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
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