (strong multiplicity one) If π = ⊗v πv and π 0 = ⊗v πv0
are two cuspidal irreducible representations of GL2 (A), if πv ∼
= πv0 , for almost
all v, then π ∼
= π0
2.3. Newform theory. The adelic analogue of classical Atkin-Lehner theory
is recalled briefly in this section, meanwhile we shall discuss a modified notion
of newform.
2.3.1. Atkin-Lehner theory. As in classical modular form case, if N 0 |N , one
may embed the space of modular forms of level N 0 into the space of modular forms of level N . In automorphic forms, one has the similar results. Let
16
Ak (N, ω) be the space of forms of weight k and level N and with central character ω, one defines the following two operators:


 $v 0 
φ 7→ π 
 φ (v - ∞),
0 1


 1 i 
φ 7→ π 
 φ (v|∞).
−i 1
The first one increases the level by order 1 at the place v, the second one
increases weight by 2 at infinite place v. Thus one obtains an embedding:
Ak0 (N 0 , ω) ,→ Ak (N, ω),
if N 0 |N and k 0 ≤ k, i.e., kv − kv0 ≥ 0, ∀v ∈ ∞. Let Ak (N, ω) be the subspace of those forms which come from lower level or lower weight, i.e., they
are obtained by applying one of these two operators. To define newform. We
need to define Hecke operators. For each finite place v, v - N , One defines
the Hecke operator
Tv to be the characteristic function of the double coset

 $v 0 
Hv := U0 (N ) 
 U0 (N ), where $v is the idele whose v-th component
0 1
is $v and 1 elsewhere. Recall that the Hecke operator acts on Ak (N, ω) by
Z
π 7→ Tv · φ(g) =
φ(gh)dh.
Hv
17
Similar to classical situation, for any ideal a, (a, N ) = 1, the Hecke operator Ta
on Ak (N, ω) is the following
 
X
Ta φ(g) =
x
αβ=a
(mod a)

  α x 
φ g 
 .
0 β
b× with
where α and β run through representatives of integral ideles modulo O
F
trivial component at places dividing N such that αβ generates a. A form
φ ∈ Ak (N, ω) is called a newform if it is an eigenform under Ta , for each ideal
a of F and there is no old form which has the same eigenvalues as φ.
In previous section, we already discussed the automorphic representation
generated by an automorphic form. A natural question is that when the representation is irreducible. One has
Lemma 2.3.1.1 Assume that φ is an eigenform for all Hecke operators Tv , v N, then the representation πφ generated by φ is irreducible.
Proof. Let Hv be the Hecke algebra of GL2 (Fv ). It’s well known, see Bump [3]
Proposition 4.6.5, that Hv is generated by Tv , Rv and R−1
v , where Rv is the
characteristic function of the double coset


 $v 0 
Hv0 = U0 (N ) 
 U0 (N ).
0 $v
Since Rv φ(g) =
R
Hv0
φ(gh)dh = ω($v )φ(g). Thus φ is also an eigenform under
Rv with eigenvalue ω($v ). Hence φ is an eigenform of Hv . Note that φ is
determined by eigenvalues of Rv and Tv .
18
Now let V be an irreducible subrepresentation inside L20 (GL2 (F )\GL2 (A), ω)
such that the projection φ0 of φ onto V is not zero. Since the projection is
GL2 (A)-equivariant, thus φ and φ0 have the same eigenvalues under Tv and Rv
at least at those places v such that both πv and πv0 are spherical representations.
Hence πv ∼
= πv0 by a well known fact in representation theory of p-adic groups,
see Bump [3] theorem 4.6.3. Finally strong multiplicity one theorem 2.2.1 implies that π ∼
= π 0 , since π is spherical representation for almost all v.
¤
Remark 2.3.1.2 If f is a classical Hilbert eigenform of level N , weight k, one
may easily show that the Hilbert modular form φf produced by f is an eigenform
of Hv for all v - N, and both have the same eigenvalues,thus corresponds to an
irreducible representation of GL2 (A).
The converse of the above lemma also holds. It is the adelic analogue of
classical Atkin-Lehner theory proved by Casselman [4]. To describe that, we
need to introduce a few notions.
I. Let F be a nonarchimedean local field with uniformizer $, (π, V ) be a
admissible irreducible representation of GL2 (F ) with central character ω. For
any c ≥ 0, one defines




¯


 a b
¯


c
¯
c
≡
0
(mod
$)
,
U0 ($ ) = 
∈
GL
(O
)

2
F ¯



 c d


 


¯

 a b
¯ a b   ∗ ∗ 


U1 ($c ) = 

≡
 ∈ GL2 (OF )¯¯ 

 c d
0 1
c d
(mod $c )





.
19
A vector v of V is said to have level $c if v is invariant under U1 ($c ).
Definition 2.3.1.3 The order o(π) of π is the minimal nonnegative integer
c such that V has nonzero vector of level $c .
Theorem 2.3.1.4 [Casselman] (1) Let




 

¯

¯   a b 
 a b 
c
c
¯
V (($ )) = f : GL2 (F ) → C¯f g 
 ∈ U0 ($ ) ,
 = ω(d)f (g), ∀ 




c d
c d



then V (($c )) is one dimensional. Let vπ be a basis.
(2) If c ≥ o(π), then the space of vectors of level $c is of dimension c−o(π)+1,
and is generated by


−i
 $
vi := π 
0
0 
 vπ ,
1
i = 0, · · · , c − o(π).
Proof. We only give the proof of the case that the representation is principal
series. See Casselman [4] for other type of representations. So assume that the
representation (π, V ) is a principal series π(µ1 , µ2 ), where µi is a quasi-character
of F × , i = 1, 2. Recall that it is the space of locally constant functions of GL2 (F )
and




 

¯

1
¯
¯

¯  a b  
¯a¯2
¯
V = f : GL2 (F ) → C¯f 
 g  = µ1 (a)µ2 (d) ¯ ¯ f (g)


d


0 d
20
Let n be the order of the representation π, using the Iwasawa decomposition of
GL2 (F ) = B(F ) · GL2 (OF ), one can write the space V (($n ))



 



¯
0
0


¯  a b   a b 
0
¯
= f : GL2 (OF ) → C¯f 
g
 = µ1 (a)µ2 (d)f (g)µ1 µ2 (d ) ,




0 d
c0 d 0


0
0
 a b 
where 
 ∈ U0 (($n )).
c0 d0
We claim that n ≥ n1 + n2 , where ni is the order of µi , i.e., ni is the minimal
nonnegative integer such that µi |1+($ni ) ≡ 1, i = 1, 2. The
Bruhat decomposi
`
 0 1 
tion of GL2 (F ) = B(F ) B(F ) · w · B(F ), where w = 
 , implies
−1 0
 

  1 x 
that a function f of V (($n )) is determined by f w 
 . So take
0 1
 

  1 x 
0 6= f ∈ V (($n )), let Φ(x) = f w 
 . We look at the action of
0 1

 
 

0 
 1 OF   a 0   1
following elements: 
,
,
 , on the function
n
0 1
0 d
($ ) 1
Φ(x). Three conditions are obtained:
(1) Φ(x) = Φ(x + b),
∀b ∈ ($n );
(2) Φ(ax) = µ2 (a)Φ(x),
∀a ∈ OF× ;
x
(3) Φ(x) = µ(cx + 1)−1 |cx + 1|−1 Φ( cx+1
), ∀c ∈ ($n ), where µ := µ1 · µ−1
2 .
We may assume that either n1 or n2 ≥ 1, since otherwise the claim is automatically true.
21
Case 1: n1 , n2 ≥ 1.
(i) Take a ∈ 1 + $n2 −1 , if x(a − 1) ∈ OF , then Φ(x) = 0, i.e., Φ(x) = 0 if
x ∈ $−n2 +1 . So we get an upper bound for ord(x), x ∈ Supp(Φ).
(ii) Φ(λx) = µ(λ)−1 |λ|−1 Φ(x),
∀λ ∈ F × , s.t., ord(λ−1 − 1) ≥ n+ ord(x). In
particular, if n+ ord(x) ≥ 0, then Φ(λx) = µ(λ)−1 |λ|−1 Φ(x),
Assume first that n+ ord(x) ≥ 0,
×
∀λ ∈ On+ord(x)
.
x ∈ Supp(Φ), then µ(λ)−1 |λ|−1 Φ(x) =
µ2 (λ)Φ(x), i.e., n+ord(x) ≥ n1 , since c(µ1 ) = n1 . which implies that n ≥
n1 + n2 . Secondly, suppose that ord(x) < −n. Recall
Φ(λx) = µ(λ)−1 |λ|−1 Φ(x),
∀λ ∈ F × , s.t., ord(λ−1 − 1) ≥ n + ord(x).
In particular, the above formula holds for any λ ∈ OF× . Hence one has
Φ(ax) = µ2 (a)Φ(x) = µ−1
1 (x)µ2 (x)Φ(x) =⇒ µ1 (x) = 1,
∀x ∈ OF× =⇒ c(µ1 ) = 0.
A contradiction!
Case 2: n1 ≥ 1,
n2 = 0.
Claim: n+ ord(x)≥ 0,
∀x ∈ Supp(Φ), then Φ(λx) = µ−1 (λ)|λ|−1 Φ(x),
λ∈
OF× =⇒ µ1 (λ) = 1, hence n1 = 0. A contradiction.
The other parts are similar. Φ(λx) = µ(λ)−1 |λ|−1 Φ(x) = µ2 (x)Φ(x),
∀λ ∈
×
Oord(x)+n
. Hence one gets ord(x) + n ≥ n1 , i.e., Supp(Φ) = $n1 −n OF . We
want: n ≥ n1 . If n < n1 , then Supp(Φ) ⊆ $. A contradiction, since Φ(x + b) =
Φ(x),
∀b ∈ OF .
Thus one can view µ1 , µ2 as characters of (OF /($n ))× , which implies that
the space V (($n )) is isomorphic to the space of functions ψ on GL2 (OF /($n ))
22
satisfying the same condition as those in V (($n )), since





¯



1 0 
¯

n
n
¯
U ($ ) = γ ∈ GL2 (OF ¯γ ≡ 
 (mod $ )




0 1
is normal in both GL2 (OF ) and U0 ($n ). We denote by B the image in GL2 (OF /($n ))
of the Borel subgroup B(OF ) and can easily show that


n
a  1 0 
GL2 (OF /($n ) =
B
 B.
i=0
$i 1


 1 0 
Therefore, a function ψ is determined by the value at B 
 B. To end the
x 1


 1 0 
proof, we just need to know what function ψ on some B 
 B satisfies
x 1
the above condition. the condition can be translated into the following one: if


 


0
0
 a b  1 0   1 0  a b 
n
=


 

 (mod $ ),
0 d
$i 1
$i 1
0 d0
then µ1 (a)µ2 (d) = µ1 (d0 )µ2 (d0 ). For given a, d, a0 , d0 there exist x, x0 for which
the equation holds if and only if the following equations have solution: d ≡ d0
(mod $i ), a ≡ a0 (mod $i ), a0 ≡ d (mod $n−i ), d−d0 = a0 −a, which is equivalent to (1) $i lies in the conductor µ1 and (2) $n−i is contained in the conductor
of µ2 . Therefore, one sees that the minimal such n is exactly n1 +n2 . For a given
c ≥ n, there are exactly c − n + 1 such distinct functions satisfying conditions,
which form a basis of V (($)).
¤
23
II. If F is an archimedean local field, we already defined the notion of weight,
which is the analogue of order for archimedean place. The weight of a representation π is the smallest nonnegative integer such that π has a nonzero
vector of weight k. In fact, from the classification of (gl2 (R), Kv )-modules, one
knows that for any integer, the space of vectors of weight n is one dimensional
of |n| > k, n ≡ k (mod 2), zero otherwise.
Back to the number field case. Thus if π = ⊗v πv is an irreducible representation of GL2 (A), since πv is irreducible, applying the above theorem, one
obtains a unique line of newforms for each place v, v - ∞. Globally, there exists a unique newvector up to a scaler, which generates the representation π.
Therefore there exists one-one correspondence between newforms of level N and
irreducible cuspidal representations of GL2 (A).
2.3.2.
Gross-Prasad theory and toric newform. We need a modified
notion of newform as well as test form theory of Gross and Prasad, which occur
in the formula. As a motivation, we first describe the theory of Waldspurger.
2.3.2.1 Theory of Waldspurger. Let F be a nonarchimedean local field, K
be a quadratic extension of F (including the split case K = F ⊕ F ). Let T
denote the torus of K × embedded in GL2 (F ). We denote G = B × /F × , where
B is the quaternion algebra over F into which K is embedded. Let (π, V ) be
an irreducible admissible, infinite dimensional representation of GL2 (F ), and
χ be a quasi-character of K × . We assume that the central character ω of π is
equal to χ−1 |F × , i.e., the subgroup ∆F × embedded diagonally in GL2 (F ) × T
24
acts trivially on V ⊗ C. One considers the space of ∆F × -invariant linear form
` : V ⊗ C → C. Using Gelfand pairings, one can show that such a space is
at most one dimensional if it exists. Waldspurger and Tunnel gave a criterion
for a nonzero ∆F × -invariant linear form to exist. To state Waldspurger and
Tunnel’s criterion, let σ1 be the 2-dimensional representation of Deligne-Weil
group of F associated to π by local Langlands correspondence, and σ2 be the
two-dimensional representation of Weil group of F which is induced from the
quasi-character χ; K × → C× . Then detσ1 = ω, and detσ2 = αK/F · χ|F × ,
where αK/F is the quadratic character associated to K/F . The four-dimensional
representation of the Deligne-Weil group has local root number ²(σ1 ⊗σ2 ) = ±1.
The condition that ²(σ1 ⊗ σ2 ) 6= αK/F · ω(−1) implies that the representation
(π, V ) is square-integrable, and K is a field, thus by local Jacquet-Langlands
correspondence, (π, V ) corresponds to an irreducible infinite dimensional representation (π 0 , V 0 ) of G(F ). Similarly one considers the representation V 0 ⊗ C
of the group G(F ) × GL1 (K), and can show that the space of ∆K × -invariant
linear form `0 : V 0 ⊗ C → C is at most one. Waldspurger and Tunnel’s criterion
for both cases is
Theorem 2.3.2.1(Waldspurger, Tunnel) There is a nonzero ∆K × - invariant
linear form ` : V ⊗ C → C, if and only if
²(σ1 ⊗ σ2 ) = αK/F · ω(−1).
25
There is a non-zero ∆K × -invariant linear form `0 : V 0 × C → C if and only if
²(σ1 ⊗ σ2 ) = −αK/F · ω(−1).
Globally, one defines the global nonzero linear form if locally it exists for each
place v. The significance of the existence of such a nonzero linear form is the
following theorem due to Waldspurger.
Theorem 2.3.2.2 There is a global nonzero linear form if and only if L( 21 , π, χ) 6=
0.
2.3.2.2 Theory of Gross-Prasad. If a local nonzero linear form ` exists, the
vector v ∈ V ⊗ C such that `(v) 6= 0 is called a test form. Gross and Prasad
gave a concrete realization of such test vector under the assumption:
Either π is a unramified principal series of GL2 (F ) or χ is an unramified
quasi-character of K × .
I. If π is an unramified principal series, then B ∼
= M2 (F ), let R be a maximal
order in M2 (F ) optimally containing the order Oc(χ) of K. In this case, their
result reads
Proposition 2.3.2.3 If (π, V ) is an umramified principal series, then there
×
is a unique line L fixed by R× × Oc(χ)
. If ` is any nonzero ∆K × -invariant linear
26
form, then `(v) 6= 0, ∀v ∈ L.
II. If χ is unramified. When ²(σ1 ⊗ σ2 ) = αK/F · ω(−1), let Rn be an order of
reduced discriminant ($)n in M2 (F ) containing OK , where n is the conductor
of π. When ²(σ1 ⊗ σ2 ) = −α · ω(−1), the condition forces n ≥ 1. Let Rn0 be an
order of reduced discriminant ($)n in B containing OK .
Proposition 2.3.2.4 Assume that χ is an unramified quasi-character of K × ,
when n(π) ≥ 2, assume further that the extension K/F is unramified.
×
If ²(σ1 ⊗σ2 ) = αK/F ·ω(−1), the open compact subgroup Rn× ×OK
fixes a unique
line L, if ` ia a nonzero ∆K × -invariant linear form, then `(v) 6= 0, ∀v ∈ L.
×
If ²(σ1 ⊗ σ2 ) = −αK/F ω(−1), the group Rn0 × × OK
fixes a unique line L0 . If `0
is a nonzero ∆K × -invariant linear form, then `0 (v) 6= 0, ∀v ∈ L0 .
If v is archimedean, then the representation π 0 is of finite dimensional and
the torus T (F ) is compact. One shows that the fixed subspace of T (F ) in π 0 is
of one dimension. And one defines a test vector to be any fixed vector up to a
scalar.
2.3.2.3 Toric newform. In the formula of level N D, there is a modified notion
of test form called toric newform, a form having character χ under the action
of T (F ). The existence and uniqueness of such a form is guaranteed by the
27
following
Lemma 2.3.2.5 [21]
0
(1) If v is non-archimedean place of F , the χ-isotypic component πv,χ
of πv0
under the action of ∆v (see [22] for definition) is one-dimensional.
(2) If v is archimedean place of F , then the subspace of forms fixed by T (Fv )
in πv0 is of one dimension.
2.4. Weil representation and Jacquet-Langlands correspondence. We
review the constructions of theta series associated to a character χ of K and
Jacquet-Langlands correspondence. For the purposes of the paper, it is sufficient to use Weil representation following Shimizu [15] to give construction
directly, even though both constructions are special cases of much more general
theory of theta lifting. So we first describe Weil representation, then give explicit constructions of theta series and Jacquet-Langlands correspondence. For
details, see [5], [15].
2.4.1. Weil representation. In this subsection, we let F denote a non-archimedean
local field. For our purposes, let V denote
(1) either a separable quadratic extension L of F equipped with a norm
map q or
(2) the unique quaternion division algebra B over F with q the reduced
norm.
28
In either case, let x → xσ denote the canonical involution of V . Then
q(x) = x · xσ ,
∀x ∈ V
and
tr(x) = x + xσ ,
∀x ∈ V.
Let’s fix a non-trivial additive character τ of F . V can be identified with its
dual by the pairing
< x, y >= τ (tr(xy)),
since (x, y) → tr(xy) is a non-degenerate bilinear form on V .
Let S(V ) denote the space of Schwartz-Bruhat functions on V . Recall that
for each Φ ∈ S(V ), the Fourier transform Φ̂ of Φ is defined to be
Z
Φ̂(x) =
Φ(y) < x, y > dy
V
where Haar measure dy is chosen so that
(Φ̂)∧ (x) = Φ(−x).
The Weil representation is associated to the pairing (q, V ). To describe it,
we first construct a representation r(s) of SL2 (F ) in S(V ). Since elements of
the form

 

 α 0   1 u 
 , and
,

−1
0 1
0 α


 0 1 


−1 0
generate SL2 (F ). It suffices to describe the actions of



 

 0 1 
 1 u   α 0 
r 

 , and r 
 , r 
−1 0
0 α
0 1
29
on S(V ). One has


 1 u 
r 
 Φ(x) = τ (uq(x))Φ(x),
0 1


1
 α 0 
r 
 Φ(x) = ω(α)|α| 2 Φ(αx),
0 α−1


 0 1 
σ
r 
 Φ(x) = γ · Φ̂(x ).
01 0
Here ω is the non-trivial character of F × /q(V × ) if V = L and the trivial
character of F × if V = B, and γ = −1 if V = B and |γ| = 1 if V = L.
The existence of such a representation is proved by Weil, Shalika, and Tanaka.
The representation may depend on the character τ . One may extend the representation to a representation of the group G+ consisting of elements in GL2 (F )
with determinant in q(V × ). This group is of index 2 or 1 in GL2 (F ) depending
on whether V = L or V = B. For a = q(h) ∈ q(V × ), set


 a 0 
r 
 Φ(x) = Φ(xh).
0 1
One can show that this gives rise to a representation denoted by r(n) of G+ .
Finally the induced representation
GL (F )
r(g) = IndG+2
(r(n))
is called the Weil representation of GL2 (F ) associated to (q, V ). A remarkable
thing is that it is independent of τ . This Weil representation is the bulk of
30
the constructions we are working on which associates to each finite dimensional
irreducible representation π 0 of V × an irreducible representation π of GL2 (F ).
So suppose that (π 0 , H) is a finite-dimensional irreducible representation of
V × . Consider the space
S(V ) ⊗ H
on which SL2 (F ) acts. Here SL2 (F ) acts on H trivially. One may view elements
of S(V ) ⊗ H as functions on V valued on H. We are interested in the subspace
{Φ ∈ S(V ) ⊗ H|Φ(xh) = π 0 (h−1 )Φ(x), ∀h ∈ V × , q(h) = 1}.
One can show the subspace is invariant under SL2 (F ). The resulting representation is denoted by rπ0 . Now following the same procedure as we did to Weil
representation, i.e., we extend the representation rπ0 to a representation of G+
by requiring


1
 a 0 
0
rπ0 
 Φ(x) = |h| 2 π (x)Φ(xh)
0 1
if a = q(h) for some h ∈ V × . And Moreover


 a 0 
r 
 = ω(a)χπ0 (a)I
0 a
for a ∈ F × and χπ0 the central character of π 0 . The induced representation,
GL (F )
still denoted by rπ0 , IndG+2
(rπ0 ) has the remarkable property.
Theorem 2.4.1. Assume that V = L.
31
(1) If there is no character λ of F × such that χ = λ · q then rπ0 (g) is a
supercuspidal representation of GL2 (F ).
(2) If χ = λ · q for some character λ of F × then rπ0 (g) is equivalent to the
principal series π(λ, λω).
Theorem 2.4.2. Assume that V = B.
(1) The representation rπ0 (g) decomposes as the direct sum of d = dim(π 0 )
mutually equivalent irreducible representations π(π 0 ) of GL2 (F );
(2) Each π(π 0 ) is supercuspidal if d > 1 and special if d = 1;
(3) All supercuspidal and special representations of GL2 (F ) are obtained in
this way. More precisely, the map
π 0 → π(π 0 )
gives a one-to-one correspondence between the equivalence classes of
finite-dimensional irreducible representations of V × and the equivalence
classes of special and supercuspidal representations of GL2 (F ).
Remark 2.4.3. In the above, we assume that F is non-archimedean. Now
assume that F is archimedean local field R.
Case 1. V = L = C.
If χ is not of the form λ · q with λ a character of R, then
χ(z) = (z z̄)r z m z̄ n
32
with r ∈ C, m and n two integers, one zero and other positive. In this case,
rχ = σ(µ1 , µ2 ). Here
µ1 (t) = |t|2r tm+n sgn(t)
and
m+n
sgn(t).
µ1 µ−1
2 (t) = t
If χ = λ · q with m + n = 0, then
rπ0 = π(µ1 , µ1 · sgn).
Case 2. V = B = H.


 a b 
Identifying H with matrices 
 , a, b ∈ C then q(h) = det(h). Every
−b̄ ā
irreducible finite-dimensional representation π 0 of H× has the form
π 0 (h) = q(h)r ρn (h)
Where r ∈ C, and ρn is the n-th symmetric tensor product of the standard
representation of GL2 (C). Let µ1 , µ2 be characters of R× defined by
1
µ1 (α) = |α|r+n+ 2
1
µ2 (α) = |α|r− 2 sgn(α)n .
define
π = σ(µ1 , µ2 ).
In particular, in our special case that the automorphic representation π =
⊗v πv is generated by Hilbert cusp newform φ of weight (2k, · · · , 2k) with trivial
central character. The local representation πv for v|∞ is a discrete series σ(p, t),
33
where p = 2k − 1, t = 0, see [5]. So one can determine n. It’s easy to see that
n = 2k − 2. Hence πv0 for v|∞ is a finite dimensional representation of SO3 (R).
Its dimension is 2k − 1.
2.4.2. Theta series and Jacquet-Langlands correspondence. Now we are able to
construct automorphic representation associated to a character of χ of K and
Jacquet-Langlands correspondence. So we assume that F is a number field.
Case 1. (V = L)
×
Let χ be a character of A×
L /L . Write
χ = ⊗w χw .
We shall attach to χ an automorphic representation
π(χ) = ⊗v πv .
of GL2 (AF ). The local representation πv is constructed as follows.
(1) If v splits in L, write v = w1 w2 in OL . Thus we may view both characters
χw1 and χw2 as characters of Fv . Define
πv = π(χw1 , χw2 ).
(2) If there is only one prime w lying above v. Then Lw is a genuine
quadratic extension of Fv . Now to χw , apply Weil representation. We
define
πv = π(χw ).
34
One can show that ⊗πv defines an irreducible unitary representation of GL2 (AF ).
The newform θχ of this representation is called the theta series associated to
χ. It is easy to see that
L(s, π(χ)) = L(s, χ).
Here the right hand of the equation is the Hecke L-series associated to χ.
Case 2. (V=B)
Let G be the algebraic group B × over F . Assume that π 0 is an irreducible
representation of G(AF ). Write
π 0 = ⊗v πv0 .
To π 0 , we may associate an irreducible representation π of GL2 (AF ).
(1) If B in unramified at v. Then Bv ∼
= M2 (Fv ). Define
πv = πv0 .
(2) If B is ramified at v. Then πv0 is finite-dimensional, since πv0 is irreducible
and G(Fv ) is compact modulo its center. Now apply Weil representation,
and define
πv = π(πv0 ).
To sum up,
Theorem 2.4.4.(Jacquet-Langlands correspondence) To each irreducible unitary representation π 0 = ⊗v πv0 of G(AF ), one associates an irreducible unitary
35
representation π = ⊗v πv of GL2 (AF ), where πv = πv0 if v is not ramified, and
πv = π(πv0 ) if v is ramified. Moreover
(1) π is cuspidal for GL2 (AF ) if π 0 is (greater than one dimensional) cuspidal for G(AF ).
(2) The mapping
π 0 → π,
restricted to the collection of (greater than one dimensional) cuspidal
representations on G(AF ) is one-to-one correspondence onto the collection of all equivalence classes of cuspidal representations ⊗v πv on
GL2 (AF ) such that πv is square-integrable for those v at which B is
ramified.
2.5. Rankin-Selberg L-functions. The aim of this subsection is to review
theory of Rankin-Selberg L-function associated to an automorphis representation and a character χ of K. Before that, we first go over the L-function
associated to a single automorphic representation. Our references are [?], [14].
2.5.1. L-function associated to an cuspidal automorphic representation. Let
π = ⊗πv be a cuspidal irreducible representation of GL2 (A). Let’s fix an
additive character ψ : A → C× . The L-series L(s, π) of π is defined as L(s, π) =
Q
Q
L
(s,
π
),
which
satisfies
functional
equation
with
²(s,
π)
=
v
v
v
v ²v (s, πv , ψv )
. So we start with local version. One defines local L-factor Lv (s, πv ) and
²v (s, φ, ψv ) for each type of local irreducible representation of GL2 (Fv ) as follows:
36
I. v is nonarchimedean place.
(1) if πv is a principal series, πv = π(µ1 , µ2 ), with µi : F × → C× , quasicharacters, then
Lv (s, πv ) = (1 − µ1 ($v )|$v |s )−1 (1 − µ2 ($v )|$v |s )−1 ,
²(s, πv , ψv ) = ²(s, µ1 , ψv )²(s, πv , ψv ).
(2)if πv is a special representation, πv = σ(µ), define
L(s, πv ) = (1 − µ($v )|$v |s )−1
L(1 − s, µ−1
1 )
²(s, πv , ψv ) = ²(s, µ1 , ψv )²(s, µ2 , ψv )
,
L(s, µ2 )
1
1
if one writes µ1 = µ · | · | 2 , and µ2 = µ · | · |− 2 .
(3) if πv is supercuspidal, one defines
L(s, πv ) = 1.
II. v is archimedean place, we assume that v is real.
(1) if πv is principal series, πv = π(µ1 , µ2 ), then
L(s, πv ) = L(s, µ1 )L(s, µ2 ),
²(s, πv ) = ²(s, µ1 , ψv )²(s, µ2 , ψv ).
(2) if πv is discrete series, πv = σ(p, t), one defines
L(s, πv ) = (2π)−s−
t+p
2
Γ(s +
t+p
),
2
²(s, πv ) = ip+1−n1 −n2 is1 +s2 ,
if µj (x) = |x|sj sgn(x)nj .
37
The L-function L(s, π) is only defined for Re(s) À 0, so one has to have
analytic continuation to the whole complex plane in order to have applications.
Theorem 2.6.1.1(Jacquet-Langlands) L(s, π) can be continued to a holomorphic function on the entire complex plane and satisfies the functional equation
L(s, π) = ²(s, π)L(1 − s, π).
Let f be a classical Hilbert newform of weight (k, · · · , k) , level N and of
character ψ, k ≥ 2. Using the type of local irreducible representation of πf
described before, one sees easily that
L(s, π) =
Y
L(s, πv ),
v
where
Lv (s, πf ) = (1 − λv |$v |s + ψ($v )|$v |2s )−1 ,
Lv (s, πf ) = (2π)−s−
where λv |$v |−
k−1
2
k−1
2
Γ(s +
k−1
),
2
if v - ∞,
if v|∞,
is the eigenvalue of Tv acting on f , ∀v - ∞.
Classically, see Shimura [15], one defines the L-series of f as
L(s, f ) =
Y
(1 − λv |$v |−
k−1
2
|$v |s + ψ($v )|$v |2s−(k−1) )−1
v
=
Y
(1 − λv |$v |s−
k−1
2
+ ψ($v )|$v |2(s−
v
=
Y
v
0
0
(1 − λv |$v |s + ψ($v )|$v |2s )−1 ,
(k−1)
)
2
)−1
38
where s0 = s −
k−1
,
2
hence one obtains
L(s, f ) = L(s −
k−1
, πf ).
2
Thus L(s, f ) yields a functional equation under s → k − s.
2.5.2. Rankin-Selberg L-functions associated to φ and χ. We only discuss the
Rankin-Selberg L-function associated to a newform φ and χ. For more general Rankin-Selberg convolution associated to automorphic representations, see
Jacquet [13] or Zhang [21].
The Rankin-Selberg L-function L(s, φ, χ) is defined by an Euler product over
primes of F :
L(s, φ, χ) =
Y
Lv (s, φ, χ),
v
where the factors is of degree ≤ 4 in |$v |s . The local factors can be defined
explicitly as follows.
For a finite place v, let write
Lv (s, φ) = (1 − α1 |$v |s )−1 (1 − α2 |$v |s )−1 ,
Y
Lw (s, χw ) = (1 − β1 |$v |s )−1 (1 − β2 |$v |s )−1 ,
w|v
then
Lv (s, φ, χ) =
Y
(1 − αi βj |$|s )−1 .
i,j
Here for a place w of K, the local factor L(s, χw ) is defined as follows:



(1 − χw ($w )|$w |s )−1 , if w - c · ∞;



L(s, χw ) =
G2 (s),
if w|∞;




 1,
if w|c(χ).
39
Here G2 (s) := 2(2π)−s Γ(s).
If v is an infinite place of F , one may write
Lv (s, φ) = G1 (s + σ1 )G1 (s + σ2 )
Lv (s, χ) = G1 (s + τ1 )G1 (s + τ2 ).
The the local L-factor Lv (s, φ, χ) is defined in the following
Lv (s, χ, χ) =
Y
G1 (s + σi + τj )
i,j
=


 G2 (s +
kv −1 2
),
2
if kv ≥ 2;

 G2 (s + itv )G2 (s − itv ), if kv = 0.
Where tv is the parameter associated to φ at place v where the weight is 0 and
G2 (s) = G1 (s)G1 (s + 1).
2.6. Unitary similitudes. Gross[6] proposes a programme unifying both special values of L(1/2, φ, χ) and L0 (1/2, φ, χ) which refines the work of Waldspurger. One of his key observations is that the group GL2 × T /∆Gm (F ) is
isomorphic to the group of F -rational points of a group of unitary similitudes
GU . We show in this subsection this isomorphism.
2.6.1. Local theory. We begin with the local theory. Let F be a local field,
and let K be an etale quadratic extension of F . Let e 7→ e be the nontrivial
involution of K fixing F . There are two cases:
(1) K is a field, then e 7→ e is the nontrivial element in Gal(K/F ).
40
(2) K is a split F -algebra. Then K is isomorphic to F [x]/(x2 − x) ' F + F .
There are two orthogonal idempotents e1 and e2 in K, with e1 + e2 = 1,
and e1 = e2 .
By local class field theory, there is a unique character ω : F × −→ {±1} whose
kernel is the norm group NK × = {ee : e ∈ K × } ⊂ F × .
Let π be an irreducible (complex) representation of GL2 (F ), with central
character ω : F × −→ C× . We assume that π is generic, or equivalently, that π
is infinite-dimensional.
Let S be the two-dimensional torus ResK/F Gm , and let χ be an irreducible
complex representation of the group S(F ) = K × . Since K has rank 2 over F ,
we have an embedding of the groups:
S(F ) ' AutK (K) −→ GL2 (F ) ' AutF (E).
We will consider the tensor product π ⊗ χ as an irreducible representation
of the group GL2 (F ) × S(F ), and wish to restrict this representation to the
diagonally embedded subgroup S(F ). The central local problem is to compute
the space of coinvariants HomS(F ) (π ⊗ χ, C). If this is non-zero, we must have
ω · χ|F × = 1
(∗)
as a character of F × . From now on, we assume that (∗) holds. Then π ⊗ χ is
an irreducible representation of the group G(F ), with
G = (GL2 × S)/∆Gm ,
41
and we wish to restrict it to the subgroup T (F ), where T is the diagonally
embedded one-dimensional torus S/Gm .The group G defined above is a group
of unitary similitudes. We explain this for general situation. Note that any
quadratic K/F can be embedded into M2 (F ) via EndK (K) ⊂ EndF (K), but
this is not true for quaternion algebra.
2.6.2.
Unitary Similitudes. Let B be a quaternion algebra over F with
a fixed embedding K ⊂ B. We will show that B × × K × /∆F × is a group of
unitary similitudes.
Proposition 2.7.2.1 Let F be a local field and K an etale quadratic algebra
over F , then there is a natural one to one correspondence between the following
two sets
(1) Quaternion algebras B with an inclusion K ⊂ B.
(2) Non-degenerate unitary space (V, φ) of dimension 2 over K, with a vector v satisfying φ(v, v) = 1.
Proof: On the one hand, assume that K ⊂ B as above. The inclusion defines
a graded algebra structure on B: B = B+ + B− with
B+ = K
B− = {b ∈ B : be = eb, for all e ∈ K}
Both B+ and B− are free K-modules of rank 1. Note that all elements in B−
have trace 0. The following pairing
φ : B × B −→ K;
(b1 , b2 ) 7→ (b1 b2 )+
42
is a non-degenerate Hermitian form on the free K module B of rank 2. The
group GU (B, φ) of unitary similitudes has F -valued points isomorphic to B × ×
K × /∆F × . Recall that by definition
n
GU (B, φ) = g ∈ GL(B) | φ(gv, gw) = λ(g)φ(v, w), for some λ(g) ∈ F
×
o
,
where λ is called a similitude factor and is a F × -valued character of GU (B, φ).
To give a specific isomorphism, we define an action of B × × K × on B by
(b, e)x = e−1 xb,
for all (b, e) ∈ B × × K × ,
x ∈ B.
Then ∆F × acts trivially on B, and the similitude factor for φ is Nb/Ne in F × .
It is easy to see that we have obtained an injective homomorphism
B × × K × /F × −→ GU (B, φ).
Now we show it is surjective. Note that
Ng(1) = φ(g(1), g(1)) = λ(g)φ(1, 1) = λ(g) ∈ F × ,
we have that b := g(1) ∈ B × . It is easy to check that h := rb−1 ◦g ∈ SU (B, φ) ⊂
GU (B, φ) and we only need to show that h has form of h(v) = e−1 ve. Let’s
now compute h:
(1) (hv)+ = φ(hv, h1) = φ(v, 1) = v+ ,
(2) Choose any element u ∈ B− , we have (hu)+ = u+ = 0, so hu ∈ B− ,
then hu = ue0 , for some e0 ∈ K = B+ . For any v ∈ B, we have
hve0 u = φ(hv, hu) = φ(v, u) = (vu)+ = v− u,
43
−1
thus (hv)− = v− e0 −1 = e−1
0 v− , and hu = ue0 = ue0 . Then Ne0 = 1,
and therefore by Hilbert 90, there exists an element e ∈ K such that
e0 = ee−1 .
Thus hv = v+ + e−1 ev− = e−1 ve.
On the other hand, if (V, φ) is a non-degenerate unitary space of dimension
2 over K, with a vector v ∈ V satisfying φ(v, v) = 1. We give V the structure
of a quaternion algebra over F , with an inclusion K ⊂ B. Indeed
V = K · v + (K · v)⊥
and we define multiplication by
(ev + u)(e0 v + u0 ) = (ee0 − φ(u, u0 ))v + (eu0 + e0 u).
The group GU (V, φ)(F ) is then isomorphic to B × × K × /∆F × , with B the
quaternion algebra so defined.
¤
44
3. Special value formula of level N D
We shall explain briefly the proof of the formula of level N D due to Hui
Xue [20] in this chapter. Using Rankin-Selberg method, Gross and Zagier
represent the L-function L(s, φ, χ) as the inner product of φ with θχ E, where
θχ is the theta series associated to χ and E certain Eisenstein series. Since θχ E
is of level N D, taking trace of θχ E from N D to N , they obtain a form of level
N:
1
X
−s
Φs (g) = TrN D/N (dF2 θχ (g)E(g)) =
1
−s
(dF2 θχ (g)E(g)),
γ∈U0 (N )/Uo (N D)
with the property that
L(s, φ, χ) = (φ, Φs )U0 (N ) = (φ, pr(Φs )),
where pr(Φs ) is the projection of Φs into the space π(φ). Hence L0 (s, φ, χ) is
the inner product of φ with Φ0s . The form Φ0s is not holomorphic. So one needs
to get the holomorphic projection Ψs of Φ0s . One has
L0 (s, φ, χ) = (φ, Ψs ) = (φ, pr(Ψs )),
where pr(Ψs ) is the projection of Ψs into the space of π(φ). By newform theory,
One has pr(Ψs ) = λφ. Thus
λ=
L( 12 , φ, χ)
.
||φ||2
On the other hand, let x be the CM-point on the modular curve X0 (N ). One
can show that the form Φ whose Fourier coefficient is given by
b
Φ(a)
= |a| < x, Ta x >,
45
is actually a cusp form of level N .
The inner product of φ and Φ gives
(xφ , xφ )||φ||2 . They show that Φ − Ψ is an old form by computing the Fourier
coefficient at a for N |a. Thus follows the formula
1
2g+1
p
L ( , χ, χ) =
||φ||2 ||xχ ||2 .
2
N (D)
0
Generalizing the formula to Hilbert modular forms of weight (2, · · · , 2), one
encounters the difficulty that the trace TrN D/N θχ E is very hard to compute
when χ is ramified as well as other geometric technical difficulties, instead
Shou-Wu Zhang works directly on level N D. So naturally one would expect
to have some other form to replace the role of φ in the level N D. Zhang uses
quasi-newform φ# , see the definition below, to replace φ. By developing a notion of geometric pairing, he computes the geometric pairing (Ta η, η) of φ and
a special CM-cycle η. The computation shows that there is a close relationship
between local Fourier coefficient of Φ 1 and local geometric pairing (Ta η, η),
2
which he calls the local Gross-Zagier formula. The special value formula of
level N D follows from this local Gross-Zagier formula.
3.1. Kernel function and quasi-newform. We have explained in last chapter that the L-function L(s, φ, χ) can be represented as inner product of φ with
θχ E, i.e.,
Z
L(s, φ, χ) = |δ|
s− 12
φ(g)θχ (g)E(s, g)dg
Z(A)GL2 (F )GL2 (A)
1
= |δ|s− 2 (φ, θχ E).
46
In order to get a more symmetric form, Zhang applies Atkin-Lehner operator
to θχ E. Let S be the set of finite places ramified in K. Recall that for each
subset T of S, the Atkin-Lehner operator
is an

 element hT in GL2 (A) whose
 0 1 
v-th component is 1 for v ∈
/ T and 
, where tv has the same order
−tv 0
as c(²) such that ²v (tv ) = 1, for v ∈ T. One can show that
γT (s)
L(s, φ, χ) =
vol(U0 (N D))
Z
Z(A)GL2 (F )\GL2 (A)
−1
φ(g)θ(gh−1
T )E(s, ghT )dg
with γ(s) certain exponential function of s. Thus if we define
1
Θ(s, g) = 2−|S| |δ|s− 2
X
−1
γT (s)θχ (gh−1
T )E(s, ghT )
T ⊂S
and call it the kernel function. Then one has
L(s, φ, χ) = (φ, Θ)U0 (N D) .
The kernel function has functional equation
Θ(s, g) = ²(s, χ)Θ(1 − s, g)
from which the functional equation of L(s, φ, χ)
L(1 − s, φ, χ) = (−1)#Σ NF/Q (N D)1−2s L(s, φ, χ)
follows. Note that the kernel function Θ(s, g) is of level N D. What Gross
and Zagier do is that they take trace of Θ(s, g) from N D to N to have a form
of level N . Because of the technical difficulty-the trace is very massy if χ is
ramified, so instead of taking trace, Zhang works directly on level N D. In this
case, one needs to find an analogue of φ for the level N D. The analogue is
47
called quasi-newform associated to χ. It is defined as follows. Let pr(Θ) be
the projection of the kernel function Θ into the space π(φ). The quasi-newform
#
φ#
s is the projection of φ into the line spanned by pr(Θ), i.e., φs is the unique
nonzero form in the space π(φ) of level N D satisfying the following identities
#
∗
#
(φ#
s , φa ) = ν (a)(φs , φs ), a|D,

1
1

s−
−s
 ν(a), if a|c(²);
Y |a|v 2 + |a|v2
∗
where ν (a) =
·

2
 0,
v∈S
otherwise.


−1
0 
 a
and φa := ρ 
 φ.
0 1
Note that the kernel function Θ(s, g) is non-holomorphic if k > 1. Thus one
needs to consider the holomorphic projection of Θ(s, g). We still denote it by
Θ(s, g). Let’s write the Fourier expansion of the kernel function Θ(s, g) as
follows:
 
Θ(s, g) = C(s, g) +
 
X
  α 0  
W s, 
 g .
×
0 1
α∈F
Since Θ(s, g) is a linear combination of the form
Θ(s, g) =
X
θi (g)Ei (g)
i
by definition. The constant and Whittaker function of Θ(s, g) can be expressed
in terms of Fourier expansions of θi and Ei .
Let
θi (g) =
X
ξ∈F
Wθi (ξ, g),
Ei (g) =
X
ξ∈F
WEi (ξ, g),
48
be Fourier expansions of θi and Ei . Then
C(s, g) =
X
C(s, ξ, g),
ξ∈F
W (s, g) =
X
W (s, ξ, g),
ξ∈F
where
C(s, ξ, g) =
X
Wθi (−ξ, g)WEi (ξ, g),
i
and
W (s, ξ, g) =
X
Wθi (1 − ξ, g)WEi (ξ, g).
i
Furthermore, one can decompose W (s, ξ, g) into
W (s, ξ, g) = ⊗v Wv (s, ξv , gv ).
The local Fourier coefficients Wv ( 21 , ξ, g) is related to certain local height pairing
of some special CM cycle via local Gross-Zagier formula.
3.2. Geometric pairing and local Gross-Zagier formula. Let B be the
quaternion algebra over F ramified at exactly in Σ. We denote by G the
algebraic group B × /F × , and by T the torus given by K × /F × embedded in G.
The set
C := T (F )\G(Af )
is called the set of CM points. For any open compact subgroup U ⊂ G(Af ),
we also denote by CU the set T (F )\G(Af )/U . Gross defines an intersection
pairing, for some fixed maximal order R of B,
(, ) : CU × CU → R,
49
such that given two points P, P 0 ∈ CU ,


 0,
if π(P ) 6= π(P 0 );
0
(P, P ) =

 #(R× ), if π(P ) = π(P 0 ).
P
Here π : CU → G(F )\G(Af )/U, and RP is the oriented order of B correspondb −1 . A vast generalization is
ing to the point P ∈ CU , i.e., RP := B ∩ P RP
introduced by Zhang via his geometric pairing. Using this geometric pairing,
Zhang proves a local version of Gross-Zagier formula which relates the local
Fourier coefficients of the kernel function to the local geometric pairing of some
special CM-cycle. This local Gross-Zagier formula is the key to proving special value formula of both L( 21 , φ, χ) and L0 ( 21 , φ, χ). We briefly review Zhang’s
theory of geometric pairing.
Let m be a real-valued locally constant function on G(Af ). In Zhang’s definition, he requires the function m first defined on G(F ) and invariant under
T (F ) such that m(γ) = m(γ −1 ), and then extend it to a function m on G(Af )
by requiring
m(γ, gf ) =


 m(γ), if gf = 1;

 0,
otherwise.
Now the kernel function
k(x, y) =
X
m(x−1 γy)
γ∈G(F )
is a function on C × C. Let S(C) denote the set of locally constant functions
with compact support and call it the space of CM-cycles. Note that C admits
a natural action of T (Af )(G(Af )) on the left (right), which induces an action
50
of T (Af ) on S(C). Since T (F )\T (Af ) is compact, the set S(C) is decomposed
as
S(C) = ⊕χ S(C, χ)
where χ runs through characters of T (F )\T (Af ). For any given CM-cycles
α, β ∈ S(C), Zhang defines
Z
< α, β > =
α(x)k(x, y)β(y)dxdy
C2
Z
= lim
U →1
α(x)kU (x, y)β(y)dxdy.
C2
where U runs over open subgroups of G(Af ) and
Z
kU (x, y) = vol(U )
−2
k(xu, yv)dudv.
C2
It’s called the geometric pairing with multiplicity function m.
Remark 3.2.1 In particular, let m be the characteristic function of the open
compact subgroup U given by a maximal order R of B considered by Gross.
For two points P, P 0 ∈ C, let α and α0 be the characteristic function of P and
P 0 respectively. Thus α, α0 ∈ S(C), then one can easily see that


 0,
if π(P ) 6= π(P 0 );
0
< α, α >=

 #(R× ), if π(P ) = π(P 0 ).
P
By identifying P (P 0 ) with α(α0 ), one thus recovers Gross’ intersection pairing.
It is not hard to see that the geometric pairing
< α, β >=
X
γ∈T (F )\G(F )/T (F )
m(γ) < α, β >γ ,
51
where < α, β >γ =
R
Tγ (F )\G(Af )
α(γy)β(y)dy and
Tγ =


 T, if γ ∈ NT ;

 1, if γ ∈
/ NT .
with NT the normalization of T in G. Since < α, β >γ only depends on the
class of γ in T (F )\G(F )/T (F ), one may pass from T (F )\G(F )/T (F ) to F by
the following embedding
ξ : T (F )\G(F )/T (F ) −→ F
a + b² −→
N(b²)
N(a + b²)
One thus defines
< α, β >ξ =


 < α, β >γ , if ξ = ξ(γ);

 0,
else.
Hence
< α, β >=
If both α =
Q
v
αv and β =
X
m(γ) < α, β >ξ .
ξ∈F
Q
v
βv are decomposable, then < α, β >ξ can be
further decomposed as
< α, β >ξ =
Y
< αv , β v > ξ
v
with < αv , βv >ξ =
R
G(Fv )
αv (γy)β v (y)dy. It is this local geometric pairing of
some special CM-cycle that is related to the local Fourier coefficient of the
kernel function Φ, which Zhang calls the local Gross-Zagier formula. To state
the formula, we need to define the special CM-cycle.
52
Let A be an order of B such that, locally for each finite place v,
Av = OK,v + λv c(χv )OK,v ,
where λv ∈ Bv with the properties that
(1) λv x = x̄λv , ∀x ∈ Kv ,
(2) ordv (detλv ) = ordv (N ).
We denote by ∆ the subgroup of G(Af ) such that
∆=
Y
×
×
A×
v Fv /Fv ·
v-c(χ)
Y
×
×
A×
v Kv /Fv .
v|c(χ)
Notice that one has a natural isomorphism
Av /λv c(χv )Av ∼
= OK,v /λv c(χv )OK,v .
Thus one may extend χ to a character, still denoted by χ, of ∆. Now the special
Q
CM-cycle is the character η = v ηv with
ηv : T (Fv )∆v −→ C×
tu −→ χv (t)χv (u).
For a ∈ A×
f an integral idéle prime to N D, the Hecke operator is defined to
be
Ta η =
Y
Tav η v
v
Z
Tav ηv (x) =
ηv (xg)dg,
H(Gv )
53
where H(Gv ) = {g ∈ M2 (Ov )||det(g)| = |av |} and we choose the measure dg
such that vol(GL2 (Ov )) = 1. The local Gross-Zagier formula is the following


−1
 av δv 0 
Proposition 3.2.3 Let g = 
 , then
0
1
1
1
1
Wv ( , ξ, g) = |c(ωv )| 2 ²(ωv , ψv )χv (u)|(1 − ξ)ξ|v2 |av |v vol(∆v )−1 < Tav ηv , ηv >ξ ,
2
where u is any trace free element in Kv .
In particular, globally one obtains
Corollary 3.2.4 Let <, > be the geometric pairing on the CM-cycle with
multiplicity m on F such that m(ξ) = 0 is ξ is not in the image of the map ξ.
Assume that δv = 1 for v|∞. Then there exist constants c1 and c2 such that for
a an integral idéle prime to N D,
1
1
|c(ω)| 2 |a| < Ta η, η >∆ = (c1 m(0) + c2 m(1))|a| 2 Wf (g)
+ ig
X
ξ∈F −{0,1}
1
1
2
|ξ(1 − ξ)∞
W f ( , ξ, g)m(ξ).
2
Now we need to choose the right multiplicity function m. For each archimedean
place v, define
m(γ, gf ) = 2CPk−1 (1 − 2ξ(γ))Wv (gv ),
54
where C =
4k−1 [(k−1)!]2
,
(2k−1)!
Pk−1 (g) is a function on G(Fv ) ∼
= SO3 (R) such that
Z
SO3 (R)
2
Pk−1
dg =
1
,
2k − 1
and Wv is the standard Whittaker function of weight 2k at v, i.e., Wv is a
function in W(πv , ψv ) such that

 

 2ak e−2πa , if a > 0;
 a 0 
Wv 
 =

 0,
0 1
else.
Here we view the multiplicity function parameterized by the continuous parameter g∞ . Now we look at the spectral decomposition of the geometric pairing
< α, β > (g∞ ) as a Whittaker function on GL2 (R). For that, it suffices to
determine the spectral decomposition of the kernel function KU (x, y).
Proposition 3.2.5 As a Whittaker function on GL2 (R),
KU (x, y) = C ·
X
Z
Wi (g∞ )φi (x)φi (y) + C
i
WM (g∞ )EM (x)E M (y)dM.
M
Where the sum runs over all cuspidal eigenforms φi of Hecke operators and
Laplace operators, and Wi is the Whittaker function of φnew
, φnew
being the
i
i
newform of weight (2k, · · · , 2k) in the representation π of PGL2 (A) corresponding to the representation πi0 of G(A) generated by φi via Jacquet-Langlands
correspondence. In particular, let α = β = η, then the form
Ψ = C|c(ω)|
1
2
X
i
Z
|(φ, η)|2
φnew
i
+ C|c(ω)|
1
2
M
new
|(EM , η)|2 dM
EM
55
has the same a-th Fourier coefficients with the kernel function Φ for a prime to
N D. Thus Φ − Ψ is an old form. On the other hand, since η has character χ
under the action of T (A), so we may assume that φi has the same character χ
under the action of T (Af ), which turns out to be exactly the toric newform φχ
defined in chapter 1.
The projection of Φ − Ψ on the space π(φ) is still an old form. By taking the
first Fourier coefficient, the desired formula is obtained, i.e.,
1
1
φ# (1)L( , φ, χ) = C|c(ω)| 2 ||φ# ||2 |(φχ , η)|2 .
2
56
4. Special value formula of level N
In this section, we shall deduce the final formula, i.e., the special value formula of level N from the formula of level N D in last section. We shall express
the special value of L(s, φ, χ) at s =
1
2
in terms of certain test form on a
Shimura variety evaluated at certain CM-point. Let’s explain the CM-divisor
Pχ occurring in the formula of level N . Recall that our formula is
1
C
e Pχ )|2 .
||φ||2 · |(φ,
L( , φ, χ) = p
2
N(D)
To define Pχ , let R be an order of B containing OK with reduced discriminant
N . One may construct such an order as follows. Choose a maximal order OB
of B containing OK and an ideal N of OK such that
NK/F (N ) · discB/F = N.
Then take R = OK + N OB . The group
Uv = Rv× /Ov×
defines an open compact subgroup of G(Fv ). Let U =
Q
v-∞
Uv . It is an open
compact subgroup of G(Af ). The Shimura variety defined by G is isomorphic
to G(F )\G(Af )/U , since B is ramified at all archimedean places. Thus it is 0dimensional. We first define a CM point Pc ∈ G(F )\G(Af )/U . Let ic ∈ G(Af )
such that
×
×
b
c
UT := ic U i−1
c ∩ T (Af ) = Oc(χ)× /OF .
57
Set Pc = [ic ] ∈ G(F )\G(Af )/U. Finally
X
Pχ =
χ(t)[tic ].
t∈T (F )\T (Af )/UT
As we see from last section that the formula of level N D has an extra term
φ# involved which is an obstruction to arithmetic applications. To deduce the
formula of level N from the formula of level N D. One rewrites the formula of
level N D in the following way:
# (1) ||φ||2 |(φ,
e Pχ )|2 1
φc
C
e Pχ )|2 .
||φ||2 |(φ,
L( , χ, φ) = p
#
2
2
N(c(χ)) ||φ || |(φχ , η)|
2
N(D)
We shall prove that
# (1) ||φ||2 |(φ,
Y
e Pχ )|2 1
φc
Qv (λv ),
L(
,
χ,
φ)
=
C(χ)
N(c(χ)) ||φ# ||2 |(φχ , η)|2 2
v|D
where λv is the parameter in the local L-factor Lv (s, φ) = (1 − λv |$v |s +
|$v |2s )−1 , and C(χ) is a constant depending only on χ and for each v|D, Qv is
a rational function in C(t) depending only on χv which takes value 1 at t = 0
1
and is regular for t ≤ |$v | 2 + |$v |
−1
2
. The idea is that we view φ as a form
varying in the space L20 (GL2 (F )\GL2 (A)), a space consisting of discrete part
(cusp forms) and continuous part (continuous sum of Eisenstein series). The
form
# (1) ||φ||2 |(φ,
e Pχ )|2 1
φc
L( , χ, φ)
N(c(χ)) ||φ# ||2 |(φχ , η)|2 2
can be viewed as a formula associated to each form in the space L20 (GL2 (F )\GL2 (A)).
The above shows exactly that
# (1) ||φ||2 |(φ,
e Pχ )|2 1
φc
L( , χ, φ)
N(c(χ)) ||φ# ||2 |(φχ , η)|2 2
58
is independent of the choices of φ. Thus one can use Eisenstein series to deterQ
mine C(χ) v|D Qv (λv ). To that end, a similar special value formula of RankinSelberg L-function associated to an Eisenstein series and χ will be deduced to
Q
explicitly determine the constant C(χ) v|D Qv (λv ). The proof consists of the
following two steps:
(1) compare two formulae of level N and N D and show they are equal up
to universal constants;
(2) obtain a special value formula of level D for Rankin-Selberg L-function
L(s, E, χ) associated to an Eisenstein series E and χ and explicitly compute the universal constants, thus prove the final formula.
4.1. Universal constants. In this subsection, we shall prove that both formulae of level N and N D are equal up to some universal constants. We have
Proposition 4.1.1 For each v|D, there exists a rational function Qv ∈ C(t),
1
1
depending only on χv , Qv (0) = 1, and being regular for t ≤ |$v | 2 + |$v |− 2 ,
such that
# (1) ||φ||2 |(φ,
Y
e Pχ )|2
φc
Qv (λv ),
=
C(χ)
N(c(χ)) ||φ# ||2 |(φχ , η)|2
v|D
where C(χ) is a constant depending only on χ and λv is the parameter in the
local L-factor
L(s, φ) = (1 − λv |$v |s + |$v |2s )−1 .
Proof. We first reduce it to local case, then show everything locally. We
need the local version of quasi-newform. Before that, Let’s fix a hermitian form
59
on the Whittaker model W(πv , ψv ). By the discussion in Chapter 1, one sees
that the irreducible cuspidal representation π = ⊗πv is unitary. Thus πv is
unitary for each v. We choose a hermitian form (, ) on W(πv , ψv ) for each v,
such that ||Wv ||2 = 1, for almost all v, where Wv is the normalized newform of
W(πv , ψv ). Hence the global hermitian form is proportional to L2 -norm. For
each v, the Whittaker model W(πv , ψv ) has the normalized newform Wv of level
ordv (N )
and the quasi-newform Wv# with respect to χv is the form of
Nv := $v
ordv (N )+ordv (D)
levelNv Dv = $v
satisfying
(Wv# , Wv,i ) = ν i (Wv# , Wv# ),
where
ν=
i = 0, 1, · · · , ordv (D),


 0,
if v|c(ω);

 χv ($K,v ), if v - c(ω).
ordv (N )
In particular, if v - D, so Wv# is of level $v
and the relation above implies
that Wv# = Wv . The uniqueness of the global quasi-newform φ# implies that
φ# = ⊗v Wv# . Now we divide the proof into three steps showing that each term
has the property in proposition.
(1)
||φ||2
.
||φ# ||2
From the above, one sees that
Q
Q
||φ||2
v (Wv , Wv )
v (Wv , Wv )
=
=
.
Q
Q
#
#
#
#
||φ# ||2
(W
,
W
)
(W
,
W
)
v
v
v
v
v
v
We now show that there exists, for each v|D, a rational function Q1,v (t) ∈ C(t),
1
1
depending only on χv , Q1,v (0) = 1, and Q1,v (t) is regular for t ≤ |$v | 2 + |$v |− 2
such that
(Wv , Wv )
(Wv# , Wv# )
= C1,v (χv )Q1,v (λv ).
60
Write Wv# =
Pordv (D)
i=0
αv,i Wv,i . The definition of quasi-newform can be trans-
lated into the system of following equations
ordv (D)
X
i=0
ordv (D)
X
(Wv,i−j , Wv )
(Wv,i , Wv )
αv,i
= νj
αv,i
, j = 1, · · · , ordv (D)
(Wv , Wv )
(Wv , Wv )
i=0
ordv (D)
ordv (D)
X
X
(Wv #, Wv )
(Wv# , Wv# )
(Wv #, Wv )
i
=
=
ν αv,i
⇔
ν i αv,i = 1.
(Wv , Wv )
(Wv , Wv )
(W
,
W
)
v
v
i=0
i=0
Hence each αv,i is a rational function in
remains to show that
Hv,j be vol(Hv,j )−1
(Wv,j ,Wv )
(Wv ,Wv )
for j = 0, · · · , ordv (D). It
(Wv,j ,Wv )
(Wv ,Wv )
is a polynomial in λv . Let Uv = GL2 (Ov ). Define


−j
 $v 0 
times the characteristic function of Uv 
 Uv and
0 1
T($v−j to be the Hecke operator corresponding to Hv,j . One obtains that
(Wv,j , Wv )
= (Wv , Wv )−1 vol(Uv )−1
(Wv , Wv )
−1
= (Wv , Wv ) vol(Hv,j )
Z
(π(u)Wv,j , π(u)Wv )du
Uv
Z
−1
(π(u)Wv , Wv )du
Hv,j
Since T ($v−j ) is generated
by usual 
Hecke operators:
Tv ,Rv and R−1
which
v




−1
0 
 $v
 $v 0 
 $v 0 
correspond to Uv 
 and Uv 
 re Uv , U v 
−1
0 1
0 $v
0 $v
spectively, See Bump [3]. The action of Rv on Wv is trivial and Tv · Wv =
λv |$v |
1−k
2
Wv . Hence T($v−j ) · Wv = Q01,v (λv )Wv , where Q01,v (t) is a polynomial
independent of πv . In fact, one has explicit relation
T($vj ) = Tv ($vj+1 ) + |$v |−1 Rv T($vj−1 ).
61
Thus Q01,v (0) is a constant depending only on χv . Finally we may simply take
Q1,v (t) = Q01,v −1 (0)Q01,v (t).
(2) φb# (1).
# (1) directly, we have
Applying the definition of φc


−1
0 
 y∞ δ
Wφ# 

0
1


φb# (1) =
 y∞ 0 
W∞ 

0 1


−1
y∞ δ v 0 
Q
#

v Wv 
0
1


=
 y∞ 0 
W∞ 

0 1


y∞ δv−1


y∞ δv−1
0  Q
0 

 · v|D Wv# 

0
1
0
1


=
 y∞ 0 
W∞ 

0 1




−1
−1
Q
 y∞ δv 0 
 y∞ δv 0  Q Pordv (D)
αv,i Wv,i 

 v|D i=0
v-D Wv 
0
1
0
1


.
=
 y∞ 0 
W∞ 

0 1
Q
#
v-D Wv 
62


−1
 a
Where the last equality is obtained, since for any a|D, φa := π 
0
is an old form, therefore φa (1) = 0.
(3)
e χ )|2
|(φ,P
.
|(φχ ,η)|2
0 
φ
1
To show that the above quantity has the desired property, one
needs to analyze both φe and φχ . First notice that the point Pχ corresponds to
e Pχ ) =
the following CM-cycle ξ : T (Af )ic U, tic u 7→ χ(t). Thus one sees that (φ,
e ξ), where the right hand side of the equality is regarded as the pairing
(φ,
between a form on G(F )\G(A) and a CM-cycle. So the idea is to compare
both CM-cycles ξ and η, more precisely, there exists h : U → C× , such that
ρ(h)η = ζ,
and then we replace η by ζ in the decomposition of the geometric pairing Φ(ζ, ζ)
to deduce the desired property. Let ξ be the function on T (Af )ic ,
ξ : T (Af )ic U → C×
tic u 7→ χ(t), ∀t ∈ T (Af ), u ∈ U.
e Pc )χ = (φ,
e ξ). We have
We first show that (φ,
Z
−1
e
e
(φ, ξ) =vol(U )
ξ(x)φ(x)dx
Z
T (F )\G(Af )
e
ξ(x)φ(x)dx
=vol(U )−1
T (F )\T (Af )ic U
Z
e c)
χ−1 (t)φ(tP
=
T (F )\T (Af )/UT
e Pc )χ .
=(φ,
63
Now we compare the CM-cycles ξ and η. Let h be the characteristic function
on U i−1
c . We claim that
ρ(h)η = ζ.
This is a local problem. For any finite place v, let hv be vol(Uv ) times the
characteristic function on Uv i−1
c,v . We want to verify that ρ(hv )ηv = ζv .
If B is ramified at v, then v|N, thus v - c(χ). We shall determine ic,v . Recall
that ic,v is an element of G(Fv ) such that
×
×
ic,v Uv i−1
c,v ∩ T (Fv ) = Ocv /Ov .
Since v - c(χ), thus Ocv = Ov . Therefore we may take ic,v = I. Hence if
1
ρ(hv )ηv (g) =
vol(Uv )
1
=
vol(Uv )
1
=
vol(Uv )
Z
ηv (gx)hv dx
Z
G(Fv )
ηv (guic,v )du
Z
Uv
ηv (gu)du 6= 0,
Uv
then gu ∈ T (Fv )∆v , hence g ∈ T (Fv )∆v = T (Fv )Uv , since ∆v = Uv , for v - c(χ),
i.e., ρ(hv )ηv is supported on T (Fv )Uv . For any g = tu ∈ T (Fv )Uv , one has
1
ρ(hv )ηv (g) =
vol(Uv )
1
=
vol(Uv )
Z
ηv (gx)dx
Z
Uv
ηv (tux)dx
Uv
1
=
χv (t)
vol(Uv )
= χv (t),
Z
χv (x)dx
Uv
64
i.e., ρ(hv )ηv = ξv .
If B splits at v. Let Uv0 = Ocv +cv λv OK,v . If ρ(hv )ηv (g) =
R
Uv
ηv (gxic,v )dx 6= 0.
Then one has gxic,v ∈ T (Fv )Uv0 , i.e., g ∈ T (Fv )∆v ic,v . Using the lemma in
Zhangc̃iteZh2, we see that ic,v Uv = Uv0 ic,v . Therefore g ∈ T (Fv )i−1
c,v Uv . Now for
−1
any g = ti−1
c,v u ∈ T (Fv )ic,v Uv ,
1
ρ(hv )ηv (g) =
vol(Uv0 )
Z
Z
Uv
ηv (ti−1
c,v uxic,v )dx
1
ηv (ti−1
c,v xic,v )dx
vol(Uv0 ) Uv
Z
=
ηv (tu)dx = χv (t),
=
Uv0
i.e., ρ(hv )ηv = ζv .
We prove the final step now. Recall that we have defined a geometric pairing
Ψ(α, β) associated to two CM-cycles α and β and have obtained the spectral decomposition of Ψ(α, β)(g∞ ) with g∞ a continuous parameter varying in
GL2 (F∞ ). We have
1
Ψ(α, β) =C|c(ω)| 2
X
i
+ C|c(ω)|
φnew
(φi , α)∆ (φi , β)∆
i
Z
1
2
M
new
EM
(EM , α)(EM , β)dM.
In particular, let α = β = ξ, we see that
1
Ψ(ξ, ξ) =C|c(ω)| 2
X
i
+ C|c(ω)|
φnew
|(φi , ξ)∆ |2
i
Z
1
2
M
new
EM
|(EM , ξ)|2 .
65
Since ξ is fixed by U , we may assume that φi is fixed by U as well. For φi
such that the representation generated by φi corresponds to φ via Jacquete Now write H(ξ, ξ) =
Langlands correspondence, this φi must be test form φ.
P new
φi |(φi , ξ)∆ |2 + continuous spectrum. Using the relation

 
−1
0 
  aδ
|a| < Ta α, β > (g∞ ) = CWH(α,β) g∞ 
 ,
0
1
and simple fact that < α, ρ(h)β >=< ρ(h∨ )α, β >, it follows that
H(ξ, ξ) = H(ρ(h∨ ∗ h)η, η).
By the decomposition S(T (F )\G(Af )) = ⊕τ S(T (F )\G(Af ), τ ), one has
h∨ ∗ h =
X
hτ ,
τ
where τ runs through all characters of T (Af ). If τ 6= χ, then it is easy to see
that
< ρ(hτ )η, η >= 0,
since η has character χ under the action of T (Af ). Therefore, we obtain
H(ρ(h∨ ∗ h)η, η) = H(ρ(hχ )η, η)
=
X
(φi , ρ(hχ )η)(φi , η) + continuous spectrum
φnew
i
i
=
X
φnew
(ρ(hχ )φi , η)(φi , η) + continuous spectrum.
i
i
Once again, observing that η has character χ under the action of T (A). We
may assume that φi has same character χ under the action of T (A). Thus φi
66
is exactly the toric newform φχ , if φnew
= φ. Now we claim that
i
ρ(hχ )φi =
Y
Q2,v (λv )φi ,
v|D
where Q2,v is certain rational function.
To that end, notice that for v - D = c(χ)2 c(²), one has Ocv = OK,v , hence
we may take ic,v = I, then h∨ ∗ h is the characteristic function of Uv , which
may be regarded as the identity element in the algebra HUv of bi-Uv invariant
functions on G(Fv ). Hence the action is trivial.
If v|D, it is easy to show that
h∨v ∗ hv ∈ HUv .
Thus
ρ(hχ )φi =
Y
Q2,v (λv )φi ,
v|D
here we use the explicit construction that πv = πv0 if B is ramified at v. Therefore finally we obtain that
Y
e Pχ )|2
|(φ,
=
C
Qv (λv ).
|(φχ , η)|2
v|D
¤
4.2. Determination of universal constants. We shall use the continuous
spectrum to compute the universal constants. First we deduce a similar special
value formula for the L-function L(s, E, χ) associated to a continuous family E
67
and character χ. The explicit construction of E allows us to compute the universal constants and finally show that the product of these universal constants
is 1. Hence the formula of level N is proved.
4.2.1. Special value formula of L(s, E, χ). we now deduce a formula for L(s, E, χ)
associated to an form E ∈ L2cont (GL2 (F )\GL2 (AF )) and χ in terms of quasinewform E # . The idea follows exactly as before. For a fixed character µ :
×
A×
→ C× , we shall use the two forms Φ and Ψ constructed in [Zh1]:
F /F
(1) the form Φ is the holomorphic projection of Θ 1 ;
2
(2) the form Ψ comes from the spectral decomposition of certain geometric
pairing.
The difference Φ - Ψ is an old form via local Gross-Zagier formula. So is the
projection of Φ - Ψ on the space Eis(µ), a subspace of L2cont (GL2 (F )\GL2 (AF ))
chosen appropriately, from which the desired formula follows.
×
Let’s fix a character µ of A×
such that µ2 is not of form | · |t , for some
F /F
0 6= t ∈ R. Let Eis(µ) be the space of forms E in L2cont (GL2 (F )\GL2 (A)) so
that
Z
∞
E(g) =
Et (g)dt,
−∞
with Et (g) := E(it, g) certain Eisenstein series in πt := π(µ| · |it , µ−1 | · |−it ), ∀t ∈
R. For two such forms E1 and E2 , one has inner product
Z
∞
(E1 , E2 ) =
(E1,t , E2,t )t dt,
−∞
where (, )t is some Hermitian form on the space π(µ| · |it , µ−1 | · |−it ).
68
For any form E(g) =
R∞
−∞
Et (g)dt ∈ Eis(µ), and a continuous function ϕ on
R, one obtains another form Eϕ twisted by ϕ by
Z
∞
Eϕ (g) =
ϕ(t)Et (g)dt.
−∞
In particular, let Etnew be the newform of the representation π(µ| · |it , µ−1 | · |−it ).
One has
Z
Eϕnew (g)
∞
=
−∞
ϕ(t)Etnew (g)dt.
We now compute the inner product (Eϕnew , Φs ). Assume that χ is not of the
form ι · NK/F , thus θ = θχ is a cusp form and the kernel function Φs is squareintegrable since its constant term has exponential decay. Hence we have
Z
(Eϕnew , Φs )
=
(Eϕnew , Θs )
=
Z
Z(A)GL2 (F )\GL2 (A)
Eϕnew Θs (g)dg
Z
∞
=
Z
Z(A)GL2 (F )\GL2 (A)
∞
Z
−∞
=
Z
−∞
Z(A)GL2 (F )\GL2 (A)
ϕ(t)Eϕnew (it, g)Θs (g)dtdg
ϕ(t)Eϕnew (it, g)Θs (g)dgdt
∞
=
ϕ(t)L(s, πt , χ)dt.
−∞
We need to compute Lv (s, πt , χ). One has
Lv (s, πt , χ) =
Y
(1 − µ($v )χ($w )|$v |s+it )−1 ·
w|v
Y
(1 − µ−1 ($v )χ($w )|$v |s−it )−1
w|v
= Lv (s + it, µK ⊗ χ) · Lv (s − it, µ−1
K ⊗ χ),
here, µK = µ · NK/F . Therefore one obtains
Z
(Eϕnew , Φs )
∞
=
−∞
ϕ(t)L(s + it, µK ⊗ χ) · L(s − it, µ−1
K ⊗ χ).
69
In particular, for s = 12 , applying the fact that χ is of finite order as well as
χ|A× ≡ 1, one has
Y
1
1
Lw ( − it, µK ⊗ χ)
L( − it, µK ⊗ χ) =
2
2
w
=
Y
1
−it −1
2
(1 − µ−1
)
K ($w )χ($w )|$w |
w
=
Y
1
(1 − µK ($wσ )χ($wσ )|$wσ | 2 +it )−1
w
1
= L( + it, µK ⊗ χ),
2
Here σ is the nontrivial automorphism in Gal(K/F ). Hence
Z
(Eϕnew , Φ)
=
∞
1
ϕ(t)|L( + it, µK ⊗ χ)|2 dt.
2
−∞
Now we need to compute the projection of both Φ and Ψ on the space Eis(µ).
Let’s start with the projection of Φ on Eis(µ) first.
Recall that the projection of Φ on Eis(µ) is the unique form pr(Φ) ∈ Eis(µ),
satisfying
(E, pr(Φ)) = (E, Φ), ∀E ∈ Eis(µ).
And the quasi-newform Et# of πt with respect to χ is the form satisfying
new
(Et# , Et,a
)t = ν(a)(Et# , Et# )t , ∀a|D,


−1
 a
new
= ρ
where Et,a
0
0 
 Etnew . We have
1
70
Lemma 4.2.1.1 The projection pr(Φ) of Φ on the space Eis(µ) is Eϕ# with
ϕ(t) =
|L( 12 + it, µK ⊗ χ)|2
||Et# ||2t
Proof. We first compute (E, Φ), ∀E =
R∞
−∞
Et dt ∈ Eis(µ). One has
Z
Z
∞
(E, Φ) = (E, Θ 1 ) =
Et (g)Θ 1 (g)dtdg
2
Z
Z(A)GL2 (F )\GL2 (A)
2
−∞
∞
=
Et (g)Θ 1 (g)dgdt.
−∞
Thus the linear map Et 7→
.
2
R
Z(A)GL2 (F )\GL2 (A)
Et (g)Θ 1 (g)dg is well defined, and
2
by Riesz representation theorem, there exists EtΘ ∈ πt such that
1
|L( + it, µK ⊗ χ)|2 =
2
Z
Z(A)GL2 (F )\GL2 (A)
Et (g)Θ 1 (g)dg = (Et , EtΘ ).
2
One easily sees that
EtΘ =
|L( 12 + it, µK ⊗ χ)|2
||Et# ||2t
Et# .
Thus
Z
∞
(E, Φ) =
−∞
ϕ(t)(Et , Et# )t dt = (E, Eϕ# ),
i.e., Et# = pr(Φ). ¤
The projection pr(Ψ) of Ψ on Eis(µ) is easier. It’s the continuous contribution
for Eis(µ) in Ψ. One has
1
pr(Ψ) = 22g |c(ω)| 2 Eψnew
71
with ψ(t) = |(Et,χ , η)∆ |2 . By local Gross-Zagier formula, Φ − Ψ is an old form.
Thus pr(Φ) − pr(Ψ) is also an old form. Hence
1
ϕ(t)Et# − 22g |c(ω)| 2 ψ(t)Etnew
is an old form too. Therefore we obtain
Proposition 4.2.1.2 Assume that χ is not of the form ι · NK/F with ι a
character of A× /F × . Then
(4.1)
1
1
c
new
, η)∆ |2 .
Et# (1)|L( + it, µK ⊗ χ)|2 = 22g |c(ε)| 2 ||Et# ||2t |(Et,χ
2
Rewrite the formula (4.1) in the following way:
c
new
new
]
]
Et# (1) ||Etnew ||2t |(E
, Pχ )|2
22 g ||Etnew ||2t |(E
, Pχ )|2
t
t
p
·
·
=
.
1
new
N(c(χ)) ||Et# ||2t |(Et,χ
, η)∆ |2
N(D) |L( 2 + it, µK ⊗ χ)|2
Then the universality of the functions Qv implies
Proposition 4.2.1.3 Assume that χ is not of form ι·NK/F with ι a character
of A× /F × , let
−it
λv (t) = µv ($v )|$v |it + µ−1
,
v ($v )|$v |
new
]
. Then
and Et∗ := ||Etnew ||t E
t
C(χ)
Y
v|D
¯
¯2
¯
¯
(Et∗ , Pχ )
¯
¯ .
Qv (λv (t)) = p
1
N(D) ¯ L( 2 + it, µK ⊗ χ) ¯
22g
72
4.2.2. Determination of universal constants. We shall finally compute the universal constants. We need to compute the pairing |(Et∗ , Pχ )|2 . By the universality of the constants, we may assume that the character µ is unramified, thus
the quaternion algebra B splits everywhere, i.e., B = M2 (F ). We first simplify
the form Et∗ .
Lemma 4.2.2.1 Let j = jf ⊗
Q
v|∞ jv
∈ G(A) such that jf ∈ G(Af ) with
b F ), and for v|∞, jv ∈ G(Fv ) with
jf−1 U (N, K)jf = GL2 (O
jv (SO2 (Fv )/ {±I})jv−1 = T (Fv ).
Then we have
Et∗ = ρ(j)Etnew .
new
]
Proof. Recall that the test form E
is a form fixed by U and normalized
t
such that ||Etnew ||2 = 1. We show first that ρ(j)Etnew is fixed by U (N, K). For
any h ∈ U (N, K),
ρ(h)(ρ(j)Etnew )(g) = Etnew (ghj)
= Etnew (gjj −1 hj)
= ρ(j)Etnew (g).
73
Since the dimension of the forms fixed by U (N, K) is one, so
ρ(j)Etnew
new
]
= ±E
.
t
||ρ(j)Etnew ||2
We may assume that
ρ(j)Etnew
new
]
=E
,
t
||ρ(j)Etnew ||2
i.e.,
new
new
]
]
ρ(j)Etnew = ||ρ(j)Etnew ||2 E
= ||Etnew ||2 E
,
t
t
since the representation π is hermitian and we only need to compute |(Et∗ , Pχ )|2 .
¤
Proposition 4.2.2.2 Assume that the character µ is unramified and χ is
not of the form µK := µ ◦ NK/F , then
r
(Et∗ , Pχ )
−g
= 2 µ(δ
−1
1
4λ
1
)|4λ/D| 4 (1+2it) 2it L( , χ · µK ),
D
2
where λ is a trace-free element of K.
X
χ−1 (σ)Pcσ . We have
Proof. Recall that Pχ =
σ∈Gal(HK /K)
Z
(Et∗ , Pχ )
=
Z
T (F )\T (A)
=
T (F )\T (A)
χ−1 (s)Et∗ (sPc )ds
χ−1 (s)Etnew (sPc j)ds.
74
here we choose the Haar measure ds = ⊗dsv such that if v is non-archimedean,
then vol(Ocv ) = 1; if v is archimedean, then vol(T (Fv )) = 1. Recall that
X
Etnew (g) =
µ−1 (δ)fΩ,t (γsPc j),
γ∈P (F )\GL2 (F )
where
Z
fΩ,t (g) = µ1 (detg)|detg|
t+ 12
A×
1+2t ×
Ω[(0, x)g]µ1 µ−1
d t
2 (x)|x|
with Ω a Schwartz-Bruhat function on A2 and µi being character of A× , i = 1, 2.
In particular, we choose the Schwartz-Bruhat function Ω = ⊗Ωv , such that
for v non-archimedean, Ωv is the characteristic function of the set Ov2 , for v
2 +y 2 )
archimedean, Ωv = e−π(x
. Therefore, we have
Z
(Et∗ , Pχ )
= µ(det(Pc j))|det(Pc j)|
Z
1
+it
2
1
χ−1 (s)µ(dets)|dets| 2 +it
T (A)
A×
· Ω((0, x)sPc j)µ2 (x)|x|1+2it d× xds
1
1
= µ(det(Pc j))|det(Pc j)| 2 +it Z( , µ · | · |it , Ω).
2
So it suffices to compute the local zeta function Zv (s, µ · | · |it , Ω) as well as
det(Pc j). We start with the local zeta function.
Since the function Ωv is supported on Ov ⊕ Ov for v non-archimedean, one
easily sees that
1
Z( , µ · | · |it , Ω) =
2
Z
1
χ−1 (x)µKv (x)|x| 2 +it d× x.
Ocv −0
75
I. If v is non-archimedean place.
(1) If v|c(χ), then Ocv = Ov . Thus
1
Z( , χ−1 µKv | · |it , Ω) =
2
Z
1
χ−1 (x)µKv (x)|x| 2 +it d× x
Ov −0
1
= L( , χ−1 µKv | · |it )
2
(2) If v - c(χ), write Ocv − 0 =
∞
[
Ocv ,n , where Ocv ,n is the set of elements of
n=0
order n in Ocv .
i) If v is inert in K, then Kv is a field. Let’s write OKv = Ov + Ov λ. We have
Z
−1
χ µK (x)|x|
Ocv −0
1
+it
2
×
d x=
∞ Z
X
n=0
1
χ−1 µK (x)|x| 2 +it d× x.
Ocv ,n
Case 1. n = 0, then OCv ,0 = Oc×v . We obtain
Z
1
Ocv ,0
χ−1 (x)µK (x)|x| 2 +it = vol(Ov× ) = 1.
Case 2. 0 < n < ordv (c(χ)), It’s easy to see that
Ocv ,n = $vn Ov× + cv OKv = $vn (Ov× + cv $v−n OKv ).
Hence
Z
1
χ−1 (x)µKv (x)|x| 2 +it d× x = 0,
Ocv ,n
since χ restricted to Ov× + cv $−n OK,v is not identically one.
×
. Now applying
Case 3. n ≥ ordv (cv ). One sees easily that Ocv ,n = $vn OK
v
¯
χ¯O× ≡ 1 and µ being unramified, we have
Kv
Z
1
χ−1 (x)µKv (x)|x| 2 +it d× x = 0.
Ocv ,n
Hence Z( 21 , χ−1 µKv | · |it , Ωv ) = 1, for v|c(χ) and v is inert in K.
76
ii) If v splits in K. Then Kv = Fv ⊕ Fv , thus OKv = Ov ⊕ Ov .
Case 1. n = 0, similarly one has
Z
1
Ocv −0
χ−1 (x)µKv (x)|x| 2 +it d× x = vol(Oc×v ) = 1.
Case 2. 0 < n < ordv (c(χ)), then Ocv ,n = $vn Ov× + cv OKv = $vn (Ov× +
cv $v−n OKv ). Hence
Z
1
χ−1 (x)µKv (x)|x| 2 +it d× x = 0,
Ocv −0
¯
since χ¯Ov× +cv $v−n O
Kv
≡
/ 1.
¯
×
Case 3. n ≥ ordv (c(χ)). The fact that Ocv ,n = $vn OK
, and χ¯O× ≡
/1 implies
v
Kv
that
Z
1
χ−1 (x)µKv (x)|x| 2 +it d× x = 0.
Ocv ,n
Again we obtain that
1
Z( , χ−1 µKv | · |it , Ωv ) = 1.
2
II. If v is archimedean place.
−1
Recall that jv ∈ GL2 (R) such that jv (SO
v = T (R). Writing K =

 2 (R)/{±I})j
1
2
√
 |λ|v 0 
F + F λ, one may simply take jv = 
 . Using the polar coordinate
0 1
×
(r, θ), the measure on C induced from the standard measure on R× and the
measure on C× /R× such that the volume of C× /R× is one has the express
drdθ
.
πr
77
2
Then the function Ωv ((0, x)sjv ) is of the form e−πr . Therefore we get
1
Z( , χ−1 µKv | · |it ) =
2
Z
2π
Z
0
Z
∞
µ(r)|r|1+2it e−πr
0
∞
2
drdθ
πr
dr
r
0
1
1 + 2it
1
= µ(π)−1 π − 2 (1+2it) Γ(
).
2
2
=2
µ(r)|r|1+2it e−πr
2
Finally We compute the det(jf Pc ) as well as det(jv ) for v archimedean. By
the definition of jf and Pc , one sees easily that
√
Ov + Ov λ = Ocv jf Pc .
Hence taking discriminant both sides, one obtains
4λ = D · det(jf Pc ).
1
For det(jv ), one has det(jv ) = |λ|v2 by the specific form we have chosen.
Summing up, We have the following formula:
à r
!
¯ 4λ 1
4λ
1
(Et∗ , Pχ ) = µ δ −1 ( ) ¯ | 4 (1+2it) 2−g · 2it L( , χ−1 · µK · | · |it ).
D
D
2
¤
We now compute the universal constants using the formula just obtained. In
particular we choose µ(x) = |x|is . Observe that
1
1
|L( + it + is, χ−1 )|2 = |L( + it + is, χ)|2 ,
2
2
since L( 12 + it + is, χ−1 ) = L( 12 − it − is, χ−1 ) and |L( 12 − it − is, χ−1 )| =
|L( 21 + it + is, χ)| by the functional equation of L( 12 + it + is, χ−1 ). Applying
78
the formula in Proposition 4.2.1.3, we have
C(χ)
Y
Qv (λv (t)) = 1.
v|D
Observe that each Qv (λ(t)) is a rational function in pnt . One can show that
Y
Qv (λ(t)) = const,
v|Dp
where Dp is the set of places v dividing D and lying over p. We claim that the
constant is one. Thus end the proof of special value formula of level N .
To that end, it’s known ([1]) that there exists a character χ0 of finite order
× ×
0
0
of A×
K /K A such that χ satisfies all properties that χ has except that χ is
unramified at w lying over p but ramified at all other places w|c(χ). Applying
the above argument to this χ0 , one obtains that
Y
Qv (λ(t)) = 1.
v|Dp
Hence we have
Theorem 4.2.2.3 The constant C(χ) = 1 and the polynomial Qv (λ(t)) = 1,
for v|D.
79
5. Appendix. Continuous spectrum of
L2(GL2(F )\GL2(A), ω)
We follow the notations in Gelbart and Jacquet [6]. Let Z be the center of
GL2 and






 1 ∗ 
 ∗ 0 
 ∗ ∗ 
P =
,N = 
.
,A = 
0 ∗
0 1
0 ∗
If ω is a (unitary) character of A× /F × , we denote by L2 (GL2 (F )\GL2 (A), ω)
the space of the functions ϕ on GL2 (A) such that
ϕ(γzg) = ω(z)ϕ(g), ∀γ ∈ GL2 (F ), z ∈ Z(A),
and
Z
|ϕ(g)|2 dg < ∞.
Z(A)GL2 (F )\GL2 (A)
If, in addition,
Z
ϕ(ng) ≡ 0, ∀g ∈ GL2 (A),
N (F )\N (A)
then we say that ϕ is a cuspidal and the subspace of cuspidal functions is denoted by L20 (GL2 (F )\GL2 (A), ω). Let ρω (ρω,0 ) be the natural representation of
GL2 (A) in L2 (GL2 (F )\GL2 (A), ω)(L20 (GL2 (F )\GL2 (A), ω)) via the right translation.
The representation ρω,0 decomposes discretely with finite multiplicity. For
details, see Bump[1].
In this section, we would like to give a description of the orthocomplement of
L20 (GL2 (F )\GL2 (A), ω) inside L2 (GL2 (F )\GL2 (A), ω) equipped with the inner
80
product
Z
(ϕ1 , ϕ2 ) =
ϕ1 (g)ϕ2 (g)dg.
GL2 (F )Z(A)\GL2 (A)
The orthocomplement can be described in terms of P -series, which we now define.
Definition 5.1If f is a function in C ∞ (N (A)P (F )\GL2 (A)) such that
(5.1.1)
f (zg) = ω(z)f (g), z ∈ Z(A).
Then the series
F (g) =
X
f (γg)
γ∈P (F )\GL2 (F )
is called a P -series.
One can show ( [6], P. 197) that if the function f is compactly supported
mod N (A)Z(A)P (F ), then the P -series is convergent. So in the rest of the
notes, we assume that function f is compact mod N (A)Z(A)P (F ).
We shall prove that the space of P -series is a dense subset of the space
L2 (GL2 (F )\GL2 (AF ), ω)⊥ .
81
We need to compute the inner product of a P -series and a cuspidal function ϕ.
Let ϕ be a function in L2 (GL2 (F )\GL2 (AF ), ω), then
Z
(ϕ, F ) =
ϕ(g)F (g)dg
Z
Z(A)GL2 (F )\GL2 (A)
=
X
ϕ(g)
Z(A)GL2 (F )\GL2 (A)
γ∈P (F )\GL2 (F )
Z
ϕ(g)f (g)dg
=
Z
Z(A)P (F )\GL2 (A)
Z
=
ϕ(ng)f (g)dn.
dg
N (A)Z(A)P (F )\GL2 (A)
Notice the
f (γg)dg
R
N (F )\N (A)
N (F )\N (A)
ϕ(ng)dn is the constant term of ϕ. Thus if ϕ is a cuspidal
form, then (ϕ, F ) = 0. Therefore F ∈ L2 (GL2 (F )\GL2 (AF ), ω)⊥ . Conversely
let ϕ be any form in L2 (GL2 (F )\GL2 (AF ), ω), if
Z
Z
(ϕ, F ) =
ϕ(ng)f (g)dn = 0
dg
N (A)Z(A)P (F )\GL2 (A)
N (F )\N (A)
for any P -series with compact support mod N (A)Z(A)P (F ), then it is easy to
R
see that the constant term N (F )\N (A) ϕ(ng)dn of ϕ is 0, i.e., ϕ is cuspidal. Thus
P -series form a dense subset of L2 (GL2 (F )\GL2 (AF ), ω)⊥ . For our purposes,
we want to express P -series as continuous sums of Eisenstein series. To define
+
be the set of ideEisenstein series, let’s introduce a Hilbert space H(s). Let F∞
les whose finite components are all 1 and whose infinite components all equal
some positive number (independent of infinite place) and A×
F,1 be the ideles of
× ∼
×
+
norm 1. One has A×
= A×
F /F
F,1 /F × F∞ .
82
Definition 5.3 H(s)is the space

 αau x
φ 
0 βav
and
Z Z
K
of functions φ ∈ C ∞ (GL2 (A)) such that

¯ u ¯s+ 1

¯ ¯ 2
φ(g),
 = ω(a) ¯ ¯
v



 a 0  
|φ|2 
 K  dadk < ∞,
F × \A×
F,1
0 1
+
where α, β ∈ F × , a ∈ A× , u, v ∈ F∞
.
The group GL2 (A) operates on H(s) via the right translation and the resulting representation is denoted by πs . The representation πs is unitary if s is
purely imaginary. One may view H(s) as a trivial fibre bundle of base C. For
any open subset U of C, the sections are functions φ(g, s) on GL2 (A) × U such
that

 
¯ u ¯s+ 1
 αau x  
¯ ¯ 2
φ(g, s).
φ 
 g  = ω(a) ¯ ¯
v
0 βav
Now we can define the Eisenstein series associated to a section of the trivial
fibre bundle H(s).
Definition 5.4. For a section φ of the trivial fibre bundle H(s), the series
E(φ(s), g) =
X
φ(s, γg)
γ∈P (F )\GL2 (F )
are called Eisenstein series.
This series converges only for Re(s) > 12 . It can be shown (c̃iteG-J, § 5) that
the Eisenstein series E(φ(s), g) can be analytically continued to the region for
which Re(s) ≥ 0.
83
To explain the relationship between P -series and Eisenstein series, we now
define the Fourier-Laplace transform of a function. For a function f satisfying
(5.1.1), one defines the Fourier-Laplace transform of f by
 

Z
 t 0   −s− 12 ×
f 
fb(g, s) =
d t.
 g  |t|
+
F∞
0 1
By our assumption that f is compactly supported mod N (A)Z(A)P (F ). then
fb defines a section of H(s). Fourier inversion implies that
1
f (g) =
2πi
Z
x−i∞
fb(g, s)ds,
x−i∞
for any x. Hence the P -series
X
F (g) =
f (γg)
γ∈P (F )\GL2 (F )
1
=
2πi
1
=
2πi
1
=
2πi
Z
X
γ∈P (F )\GL2 (F )
Z
+i∞
+i∞
fb(g, s)ds
−i∞
X
fb(g, s)ds
−i∞ γ∈P (F )\GL (F )
2
Z
+i∞
E(fb(s), g)ds.
−i∞
Here we use the analytic continuation of E(φ(s), g) to shift the integral to
the imaginary axis. In other words, any P -series are “continuous sums” of
Eisenstein series.
In the rest of the section, we shall briefly describe the relationship between
the space L2cont (GL2 (F )\GL2 (A), ω), a subspace of L2 (GL2 (F )\GL2 (AF ), ω)⊥ ,
and a subspace of sections of H(s). Let’s define L2cont (GL2 (F )\GL2 (A), ω) first.
84
Let’s denote by L2sp (GL2 (F )\GL2 (A), ω) the space spanned by characters χ
with χ2 = ω. The space L2cont (GL2 (F )\GL2 (A), ω) is the orthocomplement of
L2sp (GL2 (F )\GL2 (A), ω) in L2 (GL2 (F )\GL2 (AF ), ω)⊥ . Now let L denote the
Hilbert space of square-integrable sections a over iR satisfying
M (−it)a(−it) = a(it),
where M (s) is certain linear operator from H(s) to H(-s) which is originally
defined for Re(s) >
1
2
and can be analytically continued to the entire plane.
And let π denote the representation of GL2 (A) on L, equipped with the inner
product
1
(a1 , a2 ) =
π
Z
∞
2
(a1 (it), a2 (it))dt =
π
−∞
Z
∞
(a1 (it), a2 (it))dt,
0
given by
π(g)a(it) = πit (g)a(it).
Then one can show ( [6], § 4) that L2cont (GL2 (F )\GL2 (A), ω) is isomorphic
to L. The isomorphism can be explicitly determined. In fact, if F (g) =
X
f (γg), then
γ∈P (F )\GL2 (F )
1
a(it) = [fb(it) + M (−it)fb(−it)].
2
85
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