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
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) =
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.
→ 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 =
. If f(q) is a cusp form of weight 2, then
2πif(z)dz = f(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
∈ SL2(Z) :