III Modular Forms and L-functions

4The modular group

4 The modular group

We now move on to study the other words that appear in the title of the course,

namely modular forms. Modular forms are very special functions defined on the

upper half plane

H = {z ∈ C : Im(z) > 0}.

The main property of a modular form is that they transform nicely under

M¨obius transforms. In this chapter, we will first try to understand these M¨obius

transforms. Recall that a matrix

γ =



a b

c d



∈ GL

(C)

acts on C ∪ ∞ by

z 7→ γ(z) =

az + b

cz + d

If we further restrict to matrices in

(

), then this maps

C \R

, and

R ∪ {∞} to R ∪ {∞}.

We want to understand when this actually fixes the upper half plane. This

is a straightforward computation

Im γ(z) =



az + b

cz + d

−

a¯z + b

c¯z + d



(ad − bc)(z − ¯z)

|cz + d|

= det(γ)

Im z

|cz + d|

Thus, we know

(

)) and

(

) have the same sign iff

det

(

)

0. We write

Definition (GL

(R)

= {γ ∈ GL

(R) : det γ > 0}.

This is the group of M¨obius transforms that map H to H.

However, note that the action of

(

)

is not faithful. The kernel

is given by the subgroup

· I = R



1 0

0 1



Thus, we are naturally led to define

Definition (PGL

(R)

PGL

(R)

· I

There is a slightly better way of expressing this. Now note that we can obtain

any matrix in

(

), by multiplying an element of

(

) with a unit in

So we have

PGL

(R)

∼

(R)/{±I} ≡ PSL

(R).

What we have is thus a faithful action of

PSL

(

) on the upper half plane

From IA Groups, we know this action is transitive, and the stabilizer of

√

−1

is SO(2)/{±I}.

In fact, this group

PSL

(

) is the group of all holomorphic automorphisms

of H, and the subgroup SO(2) ⊆ SL

(R) is a maximal compact subgroup.

Theorem. The group SL

(R) admits the Iwasawa decomposition

(R) = KAN = NAK,

where

K = SO(2), A =



r 0



, N =



1 x

0 1



Note that this implies a few of our previous claims. For example, any

z = x + iy ∈ C can be written as

z = x + iy =



1 x

0 1



√

y 0

√



· i,

using the fact that K = SO(2) fixes i, and this gives transitivity.

Proof.

This is just Gram–Schmidt orthogonalization. Given

g ∈ GL

(

), we

write

= e

, ge

= e

By Gram-Schmidt, we can write

= λ

+ µe

such that

k = kf

k = 1, (f

, f

) = 0.

So we can write











0 µ



Now the left-hand matrix is orthogonal, and by decomposing the inverse of



0 µ



, we can write g =





as a product in KAN.

In general, we will be interested in subgroups Γ

≤ SL

(

), and their images

Γ in Γ ∈ PSL

(R), i.e.

Γ =

Γ ∩ {±I}

We are mainly interested in the case Γ = SL

(Z), or a subgroup of finite index.

Definition (Modular group). The modular group is

PSL

(Z) =

(Z)

{±I}

There are two particularly interesting elements of the modular group, given

S = ±



0 −1

1 0



, T = ±



1 1

0 1



Then we have

(

) =

+ 1 and

(

) =

−

. One immediately sees that

has

infinite order and S

= 1 (in PSL

(Z)). We can also compute

T S = ±



1 −1

1 0



and

(T S)

= 1.

The following theorem essentially summarizes the basic properties of the modular

group we need to know about:

Theorem. Let

D =



z ∈ H : −

≤ Re z ≤

, |z| > 1



∪ {z ∈ H : |z| = 1, Re(z) ≥ 0}.

−

−1

ρ = e

πi/3

Then

is a fundamental domain for the action of

, i.e. every orbit

contains exactly one element of D.

The stabilizer of

z ∈ D

in Γ is trivial if

z 6

i, ρ

, and the stabilizers of

and

are

= hSi

∼

= hT Si

∼

Finally, we have

Γ = hS, T i = hS, T Si.

In fact, we have

Γ = hS, T | S

= (T S)

= ei,

but we will neither prove nor need this.

The proof is rather technical, and involves some significant case work.

Proof.

Let

∗

hS, T i ⊆

. We will show that if

z ∈ H

, then there exists

γ ∈

∗

such that γ(z) ∈ D.

Since

z 6∈ R

, we know

{cz

c, d ∈ Z}

is a discrete subgroup of

C. So we know

{|cz + d| : c, d ∈ Z}

is a discrete subset of

, and is in particular bounded away from 0. Thus, we

know



Im γ(z) =

Im(z)

|cz + d|

: γ =



a b

c d



∈

∗



is a discrete subset of

and is bounded above. Thus there is some

γ ∈

∗

with

Im γ

(

) maximal. Replacing

for suitable

, we may assume

|Re γ(z)| ≤

We consider the different possible cases.

– If |γ(z)| < 1, then

Im Sγ(z) = Im

−1

γ(z)

Im γ(z)

|γ(z)|

> Im γ(z),

which is impossible. So we know

|γ

(

)

| ≥

1. So we know

(

) lives in the

closure of D.

– If Re(γ(z)) = −

, then T γ(z) has real part +

, and so T (γ(z)) ∈ D.

–

−

< Re

(

)

0 and

|γ

(

)

= 1, then

|Sγ

(

)

= 1 and 0

< Re Sγ

(

)

i.e. Sγ(z) ∈ D.

So we can move it to somewhere in D.

We shall next show that if

z, z

∈ D

, and

(

) for

γ ∈

, then

Moreover, either

– γ = 1; or

– z = i and γ = S; or

– z = ρ and γ = T S or (T S)

It is clear that this proves everything.

To show this, we wlog

Im(z

) =

Im z

|cz + d|

≥ Im z

where

γ =



a b

c d



and we also wlog c ≥ 0.

Therefore we know that |cz + d| ≤ 1. In particular, we know

1 ≥ Im(cz + d) = c Im(z) ≥ c

√

since z ∈ D. So c = 0 or 1.

– If c = 0, then

γ = ±



1 m

0 1



for some

m ∈ Z

, and this

. But this is clearly impossible. So we

must have m = 0, z = z

, γ = 1 ∈ PSL

(Z).

–

= 1, then we know

d| ≤

1. So

is at distance 1 from an integer.

As z ∈ D, the only possibilities are d = 0 or −1.

◦ If d = 0, then we know |z| = 1. So

γ =



a −1

1 0



for some a ∈ Z. Then z

= a −

. Then

∗ either a = 0, which forces z = i, γ = S; or

∗ a = 1, and z

= 1 −

, which implies z = z

= ρ and γ = T S.

◦ If d = −1, then by looking at the picture, we see that z = ρ. Then

|cz + d| = |z − 1| = 1,

and so

Im z

= Im z =

√

So we have z

= ρ as well. So

aρ + b

ρ − 1

= ρ,

which implies

− (a + 1)ρ − b = 0

So ρ = −1, a = 0, and γ = (T S)

Note that this proof is the same as the proof of reduction theory for binary

positive definite binary quadratic forms.

What does the quotient

Γ \ N

look like? Each point in the quotient can be

identified with an element in

. Moreover,

and

identify the portions of

the boundary of

. Thinking hard enough, we see that the quotient space is

homeomorphic to a disk.

An important consequence of this is that the quotient Γ

has finite invariant

measure.

Proposition. The measure

dµ =

dx dy

is invariant under

PSL

(

). If Γ

⊆ PSL

(

) is of finite index, then

(Γ

)

< ∞

Proof. Consider the 2-form associated to µ, given by

η =

dx ∧ dy

idz ∧ d¯z

2(Im z)

We now let

γ =



a b

c d



∈ SL

(R).

Then we have

Im γ(z) =

Im z

|cz + d|

Moreover, we have

dγ(z)

a(cz + d) − c(az + b)

(cz + d)

Plugging these into the formula, we see that η is invariant under γ.

Now if

Γ ≤ PSL

(

) has finite index, then we can write

PSL

(

) as a union

of cosets

PSL

(Z) =

i=1

¯γγ

where n = (PSL

(Z) :

Γ). Then a fundamental domain for

Γ is just

[

i=1

(D),

and so

µ(

Γ \ H) =

µ(γ

D) = nµ(D).

So it suffices to show that µ(D) is finite, and we simply compute

µ(D) =

dx dy

≤

x=−

y=∞

√

2/2

dx dy

< ∞.

It is an easy exercise to show that we actually have

µ(D) =

We end with a bit terminology.

Definition

(Principal congruence subgroup)

For

N ≥

1, the principal congru-

ence subgroup of level N is

Γ(N) = {γ ∈ SL

(Z) : γ ≡ I (mod N)} = ker(SL

(Z) → SL

(Z/NZ)).

Any Γ

⊆ SL

(

) containing some Γ(

) is called a congruence subgroup, and its

level is the smallest N such that Γ ⊇ Γ(N)

This is a normal subgroup of finite index.

Definition (Γ

(N), Γ

(N)). We define

(N) =



a b

c d



∈ SL

(Z) : c ≡ 0 (mod N)



and

(N) =



a b

c d



∈ SL

(Z) : c ≡ 0, d ≡ 1 (mod N)



We similarly define Γ

(

) and Γ

(

) to be the transpose of Γ

(

) and Γ

(

)

respectively.

Note that “almost all” subgroups of

(

) are not congruence subgroups.

On the other hand, if we try to define the notion of congruence subgroups in

higher dimensions, we find that all subgroups of

(

) for

n >

2 are congruence!