Cryptography Reference
In-Depth Information
(i) Show that (2
π
)
12
/
Δ=
q
−
1
n
=0
f
n
q
n
with
f
n
∈
Z
.
(j) Show that
q
+
∞
j
=
1
c
n
q
n
n
=0
with
c
n
∈
Z
.
9.2 Let
M
i
=
a
i
b
i
∈ SL
2
(
Z
)for
i
=1
,
2
,
3with
M
2
M
1
=
M
3
.Let
c
i
d
i
τ
1
∈F
.Let
τ
2
=
a
1
τ
1
+
b
1
τ
3
=
a
2
τ
2
+
b
2
c
1
τ
1
+
d
1
,
c
2
τ
2
+
d
2
.
Show that
τ
3
=
a
3
τ
1
+
b
3
c
3
τ
1
+
d
3
.
9.3 Let
k ≥
0 be an integer. Let
f
be a meromorphic function on the upper
half plane such that
f
has a
q
-expansion at
i∞
(as in Equation (9.15))
andsuchthat
f
aτ
+
b
cτ
+
d
=(
cτ
+
d
)
k
f
(
τ
)
for all
τ ∈H
and for all
ab
∈ SL
2
(
Z
). Show that
cd
2
ord
i
(
f
)+
z
ord
i∞
(
f
)+
1
3
ord
ρ
(
f
)+
1
ord
z
(
f
)=
k
12
.
=
i,ρ,i
∞
is the set of matrices
ab
9.4 The stabilizer in
SL
2
(
Z
)ofapoint
z
∈H
cd
such that (
az
+
b
)
/
(
cz
+
d
)=
z
.
(a) Show that the stabilizer of
i
has order 4.
(b) Show that the stabilizer of
ρ
has order 6.
(c) Show that the stabilizer of
i∞
consists of the matrices of the form
1
b
01
with
b
±
∈
Z
.
(d) Show that the stabilizer of each
z ∈H
has order at least 2.
It can be shown that the stabilizer of each element in the fundamental
domain
F
has order 2 except for
i
and
ρ
.
9.5 Let
E
:
y
2
=4
x
3
+
Ax
+
B
,with
A, B ∈
R
be an elliptic curve defined
over
R
. We know by Theorem 9.21 that
E
(
C
)
C
/L
for some lattice
L
. The goal of this exercise is to show that
L
has one of the two shapes
giveninpart(i)below.
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