1.4. Remarks on Congruence Subgroups 7

in powers of the function

q1/H

=

e2πiz/H

. We say that f is holomorphic at

∞ if in (1.3.2) we have m ≥ 0.

What about the other cusps α ∈

P1(Q)?

By Lemma 1.12 there is a

γ ∈ SL2(Z) such that γ(∞) = α. We declare f to be holomorphic at the

cusp α if the weakly modular function f

[γ]k

is holomorphic at ∞.

Definition 1.15 (Modular Form). A modular form of integer weight k for

a congruence subgroup Γ is a weakly modular function f : h → C that is

holomorphic on

h∗.

We let Mk(Γ) denote the space of weight k modular

forms of weight k for Γ.

Proposition 1.16. If a weakly modular function f is holomorphic at a set

of representative elements for C(Γ), then it is holomorphic at every element

of

P1(Q).

Proof. Let c1,...,cn ∈

P1(Q)

be representatives for the set of cusps for

Γ. If α ∈

P1(Q),

then there is γ ∈ Γ such that α = γ(ci) for some i. By

hypothesis f is holomorphic at ci, so if δ ∈ SL2(Z) is such that δ(∞) = ci,

then f

[δ]k

is holomorphic at ∞. Since f is a weakly modular function for Γ,

(1.3.3) f

[δ]k

= (f

[γ]k )[δ]k

= f

[γδ]k

.

But γ(δ(∞)) = γ(ci) = α, so (1.3.3) implies that f is holomorphic at α.

1.4. Remarks on Congruence Subgroups

Recall that a congruence subgroup is a subgroup of SL2(Z) that contains

Γ(N) for some N. Any congruence subgroup has finite index in SL2(Z),

since Γ(N) does. What about the converse: is every finite index subgroup

of SL2(Z) a congruence subgroup? This is the congruence subgroup problem.

One can ask about the congruence subgroup problem with SL2(Z) replaced

by many similar groups. If p is a prime, then one can prove that every finite

index subgroup of SL2(Z[1/p]) is a congruence subgroup (i.e., contains the

kernel of reduction modulo some integer coprime to p), and for any n 2, all

finite index subgroups of SLn(Z) are congruence subgroups (see [Hum80]).

However, there are numerous finite index subgroups of SL2(Z) that are not

congruence subgroups. The paper [Hsu96] contains an algorithm to decide

if certain finite index subgroups are congruence subgroups and gives an

example of a subgroup of index 12 that is not a congruence subgroup.

One can consider modular forms even for noncongruence subgroups. See,

e.g., [Tho89] and the papers it references for work on this topic. We will not

consider such modular forms further in this book. Note that modular sym-

bols (which we define later in this book) are computable for noncongruence

subgroups.