Classical Modular Formsusers.ictp.it/~pub_off/lectures/lns021/Venkataramana/Venkataramana.pdf · crucial step in the proof of Fermat’s Last Theorem by Andrew Wiles), etc. The simplest

Classical Modular Forms

T.N. Venkataramana∗

School of Mathematics, Tata Institute of Fundamental Research,

Colaba, Mumbai, India

Lectures given at the

School on Automorphic Forms on GL(n)

Trieste, 31 July - 18 August 2000

LNS0821002

∗[email protected]

Contents

1 Introduction 43

2 A Fundamental Domain for SL(2, Z) 43

3 Modular Forms; Definition and Examples 49

4 Modular Forms and Representation Theory 54

5 Modular Forms and Hecke Operators 62

6 L-functions of Modular Forms 70

Classical Modular Forms 43

1 Introduction

The simplest kind of automorphic forms (apart from Grossencharacters,

which will also be discussed in this conference) are the “elliptic modular

forms”. We will study modular forms and their connection with automor-

phic forms on GL(2), in the sense of representation theory.

Modular forms arise in many contexts in number theory, e.g. in questions

involving representations of integers by quadratic forms, and in expressing

elliptic curves over Q as quotients Jacobians of modular curves (this was the

crucial step in the proof of Fermat’s Last Theorem by Andrew Wiles), etc.

The simplest modular forms are those on the modular group SL(2, Z)

and we will first define modular forms on SL(2, Z).

To begin with, in section 2, we will describe a fundamental domain for

the action of SL(2, Z) on the upper half plane h. The fundamental domain

will be seen to parametrise isomorphism classes of elliptic curves.

In section 3, we will define modular forms for SL(2, Z) and construct

some modular forms, by using the functions E4 and E6 which we encounter

already in the section on elliptic curves.

In section 4, a representation theoretic interpretation of modular forms

will be given, which will enable us to think of them as automorphic forms

on GL(2, R).

In section 5, we will give an adelic interpretation of modular forms. This

will enable us to think of Hecke operators as convolution operators in the

Hecke algebra; using this, we show the commutativity of the Hecke operators.

We will also prove a special case of the Multiplicity One theorem.

2 A Fundamental Domain for SL(2, Z)

Notation 2.1 Denote by h the “Poincare upper half-plane” i.e. the space

of complex numbers whose imaginary part is positive:

h = {z ∈ C; z = x + iy, x, y ∈ R, y > 0}.

If z ∈ C, denote respectively by Re(z) and Im(z) the real and imaginary

parts of z.

On the upper half plane h, the group GL(2, R)+ of real 2 × 2 matrices

with positive determinant operates as follows: let g =

(a bc d

)∈ GL(2, R)+,

and let z ∈ h. Set g(z) = (az + b)/(cz + d). Notice that if cz + d = 0

44 T.N. Venkataramana

and c 6= 0 then, z = −d/c is real, which is impossible since z has positive

imaginary part. Thus, the formula for g(z) makes sense. Observe that

Im(g(z)) = Im(z)(det(g))/ | cz + d |2 . (1)

The equation (1) shows that the map (g, z) 7→ g(z) takes GL(2, R)+ ×h into

h. One checks immediately that this map gives an action of GL(2, R)+ on

the upper half plane h. Note also that

| cz + d |2= c2y2 + (cx + d)2. (2)

Therefore, | cz + d |2≥ y2 or 1 according as | c |= 0 or nonzero. Therefore,

Im(γ(z)) ≤ y/min{1, y2} ∀γ ∈ Γ0 ⊂ SL(2, Z), (3)

where min{1, y2} denotes the minimum of 1 and y2 and Γ0 ⊂ SL(2, Z) is

the group generated by the elements T = ( 1 10 1 ) and S = ( 0 1

−1 0 ). Later we

will see that Γ0 is actually SL(2, Z). The element T acts on the upper half

plane h by translation by 1:

T (z) =

(1 10 1

)(z) = z + 1 ∀z ∈ h. (4)

Similarly, the element S acts by inversion:

S(z) =

(0 1

−1 0

)(z) = −1/z ∀z ∈ h. (5)

Consider the set

F = {z ∈ h;−1/2 < Re(z) ≤ 1/2, | z |≥ 1, and 0 ≤ Re(x)} if | z |= 1.

Theorem 2.2 Given z ∈ h there is a unique point z0 ∈ F and an element

γ ∈ SL(2, Z) such that γ(z) = z0. Moreover, given γ ∈ SL(2, Z), we have

γ(F ) ∩ F = φ unless γ lies in a finite set ( of elements of SL(2, Z) which

fix the point ω = 1/2 + i31/2/2 ∈ h or i ∈ h). [one then says that F is a

fundamental domain for the action of SL(2, Z) on the upper half plane

h].

Proof We will first show that any point z on the upper half plane can

be translated by an element of the subgroup Γ0 of SL(2, Z) (generated by

T ( 1 10 1 ) and S = ( 0 1

−1 0 )) into a point in the “fundamental domain” F .


Now, given a real number x, there exists an integer k such that −1/2 <

x + k ≤ 1/2. Therefore, equation (4) shows that given z ∈ h there exists

an integer k such that the real part x′ of T k(z) satisfies the inequalities

−1/2 < x′ ≤ 1/2.

Let y denote the imaginary part of z and denote by Sz the set

Sz = {γ(z); γ ∈ Γ0, Im(γ(z)) ≥ y ,−1/2 < Re(γ(z)) ≤ 1/2} .

We will first show that Sz is nonempty and finite. Let k be as in the pre-

vious paragraph. Then −1/2 < Re(T k(z)) ≤ 1/2 and Im(T k(z) = Im(z);

therefore, T k(z) lies in Sz and Sz is nonempty.

Now, equation (3) shows that the imaginary parts of elements of the set

Sz are all bounded from above by y/min 1, y2. By definition, the imaginary

parts of points on Sz are bounded from below by y. The definition of Sz

shows that Sz is a relatively compact subset of h. We get from (3) that

| cz + d |2≤ 1; now (2) shows that | c |≤ 1/y2. Suppose γ ∈ γ0 = ( a bc d ) is

such that γ(z) ∈ Sz then, c is bounded by 1/y2 and is in a finite set. The

fact that cz+d is bounded now shows that d also lies in a finite set. Since Sz

is relatively compact in h, it follows that γ(z) = (az + b)/(cz +d) is bounded

for all γ(z) ∈ Sz; therefore, az + b is bounded as well, and hence a and b run

through a finite set. We have therefore proved that Sz is finite.

Let y0 be the supremum of the imaginary parts of the elements of the

finite set Sz; let S1 = {z′ ∈ Sz; Im(z′) = y0} and let z0 ∈ S1 be an element

whose real part is maximal among elements of S1. We claim that z0 ∈ F .

First observe that if z′ ∈ Sz then S(z′) = −1/z′ has imaginary part y0/ |

z |2= Im(z′)/ | z′ |2≤ y0 whence | z′ |2≥ 1. If | z0 |> 1, then it is immediate

from the definitions of F and Sz that z0 ∈ F . Suppose that | z0 |= 1. Then,

S(z0) = −1/z0 also has absolute value 1, its imaginary part is y0 and its real

part is the negative of Re(z0); hence S(z0) ∈ S1. The maximality of the real

part of z0 among elements of S1 now implies that Re(z0) ≥ 0. Therefore,

z0 ∈ F . We have proved that every element z0 may be translated by an

element of Γ0 into a point in the fundamental domain F .

Suppose now that z ∈ γ−1(F )∩F for some γ ∈ SL(2, Z). Write γ = ( a bc d )

with a, b, c, d ∈ Z and ad − bc = 1. Suppose that Im(γ(z)) ≥ Im(z) = y

(otherwise, replace z by γ(z)). Then, by (3) one gets

(cx + d)2 + c2y2 ≤ 1. (6)

Since z ∈ F , we have x2 + y2 ≥ 1 and 0 ≤ x ≤ 1/2. Therefore y2 ≥ 3/4 and

(1 ≥)c2y2 ≥ c24/3. (7)


This shows that c2 ≤ 1 since c is an integer.

Suppose c = 0. Then, ad = 1, a, d ∈ Z and we may assume (by

multiplying by the matrix −Id [minus identity] if necessary) that d = 1.

Hence γ = ( 1 b0 1 ). Then, γ(z) = z + b ∈ D which means that 0 ≤ x+ b ≤ 1/2

and 0 ≤ x ≤ 1/2. Thus, −1/2 ≤ b ≤ 1/2, i.e. b = 0 and γ is the identity

matrix.

The other possibility is c2 = 1, and by multiplying by the matrix −Id

(minus identity) we may assume that c = 1. Suppose first that d = 0. Then,

bc = −1 whence b = −1. Now, (7) shows that x2 + y2 ≤ 1. Moreover,

γ(z) = az + b/z = a + bz/ | z |2= a − z

whence its real part is a− x which lies between 0 and 1/2. Since 0 ≤ x ≤ 1,

it follows that 0 ≤ a ≤ 1. If a = 0 then γ = ( 0 −11 0 ) and lies in the isotropy

of the point i ∈ h. If a = 1 then, γ = ( 1 −11 0 ) which lies in the isotropy of the

point ω = 1/2 + i31/2/2.

We now examine the remaining case of c = 1 and d 6= 0. From (6) we

get (x + d)2 + y2 ≤ 1. If d ≥ 1 then the inequality 0 ≤ x ≤ 1/2 shows that

1 ≤ d ≤ x + d which contradicts the inequality (x + d)2 + y2 ≤ 1, which

is impossible. Thus, d ≤ −1; then the inequality 0 ≤ x ≤ 1/2 implies that

x + d ≤ 1/2 + (−1) = −1/2 whence (x + d)2 ≥ 1/4. Since y2 ≥ 3/4 the

inequality (x+d)2+y2 ≤ 1 implies that equalities hold everywhere: y2 = 3/4,

x = 1/2 and d = −1. Thus, z = ω and z−1 = z2. Since 1 = ad−bc = −a−b

(d = −1 and c = 1), and

γ(z) = (az+b)/(z−1) = (az+b)/z2 = −(az+b)z = a+(−a−b)z = a+z ∈ D,

the real part of γ(z) is a + x = a + 1/2 and is between 0 and 1/2, i.e.

−1/2 ≤ a ≤ 0 i.e. a = 0 and b = −1. Therefore, γ = ( 0 −11 −1 ) lies in the

isotropy of ω. This completes the proof of Theorem (2.2).

Corollary 2.3 The group SL(2, Z) is generated by the matrices T = ( 1 10 1 )

and S = ( 0 1−1 0 ).

Proof In the proof of Theorem (2.2), a point on the upper half plane is

brought into the fundamental domain F by applying only the transforma-

tions generated by S and T . The fact that the points on the fundamen-

tal domain are inequivalent under the action of SL(2, Z) now implies that

SL(2, Z) is generated by S and T .


(The Corollary can also be proved directly by observing that ST−1S−1 =

( 1 01 1 ). Now, the usual row-column reduction of matrices with integral entries

implies that T and STS−1 generate SL(2, Z)).

Notation 2.4 Elliptic Functions. We recall briefly some facts on elliptic

functions (for a reference to this subsection, see Ahlfors’ book on Complex

Analysis). Given a point τ on the upper half plane h, the space Γτ = Z⊕Zτ

of integral linear combinations of 1 and τ forms a discrete subgroup of C

with compact quotient. The quotient

Eτ = C/Γτ

may be realised as the curve in P2(C) whose intersection with the comple-

ment of the plane at infinity is given by

y2 = 4x3 − g2x − g3 (8)

The curve Eτ = C/Γτ is called an “elliptic curve”.

The map of C/Γτ to P2 is given by z 7→ (℘′(z), ℘(z), 1) for z ∈ C. Recall

the definition of ℘: if z ∈ C and does not lie in the lattice Γτ , then write

℘(z) = 1/z2 +

′∑(1/(z + w)2 − 1/w2),

where∑′

is the sum over all the non-zero points w in the lattice Γτ . The

derivative ℘′(z) of ℘(z) is then given by

℘′(z) =∑

1/(z + w)3,

where the sum is over all the points of the lattice Γτ . One has the equation

(cf. equation (8))

℘′(z)2 = 4℘(z)3 − g2(τ)℘(z) − g3(τ). (9)

If γ =

(a bc d

)∈ SL(2, Z) and τ ∈ h, then the elliptic curve Eγ(τ) is

isomorphic as an algebraic group (which is also a projective variety) to the

elliptic curve Eτ . The explicit isomorphism on C is given by z 7→ z/(cτ +d).

It is also possible to show that if Eτ and Eτ ′ are isomorphic elliptic curves,

then τ ′ is a translate of τ by an element of SL(2, Z).

Thus the fundamental domain F which was constructed in The-

orem (2.2) parametrises isomorphism classes of elliptic curves.


In equation (9), recall that the coefficients g2 and g3 are given by

g2(τ) = 60G4(τ) = 60

′∑(mτ + n)−4

and

g3(τ) = 140G6(τ) = 140

′∑(mτ + n)−6

where∑′

is the sum over all the pairs of integers (m,n) such that not both

m and n are zero. The discriminant of the cubic equation in (9) is given by

1/(16)∆(τ) where

∆(τ) = g32 − 27g2

3 . (10)

It is well known and easily proved that ℘′(z) has a simple zero at all the

2-division points 1/2,τ/2 and (1+τ)/2 and that ℘(1/2),℘(τ/2) and ℘((1+

τ)/2) are all distinct. Thus equation (9) transforms to

℘′(z)2 = 4(℘(z) − ℘(1/2))(℘(z) − ℘(τ/2))(℘(z) − ℘((1 + τ)/2)) (11)

Thus the discriminant of the (nonsingular) cubic in equation (9) is non-zero

and so we obtain that

∆(τ) 6= 0 (12)

for all τ ∈ h.

Notation 2.5 On the upper half plane h, there is a measure denoted y−2dxdy,

as z = x + iy varies in h. This measure is easily seen to be invariant under

the action of elements of the group GL(2, R)+ of nonsingular matrices with

positive determinant.

Lemma 2.6 With respect to this measure, the fundamental domain F has

finite volume.

Proof We compute the volume of F . Note that if z = x + iy lies in F ,

then, −1/2 ≤ x ≤ 1/2 and 1/(1 − x2)1/2 ≤ y < ∞. Thus the volume of F is

the integral ∫

Fdxdy/y2 =

∫ 1/2

−1/2dx(

∫ ∞

(1−x2)1/2

dy/y2)

which is easily seen to be π/3. In particular, F has finite volume.


Notation 2.7 Let S denote the inverse image of the fundamental domain

F ⊂ h under the quotient map GL(2, R) → GL(2, R)/O(2)Z = h. Then,

we have proved that GL(2, Z)S = GL(2, R). The set S is called a Siegel

Fundamental Domain.

3 Modular Forms; Definition and Examples

Notation 3.1 Given z ∈ h (h is the upper half plane) and an element

g = ( a bc d ), write

j(g, z) = cz + d.

Note that if j(g, z) = 0, then by comparing the real parts and imaginary

parts we get c = 0 and d = 0 which is impossible since ad − bc 6= 0. Thus,

j(g, z) is never zero.

Definition 3.2 A function f : h → C is weakly modular of weight w if

the following two conditions hold.

(1) f is holomorphic on the upper half plane.

(2) for all γ ∈ SL(2, Z), with γ = ( a bc d ), we have the equation

f((az + b)/(cz + d)) = (cz + d)wf(z). (13)

Given g = ( a bc d ) and a function f on the upper half plane h, define

g−1 ∗ f(z) = (cz + d)−wf(g(z)) ∀z ∈ h.

Then, it is easily checked that the map (g, f) → g−1 ∗ f defines an action

of GL(2, R) on the space of functions on h. Thus, the condition (2) above

is that the function f there is invariant under this action by SL(2, Z). Now

by Corollary (2.3), SL(2, Z) is generated by the matrices S = ( 0 1−1 0 ) and

T = ( 1 10 1 ). Thus condition (2) is equivalent to saying that γ−1 ∗ f = f for

γ = S, T . This amounts to saying that

f(−1/z) = zwf(z) (14)

and

f(z + 1) = f(z). (15)

Note that the invariance of f under the action of −1 where 1 is the

identity matrix in SL(2, Z) implies that f is zero of w is odd: f(z) =

(−1)wf(z). Therefore, we assume from now on (while considering modular

forms for the group SL(2, Z) ) that w = 2k where k is an integer.


Definition 3.3 The map exp : h → D∗ given by z 7→ e2πiz = q is easily seen

to be a covering map of the upper half plane h onto the set D∗ of non-zero

complex numbers of modulus less than one. The covering transformations

are generated by T (z) = z + 1. A weakly modular function f is invariant

under T and therefore yields a holomorphic map f∗ : D∗ → C given by

f∗(q) = f(z) for all z ∈ h. We say that a weakly modular function of weight

w is a modular function of weight w if f∗ extends to a holomorphic

function of D (the set of complex numbers of modulus less than one) i.e. f∗

extends to 0 ∈ D.

Let f be a weakly modular function on h. Then, f is a modular func-

tion if and only if the function f∗ has the “Fourier expansion” (or the “q-

expansion”)

f∗(q) =∑

n≥0

anqn, (16)

where an are complex numbers and the summation is over all non-negative

integers n. Observe that a weakly modular function is modular if and only

if it is bounded in the fundamental domain F .

We will say that a modular form is a cusp form if the constant term of

its q-expansion is zero: i.e. a0 = 0 in the notation of equation (15).

Notation 3.4 Examples of modular forms.

First we note that if f and g are modular forms of weights w and w′

then, the product function fg is a modular form of weight ww′.

We will first prove that for the modular group SL(2, Z), there are no

non-constant “weight zero” modular forms. First note that if f is a weight

zero modular form, then the function f∗ extends to 0 and hence is bounded

in a disc of radius r < 1. Its inverse image under exp : F → D∗ is precisely

the set A = {z = x + iy ∈ F ; y > − log r} and f is bounded on the set A.

The complement of the set A in the fundamental domain F is compact, and

f is bounded there as well, whence f is bounded on all of the fundamental

domain F as well as at “infinity”. By the maximum principle, f is constant.

We will now show that there are no modular forms of weight two on

SL(2, Z). Suppose f is one and let F (z) be its integral from z0 to z for

some fixed z0 ∈ h. The modularity of f shows that γ 7→ F (γ(z0)) gives a

homomorphism from SL(2, Z) to C. But, SL(2, Z) is generated by the finite

order elements S and ST whence, this homomorphism is identically zero.

This and the modularity of f shows that the integral F is invariant under

SL(2, Z). It is easy to show that F ∗ is holomorphic at 0 (integrate both


sides of equation (15)), and use the invariance of F under T ). Hence F is a

modular form of weight zero. By the foregoing paragraph, f is a constant,

i.e. f = 0.

Fix an even positive integer 2k, with k ≥ 2. We will construct a modular

form of degree k as follows. Let τ ∈ h and write (compare the definition of

g2 and g4 in section (2.4))

G2k(τ) =∑

′

(mτ + n)−2k, (17)

where∑′

is the sum over all the pairs of integers (m,n) not both of which are

zero. Then, G2k is easily shown to be a weakly modular function of weight

2k on the upper-half plane. If τ is varying in the fundamental domain and

its imaginary part tends to infinity, then it is clear from the formula for G2k

that G2k(τ) tends to∑′

n−2k = 2ζ(2k) where

ζ(s) =∑

n−s

is the Riemann zeta function (the sum is over all the positive integers n and

in the sum, the real part of s exceeds 1). Consequently, G2k is a modular

form of weight 2k. We will now outline a derivation of the q-expansion of

G2k. Start with the partial fraction expansion

πcot(πz) = z−1 +∑

(z + n)−1 + (z − n)−1 (18)

where the sum is over all positive integers n. This series converges uniformly

on compact subsets of the complement of Z in C.

Write q = e2πiz (where i ∈ h and i2 = −1). Then one has the q-expansion

πcot(πz) = πi(q + 1)/(q − 1) = −πi − 2πi∑

n≥1

qn (19)

Differentiate 2k- times, the right-hand sides of equations (17) and (18)

with respect to z. We then get the equality∑

n∈Z

(z + n)−2k = ((2k − 1)!)−1(2πi)2k∑

n≥1

n2k−1qn (20)

Fix m and in equation (19) take for z the complex number mτ . Then sum

over all m. We obtain by equations (16) and (18), the q-expansion

G2k(τ) = 2ζ(2k) + ((2k − 1)!)−1(2πi)2k∑

n≥1

σ2k−1qn (21)


where for an integer r and n ≥ 1, σr(n) is defined to be the sum∑

dr where

d runs over the positive divisors of n.

By using the power series expansion

(1 + x)−2 =∑

n≥1

nxn−1

and equation (17) one has the power series identity

πcot(πz) = z−1 + 2∑

n≥1

∑

j≥1

n2jz2j−1 = z−1 + 2∑

j≥1

ζ(2j)z2j−1 (22)

By comparing the power series expansions cos(x) =∑

m≥0((2m)!)−1x2m

and sin(x) =∑

m≥0((2m+1)!)−1x2m+1 with the right-hand side of equation

(21) one obtains

ζ(2) = π2/6, ζ(4) = π4/90 andζ(6) = π6/(33.5.7). (23)

Using (20) and (22) we get

g2 = 60G4 = (4/3)π4 + 160π4(q + · · · ) (24)

where the expression q + · · · is a power series in q with integral coefficients

with the coefficient of q being 1. Similarly, we get (again from (20) and (22))

g3 = 140G6 = (8/27)π6 − 25.7π6/3(q + · · · ) (25)

Therefore, we get, after some calculation, that for all z ∈ h,

∆(z) = g2(z)3 − 27g3(z)2 = 211π12(q +∑

n≥2

τ(n)qn) (26)

where the τ(n) are integers. We recall that ∆(z) is never zero on the upper-

half plane (section (2.4)). The equation (26) shows that the coefficient of q

in q-expansion of ∆ is non-zero, (and that its constant term is zero).

Lemma 3.5 There are no modular forms of negative weight.

Proof Suppose that f is a modular form of weight −l with l > 0. Form

the product g = f12∆l. Since f and ∆ are modular forms, so is the product.

Since its weight is zero, g is a constant (see the beginning of this subsection).

But, (26) shows that the q-expansion of g has no constant term. Hence g = 0

whence, f = 0.


Lemma 3.6 Suppose that f is a cusp form. Then ∆ divides f i.e., there

is a modular form g such that f = ∆g. In particular, the weight of f is at

least 12.

Proof Consider the quotient g = f/∆. Since ∆ has no zero in h, it follows

that g is holomorphic in h. Clearly, g is weakly modular of weight =weight

of f -12. Now the q-expansion of f (and also ∆), has no constant term; and

the coefficient of q in the q-expansion of ∆ is non-zero. Therefore, g∗extends

to a holomorphic function in a neighbourhood of 0. That is, g is a modular

function. Since the weight of g is non-negative (by Lemma (3.5)), it follows

that the weight of f must be at least that of ∆, namely, 12.

Corollary 3.7 The space of cusp forms of weight 12 (for SL(2, Z)), is one

dimensional.

Proof If f is a cusp form of weight 12, then f/∆ is a modular form of

weight zero, hence is a constant. That is, the space of cusp forms of weight

12 is spanned by ∆.

Theorem 3.8 The space of modular forms of weight 2k with k ≥ 0 is

spanned by the modular forms Gm4 Gn

6 with 4m + 6n = 2k.

Proof Argue by induction on k. We have already excluded the possibilities

k < 0 and k = 0 and k = 1.

Suppose that k ≥ 2 and that f is modular of weight 2k. First observe

that any integer k ≥ 2 may be written as 2m + 3n for non-negative integers

m and n. Now, the q-expansion of G4 and G6 have non-zero constant term.

Hence h = f−λGm4 Gn

6 ) for a suitable constant λ, has no constant term in its

q-expansion, and is a cusp form. Now, Lemma (3.6) shows that g = h/∆ is a

modular form of weight 2k−12. By induction, g is a linear combination of the

modular forms Ga4G

b6 with k − 6 = 2a + 3b whence, h is a sum of monomials

of the form Gp4G

q6 with 2p + 3q = k (recall that ∆ is (60G4)

3 − 27(140G6)2).

Therefore, so is f .

Notation 3.9 Define E2k(z) = G2k/2ζ(2k). Then, it follows from the

Fourier expansion of G2 and G6 that the modular forms E4 and E6 have in-

tegral Fourier coefficients. One sometimes writes ∆(z) = q +∑

n≥2 τ(n)qn.

Then, ∆ has integral Fourier coefficients as well. We now consider the Z-

module spanned by Em4 En

6 with 4m + 6n = 2k. We get an integral lattice


of modular forms of weight 2k. We will see later that this integral lattice is

stable under the Hecke operators.

4 Modular Forms and Representation Theory

Notation 4.1 We will begin with some calculations on the Lie algebra g of

the group GL(2, R). Write,

X =

(0 10 0

), Y =

(1 00 0

), Z =

(1 00 1

), and H =

(0 1

−1 0

). (27)

The complexified Lie algebra of GL(2, R) is M2(C) the space of 2 × 2

matrices with complex entries; the Lie algebra structure is given by (a, b) 7→

[a, b] = ab−ba; M2(C) is spanned by X,Y,Z and A. Write A = −iH (where

i ∈ h is the unique element whose square is -1). Then, A acts semisimply

(under the adjoint action) on g with real eigenvalues. Write

g = CE+ ⊕ CE− ⊕ CZ ⊕ CA (28)

where

E− = X + iY − (i/2)A− (i/2)Z and E+ = X − iY − (i/2)A+ (i/2)Z. (29)

Then E− and E+ are eigenvectors for A with eigenvalues −2 and 2 respec-

tively. Of course, on A and Z, A acts by 0. Thus, the complex Lie algebra

spanned by E+, E− and A is isomorphic to sl2(C).

Definition 4.2 Fix the subgroup K∞ = O(2) of GL(2, R). This is the

group generated by

SO(2) =

{Rθ =

(cosθ sinθ

−sinθ cosθ

): θ ∈ R

}(30)

and

ι =

(−1 0

0 1

). (31)

Then, O(2) is a maximal compact subgroup of GL(2, R). Suppose that (π, V )

is a module for g as well as for O(2) such that the module structures are

compatible. That is, suppose that v ∈ V and ξ ∈ g, and σ ∈ O(2). Then,

π(σ)π(ξ)(v) = π(σ(ξ))(v)


where σ(ξ) is the inner conjugation action of O(2) on the Lie algebra g. One

then says that (π, V ) is a (g,K∞)-module. If, as a K∞-module, (π, V ) is a

direct sum of irreducible representations of K∞ with each irreducible repre-

sentation occurring only finitely many times, then one says that the (g,K∞)-

module is admissible. One then sees at once that a (g,K∞)-submodule (or

a quotient module) of an admissible module is also admissible. One says

that a vector v ∈ V generates (π, V ) as a (g, O(2))-module, if the smallest

submodule of V containing v is all of V .

Notation 4.3 The Tensor Algebra. Given a (complex) vector space

V , denote by T n(V ) = V ⊗n the n-th tensor power of V . This is of

course, spanned by the pure tensors, i.e. vectors of the form v1 ⊗ · · · ⊗ vn

with vi ∈ V . By definition, T 0(V ) = C and T 1(V ) = V . Denote by

T (V ) = ⊕T n(V ) where the direct sum is over all the non-negative integers n.

Given non-negative integers m and n, there exists a linear map

Tm(V ) ⊗ T n(V ) → Tm+n(V )

which on pure tensors is the map

(v1 ⊗ · · · ⊗ vm) ⊗ (w1 ⊗ · · · ⊗ wn) 7→ (v1 ⊗ · · · ⊗ vm ⊗ w1 ⊗ · · ·wn).

This extends by linearity to all of Tm(V ) ⊗ T n(V ) and thence to all of the

direct sum T (V ). Under this “multiplication”, T (V ) becomes an associa-

tive algebra, and is called the tensor algebra of the vector space V . The

subspace T 0(V ) = C acts simply by scalar multiplication.

Notation 4.4 The Universal Enveloping Algebra. Given now a Lie

algebra g, let u(g) denote the quotient of the tensor algebra T (g) of g, by the

two sided ideal generated by the elements x⊗y−y⊗x−[x, y], as x and y vary

over the elements of the Lie algebra g. Here, ⊗ denotes the multiplication

in the tensor algebra T (g) and the bracket [x, y] denotes the Lie bracket in g.

The algebra u(g) is called the universal enveloping algebra of g. Note

that g is a subspace of u(g), with [x, y] = xy − yx. Here, x, y are the images

of x, y ∈ g = T 1(g) under the quotient map T (g) → u(g).

Suppose that u is some algebra over C and f : g → u a linear map such

that f([x, y]) = f(x)f(y)−f(y)f(x) for all elements x, y ∈ g. Here f(x)f(y)

refers to the product of the two elements in the algebra u. Then, there exists


a unique algebra map F : u(g) → u which extends f . This is why u(g) is

called the universal enveloping algebra of g.

In particular, if V is a module over g, we have a map f : g → End(V ) of

Lie algebras, and we have f([x, y]) = [f(x), f(y)] where the bracket structure

on End(V ) is simply the commutator in the algebra End(V ). Therefore, by

the last paragraph, we get a unique extension F : u(g) → End(V ) , with F

an algebra map. In other words, the g module V is naturally a module over

u(g) as well.

Theorem 4.5 The Poincare-Birkhoff-Witt Theorem: Let a, b and c

be subalgebras of a Lie algebra g such that g = a ⊕ b ⊕ c. Then, one has the

decomposition

u(g) = u(a) ⊗ u(b) ⊗ u(c)

Proof We must prove that every element of u(g) lies in the subspace of

the right-hand side of the above equation. Argue by an induction on the

degree of an element ξ ∈ T n(g). If we have an element yx for example, with

y ∈ c and x ∈ c, then, we may write it as xy − [x, y]. Now xy is in the

above subspace, and since g is by assumption a direct sum of a, b and c, the

element [x, y] also lies in the relevant subspace. We omit the details, since

this would be rather technical, and the reader can easily supply the details.

We now return to the group G = GL(2, R) and its Lie algebra g.

We will now prove the basic fact from representation theory which we

will use.

Theorem 4.6 Let (π, V ) be a (g,K∞)-module. Suppose that v ∈ V has the

following properties:

(1) v generates V .

(2) The connected component SO(2) of O(2) acts on v by the character

determined by Rθ(v) = e2πiθmv, for some positive integer m (i.e. v is an

eigenvector for A with eigenvalue m).

(3) E−(v) = 0 and Z(v) = 0.

Then the (π, V ) is admissible and irreducible.

Proof Let u(g) denote the universal enveloping algebra of the Lie-algebra

g. One has the decomposition (the Poincare-Birkhoff-Witt Theorem)

u(g) = u(g).[E−] + u(g).[Z] ⊕ C[E+] ⊗ C[A] (32)


where C[ξ] denotes the algebra generated by the operator ξ. Therefore, if

(as in (31))

ι =

(−1 0

0 1

)

then by assumptions (1) and (2) of the Theorem,

V = C[E+](v) ⊕ ιC[E+](v). (33)

On E+ the element A acts by the eigenvalue 2. Therefore, for an integer

p ≥ 0, the element (E+)p(v) is an eigenvector for A with eigenvalue (2p +

m), and ι(E+)p(v) is an eigenvector with eigenvalue (−2p − m) (note that

under the conjugation action of ι, the element A goes to −A, hence ι takes

an r-eigenspace for A into the −r-eigenspace). Since all these weights are

different, equation (33) shows that V is admissible as an SO(2) module (A

generates the complexified Lie algebra of SO(2)). In fact, equation (33)

shows that the multiplicity of an irreducible representation of SO(2) in V is

at most one, i.e. V is admissible.

Suppose that W ⊂ V is a submodule. In the last paragraph, we saw that

the action of A on V is completely reducible; hence the same holds for W .

Suppose that w is a weight vector in W of weight j, say. By replacing w

by ι(w) if necessary, we may assume that j > 0. The last paragraph shows

that j = 2p + m for some p ≥ 0 and also that (E+)p(v) = w (up to scalar

multiples). We may assume that p is the smallest non-negative integer such

that W contains the eigenvector (E+)p(v) = w with eigenvalue 2p+m. The

minimality of p implies that E−(w) = 0. Let W ′ be the submodule of W

generated by the vector w. To prove the irreducibility, it is enough to show

that W ′ = V . We may assume then that W = W ′.

Since v generates V and Z annihilates v, it follows that Z acts by zero

on all of V . Therefore, the vector w satisfies all the properties that v does

in the assumptions of the Theorem (except that in (2) the eigencharacter is

2p + m). Therefore, cf. equation (33), we have

W = C[E+](w) ⊕ ιC[E+](w) = C[E+](E+)p(v) ⊕ ιC[E+](E+)p(v). (34)

Now the equations (33) and (34) show that the codimension of W in V

is finite: dim(V/W ) < ∞. Hence V/W also satisfies the assumptions of

the Theorem (with v ∈ V replaced by its image v ∈ V/W ), but is finite

dimensional. This is impossible by the finite dimensional representation

theory of sl(2, C): a lowest weight vector (i.e. one killed by E− of sl(2))


cannot have positive weight for A. But v has exactly this property in

V/W . This shows that V/W = 0 i.e. W = V .

Proposition 4.7 Given m > 0, there is a unique irreducible (g,K∞)-

module ρm which satisfies the properties of 4.6.

Proof The uniqueness follows easily from the above proof of Theorem (4.6).

Let χm denote the one dimensional complex vector space on which the

group SO(2) acts by the character Rθ 7→ e2πimθ (where Rθ is, as in (30), the

rotation by θ in 2-space). Consider the space

u(g) ⊗ χm.

This is a representation for SO(2) (as well as for the universal enveloping

algebra u(g)). Let ρm be the O(2)-module induced from this SO(2)-module.

Then, ρm satisfies the properties of Theorem (4.6) and is therefore irre-

ducible. Moreover, it is clear that any module V of the type considered in

4.6 is a quotient of ρm. By irreducibility, V = ρm.

Remark 4.8 The modules ρ2k are called the discrete series representa-

tions of weight 2k of (g, O(2)). This means the following. Suppose there

exists an irreducible unitary representation of the group GL(2, R), call it ρ.

Suppose that this occurs discretely (i.e. is a closed subspace of ) in a space

of functions L2(G,ω) which transform according to the unitary character ω

of the centre Z of GL(2, R) and which are square summable with respect to

the Haar measure on the quotient GL(2, R)/Z. Given such a unitary module

ρ, consider the space of vectors whose translates under the compact group

O(2) form a finite dimensional vector space. This is the Harish-Chandra

module of the unitary representation ρ and is a (g, O(2))-module. The

representations ρ2k are the Harish-Chandra modules of discrete series repre-

sentations of even weight. We will show in the next section, that these are

closely related to modular forms of weight 2k.

There are also the discrete series representations of odd weight, which

we will not discuss, since we are dealing with the group SL(2, Z) and it has

no modular forms of odd weight.

We are now in a position to state the precise relationship of modular

forms with representation theory.


Notation 4.9 Let f be a modular form of weight 2k with k > 0. We will

now construct a function on the group G+ = GL(2, R)+ as follows. Set

Ff (g) = j(g, i)−2kf(g(i))det(g)k

where i ∈ h is the point whose isotropy is the group SO(2) as in equation

(30). As before, j(g, i) = cz + d, where

g = ( a bc d ).

By using the modularity of f and the equation j(gh, z) = j(g, h(z))j(h, z)

for the “automorphy factor” j(g, z), it is easy to see that Ff is invariant

under left translation by elements of SL(2, Z) and also under the centre Z∞

of GL(2, R).

We will now check that the (g, O(2))-module generated by Ff is isomor-

phic to ρ2k, with ρ2k as in 4.7. Note that Ff is contained in the space

C∞(Z∞GL(2, Z)\GL(2, R),

the space of smooth functions on the relevant space and that the latter is a

(g, O(2))-module under right translation by elements of GL(2, R). Moreover,

for all y > 0 and x ∈ R we have

Ff

(y x0 1

)= ykf(x + iy) (35)

The function g 7→ f(g(i)) is right invariant under the action of SO(2)

since i is the isotropy of SO(2). Using the fact that j(Rθ, i) = e−iθ (where

Rθ is as in (30)) one checks that j(gRθ , i) = j(g, i)(e−2iθ). Therefore, it

follows that

Ff (gRθ) = Ff (g)e2ikθ . (36)

This equation implies that under the action of the element A = i( 0 1−1 0 ) (A

generates the Lie algebra of SO(2)), Ff is an eigenvector with eigenvalue 2k.

Compute the action of E− (E− as in (29)) on Ff . Using the invariance

of of Ff under Z∞ and that it is an eigenvector of A with eigenvalue 4k, one

sees that

E−(Ff ) = (X + iY )Ff − ikFf .

Now use equation (35) to conclude that E−Ff = y2k(∂f/∂z). Since f is

holomorphic, one obtains that E−Ff = 0.

The (g, O(2)) module generated by Ff satisfies the conditions of 4.6.


Notation 4.10 Growth Properties of Ff . Consider now the growth

properties of Ff . We have the quotient map

GL(2, R)+ → h(⊃ F )

where F is the fundamental domain constructed in section 2. Let S be the

pre-image of F under this quotient map. Then, we have the inclusion

S ⊂ Z∞O(2)

{(y x0 1

): y2 > 3/4 and − 1/2 < x < 1/2

}

and the latter is a “Siegel set”. Now,

Ff

(y x0 1

)= ykf(x + iy).

From the modularity property of f , it follows that f is “bounded at infinity”,

which means that there exists a constant C > 0 such that on the fundamental

domain F of SL(2, Z), the function z 7→ f(z) = f(x + iy) is bounded by C:

| f(z) |≤ C ∀z ∈ F.

Therefore, on the Siegel set S, we have

Ff

(y x0 1

)≤ Cyk,

i.e. Ff has moderate growth on the Siegel Set.

Suppose now that f is a cusp form. Then, the Fourier expansion at

infinity of f is of the form

f(z) =

n=∞∑

n=1

anexp(2πinz)

where a(n) are the Fourier coefficients. The function

∑anqn−1

is clearly bounded in a neighbourhood of infinity in the fundamental domain

F and the complement of a neighbourhood of infinity being compact, it is

bounded on all of F too, by a constant C.


This shows the existence of a constant C > 0 such that for all z ∈ F ,

one has

f(x + iy) ≤ Cexp(−y).

The Haar measure on the group GL(2, R) is the product of the Haar

measure on the group O(2) and the invariant measure dxdy/y2 on the upper

half plane h = GL(2, R)/O(2) constructed at the end of section 2. This is

an easy exercise.

Thus, the square of the absolute value of Ff integrated on S = π−1(F )

(where π is the quotient map GL(2, R) → h ) is simply the integral of the

square of the absolute value of f(z) on the domain F . The above estimate

for f shows that this integral over F is finite:

∫ 1/2

−1/2dx

(∫ ∞

(1−x2)1/2

(dy/y2)exp(−y)

)< ∞

Definition 4.11 Automorphic Forms. Recall the definition of automor-

phic forms on GL(2, Z). These are smooth functions φ on the quotient

GL(2, Z)\GL(2, R), which are

(1) K-finite. That is, the space of right translates of φ under the compact

group K = O(2) forms a finite dimensional vector space.

(2) The function φ has moderate growth on the Siegel set St,1/2, i.e.

St,1/2 is a set of the form N1/2AtKZ where, Z is the centre of GL(2, R),

N1/2 is the set of matrices of the form n = ( 1 x0 1 ) with −1/2 ≤ x ≤ 1/2, and

At is the set of diagonal matrices of the form a = ( y 00 1 ) with (0 <)t < y, and

there exists a constant C > 0 such that in the above notation,

φ(nakz) ≤ CyN

for some integer N and for all elements nakz ∈ St,1/2 in the Siegel set.

(3) There is an ideal I of finite codimension in the centre of the universal

enveloping algebra U(g) which annihilates the smooth function φ.

The last few paragraphs imply the following

Theorem 4.12 Let f be a modular form of weight 2k. Let Ff be the associ-

ated function on GL(2, Z)\GL(2, R). Then, Ff is an automorphic form.

Moreover, the (g, O(2)) module generated by Ff is isomorphic to ρ2k with

ρ2k as in 4.7


Moreover, if f is a cusp form, then, the associated function Ff is rapidly

decreasing on the Siegel domain, and is therefore square summable on the

quotient space Z∞GL(2, Z)\GL(2, R).

Proof We need only check that an ideal I of finite codimension in the

centre z of the universal enveloping algebra of g annihilates Ff . But, the

module generated by Ff is ρ2k by 4.6 (and 4.7). Now, the 2k eigenspace of

the operator A in the representation ρ2k is one dimensional (and is generated

by Ff ), and z commutes with the action of A (and in fact with all of u(g) as

well). Therefore, the annihilator of Ff in z is an ideal I of codimension one.

Theorem 4.13 The space M2k of modular forms of weight 2k for the group

SL(2, Z) may be identified with the isotypical subspace of the irreducible

(g, O(2)) module ρ2k in the space

C∞(Z∞GL(2, Z)\GL(2, R)).

The isomorphism is obtained by sending a modular form f to the span

of the function Ff under the action of (g, O(2)) (the latter (g, O(2))-module

is isomorphic to ρ2k).

5 Modular Forms and Hecke Operators

Notation 5.1 Let Af be the ring of finite adeles over Q. Recall that this is

the direct limit (the maps are inclusion maps) as the finite set S of primes

varies, of the product

AS =∏

p∈S

Qp ×∏

p/∈S

Zp.

The group of units of Af is the group A∗f of ideles and is the direct limit as

S varies, of

A∗S =

∏

p∈S

Q∗p ×

∏

p/∈S

Z∗p,

(where ∗ denotes the group of units of the ring under consideration).

There is a natural inclusion of Q in Af (and hence of Q∗ in A∗f and

of GL(2, Q) in GL(2, Af )). Denote by P the set of primes. The Strong

Approximation Theorem (Chinese Remainder Theorem) implies that

Af = Q +∏

p∈P

Zp. (37)


This, and the fact that Z is a principal ideal domain imply that

A∗f = Q∗.

∏

p∈P

Z∗p. (38)

From this it is not difficult to deduce that

GL(2, Af ) = GL(2, Q).∏

p∈P

GL(2, Zp) (39)

Note that the intersection of GL(2, Q) with Kf =∏

GL(2, Zp) is precisely

GL(2, Z).

Let A = R × Af be the ring of adeles over Q. Then, Q is diago-

nally imbedded in A. Hence there is a diagonal imbedding of GL(2, Q)

in GL(2, A) = GL(2, R)×GL(2, Af ). Then, GL(2, Q) is a discrete subgroup

of GL(2, A). Now equation (39) (a consequence of strong approximation)

implies that

GL(2, A) = GL(2, Q)(GL(2, R) ×∏

p∈P

GL(2, Zp)). (40)

Now, equation (40) and the last sentence of the previous paragraph imply

that the quotient

GL(2, Q)\GL(2, A) = GL(2, Z)\(GL(2, R) ×∏

p∈P

GL(2, Zp)). (41)

Note that GL(2, A) acts by right translations on the left-hand side of the

equation (41).

Notation 5.2 A representation (π,W ) of GL(2, Af ) is said to be smooth

if the isotropy of any vector in W is an open subgroup of GL(2, Af ). Define

the “Hecke algebra” H of GL(2, Af ) as the space of compactly supported

locally constant functions on GL(2, Af ). If W is a smooth representation of

GL(2, Af ), then the Hecke Algebra H also operates on W by “convolutions”:

if µ is a Haar measure on GL(2, Af ), φ ∈ H, and w ∈ W is a vector, then the

W valued function g 7→ φ(g)π(g)w is a locally constant compactly supported

function and hence can be integrated with respect to the Haar measure µ.

Define

φ ∗ w = π(φ)(w) =

∫φ(g)π(g)(w)dµ(g) (42)


This gives the GL(2, Af )-module π, the structure of an H-module. As is well

known, the category of smooth representations of GL(2, Af ) is isomorphic

to the category of representations of the Hecke algebra H, the isomorphism

arising from the foregoing action of the Hecke algebra on the smooth module

π.

Notation 5.3 The group K0 = GL(2, Z) is an open compact subgroup of

GL(2, Af ) and is the product over all primes p of the groups GL(2, Zp).

Given g ∈ GL(2, Af ), consider the characteristic function χg of the double

coset set K0gK0. Then χg is an element of the Hecke algebra and elements

of H which are bi-invariant under H are finite linear combinations of the

functions χg as g varies. We will refer to the subalgebra generated by these

elements as the ‘unramified Hecke algebra and denote it by H0.

Under convolution, H is an algebra and H0 is a commutative subalgebra.

Fix a prime p. Let H0(p) be the subalgebra generated by the elements χMp

and χNp where Mp = ( p 00 1 ) and Np = ( p 0

0 p ). It is easily proved that for

varying p, the algebras H0(p) generate the unramified Hecke algebra H0.

Notation 5.4 The equation (41) implies that the space of smooth functions

on Z∞SL(2, Z)\GL(2, R)+ is isomorphic to the space V0 of K0-invariant

smooth functions on the quotient GL(2, Q)Z(A)\GL(2, A). On V0 the un-

ramified Hecke algebra operates. Suppose S denotes the image of F ×K0 in

Z(A)\GL(2, A). Then, S is contained in a Siegel set S0 whose elements are

of the form

z∞

(y x0 1

)× k0

where z∞ ∈ Z∞, k0 ∈ K0, | x |< 1/2 and y2 > 3/4. Suppose that f is a

cusp form for SL(2, Z) and Ff be as in section (4.5). Given g ∈ GL(2, A) =

GL(2, R) × GL(2, Af ), write g = (g∞, gf ) accordingly. Define the function

Φf on GL(2, Q)\GL(2, A) as follows. Set Φf (g∞, gf ) = Ff (g∞) if gf ∈ K0

and extend to G(A) by demanding that Φf be GL(2, Q)-invariant. The

SL(2, Z)-invariance of Ff implies that Φf is well defined. Now, 4.12 shows

that Φf is an automorphic form on GL(2, A).

By 4.13, Ff is rapidly decreasing on S0; moreover, Ff is a cuspidal

automorphic form in the sense that for all g ∈ G(A), the following holds.

∫

U(Q)\U(A)Φf (ng)dn = 0 (43)


where U is the group of unipotent upper triangular matrices in GL(2) with

ones on the diagonal and dn is the Haar measure on U(A). To prove this,

we note that the vanishing of the integral is unaltered by changing g on the

(1) right by an element of Z∞O(2) × K0, since Φf is an eigenvector for

the right translation action by K∞Z∞ × K0. We may hence assume that

g = (g∞, gf ). Now, up to elements of K0 = GL(2, Z) an element of G(Af )

is upper triangular;

(2) left by an element of B(Q) since G(Q) normalises U(A) and U(Q),

and preserves the Haar measure on U(A). Note that (the Iwasawa decom-

position) the double coset B(Q)\G(Af )/K0Z(A) is a singleton. Hence, we

may assume that g = g∞ = ( y x0 1 ). Then the above integral is the same as

∫

U(Z)\U(R)f(x + n + iy)dn

which is nothing but the zero-th Fourier coefficient of f , and by the cuspi-

dality of f , this is zero.

On the Siegel domain, the modular function f satisfies an estimate of

the form

|f(x + iy)| < Cexp(−y)

where C is some constant. This implies that on the Siegel set S, the function

Φf satisfies an estimate of the form

Φf (g) = O(|g|−N )

for some positive integer N . This can be shown to imply that the function

Φf is square summable on the quotient Z(A)GL(2, Q)\GL(2, A) with respect

to the Haar measure. Further, one has the L2-metric < , > on the space

of cuspidal automorphic forms which translates to the “Petersson” metric

< f, g >=

∫

Ff(z)g(z)y2k(y−2dxdy)

for cusp forms f and g of weight k. As before, F is the fundamental domain

for SL(2, Z).

Notation 5.5 From now on, we will fix our attention on cusp forms. We

have the natural inclusion of GL(2, Q) in GL(2, Af ). Let p > 0 be a prime

and let gp = ( p 00 1 ) be thought of as an element in GL(2, Af ) under the


foregoing inclusion. Let Xp denote the characteristic function of the double

coset of K0 through the element gp.

If χp denotes the characteristic function Xp, and f is a cuspidal mod-

ular form of weight 2k, then Φ′ = Φf ∗ R(χp) (where R(φ) denotes the

right convolution by the function φ) is a smooth function on the quotient

GL(2, Q)Z(A)\GL(2, A) whose “infinite” component is still ρ2k (since χp

commutes with the right action of GL(2, R) on the above quotient). Since

χp is K0 invariant, it follows that Φ′ is also right K0 invariant. Therefore,

it corresponds to a modular form g, i.e. Φ′ = Φg. It is easy to show that

Φ′ is cuspidal (the space of cusp forms is stable under right convolutions).

Therefore, g is a cusp form of weight 2k as well. Denote g = T (p)(f). Then,

T (p) is called the Hecke operator corresponding to the prime p. By noting

that convolution by χp is self-adjoint for the L2 metric on cuspidal automor-

phic functions on GL(2) one immediately sees that the operators T (p) are

self-adjoint for the Petersson metric on the space of cusp forms of weight

2k. The commutativity of the unramified Hecke algebra implies that the

operators T (p) (as p varies) commute as well.

Definition 5.6 Now a commuting family of self-adjoint operators on a fi-

nite dimensional complex vector space can be simultaneously diagonalised.

Consequently, there exists a basis of cusp forms of weight 2k which are si-

multaneous eigenfunctions for all the Hecke operators T (p); these are called

Hecke eigenforms. If f is a Hecke eigenform for SL(2, Z) and has constant

term 1, then it is called a normalised Hecke eigenform.

Theorem 5.7 The Iwasawa Decomposition: Any matrix in GL(2, Af )

may be written as a product bk with b ∈ B(Af ) (the group of upper triangular

matrices), and k ∈ K0 = GL(2, Z).

Proof This is an easy application of the elementary divisors theorem. By

identifying B\G with the projective line P1, we see that the Iwasawa decom-

position amounts to the transitivity of the action of GL(2, Z) on P1(Af ).

But, any element of P1(Af ) may be written as a vector (x, y) ∈ A2f where

for every prime p, the p-th components (xp, yp) are not both zero.

By changing (x, y) by an element of A∗f if necessary, (x, y) may be as-

sumed to be in Z2. Further, x, y may be assumed to be coprime, in the sense

that for every prime p, the p-adic components xp, yp of x, y are coprime.

Now, by writing everything in the notation of row vectors, we want to solve


for g ∈ GL(2, Z) the equation

(x, y) = (0, 1)g

(note that the isotropy at (0, 1) is precisely B(Af )). This amounts to finding

(z, t) ∈ Z2 to a basis of Z2, which can be done precisely because x, y are

coprime.

Notation 5.8 This implies that for a prime p, we have

Xp = ∪( p x0 1 )K0 ∪ ( 1 0

0 p )K0

where the union is a disjoint union, and 0 ≤ x ≤ p−1. Here K0 = GL(2, Z),

as before.

Notation 5.9 We will now state without proof the computation of T (p) for

a prime p. Note that by strong approximation, the K0 invariant function

Φf on the quotient Z(A)GL(2, Q)\GL(2, A) is completely determined by

its values on elements of the form ( y x0 1 ) with y > 0, in the quotient. We

compute (using the description of Xp in the previous section)

Φf ∗ R(χp)(y x0 1 )

and find that this is equal, to

p2k−1Φg(y x0 1 )

where

g(z) = (1/p)∑

0≤m≤p−1

f((z + m)/p) + p2k−1f(pz) = T (p)(f)(z).

The Fourier coefficients of g at infinity are given by

g(m) = a(mp)

if m is coprime to p and

g(m) = a(mp) + p2k−1a(m/p)

if p divides m.


Notation 5.10 In particular, if f is an eigenfunction for all the T (p) with

eigenvalue λ(p) say, the equation T (p)f = λ(p)f implies, by comparing the

p-th Fourier coefficients, that a(p) = λ(p)a(1) for each p.

Remark 5.11 In particular, we get from the last two paragraphs, that if f

is an eigenform whose first Fourier coefficient a(1) is zero, then, a(m) = 0 for

all positive m. This easily follows from induction and the formula a(mp) =

λ(p)a(p) if m and p are coprime, and a(m)λ(p) = a(mp) + p2k−1a(m/p) if p

divides m. Hence f = 0. Thus, we have proved that every Hecke eigenform

is a nonzero multiple of a normalised Hecke eigenform.

Theorem 5.12 (The Multiplicity 1 Theorem): Let f1 and f2 be two

normalised Hecke eigenforms for the action of the Hecke operators T (p) with

the same eigenvalues λ(p) for every prime p. Then, f1 = f2.

Proof We will prove this by showing that the Fourier coefficients of f1

and f2 are the same. This will imply, by the Fourier expansion for modular

forms, that f1 = f2. Write f = f1 − f2.

Now, the first Fourier coefficient of f is zero, since f1 and f2 are nor-

malised. Further, f is also a Hecke eigenform, since f1 and f2 are so, and

with the same eigenvalues. Therefore, by the previous remark, f = 0.

Recall that we have identified [representations π of GL(2, A) whose infi-

nite component π∞ is ρ2k and whose finite component πf contains a non-zero

GL(2, Af ) invariant vector], with [normalised eigenforms f of weight 2k for

the group GL(2, Z)].

Therefore, we have proved that the multiplicity of such a π in the

space of cusp forms on GL(2, A) is one.

Remark 5.13 Later, Cogdell will prove that the multiplicity of a cuspidal

automorphic representation of GL(n) is always 1. This is the famous mul-

tiplicity 1 theorem due to Jacquet-Langlands for GL(2) and to Piatetskii-

Shapiro and Shalika in general.

What we have proved is therefore a very special case when n = 2, the

infinite component is ρ2k, and the representation is unramified at all the

local places.


Definition 5.14 We now define inductively, the operators T (n) as follows.

If m and n are coprime, we define T (mn) = T (m)T (n). This reduces us to

defining T (pm) where p is a prime. Define recursively the operators T (pm)

by the formula T (p)T (pm) = T (pm+1) + pT (pm−1).

This now implies (by the similarity of the recursive formulae for T (n)

and the Fourier coefficients a(n)), that if f =∑

a(n)qn is a normalised

Hecke eigenform, then T (n)f = a(n)f for all n. These T (n)’s are the classi-

cal Hecke operators. By construction, they commute, one has T (mn) =

T (m)T (n) if m,n are coprime, and they are self-adjoint for the Petersson

inner product on modular forms of weight 2k.

Theorem 5.15 If f is a normalised Hecke eigenform for SL(2, Z), then,

all its Fourier coefficients are algebraic integers.

Proof We consider the action of the Hecke operators T (n) on the space

M02k of cups forms. Note that the space of cusp forms contains the (ad-

ditive) subgroup L of those cusp forms whose Fourier coefficients are ra-

tional integers. This subgroup is stable under the action of the operators

T (n). To see this, first suppose that n = p is a prime. By the formula

aT (p)f (m) = a(pm) + p2k−1a(m/p) if p divides m and aT (p)f (m) = a(pm)

otherwise, we see that T (p) stabilises the subgroup L. Since the T (p) gen-

erate T (n), the operator T (n) also stabilises L.

By induction on k, we see that the space M2k of modular forms of

weight 2k has a basis whose Fourier coefficients are integral: M2k = CE2k ⊕

M2k−12∆, and ∆ and E2k have integral Fourier coefficients. This shows that

the subgroup L of the last paragraph contains a basis of the space of cusp

forms M02k.

Since the operator T (n) is self adjoint with respect to a suitable metric

on the space M02k, it follows that the eigenvalues of T (n) are all real and are

all algebraic integers.

Now the Fourier coefficients a(n) of the eigenform f are nothing but

the eigenvalue λ(n) of T (n) corresponding to the eigenvector f , by the last

paragraph of the previous section. Consequently, the Fourier coefficients of

f are all real algebraic integers.


6 L-functions of Modular Forms

In this section, we define the L-function of a cusp form for SL(2, Z) and

prove that it has analytic continuation to the entire plane and has a nice

functional equation. Later, Cogdell will prove analogous statements for cus-

pidal automorphic representations for GL(n).

To begin with, we prove an estimate –due to Hecke– for the n-th Fourier

coefficient of a cusp form of weight 2k.

Lemma 6.1 (Hecke) Let f =∑n=∞

n=1 a(n)qn be a cusp form of weight 2k.

Then, there exists a constant C > 0 such that

a(n) ≤ Cnk/2 ∀n ≥ 1.

Proof Consider the function φ defined and continuous on the upper half

plane h, given by

φ(z) = yk/2|f(z)|.

If γ = ( a bc d ) ∈ SL(2, Z) and z′ = γ(z) = x′ + iy′, then recall that y′ =

y/(|cz + d|2). Therefore, we obtain from the modularity property of f , that

φ(γ(z)) = φ(z), i.e. φ is invariant under SL(2, Z). Hence, φ is determined

by its restriction to the fundamental domain F . As z tends to infinity in F ,

the cuspidality condition of f shows that f(z) = O(exp(−2πy)). Therefore,

φ is bounded on F and hence on all of the upper half plane h. Thus, there

exists a constant C1 such that

|f(z)| ≤ C1y−k/2 ∀z ∈ h.

Now consider the n-th Fourier coefficient a(n) of f . Clearly,

a(n) = e2πny

∫ 1

0f(x + iy)e−2iπnxdx.

By applying the foregoing estimate for f to this equation, we obtain

|a(n)| ≤ C1e2πnyyk/2

for all y > 0. Take y = 1/n. We then get

a(n) ≤ (C1e2π)nk/2.


Remark 6.2 Deligne has proved that for all primes p, |a(p)| = O(p((k−1)/2)),

by linking these estimates with the “Weil Conjectures” for the number of

rational points of algebraic varieties over finite fields.

Definition 6.3 If f(z) =∑

a(n)qn is a cusp form of weight 2k, define for

a complex variable s, the Dirichlet Series L(f, s) by the formula

L(f, s) =

∞∑

n=1

a(n)/ns.

From Lemma (6.1), it follows that the series converges and is holomorphic in

the region Re(s) > 1 + (k/2). The function L(f, s) is called the L-function

of the cusp form f .

Notation 6.4 an integral expression for L(f, s)). We will now write an

integral formula for the L-function of f . First consider the integral∫ ∞

0f(iy)ys(dy/y).

Since f(iy) = O(exp(−2πy)) for all y > 0 (cf. the proof of Lemma (6.1)), it

follows that if Re(s) > 0, then the integral converges. Let σ be the real part

of s. Then, for each n ≥ 1 the integral∫ ∞

0|a(n)|e−2πnyyσ(dy/y)

converges, and is equal to (|a(n)|/nσ)(2π)−σΓ(σ) where Γ is the classical

Γ-function:

Γ(z) =

∫ ∞

0e−ttz(dt/t).

From the Hecke estimate a(n) = O(nk/2) of Lemma (6.1), it follows that the

infinite sum of these integrals also converges, provided σ > k/2 + 1. Thus,

by the Dominated Convergence Theorem (to justify the interchange of sum

and integral), we obtain the equation

∫ ∞

0f(iy)ys(dy/y) = (2π)−sΓ(s)

∞∑

n=1

a(n)n−s.

We finally obtain the integral expression:∫ ∞

0f(iy)ys(dy/y) = (2π)−sΓ(s)L(f, s).


Remark 6.5 Notice that the integral expression says that the function

L(f, s) can be analytically continued to the region Re(s) > 0, since the

left-hand side is analytic there, and the function (2π)−sΓ(s) has no zero’s

on the complex plane.

Notation 6.6 The functional Equation. The L-Function and the in-

tegral expression could have been defined for any holomorphic function

f(z) =∑

a(n)qn (q = e2πiz), provided a(n) satisfy a Hecke estimate. We

will now prove a functional equation for L(f, s), by using the modularity

property, especially that f(−1/z) = (−1)kz2kf(z).

Consider now the integral I(s) =∫∞0 f(iy)ys(dy/y), which converges for

Re(s) > 0. We write the integral as a sum of the integral from 1 to ∞ and

the integral from from 0 to 1. By making a change of variable y 7→ 1/y we

get, ∫ 1

0f(iy)ys(dy/y) =

∫ ∞

1f(−1/(iy))y−s(dy/y).

Note that as f(1/(iy)) is bounded on the interval [1,∞], and Re(s) > 0, the

integral on the right side converges. Therefore, we get, using the functional

equation f(−1/iy) = (−1)ky2kf(iy), that

∫ 1

0f(iy)ys(dy/y) = (−1)k

∫ ∞

1f(iy)y2k−s(dy/y).

We then get

I(s) =

∫ ∞

1f(iy)(ys + (−1)ky2k−s)(dy/y).

This holds for all s, with Re(s) > 0. We now make the change of variable

s 7→ 2k − s, for s in the region 0 < Re(s) < 2k. Then the above expression

for I(s) shows that I(2k − s) = (−1)kI(s). This is the functional equation

for L(f, s):

(2π)−sΓ(s)L(f, s) = (−1)k(2π)2k−sΓ(2k − s)L(f, 2k − s)

for all s in the region 0 < Re(s) < 2k. The left side of this equation is

analytic in the region Re(s) > 0 and the right side is analytic in the region

Re(s) < 2k. Using the functional equation (and the fact the (2π)−sΓ(s)

never vanishes on the complex plane), we now see that L(f, s) has an analytic

continuation over the entire complex plane.


Notation 6.7 Euler Factors. In the last few sections, we derived the

functional equation and analytic continuation of a cusp form for SL(2, Z).

We now derive an Euler product, for the L-function of a normalised Hecke

eigenform f .

Let then f =∑∞

n=1 a(n)qn be a normalised Hecke eigenform of weight

2k. Fix a prime p and consider the infinite sum

Lp(f, s) =

∞∑

m=0

a(pm)/pms

where Re(s) > k/2+1. It converges, by the Hecke estimate a(n) = O(nk/2).

We now use the relations a(pm)a(p) = a(pm+1)+p2k−1a(pm−1). Multiplying

these equations by 1/p(m+1)s and then summing over all m ≥ 0 we get

a(p)Lp(f, s) = (Lp(f, s) − 1)ps + (p2k−1/p2s)Lp(f, s).

That is,

Lp(f, s)−1 = 1 − a(p)/ps + p2k−1/p2s.

We now use the fact that a(mn) = a(m)a(n) if m,n are coprime, since

f is a normalised Hecke eigenform. Form the product over all primes p of

these Lp(f, s). We then get (by using the Dominated Convergence Theorem

to justify interchanges) the equation∏

Lp(f, s) =∑

a(n)/ns = L(f, s).

Thus we have the infinite product expansion (the product being over all

primes p)

L(f, s) =∏

p

1/(1 − a(p)/ps + p2k−1/p2s).

From now on we consider the function L∗(f, s) = (2π)−sΓ(s)L(f, s) and

refer to this as the L-function of f .

We have thus proved the following Theorem.

Theorem 6.8 Let f =∑∞

n=1 a(n)qn be a normalised Hecke eigenform of

weight 2k for SL(2, Z). Then, the L-function L∗(f, s) = (2π)−sΓ(s)∑

a(n)/ns

converges for Re(s) > k/2, has an analytic continuation to the entire com-

plex plane and satisfies the functional equation

L∗(f, s) = (−1)kL∗(f, 2k − s).

Moreover, in the region Re(s) > k/2, one has the Euler product

L∗(f, s) = (2π)−sΓ(s)∏

p

1/(1 − a(p)/ps + p2k−1/p2s).


Remark 6.9 Recall that to each Hecke eigenform f of weight 2k, there

corresponds an irreducible cuspidal automorphic representation π(f) = π =

π∞ ⊗p πp of the (restricted direct product) group GL(2, A) = GL(2, R) ×∏p GL(2, Qp) such that π∞ is the discrete series representation ρ2k and each

πp is an irreducible unramified representation of GL(2, Qp).

In the lectures of Cogdell, you will see that each cuspidal automorphic

representation π of GL(n, A) has an L-function attached to it –denoted

L(π, s)– which satisfies a functional equation, has an analytic continuation

to the entire plane, and has an Euler product comprising of terms which are

monic polynomials in p−s of degree n.

It turns out that for the representation π(f) = π attached to the Hecke

eigenform f of weight 2k, the L-function is nothing but L(π, s) = L∗(f, s +

(k − 1/2)), which can easily be seen to satisfy the equation

L(π, s) = (−1)kL(π, 1 − s).

Moreover, the local factors are of the form

L(πp, s)−1 = (1 − (a(p)/pk−1/2)/ps + 1/p2s,

a monic polynomial in in p−s of degree two.

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