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.
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