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 = 12

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 ND) 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 ND.

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 ND 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 general-

ization 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. Assum-

ing 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 0-

dimensional Shimura variety, namely, an (torus) integral of certain form on the

torus given by K×. His results generalize Gross’ earlier results and give an re-

finement 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 ND [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×K/K× with conductor c(χ),

i.e., χ : A×K/K×A×F → 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)#ΣNF/Q(ND)1−2sL(1− s, φ, χ),

where D := c(χ)2c(ω) and Σ is the following set of places of F

Σ = v|v|∞, or v -∞ and ωv(N) = −1 .

The point s = 12

is the central critical point. We consider the special value of

L(s, φ, χ) at s = 12

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 ND 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.

and φa := ρ

a−1 0

0 1

φ;

(2) a test form φ,

an automorphic form on MU such that for each finite place v not dividing

ND, φ 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) φ#(1)L(1

2, φ, χ) =

C√NF/Q(c(ω))

||φ#||2U0(ND) · |(φχ, η)|2,

where C = (2·4k−1[(k−1)!]2

(2k−1)!)g, φ#(1) the first Fourier coefficient of the quasi-

newform φ#, φχ a toric newform on MU suitably normalized, and ||φ#||2U0(ND)

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(ND)) = 1.

In [8], Gross conjectures that the special value may be expressed in terms

of the test form φ in viewing of earlier work of Waldspurger [19] and taking

advantage of φ 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 φ is a form in the space

of π′ = ⊗vπ′v, the representation of G(A) corresponding to π via Jacquet-

Langlands correspondence. In the special case that k = 1, π′v 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 π′v 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) L(1

2, φ, χ) =

C√NF/Q(D)

||φ||2U0(N)|(φ, Pχ)|2,

Here Pχ :=∑

σ∈Gal(Hc/K)

χ−1(σ)[P σc ] with Pc a CM point in G(F )\G(Af )/U.

To explain the idea of the proof, one rewrites the formula (1.1) in the following

way:

φ#(1)

NF/Qc(χ)

||φ||2U0(N)

||φ#||2U0(ND)

|(φ, Pχ)|2|(φχ, η)|2 L(

1

2, φ, χ) =

C√N(D)

||φ||2U0(N)|(φ, Pχ)|2.

We define the archimedean components of the test form φ 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

(1.3)φ#(1)

NF/Qc(χ)

||φ||2U0(N)

||φ#||2U0(ND)

|(φ, Pχ)|2|(φχ, η)|2 = 1.

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

only on χv, Qv(0) = 1, and Qv(t) is regular for |t| ≤ |$v| 12 + |$v|− 12 such that

φ#(1)

NF/Qc(χ)

||φ||2U0(N)

||φ#||2U0(ND)

|(φ, Pχ)|2|(φχ, η)|2 = C(χ)

∏

v|DQv(λv),

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 |(eφ,Pχ)|2|(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 mod-

ular forms and automorphic representations, adelic newform theory, Weil repre-

sentation 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 ND 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 = ⊗vψv, and δv ∈ F×v

denotes the conductor of ψv, i.e., if v is finite, then δ−1v Ov is the maximal

fractional ideal of Fv such that ψv|δ−1v Ov

≡ 1, if v is infinite, ψv(x) = e2πδvx. Set

7

δ =∏

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 = ⊗vdxv

such that dxv is self-dual with respect to ψv, and the multiplicative measure

dx× = ⊗vdx×v has the property that

dx×v =dxv

xv

if v|∞, and

vol(O×v ) = 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 = ⊗vdgv

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 = ⊗vdgv, 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 Poincare upper half plane.

Let m = k− 2t and we choose v ∈ ZI such that wv ≥ 0, wv = 0 for some v and

m + 2w = µt for some µ ∈ Z≥0 For g =

a b

c d

∈ GL2(F∞) and z ∈ HI ,

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 |ku)(x) := jk,w(u∞, z0)−1f(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

∫

F\Af

1 a

0 1

g

da = 0

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 SO2(Fv)-finite, i.e., φ(gr(θ))

form a finite dimensional vector space, where r(θ) =

cos θ sin 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

dtφ(g exp(tX)) |t=0

• φ has moderate growth, i.e., for any compact subset Ω there exist positive

numbers C, t, such that

∣∣∣∣∣∣∣φ

a 0

0 1

g

∣∣∣∣∣∣∣< C(|a|+ |a|−1)t,∀g ∈ Ω.

Let A(GL2(F )\GL2(A), ω) denote the space of automorphic forms with central

character ω.

In addition, if

∫

F\Aφ

1 a

0 1

g

= 0,∀g ∈ GL2(A),

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 spaceA(GL2(F )\GL2(A), ω),

since π(g)φ may not be SO2(Fv)-finite, for g ∈ GL2(F∞). But we do have ac-

tions of both SO2(Fv) and gl2(Fv). For X ∈ gl2(Fv), define

Xφ(g) :=d

dtφ(g exp(tX)) |t=0

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 represen-

tation of GL2(Af ) and a commuting (SO2(Fv),gl2(Fv))-module is called an auto-

morphic 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

U0(N) =

a b

c d

∈ GL2(OF )

∣∣∣∣c ≡ 0 (mod N)

,

12

U1(N) =

a b

c d

∈ U0(N)

∣∣∣∣d ≡ 1 (mod N)

,

For each v|∞, let r(θ) =

cos θ sin θ

− sin θ cos θ

. one easily checks that the Hilbert

modular form f satisfies the following property:

f(gr(θ)) = f(g)ekvθi,

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)ekvθi.

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 ∼= ⊗vπv.

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 quasi-

character 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)

B(µ1, µ2) = IndGL2B (µ1µ2)

=

f : GL2(F ) → C∣∣∣∣f

a b

0 d

g

= µ1(a)µ2(d)

∣∣∣∣d

a

∣∣∣∣12

f(g)

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

where B(Fv) =

a b

0 d

∣∣∣∣a, d ∈ F×v , b ∈ Fv

and Kv = GL2(Ov), one sees

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 irre-

ducible 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):

B(µ1, µ2) =

f : GL2(R) → C∣∣∣∣f

a b

0 d

g

= µ1(a)µ2(d)

∣∣∣∣d

a

∣∣∣∣12

f(g)

.

The Lie algebra gl2(R) acts on B(µ1, µ2) by

X · φ(g) =d

dtφ(g exp(tx))|t=0,

15

where X ∈ gl2(R). The compact group Kv acts via the right translation, thus

produces a (gl2(R), Kv)-module. If µ(x) := µ1µ−12 (x) 6= sgn(x)ε|x|k−1, where

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 applica-

tions.

Theorem 2.2.1 (strong multiplicity one) If π = ⊗vπv and π′ = ⊗vπ′v

are two cuspidal irreducible representations of GL2(A), if πv∼= π′v, for almost

all v, then π ∼= π′

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 ′|N , one

may embed the space of modular forms of level N ′ into the space of modu-

lar 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 char-

acter ω, one defines the following two operators:

φ 7→ π

$v 0

0 1

φ (v -∞),

φ 7→ π

1 i

−i 1

φ (v|∞).

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:

Ak′(N′, ω) → Ak(N, ω),

if N ′|N and k′ ≤ k, i.e., kv − k′v ≥ 0,∀v ∈ ∞. Let Ak(N, ω) be the sub-

space 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

Hv := U0(N)

$v 0

0 1

U0(N), where $v is the idele whose v-th component

is $v and 1 elsewhere. Recall that the Hecke operator acts on Ak(N,ω) by

π 7→ Tv · φ(g) =

∫

Hv

φ(gh)dh.

17

Similar to classical situation, for any ideal a, (a, N) = 1, the Hecke operator Ta

on Ak(N,ω) is the following

Taφ(g) =∑

αβ=ax (mod a)

φ

g

α x

0 β

.

where α and β run through representatives of integral ideles modulo O×F with

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 repre-

sentation 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. LetHv 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−1v , where Rv is the

characteristic function of the double coset

H ′v = U0(N)

$v 0

0 $v

U0(N).

Since Rvφ(g) =∫

H′vφ(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 φ′ of φ onto V is not zero. Since the projection is

GL2(A)-equivariant, thus φ and φ′ have the same eigenvalues under Tv and Rv

at least at those places v such that both πv and π′v are spherical representations.

Hence πv∼= π′v 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 im-

plies that π ∼= π′, 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

U0($c) =

a b

c d

∈ GL2(OF )

∣∣∣∣c ≡ 0 (mod $)

,

U1($c) =

a b

c d

∈ GL2(OF )

∣∣∣∣

a b

c d

≡

∗ ∗0 1

(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

V (($c)) =

f : GL2(F ) → C∣∣∣∣f

g

a b

c d

= ω(d)f(g), ∀

a b

c d

∈ U0($

c)

,

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

vi := π

$−i 0

0 1

vπ, 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

V =

f : GL2(F ) → C∣∣∣∣f

a b

0 d

g

= µ1(a)µ2(d)

∣∣∣ad

∣∣∣12f(g)

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))

=

f : GL2(OF ) → C∣∣∣∣f

a b

0 d

g

a′ b′

c′ d′

= µ1(a)µ2(d)f(g)µ1µ2(d

′)

,

where

a′ b′

c′ d′

∈ U0(($

n)).

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-

tion of GL2(F ) = B(F )∐

B(F ) · w · B(F ), where w =

0 1

−1 0

, implies

that a function f of V (($n)) is determined by f

w

1 x

0 1

. So take

0 6= f ∈ V (($n)), let Φ(x) = f

w

1 x

0 1

. We look at the action of

following elements:

1 OF

0 1

,

a 0

0 d

,

1 0

($n) 1

, on the function

Φ(x). Three conditions are obtained:

(1) Φ(x) = Φ(x + b), ∀b ∈ ($n);

(2) Φ(ax) = µ2(a)Φ(x), ∀a ∈ O×F ;

(3) Φ(x) = µ(cx + 1)−1|cx + 1|−1Φ( xcx+1

),∀c ∈ ($n), where µ := µ1 · µ−12 .

We may assume that either n1 or n2 ≥ 1, since otherwise the claim is auto-

matically 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|−1Φ(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|−1Φ(x), ∀λ ∈ O×n+ord(x).

Assume first that n+ ord(x) ≥ 0, x ∈ Supp(Φ), then µ(λ)−1|λ|−1Φ(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|−1Φ(x), ∀λ ∈ F×, s.t., ord(λ−1 − 1) ≥ n + ord(x).

In particular, the above formula holds for any λ ∈ O×F . Hence one has

Φ(ax) = µ2(a)Φ(x) = µ−11 (x)µ2(x)Φ(x) =⇒ µ1(x) = 1, ∀x ∈ O×

F =⇒ 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|−1Φ(x), λ ∈O×

F =⇒ µ1(λ) = 1, hence n1 = 0. A contradiction.

The other parts are similar. Φ(λx) = µ(λ)−1|λ|−1Φ(x) = µ2(x)Φ(x), ∀λ ∈O×

ord(x)+n. Hence one gets ord(x) + n ≥ n1, i.e., Supp(Φ) = $n1−nOF . 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

U($n) =

γ ∈ GL2(OF

∣∣∣∣γ ≡

1 0

0 1

(mod $n)

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

GL2(OF /($n) =n∐

i=0

B

1 0

$i 1

B.

Therefore, a function ψ is determined by the value at B

1 0

x 1

B. To end the

proof, we just need to know what function ψ on some B

1 0

x 1

B satisfies

the above condition. the condition can be translated into the following one: if

a b

0 d

1 0

$i 1

=

1 0

$i 1

a′ b′

0 d′

(mod $n),

then µ1(a)µ2(d) = µ1(d′)µ2(d

′). For given a, d, a′, d′ there exist x, x′ for which

the equation holds if and only if the following equations have solution: d ≡ d′

(mod $i), a ≡ a′ (mod $i), a′ ≡ d (mod $n−i), d−d′ = a′−a, which is equiva-

lent 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 rep-

resentation 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 = ⊗vπv is an irreducible repre-

sentation 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 ex-

ists 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 ε(σ1⊗σ2) = ±1.

The condition that ε(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 rep-

resentation (π′, V ′) of G(F ). Similarly one considers the representation V ′⊗Cof the group G(F ) × GL1(K), and can show that the space of ∆K×-invariant

linear form `′ : V ′⊗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

ε(σ1 ⊗ σ2) = αK/F · ω(−1).

25

There is a non-zero ∆K×-invariant linear form `′ : V ′ × C→ C if and only if

ε(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(12, 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××O×c(χ). If ` is any nonzero ∆K×-invariant linear

26

form, then `(v) 6= 0,∀v ∈ L.

II. If χ is unramified. When ε(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 ε(σ1 ⊗ σ2) = −α · ω(−1), the condition forces n ≥ 1. Let R′n 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 ε(σ1⊗σ2) = αK/F ·ω(−1), the open compact subgroup R×n ×O×

K fixes a unique

line L, if ` ia a nonzero ∆K×-invariant linear form, then `(v) 6= 0,∀v ∈ L.

If ε(σ1 ⊗ σ2) = −αK/F ω(−1), the group R′n× ×O×

K fixes a unique line L′. If `′

is a nonzero ∆K×-invariant linear form, then `′(v) 6= 0,∀v ∈ L′.

If v is archimedean, then the representation π′ is of finite dimensional and

the torus T (F ) is compact. One shows that the fixed subspace of T (F ) in π′ 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 ND, 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]

(1) If v is non-archimedean place of F , the χ-isotypic component π′v,χ of π′v

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 π′v 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 suf-

ficient 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 ex-

plicit 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 Φ of Φ is defined to be

Φ(x) =

∫

V

Φ(y) < x, y > dy

where Haar measure dy is chosen so that

(Φ)∧(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

0 α−1

,

1 u

0 1

, and

0 1

−1 0

generate SL2(F ). It suffices to describe the actions of

r

1 u

0 1

, r

α 0

0 α

, and r

0 1

−1 0

29

on S(V ). One has

r

1 u

0 1

Φ(x) = τ(uq(x))Φ(x),

r

α 0

0 α−1

Φ(x) = ω(α)|α| 12 Φ(αx),

r

0 1

01 0

Φ(x) = γ · Φ(xσ).

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 repre-

sentation 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

r

a 0

0 1

Φ(x) = Φ(xh).

One can show that this gives rise to a representation denoted by r(n) of G+.

Finally the induced representation

r(g) = IndGL2(F )G+

(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′ of V × an irreducible representation π of GL2(F ).

So suppose that (π′, 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′(h−1)Φ(x), ∀h ∈ V ×, q(h) = 1.

One can show the subspace is invariant under SL2(F ). The resulting represen-

tation is denoted by rπ′ . Now following the same procedure as we did to Weil

representation, i.e., we extend the representation rπ′ to a representation of G+

by requiring

rπ′

a 0

0 1

Φ(x) = |h| 12 π′(x)Φ(xh)

if a = q(h) for some h ∈ V ×. And Moreover

r

a 0

0 a

= ω(a)χπ′(a)I

for a ∈ F× and χπ′ the central character of π′. The induced representation,

still denoted by rπ′ , IndGL2(F )G+

(rπ′) 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 rπ′(g) is a

supercuspidal representation of GL2(F ).

(2) If χ = λ · q for some character λ of F× then rπ′(g) is equivalent to the

principal series π(λ, λω).

Theorem 2.4.2. Assume that V = B.

(1) The representation rπ′(g) decomposes as the direct sum of d = dim(π′)

mutually equivalent irreducible representations π(π′) of GL2(F );

(2) Each π(π′) 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′ → π(π′)

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) = (zz)rzmzn

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|2rtm+nsgn(t)

and

µ1µ−12 (t) = tm+nsgn(t).

If χ = λ · q with m + n = 0, then

rπ′ = π(µ1, µ1 · sgn).

Case 2. V = B = H.

Identifying H with matrices

a b

−b a

, a, b ∈ C then q(h) = det(h). Every

irreducible finite-dimensional representation π′ of H× has the form

π′(h) = q(h)rρn(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

µ1(α) = |α|r+n+ 12

µ2(α) = |α|r− 12 sgn(α)n.

define

π = σ(µ1, µ2).

In particular, in our special case that the automorphic representation π =

⊗vπv 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 π′v 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 = ⊗wχw.

We shall attach to χ an automorphic representation

π(χ) = ⊗vπv.

of GL2(AF ). The local representation πv is constructed as follows.

(1) If v splits in L, write v = w1w2 inOL. 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′ is an irreducible

representation of G(AF ). Write

π′ = ⊗vπ′v.

To π′, we may associate an irreducible representation π of GL2(AF ).

(1) If B in unramified at v. Then Bv∼= M2(Fv). Define

πv = π′v.

(2) If B is ramified at v. Then π′v is finite-dimensional, since π′v is irreducible

and G(Fv) is compact modulo its center. Now apply Weil representation,

and define

πv = π(π′v).

To sum up,

Theorem 2.4.4.(Jacquet-Langlands correspondence) To each irreducible uni-

tary representation π′ = ⊗vπ′v of G(AF ), one associates an irreducible unitary

35

representation π = ⊗vπv of GL2(AF ), where πv = π′v if v is not ramified, and

πv = π(π′v) if v is ramified. Moreover

(1) π is cuspidal for GL2(AF ) if π′ is (greater than one dimensional) cusp-

idal for G(AF ).

(2) The mapping

π′ → π,

restricted to the collection of (greater than one dimensional) cuspidal

representations on G(AF ) is one-to-one correspondence onto the col-

lection of all equivalence classes of cuspidal representations ⊗vπv 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 represen-

tation 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) =

∏v Lv(s, πv), which satisfies functional equation with ε(s, π) =

∏v εv(s, πv, ψv)

. So we start with local version. One defines local L-factor Lv(s, πv) and

εv(s, φ, ψv) for each type of local irreducible representation of GL2(Fv) as fol-

lows:

36

I. v is nonarchimedean place.

(1) if πv is a principal series, πv = π(µ1, µ2), with µi : F× → C×, quasi-

characters, then

Lv(s, πv) = (1− µ1($v)|$v|s)−1(1− µ2($v)|$v|s)−1,

ε(s, πv, ψv) = ε(s, µ1, ψv)ε(s, πv, ψv).

(2)if πv is a special representation, πv = σ(µ), define

L(s, πv) = (1− µ($v)|$v|s)−1

ε(s, πv, ψv) = ε(s, µ1, ψv)ε(s, µ2, ψv)L(1− s, µ−1

1 )

L(s, µ2),

if one writes µ1 = µ · | · | 12 , and µ2 = µ · | · |− 12 .

(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),

ε(s, πv) = ε(s, µ1, ψv)ε(s, µ2, ψv).

(2) if πv is discrete series, πv = σ(p, t), one defines

L(s, πv) = (2π)−s− t+p2 Γ(s +

t + p

2),

ε(s, πv) = ip+1−n1−n2is1+s2 ,

if µj(x) = |x|sjsgn(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 holo-

morphic function on the entire complex plane and satisfies the functional equa-

tion

L(s, π) = ε(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) =∏

v

L(s, πv),

where

Lv(s, πf ) = (1− λv|$v|s + ψ($v)|$v|2s)−1, if v -∞,

Lv(s, πf ) = (2π)−s− k−12 Γ(s +

k − 1

2), if v|∞,

where λv|$v|− k−12 is the eigenvalue of Tv acting on f , ∀v -∞.

Classically, see Shimura [15], one defines the L-series of f as

L(s, f) =∏

v

(1− λv|$v|− k−12 |$v|s + ψ($v)|$v|2s−(k−1))−1

=∏

v

(1− λv|$v|s− k−12 + ψ($v)|$v|2(s− (k−1)

2))−1

=∏

v

(1− λv|$v|s′ + ψ($v)|$v|2s′)−1,

38

where s′ = s− k−12

, hence one obtains

L(s, f) = L(s− k − 1

2, πf ).

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 gen-

eral 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) =∏

v

Lv(s, φ, χ),

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,

∏

w|vLw(s, χw) = (1− β1|$v|s)−1(1− β2|$v|s)−1,

then

Lv(s, φ, χ) =∏i,j

(1− αiβj|$|s)−1.

Here for a place w of K, the local factor L(s, χw) is defined as follows:

L(s, χw) =

(1− χw($w)|$w|s)−1, if w - c · ∞;

G2(s), if w|∞;

1, if w|c(χ).

39

Here G2(s) := 2(2π)−sΓ(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) =∏i,j

G1(s + σi + τj)

=

G2(s + kv−12

)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 spe-

cial values of L(1/2, φ, χ) and L′(1/2, φ, χ) which refines the work of Wald-

spurger. 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/FGm, 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 vec-

tor 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→ (b1b2)+

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

GU(B, φ) =

g ∈ GL(B) | φ(gv, gw) = λ(g)φ(v, w), for some λ(g) ∈ F×

,

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−1xb, 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−1ve. 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

hve0u = φ(hv, hu) = φ(v, u) = (vu)+ = v−u,

43

thus (hv)− = v−e0−1 = e−1

0 v−, and hu = ue0 = ue0−1. Then Ne0 = 1,

and therefore by Hilbert 90, there exists an element e ∈ K such that

e0 = ee−1.

Thus hv = v+ + e−1ev− = e−1ve.

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)(e′v + u′) = (ee′ − φ(u, u′))v + (eu′ + e′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 ND

We shall explain briefly the proof of the formula of level ND 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 θχE, where

θχ is the theta series associated to χ and E certain Eisenstein series. Since θχE

is of level ND, taking trace of θχE from ND to N , they obtain a form of level

N :

Φs(g) = TrND/N(d12−s

F θχ(g)E(g)) =∑

γ∈U0(N)/Uo(ND)

(d12−s

F θχ(g)E(g)),

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 L′(s, φ, χ) is

the inner product of φ with Φ′s. The form Φ′

s is not holomorphic. So one needs

to get the holomorphic projection Ψs of Φ′s. One has

L′(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(1

2, 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

Φ(a) = |a| < x, Tax >,

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

L′(1

2, χ, χ) =

2g+1

√N(D)

||φ||2||xχ||2.

Generalizing the formula to Hilbert modular forms of weight (2, · · · , 2), one

encounters the difficulty that the trace TrND/NθχE is very hard to compute

when χ is ramified as well as other geometric technical difficulties, instead

Shou-Wu Zhang works directly on level ND. So naturally one would expect

to have some other form to replace the role of φ in the level ND. Zhang uses

quasi-newform φ#, see the definition below, to replace φ. By developing a no-

tion of geometric pairing, he computes the geometric pairing (Taη, η) of φ and

a special CM-cycle η. The computation shows that there is a close relationship

between local Fourier coefficient of Φ 12

and local geometric pairing (Taη, η),

which he calls the local Gross-Zagier formula. The special value formula of

level ND follows from this local Gross-Zagier formula.

3.1. Kernel function and quasi-newform. We have explained in last chap-

ter that the L-function L(s, φ, χ) can be represented as inner product of φ with

θχE, i.e.,

L(s, φ, χ) = |δ|s− 12

∫

Z(A)GL2(F )GL2(A)

φ(g)θχ(g)E(s, g)dg

= |δ|s− 12 (φ, θχE).

46

In order to get a more symmetric form, Zhang applies Atkin-Lehner operator

to θχE. 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

v-th component is 1 for v /∈ T and

0 1

−tv 0

, where tv has the same order

as c(ε) such that εv(tv) = 1, for v ∈ T. One can show that

L(s, φ, χ) =γT (s)

vol(U0(ND))

∫

Z(A)GL2(F )\GL2(A)

φ(g)θ(gh−1T )E(s, gh−1

T )dg

with γ(s) certain exponential function of s. Thus if we define

Θ(s, g) = 2−|S||δ|s− 12

∑T⊂S

γT (s)θχ(gh−1T )E(s, gh−1

T )

and call it the kernel function. Then one has

L(s, φ, χ) = (φ, Θ)U0(ND).

The kernel function has functional equation

Θ(s, g) = ε(s, χ)Θ(1− s, g)

from which the functional equation of L(s, φ, χ)

L(1− s, φ, χ) = (−1)#ΣNF/Q(ND)1−2sL(s, φ, χ)

follows. Note that the kernel function Θ(s, g) is of level ND. What Gross

and Zagier do is that they take trace of Θ(s, g) from ND 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 ND. In this

case, one needs to find an analogue of φ for the level ND. 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., φ#

s is the unique

nonzero form in the space π(φ) of level ND satisfying the following identities

(φ#s , φa) = ν∗(a)(φ#

s , φ#s ), a|D,

where ν∗(a) =∏v∈S

|a|s−12

v + |a|12−s

v

2·

ν(a), if a|c(ε);0, otherwise.

and φa := ρ

a−1 0

0 1

φ.

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) +∑

α∈F×W

s,

α 0

0 1

g

.

Since Θ(s, g) is a linear combination of the form

Θ(s, g) =∑

i

θi(g)Ei(g)

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) =∑

ξ∈F

Wθi(ξ, g), Ei(g) =

∑

ξ∈F

WEi(ξ, g),

48

be Fourier expansions of θi and Ei. Then

C(s, g) =∑

ξ∈F

C(s, ξ, g),

W (s, g) =∑

ξ∈F

W (s, ξ, g),

where

C(s, ξ, g) =∑

i

Wθi(−ξ, g)WEi

(ξ, g),

and

W (s, ξ, g) =∑

i

Wθi(1− ξ, g)WEi

(ξ, g).

Furthermore, one can decompose W (s, ξ, g) into

W (s, ξ, g) = ⊗vWv(s, ξv, gv).

The local Fourier coefficients Wv(12, 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 ′ ∈ CU ,

(P, P ′) =

0, if π(P ) 6= π(P ′);

#(R×P ), if π(P ) = π(P ′).

Here π : CU → G(F )\G(Af )/U, and RP is the oriented order of B correspond-

ing to the point P ∈ CU , i.e., RP := B ∩ PRP−1. A vast generalization is

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 spe-

cial value formula of both L(12, φ, χ) and L′(1

2, 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 def-

inition, 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) =∑

γ∈G(F )

m(x−1γy)

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) = ⊕χS(C, χ)

where χ runs through characters of T (F )\T (Af ). For any given CM-cycles

α, β ∈ S(C), Zhang defines

< α, β > =

∫

C2

α(x)k(x, y)β(y)dxdy

= limU→1

∫

C2

α(x)kU(x, y)β(y)dxdy.

where U runs over open subgroups of G(Af ) and

kU(x, y) = vol(U)−2

∫

C2

k(xu, yv)dudv.

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 ′ ∈ C, let α and α′ be the characteristic function of P and

P ′ respectively. Thus α, α′ ∈ S(C), then one can easily see that

< α, α′ >=

0, if π(P ) 6= π(P ′);

#(R×P ), if π(P ) = π(P ′).

By identifying P (P ′) with α(α′), one thus recovers Gross’ intersection pairing.

It is not hard to see that the geometric pairing

< α, β >=∑

γ∈T (F )\G(F )/T (F )

m(γ) < α, β >γ,

51

where < α, β >γ=∫

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 + bε −→ N(bε)

N(a + bε)

One thus defines

< α, β >ξ=

< α, β >γ, if ξ = ξ(γ);

0, else.

Hence

< α, β >=∑

ξ∈F

m(γ) < α, β >ξ .

If both α =∏

v αv and β =∏

v βv are decomposable, then < α, β >ξ can be

further decomposed as

< α, β >ξ=∏

v

< αv, βv >ξ

with < αv, βv >ξ=∫

G(Fv)αv(γy)βv(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 + λvc(χv)OK,v,

where λv ∈ Bv with the properties that

(1) λvx = xλv,∀x ∈ Kv,

(2) ordv(detλv) = ordv(N).

We denote by ∆ the subgroup of G(Af ) such that

∆ =∏

v-c(χ)

A×v F×

v /F×v ·

∏

v|c(χ)

A×v K×

v /F×v .

Notice that one has a natural isomorphism

Av/λvc(χv)Av∼= OK,v/λvc(χv)OK,v.

Thus one may extend χ to a character, still denoted by χ, of ∆. Now the special

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 idele prime to ND, the Hecke operator is defined to

be

Taη =∏

v

Tavηv

Tavηv(x) =

∫

H(Gv)

ηv(xg)dg,

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

Proposition 3.2.3 Let g =

avδ−1v 0

0 1

, then

Wv(1

2, ξ, g) = |c(ωv)| 12 ε(ωv, ψv)χv(u)|(1− ξ)ξ|

12v |av|vvol(∆v)

−1 < Tavηv, ηv >ξ,

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 idele prime to ND,

|c(ω)| 12 |a| < Taη, η >∆ = (c1m(0) + c2m(1))|a| 12 Wf (g)

+ ig∑

ξ∈F−0,1|ξ(1− ξ)

12∞W f (

1

2, ξ, g)m(ξ).

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

∫

SO3(R)

P 2k−1dg =

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

Wv

a 0

0 1

=

2ake−2πa, if a > 0;

0, else.

Here we view the multiplicity function parameterized by the continuous param-

eter 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 ·∑

i

Wi(g∞)φi(x)φi(y) + C

∫

M

WM(g∞)EM(x)EM(y)dM.

Where the sum runs over all cuspidal eigenforms φi of Hecke operators and

Laplace operators, and Wi is the Whittaker function of φnewi , φnew

i being the

newform of weight (2k, · · · , 2k) in the representation π of PGL2(A) correspond-

ing to the representation π′i of G(A) generated by φi via Jacquet-Langlands

correspondence. In particular, let α = β = η, then the form

Ψ = C|c(ω)| 12∑

i

φnewi |(φ, η)|2 + C|c(ω)| 12

∫

M

EnewM |(EM, η)|2dM

55

has the same a-th Fourier coefficients with the kernel function Φ for a prime to

ND. 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.,

φ#(1)L(1

2, φ, χ) = C|c(ω)| 12 ||φ#||2|(φχ, η)|2.

56

4. Special value formula of level N

In this section, we shall deduce the final formula, i.e., the special value for-

mula of level N from the formula of level ND in last section. We shall express

the special value of L(s, φ, χ) at s = 12

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

L(1

2, φ, χ) =

C√N(D)

||φ||2 · |(φ, Pχ)|2.

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 +NOB. The group

Uv = R×v /O×

v

defines an open compact subgroup of G(Fv). Let U =∏

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 0-

dimensional. We first define a CM point Pc ∈ G(F )\G(Af )/U . Let ic ∈ G(Af )

such that

UT := icUi−1c ∩ T (Af )

× = Oc(χ)×/OF

×.

57

Set Pc = [ic] ∈ G(F )\G(Af )/U. Finally

Pχ =∑

t∈T (F )\T (Af )/UT

χ(t)[tic].

As we see from last section that the formula of level ND 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 ND. One rewrites the formula of

level ND in the following way:

φ#(1)

N(c(χ))

||φ||2||φ#||2

|(φ, Pχ)|2|(φχ, η)|2 L(

1

2, χ, φ) =

C√N(D)

||φ||2|(φ, Pχ)|2.

We shall prove that

φ#(1)

N(c(χ))

||φ||2||φ#||2

|(φ, Pχ)|2|(φχ, η)|2 L(

1

2, χ, φ) = C(χ)

∏

v|DQv(λv),

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

and is regular for t ≤ |$v| 12 + |$v|−12 . 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

φ#(1)

N(c(χ))

||φ||2||φ#||2

|(φ, Pχ)|2|(φχ, η)|2 L(

1

2, χ, φ)

can be viewed as a formula associated to each form in the space L20(GL2(F )\GL2(A)).

The above shows exactly that

φ#(1)

N(c(χ))

||φ||2||φ#||2

|(φ, Pχ)|2|(φχ, η)|2 L(

1

2, χ, φ)

58

is independent of the choices of φ. Thus one can use Eisenstein series to deter-

mine C(χ)∏

v|D Qv(λv). To that end, a similar special value formula of Rankin-

Selberg L-function associated to an Eisenstein series and χ will be deduced to

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 ND 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 com-

pute the universal constants, thus prove the final formula.

4.1. Universal constants. In this subsection, we shall prove that both for-

mulae of level N and ND 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),

depending only on χv, Qv(0) = 1, and being regular for t ≤ |$v| 12 + |$v|− 12 ,

such that

φ#(1)

N(c(χ))

||φ||2||φ#||2

|(φ, Pχ)|2|(φχ, η)|2 = C(χ)

∏

v|DQv(λv),

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

Nv := $ordv(N)v and the quasi-newform W#

v with respect to χv is the form of

levelNvDv = $ordv(N)+ordv(D)v satisfying

(W#v ,Wv,i) = νi(W#

v ,W#v ), i = 0, 1, · · · , ordv(D),

where

ν =

0, if v|c(ω);

χv($K,v), if v - c(ω).

In particular, if v - D, so W#v is of level $

ordv(N)v and the relation above implies

that W#v = Wv. The uniqueness of the global quasi-newform φ# implies that

φ# = ⊗vW#v . 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

||φ||2||φ#||2 =

∏v(Wv,Wv)∏

v(W#v ,W#

v )=

∏v(Wv,Wv)∏

v(W#v ,W#

v ).

We now show that there exists, for each v|D, a rational function Q1,v(t) ∈ C(t),

depending only on χv, Q1,v(0) = 1, and Q1,v(t) is regular for t ≤ |$v| 12 + |$v|− 12

such that

(Wv,Wv)

(W#v ,W#

v )= C1,v(χv)Q1,v(λv).

60

Write W#v =

∑ordv(D)i=0 αv,iWv,i. The definition of quasi-newform can be trans-

lated into the system of following equations

ordv(D)∑i=0

αv,i(Wv,i−j,Wv)

(Wv,Wv)= νj

ordv(D)∑i=0

αv,i(Wv,i,Wv)

(Wv,Wv), j = 1, · · · , ordv(D)

(Wv#,Wv)

(Wv,Wv)=

(W#v ,W#

v )

(Wv,Wv)=

ordv(D)∑i=0

νiαv,i(Wv#,Wv)

(Wv,Wv)⇔

ordv(D)∑i=0

νiαv,i = 1.

Hence each αv,i is a rational function in(Wv,j ,Wv)

(Wv,Wv)for j = 0, · · · , ordv(D). It

remains to show that(Wv,j ,Wv)

(Wv ,Wv)is a polynomial in λv. Let Uv = GL2(Ov). Define

Hv,j be vol(Hv,j)−1 times the characteristic function of Uv

$−jv 0

0 1

Uv and

T($−jv to be the Hecke operator corresponding to Hv,j. One obtains that

(Wv,j,Wv)

(Wv,Wv)= (Wv,Wv)

−1vol(Uv)−1

∫

Uv

(π(u)Wv,j, π(u)Wv)du

= (Wv,Wv)−1vol(Hv,j)

−1

∫

Hv,j

(π(u)Wv,Wv)du

Since T ($−jv ) is generated by usual Hecke operators: Tv, Rv and R−1

v which

correspond to Uv

$v 0

0 1

Uv, Uv

$v 0

0 $v

and Uv

$−1v 0

0 $−1v

re-

spectively, See Bump [3]. The action of Rv on Wv is trivial and Tv · Wv =

λv|$v| 1−k2 Wv. Hence T($−j

v ) ·Wv = Q′1,v(λv)Wv, where Q′

1,v(t) is a polynomial

independent of πv. In fact, one has explicit relation

T($jv) = Tv($

j+1v ) + |$v|−1RvT($j−1

v ).

61

Thus Q′1,v(0) is a constant depending only on χv. Finally we may simply take

Q1,v(t) = Q′1,v−1(0)Q′

1,v(t).

(2) φ#(1). Applying the definition of φ#(1) directly, we have

φ#(1) =

Wφ#

y∞δ−1 0

0 1

W∞

y∞ 0

0 1

=

∏v W#

v

y∞δ−1v 0

0 1

W∞

y∞ 0

0 1

=

∏v-D W#

v

y∞δ−1v 0

0 1

·∏v|D W#

v

y∞δ−1v 0

0 1

W∞

y∞ 0

0 1

=

∏v-D Wv

y∞δ−1v 0

0 1

∏

v|D∑ordv(D)

i=0 αv,iWv,i

y∞δ−1v 0

0 1

W∞

y∞ 0

0 1

.

62

Where the last equality is obtained, since for any a|D, φa := π

a−1 0

0 1

φ

is an old form, therefore φa(1) = 0.

(3) |(eφ,Pχ)|2|(φχ,η)|2 . To show that the above quantity has the desired property, one

needs to analyze both φ and φχ. First notice that the point Pχ corresponds to

the following CM-cycle ξ : T (Af )icU, ticu 7→ χ(t). Thus one sees that (φ, Pχ) =

(φ, ξ), where the right hand side of the equality is regarded as the pairing

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 )icU → C×

ticu 7→ χ(t), ∀t ∈ T (Af ), u ∈ U.

We first show that (φ, Pc)χ = (φ, ξ). We have

(φ, ξ) =vol(U)−1

∫

T (F )\G(Af )

ξ(x)φ(x)dx

=vol(U)−1

∫

T (F )\T (Af )icU

ξ(x)φ(x)dx

=

∫

T (F )\T (Af )/UT

χ−1(t)φ(tPc)

=(φ, Pc)χ.

63

Now we compare the CM-cycles ξ and η. Let h be the characteristic function

on Ui−1c . 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 Uvi−1c,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,vUvi−1c,v ∩ T (Fv) = O×

cv/O×

v .

Since v - c(χ), thus Ocv = Ov. Therefore we may take ic,v = I. Hence if

ρ(hv)ηv(g) =1

vol(Uv)

∫

G(Fv)

ηv(gx)hvdx

=1

vol(Uv)

∫

Uv

ηv(guic,v)du

=1

vol(Uv)

∫

Uv

ηv(gu)du 6= 0,

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

ρ(hv)ηv(g) =1

vol(Uv)

∫

Uv

ηv(gx)dx

=1

vol(Uv)

∫

Uv

ηv(tux)dx

=1

vol(Uv)χv(t)

∫

Uv

χv(x)dx

= χv(t),

64

i.e., ρ(hv)ηv = ξv.

If B splits at v. Let U ′v = Ocv+cvλvOK,v. If ρ(hv)ηv(g) =

∫Uv

ηv(gxic,v)dx 6= 0.

Then one has gxic,v ∈ T (Fv)U′v, i.e., g ∈ T (Fv)∆vic,v. Using the lemma in

ZhangciteZh2, we see that ic,vUv = U ′vic,v. Therefore g ∈ T (Fv)i

−1c,vUv. Now for

any g = ti−1c,vu ∈ T (Fv)i

−1c,vUv,

ρ(hv)ηv(g) =1

vol(U ′v)

∫

Uv

ηv(ti−1c,vuxic,v)dx

=1

vol(U ′v)

∫

Uv

ηv(ti−1c,vxic,v)dx

=

∫

U ′v

ηv(tu)dx = χv(t),

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 spec-

tral decomposition of Ψ(α, β)(g∞) with g∞ a continuous parameter varying in

GL2(F∞). We have

Ψ(α, β) =C|c(ω)| 12∑

i

φnewi (φi, α)∆(φi, β)∆

+ C|c(ω)| 12∫

MEnewM (EM, α)(EM, β)dM.

In particular, let α = β = ξ, we see that

Ψ(ξ, ξ) =C|c(ω)| 12∑

i

φnewi |(φi, ξ)∆|2

+ C|c(ω)| 12∫

MEnewM |(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 Jacquet-

Langlands correspondence, this φi must be test form φ. Now write H(ξ, ξ) =

∑φnew

i |(φi, ξ)∆|2 + continuous spectrum. Using the relation

|a| < Taα, β > (g∞) = CWH(α,β)

g∞

aδ−1 0

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 )) = ⊕τS(T (F )\G(Af ), τ), one has

h∨ ∗ h =∑

τ

hτ ,

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)

=∑

i

φnewi (φi, ρ(hχ)η)(φi, η) + continuous spectrum

=∑

i

φnewi (ρ(hχ)φi, η)(φi, η) + continuous spectrum.

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 φnewi = φ. Now we claim that

ρ(hχ)φi =∏

v|DQ2,v(λv)φi,

where Q2,v is certain rational function.

To that end, notice that for v - D = c(χ)2c(ε), 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 =∏

v|DQ2,v(λv)φi,

here we use the explicit construction that πv = π′v if B is ramified at v. There-

fore finally we obtain that

|(φ, Pχ)|2|(φχ, η)|2 = C

∏

v|DQv(λv).

¤

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 uni-

versal 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 quasi-

newform E#. The idea follows exactly as before. For a fixed character µ :

A×F /F× → C×, we shall use the two forms Φ and Ψ constructed in [Zh1]:

(1) the form Φ is the holomorphic projection of Θ 12;

(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×F /F× such that µ2 is not of form | · |t, for some

0 6= t ∈ R. Let Eis(µ) be the space of forms E in L2cont(GL2(F )\GL2(A)) so

that

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

(E1, E2) =

∫ ∞

−∞(E1,t, E2,t)tdt,

where (, )t is some Hermitian form on the space π(µ| · |it, µ−1| · |−it).

68

For any form E(g) =∫∞−∞ Et(g)dt ∈ Eis(µ), and a continuous function ϕ on

R, one obtains another form Eϕ twisted by ϕ by

Eϕ(g) =

∫ ∞

−∞ϕ(t)Et(g)dt.

In particular, let Enewt be the newform of the representation π(µ| · |it, µ−1| · |−it).

One has

Enewϕ (g) =

∫ ∞

−∞ϕ(t)Enew

t (g)dt.

We now compute the inner product (Enewϕ , 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 square-

integrable since its constant term has exponential decay. Hence we have

(Enewϕ , Φs) = (Enew

ϕ , Θs) =

∫

Z(A)GL2(F )\GL2(A)

Enewϕ Θs(g)dg

=

∫

Z(A)GL2(F )\GL2(A)

∫ ∞

−∞ϕ(t)Enew

ϕ (it, g)Θs(g)dtdg

=

∫ ∞

−∞

∫

Z(A)GL2(F )\GL2(A)

ϕ(t)Enewϕ (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) =∏

w|v(1− µ($v)χ($w)|$v|s+it)−1 ·

∏

w|v(1− µ−1($v)χ($w)|$v|s−it)−1

= Lv(s + it, µK ⊗ χ) · Lv(s− it, µ−1K ⊗ χ),

here, µK = µ · NK/F . Therefore one obtains

(Enewϕ , 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

L(1

2− it, µK ⊗ χ) =

∏w

Lw(1

2− it, µK ⊗ χ)

=∏w

(1− µ−1K ($w)χ($w)|$w| 12−it)−1

=∏w

(1− µK($σw)χ($σ

w)|$σw|

12+it)−1

= L(1

2+ it, µK ⊗ χ),

Here σ is the nontrivial automorphism in Gal(K/F ). Hence

(Enewϕ , Φ) =

∫ ∞

−∞ϕ(t)|L(

1

2+ it, µK ⊗ χ)|2dt.

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 E#t of πt with respect to χ is the form satisfying

(E#t , Enew

t,a )t = ν(a)(E#t , E#

t )t, ∀a|D,

where Enewt,a = ρ

a−1 0

0 1

Enew

t . We have

70

Lemma 4.2.1.1 The projection pr(Φ) of Φ on the space Eis(µ) is E#ϕ with

ϕ(t) =|L(1

2+ it, µK ⊗ χ)|2||E#

t ||2t.

Proof. We first compute (E, Φ),∀E =∫∞−∞ Etdt ∈ Eis(µ). One has

(E, Φ) = (E, Θ 12) =

∫

Z(A)GL2(F )\GL2(A)

∫ ∞

−∞Et(g)Θ 1

2(g)dtdg

=

∫ ∞

−∞Et(g)Θ 1

2(g)dgdt.

Thus the linear map Et 7→∫

Z(A)GL2(F )\GL2(A)Et(g)Θ 1

2(g)dg is well defined, and

by Riesz representation theorem, there exists EΘt ∈ πt such that

|L(1

2+ it, µK ⊗ χ)|2 =

∫

Z(A)GL2(F )\GL2(A)

Et(g)Θ 12(g)dg = (Et, E

Θt ).

One easily sees that

EΘt =

|L(12

+ it, µK ⊗ χ)|2||E#

t ||2tE#

t .

Thus

(E, Φ) =

∫ ∞

−∞ϕ(t)(Et, E

#t )tdt = (E, E#

ϕ ),

i.e., E#t = 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

pr(Ψ) = 22g|c(ω)| 12 Enewψ

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

ϕ(t)E#t − 22g|c(ω)| 12 ψ(t)Enew

t

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) E#t (1)|L(

1

2+ it, µK ⊗ χ)|2 = 22g|c(ε)| 12 ||E#

t ||2t |(Enewt,χ , η)∆|2.

Rewrite the formula (4.1) in the following way:

E#t (1)

N(c(χ))· ||E

newt ||2t

||E#t ||2t

· |(Enewt , Pχ)|2

|(Enewt,χ , η)∆|2 =

22g√N(D)

||Enewt ||2t |(Enew

t , Pχ)|2|L(1

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

λv(t) = µv($v)|$v|it + µ−1v ($v)|$v|−it,

and E∗t := ||Enew

t ||tEnewt . Then

C(χ)∏

v|DQv(λv(t)) =

22g

√N(D)

∣∣∣∣(E∗

t , Pχ)

L(12

+ it, µK ⊗ χ)

∣∣∣∣2

.

72

4.2.2. Determination of universal constants. We shall finally compute the uni-

versal constants. We need to compute the pairing |(E∗t , Pχ)|2. By the univer-

sality 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 E∗t .

Lemma 4.2.2.1 Let j = jf ⊗∏

v|∞ jv ∈ G(A) such that jf ∈ G(Af ) with

j−1f U(N,K)jf = GL2(OF ), and for v|∞, jv ∈ G(Fv) with

jv(SO2(Fv)/ ±I)j−1v = T (Fv).

Then we have

E∗t = ρ(j)Enew

t .

Proof. Recall that the test form Enewt is a form fixed by U and normalized

such that ||Enewt ||2 = 1. We show first that ρ(j)Enew

t is fixed by U(N, K). For

any h ∈ U(N,K),

ρ(h)(ρ(j)Enewt )(g) = Enew

t (ghj)

= Enewt (gjj−1hj)

= ρ(j)Enewt (g).

73

Since the dimension of the forms fixed by U(N, K) is one, so

ρ(j)Enewt

||ρ(j)Enewt ||2 = ±Enew

t .

We may assume that

ρ(j)Enewt

||ρ(j)Enewt ||2 = Enew

t ,

i.e.,

ρ(j)Enewt = ||ρ(j)Enew

t ||2Enewt = ||Enew

t ||2Enewt ,

since the representation π is hermitian and we only need to compute |(E∗t , 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

(E∗t , Pχ) = 2−gµ(δ−1

√4λ

D)|4λ/D| 14 (1+2it)2itL(

1

2, χ · µK),

where λ is a trace-free element of K.

Proof. Recall that Pχ =∑

σ∈Gal(HK/K)

χ−1(σ)P σc . We have

(E∗t , Pχ) =

∫

T (F )\T (A)

χ−1(s)E∗t (sPc)ds

=

∫

T (F )\T (A)

χ−1(s)Enewt (sPcj)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

Enewt (g) =

∑

γ∈P (F )\GL2(F )

µ−1(δ)fΩ,t(γsPcj),

where

fΩ,t(g) = µ1(detg)|detg|t+ 12

∫

A×Ω[(0, x)g]µ1µ

−12 (x)|x|1+2td×t

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 O2v, for v

archimedean, Ωv = e−π(x2+y2). Therefore, we have

(E∗t , Pχ) = µ(det(Pcj))|det(Pcj)| 12+it

∫

T (A)

∫

A×χ−1(s)µ(dets)|dets| 12+it

· Ω((0, x)sPcj)µ2(x)|x|1+2itd×xds

= µ(det(Pcj))|det(Pcj)| 12+itZ(1

2, µ · | · |it, Ω).

So it suffices to compute the local zeta function Zv(s, µ · | · |it, Ω) as well as

det(Pcj). We start with the local zeta function.

Since the function Ωv is supported on Ov ⊕ Ov for v non-archimedean, one

easily sees that

Z(1

2, µ · | · |it, Ω) =

∫

Ocv−0

χ−1(x)µKv(x)|x| 12+itd×x.

75

I. If v is non-archimedean place.

(1) If v|c(χ), then Ocv = Ov. Thus

Z(1

2, χ−1µKv | · |it, Ω) =

∫

Ov−0

χ−1(x)µKv(x)|x| 12+itd×x

= L(1

2, χ−1µKv | · |it)

(2) If v - c(χ), write Ocv − 0 =∞⋃

n=0

Ocv ,n, where Ocv,n is the set of elements of

order n in Ocv .

i) If v is inert in K, then Kv is a field. Let’s write OKv = Ov +Ovλ. We have

∫

Ocv−0

χ−1µK(x)|x| 12+itd×x =∞∑

n=0

∫

Ocv,n

χ−1µK(x)|x| 12+itd×x.

Case 1. n = 0, then OCv,0 = O×cv

. We obtain

∫

Ocv,0

χ−1(x)µK(x)|x| 12+it = vol(O×v ) = 1.

Case 2. 0 < n < ordv(c(χ)), It’s easy to see that

Ocv ,n = $nvO×

v + cvOKv = $nv (O×

v + cv$−nv OKv).

Hence ∫

Ocv,n

χ−1(x)µKv(x)|x| 12+itd×x = 0,

since χ restricted to O×v + cv$

−nOK,v is not identically one.

Case 3. n ≥ ordv(cv). One sees easily that Ocv ,n = $nvO×

Kv. Now applying

χ∣∣O×Kv

≡ 1 and µ being unramified, we have

∫

Ocv,n

χ−1(x)µKv(x)|x| 12+itd×x = 0.

Hence Z(12, χ−1µKv | · |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

∫

Ocv−0

χ−1(x)µKv(x)|x| 12+itd×x = vol(O×cv

) = 1.

Case 2. 0 < n < ordv(c(χ)), then Ocv ,n = $nvO×

v + cvOKv = $nv (O×

v +

cv$−nv OKv). Hence

∫

Ocv−0

χ−1(x)µKv(x)|x| 12+itd×x = 0,

since χ∣∣O×v +cv$−n

v OKv≡/ 1.

Case 3. n ≥ ordv(c(χ)). The fact that Ocv ,n = $nvO×

Kv, and χ

∣∣O×Kv

≡/1 implies

that

∫

Ocv,n

χ−1(x)µKv(x)|x| 12+itd×x = 0.

Again we obtain that

Z(1

2, χ−1µKv | · |it, Ωv) = 1.

II. If v is archimedean place.

Recall that jv ∈ GL2(R) such that jv(SO2(R)/±I)j−1v = T (R). Writing K =

F + F√

λ, one may simply take jv =

|λ|

12v 0

0 1

. Using the polar coordinate

(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 drdθπr

.

77

Then the function Ωv((0, x)sjv) is of the form e−πr2. Therefore we get

Z(1

2, χ−1µKv | · |it) =

∫ 2π

0

∫ ∞

0

µ(r)|r|1+2ite−πr2 drdθ

πr

= 2

∫ ∞

0

µ(r)|r|1+2ite−πr2 dr

r

=1

2µ(π)−1π−

12(1+2it)Γ(

1 + 2it

2).

Finally We compute the det(jfPc) as well as det(jv) for v archimedean. By

the definition of jf and Pc, one sees easily that

Ov +Ov

√λ = OcvjfPc.

Hence taking discriminant both sides, one obtains

4λ = D · det(jfPc).

For det(jv), one has det(jv) = |λ|12v by the specific form we have chosen.

Summing up, We have the following formula:

(E∗t , Pχ) = µ

(δ−1

√(4λ

D)

)∣∣4λD| 14 (1+2it)2−g · 2itL(

1

2, χ−1 · µK · | · |it).

¤

We now compute the universal constants using the formula just obtained. In

particular we choose µ(x) = |x|is. Observe that

|L(1

2+ it + is, χ−1)|2 = |L(

1

2+ it + is, χ)|2,

since L(12

+ it + is, χ−1) = L(12− it − is, χ−1) and |L(1

2− it − is, χ−1)| =

|L(12

+ 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)∏

v|DQv(λv(t)) = 1.

Observe that each Qv(λ(t)) is a rational function in pnt. One can show that

∏

v|Dp

Qv(λ(t)) = const,

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 χ′ of finite order

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 χ′, one obtains that

∏

v|Dp

Qv(λ(t)) = 1.

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

P =

∗ ∗0 ∗

, A =

∗ 0

0 ∗

, N =

1 ∗0 1

.

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(A)GL2(F )\GL2(A)

|ϕ(g)|2dg < ∞.

If, in addition,∫

N(F )\N(A)

ϕ(ng) ≡ 0,∀g ∈ GL2(A),

then we say that ϕ is a cuspidal and the subspace of cuspidal functions is de-

noted 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 trans-

lation.

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

(ϕ1, ϕ2) =

∫

GL2(F )Z(A)\GL2(A)

ϕ1(g)ϕ2(g)dg.

The orthocomplement can be described in terms of P -series, which we now de-

fine.

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) =∑

γ∈P (F )\GL2(F )

f(γg)

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

(ϕ, F ) =

∫

Z(A)GL2(F )\GL2(A)

ϕ(g)F (g)dg

=

∫

Z(A)GL2(F )\GL2(A)

ϕ(g)∑

γ∈P (F )\GL2(F )

f(γg)dg

=

∫

Z(A)P (F )\GL2(A)

ϕ(g)f(g)dg

=

∫

N(A)Z(A)P (F )\GL2(A)

dg

∫

N(F )\N(A)

ϕ(ng)f(g)dn.

Notice the∫

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

(ϕ, F ) =

∫

N(A)Z(A)P (F )\GL2(A)

dg

∫

N(F )\N(A)

ϕ(ng)f(g)dn = 0

for any P -series with compact support mod N(A)Z(A)P (F ), then it is easy to

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

Eisenstein series, let’s introduce a Hilbert space H(s). Let F+∞ be the set of ide-

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×F /F× ∼= A×F,1/F× × F+

∞.

82

Definition 5.3 H(s)is the space of functions φ ∈ C∞(GL2(A)) such that

φ

αau x

0 βav

= ω(a)

∣∣∣uv

∣∣∣s+ 1

2φ(g),

and∫

K

∫

F×\A×F,1

|φ|2

a 0

0 1

K

dadk < ∞,

where α, β ∈ F×, a ∈ A×, u, v ∈ F+∞.

The group GL2(A) operates on H(s) via the right translation and the result-

ing 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

φ

αau x

0 βav

g

= ω(a)

∣∣∣uv

∣∣∣s+ 1

2φ(g, s).

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) =∑

γ∈P (F )\GL2(F )

φ(s, γg)

are called Eisenstein series.

This series converges only for Re(s) > 12. It can be shown (citeG-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

f(g, s) =

∫

F+∞f

t 0

0 1

g

|t|−s− 1

2 d×t.

By our assumption that f is compactly supported mod N(A)Z(A)P (F ). then

f defines a section of H(s). Fourier inversion implies that

f(g) =1

2πi

∫ x−i∞

x−i∞f(g, s)ds,

for any x. Hence the P -series

F (g) =∑

γ∈P (F )\GL2(F )

f(γg)

=1

2πi

∑

γ∈P (F )\GL2(F )

∫ +i∞

−i∞f(g, s)ds

=1

2πi

∫ +i∞

−i∞

∑

γ∈P (F )\GL2(F )

f(g, s)ds

=1

2πi

∫ +i∞

−i∞E(f(s), g)ds.

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) > 12

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

(a1, a2) =1

π

∫ ∞

−∞(a1(it), a2(it))dt =

2

π

∫ ∞

0

(a1(it), a2(it))dt,

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) =∑

γ∈P (F )\GL2(F )

f(γg), then

a(it) =1

2[f(it) + M(−it)f(−it)].

85

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