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78
LECTURE 8. THE MODULAR CURVE
Example
8.1
.
Let Γ = Γ(1). Let us show that there exists a holomorphic isomorphism
H
/
SL(2
,
Z
)
∼
=
C
.
This shows that the set of isomorphism classes of elliptic curves has a natural structure
of a complex manifold of dimension 1 isomorphic to the complex plane
C
. Since
g
3
2
and Δ are of the same weight, the map
H →
P
1
(
C
)
,
τ
→
(
g
2
(
τ
)
3
,
Δ(
τ
))
is a well deﬁned holomorphic map. Obviously it is constant on any orbit of Γ(1), hence
factors through a holomorphic map
f
:
H
/
Γ(1)
→
P
1
(
C
)
.
Since Δ does not vanish on
H
, its image is contained in
P
1
(
C
)
\ {∞}
=
C
. I claim
that
f
is onetoone onto
C
. In fact, for any complex number
c
the modular form
f
=
g
3
2

c
Δ is of weight 6. It follows from Lemma 7.1 that
f
has either one simple
zero, or one zero of multiplicity 2 at the elliptic point of order 2, or a triple zero at
the elliptic point of index 3. This shows that each
c
∈
Z
occurs in the image of
j
on
H
/
Γ and only once.
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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

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