6 1. Modular Forms

See [DS05, §3.8] and Algorithm 8.12 below for more discussion of cusps

and results relevant to their enumeration.

In order to define modular forms for general congruence subgroups, we

next explain what it means for a function to be holomorphic on the extended

upper half plane

h∗

= h ∪

P1(Q).

See [Shi94, §1.3–1.5] for a detailed description of the correct topology

to consider on

h∗.

In particular, a basis of neighborhoods for α ∈ Q is given

by the sets {α}∪ D, where D is an open disc in h that is tangent to the real

line at α.

Recall from Section 1.2 that a weakly modular function f on SL2(Z) is

holomorphic at ∞ if its q-expansion is of the form

∑∞

n=0

anqn.

In order to make sense of holomorphicity of a weakly modular function f

for an arbitrary congruence subgroup Γ at any α ∈ Q, we first prove a lemma.

Lemma 1.14. If f : h → C is a weakly modular function of weight k for

a congruence subgroup Γ and if δ ∈ SL2(Z), then f

[δ]k

is a weakly modular

function for

δ−1Γδ.

Proof. If s =

δ−1γδ

∈

δ−1Γδ,

then

(f

[δ]k )[s]k

= f

[δs]k

= f

[δδ−1γδ]k

= f

[γδ]k

= f

[δ]k

.

Fix a weakly modular function f of weight k for a congruence subgroup

Γ, and suppose α ∈ Q. In Section 1.2 we constructed the q-expansion of

f by using that f(z) = f(z + 1), which held since T =

1 1

0 1

∈ SL2(Z).

There are congruence subgroups Γ such that T ∈ Γ. Moreover, even if we

are interested only in modular forms for Γ1(N), where we have T ∈ Γ1(N)

for all N, we will still have to consider q-expansions at infinity for modular

forms on groups

δ−1Γ1(N)δ,

and these need not contain T . Fortunately,

T N =

(

1 N

0 1

)

∈ Γ(N), so a congruence subgroup of level N contains T N .

Thus we have f(z + H) = f(H) for some positive integer H, e.g., H = N

always works, but may be a smaller choice of H. The minimal choice of

H 0 such that

(there

1 H

0 1

)

∈

δ−1Γδ,

where δ(∞) = α, is called the width of the

cusp α relative to the group Γ (see Section 1.4.1). When f is meromorphic

at infinity, we obtain a Fourier expansion

(1.3.2) f(z) =

∞

n=m

anqn/H