1.2. Modular Forms of Level 1 3

Modular forms are holomorphic functions on h that transform in a par-

ticular way under a certain subgroup of SL2(Z). Before defining general

modular forms, we define modular forms of level 1.

1.2. Modular Forms of Level 1

Definition 1.4 (Weakly Modular Function). A weakly modular function of

weight k ∈ Z is a meromorphic function f on h such that for all γ =

(

a b

c d

)

∈

SL2(Z) and all z ∈ h we have

(1.2.1) f(z) = (cz +

d)−kf(γ(z)).

The constant functions are weakly modular of weight 0. There are no

nonzero weakly modular functions of odd weight (see Exercise 1.4), and it

is not obvious that there are any weakly modular functions of even weight

k ≥ 2 (but there are, as we will see!). The product of two weakly modular

functions of weights k1 and k2 is a weakly modular function of weight k1 +k2

(see Exercise 1.3).

When k is even, (1.2.1) has a possibly more conceptual interpretation;

namely (1.2.1) is the same as

f(γ(z))(d(γ(z)))k/2

=

f(z)(dz)k/2.

Thus (1.2.1) simply says that the weight k “differential form”

f(z)(dz)k/2

is

fixed under the action of every element of SL2(Z).

By Theorem 1.2, the group SL2(Z) is generated by the matrices S and

T of (1.1.2), so to show that a meromorphic function f on h is a weakly

modular function, all we have to do is show that for all z ∈ h we have

(1.2.2) f(z + 1) = f(z) and f(−1/z) =

zkf(z).

Suppose f is a weakly modular function of weight k. A Fourier expansion

of f, if it exists, is a representation of f as f(z) =

∑

∞

n=m

ane2πinz,

for all

z ∈ h. Let q = q(z) =

e2πiz,

which we view as a holomorphic function on

C. Let D be the open unit disk with the origin removed, and note that

q defines a map h → D . By (1.2.2) we have f(z + 1) = f(z), so there is

a function F : D → C such that F(q(z)) = f(z). This function F is a

complex-valued function on D , but it may or may not be well behaved at 0.

Suppose that F is well behaved at 0, in the sense that for some m ∈ Z

and all q in a neighborhood of 0 we have the equality

(1.2.3) F(q) =

∞

n=m

anqn.