1.2. INTEGER WEIGHT MODULAR FORMS 3
1.2. Integer weight modular forms
(R) = γ =
: a, b, c, d ∈ R and ad − bc 0
acts on functions f(z) : H → C. In particular, suppose that γ =
(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
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
az + b
cz + d
= (cz +
for all z ∈ H and all
(2) If γ0 ∈ SL2(Z), then (f|kγ0) (z) has a Fourier expansion of the form
(f|kγ0) (z) =
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
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
∈ 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.