This preview shows page 1. Sign up to view the full content.
50
LECTURE 6. MODULAR FORMS
of Γ in the space of holomorphic functions on
H
deﬁned by
ρ
(
g
)(
φ
(
z
)) =
φ

k
g

1
.
(6.6)
Note that we switched here to
g

1
in order to get
ρ
(
gg
0
) =
ρ
(
g
)
◦
ρ
(
g
0
)
.
It follows from the above that to check (6.1) for some subgroup Γ it is enough to
verify it only for generators of Γ. Now we use the following:
Lemma 6.1.
The group
G
=
P
SL(2
,
Z
) = SL(2
,
Z
)
/
{±}
is generated by the matrices
S
=
„
0

1
1
0
«
,
T
=
„
1
1
0
1
«
.
These matrices satisfy the relations
S
2
= 1
,
(
ST
)
3
= 1
.
Proof.
We know that the modular ﬁgure
D
(more exactly its subset
D
0
) is a fundamen
tal domain for the action of
G
in the upper halfplane
H
by Moebius transformations.
Take some interior point
z
0
∈ D
and any
g
∈
G
. Let
G
0
be the subgroup of
G
gener
ated by
S
and
T
. If we ﬁnd an element
g
0
∈
G
such that
g
0
g
·
z
0
belongs to
D
, then
g
0
g
= 1 and hence
g
∈
G
0
. Let us do it. First ﬁnd
g
0
∈
G
0
such that Im (
g
0
·
(
g
·
z
0
)) is
maximal possible. We have, for any
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 01/08/2012 for the course MATH 300 taught by Professor Ontonkong during the Fall '09 term at SUNY Stony Brook.
 Fall '09
 ONTONKONG

Click to edit the document details