1.2. INTEGER WEIGHT MODULAR FORMS 3

1.2. Integer weight modular forms

The group

GL2

+

(R) = γ =

a b

c d

: a, b, c, d ∈ R and ad − bc 0

acts on functions f(z) : H → C. In particular, suppose that γ =

a b

c d

∈

GL2

+

(R). If f(z) is a meromorphic function on H and k is an integer, then define

the “slash” operator |k by

(1.3) (f|kγ) (z) := (det

γ)k/2(cz

+

d)−kf(γz),

where

γz :=

az + b

cz + d

.

Definition 1.8. Suppose that f(z) is a meromorphic function on H, that

k ∈ Z, and that Γ is a congruence subgroup of level N. Then f(z) is called a

meromorphic modular form with integer weight k on Γ if the following hold:

(1) We have

f

az + b

cz + d

= (cz +

d)kf(z)

for all z ∈ H and all

a b

c d

∈ Γ.

(2) If γ0 ∈ SL2(Z), then (f|kγ0) (z) has a Fourier expansion of the form

(f|kγ0) (z) =

n≥nγ0

aγ0 (n)qN

n

,

where qN := e2πiz/N and aγ0 (nγ0 ) = 0.

If k = 0, then f(z) is known as a modular function on Γ.

Remark 1.9. Note that condition (2) in Definition 1.8 can refer to a Fourier

series involving fractional powers of qN . More specifically, for “irregular cusps”

these series can be thought of as expansions in

qN/2. 1

The reader should consult

page 29 of [Shi1] for a detailed discussion.

Remark 1.10. Condition (2) of Definition 1.8 means that f(z) is meromorphic

at the cusps of Γ. If nγ0 ≥ 0 (resp. nγ0 0) for each γ0 ∈ SL2(Z), then we say

that f(z) is holomorphic (resp. vanishes) at the cusps of Γ.

Remark 1.11. Since

−1 0

0 −1

∈ Γ0(N), there are no nonzero meromorphic

modular forms of odd weight k on Γ0(N).

Definition 1.12. Suppose that f(z) is an integer weight meromorphic modular

form on a congruence subgroup Γ. We say that f(z) is a holomorphic modular (resp.

cusp) form if f(z) is holomorphic on H and is holomorphic (resp. vanishes) at the

cusps of Γ. We say that f(z) is a weakly holomorphic modular form if its poles (if

there are any) are supported at the cusps of Γ.