Lecture 8
The Modular Curve
8.1
In this lecture we shall give an explicit formula for the dimension of the spaces
M
k
(Γ), where Γ is any subgroup of finite index in SL(2
,
Z
). For this we have to apply
some techinique from algebraic geometry. We shall start with equipping
H
*
/
Γ with a
structure of a compact Riemann surface.
Let Γ be a subgroup of SL(2
,
R
). We say that Γ is a
discrete subgroup
if the usual
topology in SL(2
,
R
) (considered as a subset of
R
4
) induces a discrete topology in
Γ. The latter means that any point of Γ is an open subset in the induced topology.
Obviously SL(2
,
Z
) is a discrete subgroup of SL(2
,
R
). We shall consider the natural
action of SL(2
,
R
) on the upper halfplane
H
by Moebius transformations.
Lemma 8.1.
Any discrete subgroup
Γ
of
SL(2
,
R
)
acts on
H
properly discontinuously.
Proof.
Observe that the group SL(2
,
R
) acts transitively on
H
(view the latter as a
subset of
R
2
of vectors with positive second coordinate).
For any point
z
∈ H
the
stabilizer group is conjugate to the stabilizer of say
z
=
i
.
The latter consists of
matrices (
a b
c d
)
∈
SL(2
,
R
) such that
a
=
d, b
=

c
.
It follows that this group is
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 Fall '09
 ONTONKONG
 Algebra, Geometry, Topology, γ, General topology, discrete subgroup

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