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.
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