1.3. Modular Forms of Any Level 5
and
Γ0(N) =
a b
c d
SL2(Z) :
a b
c d


0
(mod N) ,
where means any element. Both groups have level N (see Exercise 1.6).
Let k be an integer. Define the weight k right action of GL2(Q) on the
set of all functions f : h C as follows. If γ =
(
a b
c d
)
GL2(Q), let
(1.3.1) (f
[γ]k
)(z) =
det(γ)k−1(cz
+
d)−kf(γ(z)).
Proposition 1.10. Formula (1.3.1) defines a right action of GL2(Z) on the
set of all functions f : h C; in particular,
f
[γ1γ2]k
= (f
[γ1]k )[γ2]k
.
Proof. See Exercise 1.7.
Definition 1.11 (Weakly Modular Function). A weakly modular function of
weight k for a congruence subgroup Γ is a meromorphic function f : h C
such that f
[γ]k
= f for all γ Γ.
A central object in the theory of modular forms is the set of cusps
P1(Q)
= Q {∞}.
An element γ =
(
a b
c d
)
SL2(Z) acts on
P1(Q)
by
γ(z) =
az+b
cz+d
if z = ∞,
a
c
if z = ∞.
Also, note that if the denominator c or cz + d is 0 above, then
γ(z) =
P1(Q).
The set of cusps for a congruence subgroup Γ is the set C(Γ) of Γ-orbits
of
P1(Q).
(We will often identify elements of C(Γ) with a representative
element from the orbit.) For example, the lemma below asserts that if
Γ = SL2(Z), then there is exactly one orbit, so C(SL2(Z)) = {[∞]}.
Lemma 1.12. For any cusps α, β
P1(Q)
there exists γ SL2(Z) such
that γ(α) = β.
Proof. This is Exercise 1.8.
Proposition 1.13. For any congruence subgroup Γ, the set C(Γ) of cusps
is finite.
Proof. This is Exercise 1.9.
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