4 1. Modular Forms

If this is the case, we say that f is meromorphic at ∞. If, moreover, m ≥ 0,

we say that f is holomorphic at ∞. We also call (1.2.3) the q-expansion of f

about ∞.

Definition 1.5 (Modular Function). A modular function of weight k is a

weakly modular function of weight k that is meromorphic at ∞.

Definition 1.6 (Modular Form). A modular form of weight k (and level 1)

is a modular function of weight k that is holomorphic on h and at ∞.

If f is a modular form, then there are numbers an such that for all z ∈ h,

(1.2.4) f(z) =

∞

n=0

anqn.

Proposition 1.7. The above series converges for all z ∈ h.

Proof. The function f(q) is holomorphic on D, so its Taylor series converges

absolutely in D.

Since

e2πiz

→ 0 as z → i∞, we set f(∞) = a0.

Definition 1.8 (Cusp Form). A cusp form of weight k (and level 1) is a

modular form of weight k such that f(∞) = 0, i.e., a0 = 0.

Let C[[q]] be the ring of all formal power series in q. If k = 2, then

dq = 2πiqdz, so dz =

1

2πi

dq

q

. If f(q) is a cusp form of weight 2, then

2πif(z)dz = f(q)

dq

q

=

f(q)

q

dq ∈ C[[q]]dq.

Thus the differential 2πif(z)dz is holomorphic at ∞, since q is a local pa-

rameter at ∞.

1.3. Modular Forms of Any Level

In this section we define spaces of modular forms of arbitrary level.

Definition 1.9 (Congruence Subgroup). A congruence subgroup of SL2(Z)

is any subgroup of SL2(Z) that contains

Γ(N) = Ker(SL2(Z) → SL2(Z/NZ))

for some positive integer N. The smallest such N is the level of Γ.

The most important congruence subgroups in this book are

Γ1(N) =

a b

c d

∈ SL2(Z) :

a b

c d

≡

1 ∗

0 1

(mod N)