12 1. Modular Forms

(c) If f and g are modular functions, show that fg is a modular

function for Γ.

(d) If f and g are modular forms, show that fg is a modular form

for Γ.

1.4 Suppose f is a weakly modular function of odd weight k and level

Γ0(N) for some N. Show that f = 0.

1.5 Prove that SL2(Z) = Γ0(1) = Γ1(1) = Γ(1).

1.6 (a) Prove that Γ1(N) is a group.

(b) Prove that Γ1(N) has finite index in SL2(Z) (Hint: It contains

the kernel of the homomorphism SL2(Z) → SL2(Z/NZ).)

(c) Prove that Γ0(N) has finite index in SL2(Z).

(d) Prove that Γ0(N) and Γ1(N) have level N.

1.7 Let k be an integer, and for any function f :

h∗

→ C and γ =

(

a b

c d

)

∈ GL2(Q), set f

[γ]k

(z) =

det(γ)k−1

· (cz +

d)−k

· f(γ(z)).

Prove that if γ1,γ2 ∈ GL2(Z), then for all z ∈

h∗

we have

f

[γ1γ2]k

(z) = ((f

[γ1]k )[γ2]k

)(z).

1.8 Prove that for any α, β ∈

P1(Q),

there exists γ ∈ SL2(Z) such that

γ(α) = β.

1.9 Prove Proposition 1.13, which asserts that the set of cusps C(Γ),

for any congruence subgroup Γ, is finite.

1.10 Use Algorithm 1.19 to give an example of a group Γ and cusp α

with width 2.