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86
LECTURE 8. THE MODULAR CURVE
function
φ
(
N
). The number of elements
b
is
N
. This gives the index of Γ(
N
) in
Γ
0
(
N
). The index of Γ
0
(
N
) in Γ(1) is equal to the number of elements in the orbit of
(1
,
0). It is the set of pairs (
a,b
)
∈
Z
/N
which are coprime modulo
N
. This is easy to
compute.
Lemma 8.6.
There are no elliptic points for
Γ(
N
)
if
N >
1
. The number of cusps is
equal to
μ
N
/N
. Each of them is of order
N
.
Proof.
The subgroup Γ = Γ(
N
) is normal in Γ(1). If Γ
τ
6
=
{±
1
}
, then
g
Γ
g

1
= Γ
i
for any
g
∈
Γ(1) which sends
τ
to
i
. Similarly for elliptic points of order 3 we get a
subgroup of Γ ﬁxing
e
2
πi/
3
. It is easy to see that only the matrices 1 or

1 , if
N
= 2,
from Γ(
N
) satisfy this property. We leave to the reader to prove the assertion about
the cusps.
Next computation will be given without proof. The reader is referred to
[Shimura]
.
Lemma 8.7.
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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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