1.3. Modular Forms of Any Level 5

and

Γ0(N) =

a b

c d

∈ SL2(Z) :

a b

c d

≡

∗ ∗

0 ∗

(mod N) ,

where ∗ means any element. Both groups have level N (see Exercise 1.6).

Let k be an integer. Define the weight k right action of GL2(Q) on the

set of all functions f : h → C as follows. If γ =

(

a b

c d

)

∈ GL2(Q), let

(1.3.1) (f

[γ]k

)(z) =

det(γ)k−1(cz

+

d)−kf(γ(z)).

Proposition 1.10. Formula (1.3.1) defines a right action of GL2(Z) on the

set of all functions f : h → C; in particular,

f

[γ1γ2]k

= (f

[γ1]k )[γ2]k

.

Proof. See Exercise 1.7.

Definition 1.11 (Weakly Modular Function). A weakly modular function of

weight k for a congruence subgroup Γ is a meromorphic function f : h → C

such that f

[γ]k

= f for all γ ∈ Γ.

A central object in the theory of modular forms is the set of cusps

P1(Q)

= Q ∪ {∞}.

An element γ =

(

a b

c d

)

∈ SL2(Z) acts on

P1(Q)

by

γ(z) =

az+b

cz+d

if z = ∞,

a

c

if z = ∞.

Also, note that if the denominator c or cz + d is 0 above, then

γ(z) = ∞ ∈

P1(Q).

The set of cusps for a congruence subgroup Γ is the set C(Γ) of Γ-orbits

of

P1(Q).

(We will often identify elements of C(Γ) with a representative

element from the orbit.) For example, the lemma below asserts that if

Γ = SL2(Z), then there is exactly one orbit, so C(SL2(Z)) = {[∞]}.

Lemma 1.12. For any cusps α, β ∈

P1(Q)

there exists γ ∈ SL2(Z) such

that γ(α) = β.

Proof. This is Exercise 1.8.

Proposition 1.13. For any congruence subgroup Γ, the set C(Γ) of cusps

is finite.

Proof. This is Exercise 1.9.