Cryptography Reference
In-Depth Information
Modular curves
Recall that
SL
2
(
Z
) acts on the upper half plane
H
by linear fractional
transformations:
ab
cd
τ
=
aτ
+
b
cτ
+
d
.
The fundamental domain
F
for this action is described in Section 9.3. The
subgroup Γ
0
(
N
) (defined by the condition that
c ≡
0(mod
N
)) also acts
on
H
.The
modular curve
X
0
(
N
) is defined over
C
by taking the upper
half plane modulo the action of Γ
0
(
N
), and then adding finitely many points,
called cusps, to make
X
0
(
N
) compact. We obtain a fundamental domain
D
for Γ
0
(
N
)bywriting
∪
i
γ
i
Γ
0
(
N
)
for some coset representatives
γ
i
and letting
D
=
∪
i
γ
−
1
SL
2
(
Z
)=
F
. Certain edges of
this fundamental domain are equivalent under the action of Γ
0
(
N
). When
equivalent edges are identified, the fundamental domain gets bent around to
form a surface. There is a hole in the surface corresponding to
i
i
,andthere
are also finitely many holes corresponding to points where the fundamental
domain touches the real axis. These holes are filled in by points, called
cusps
,
to obtain
X
0
(
N
). It can be shown that
X
0
(
N
) can be represented as an
algebraic curve defined over
Q
.
Figure 15.1 gives a fundamental domain for Γ
0
(2). The three pieces are
obtained as
γ
−
1
∞
F
,where
γ
1
=
10
i
,
2
=
0
,
3
=
11
.
1
10
−
01
−
10
The modular curve
X
0
(
N
) has another useful description, which works over
arbitrary fields
K
with the characteristic of
K
not dividing
N
. Consid
er
pairs
(
E,C
), where
E
is an elliptic
cu
rve (defined over the algebraic closure
K
)and
C
is a cyclic subgroup of
E
(
K
)oforder
N
. The set of such
pa
irs is in one-
to-one correspondence with the noncuspidal points of
X
0
(
N
)(
K
). Of course,
it is not obvious that this collection of pairs can be given the structure of an
algebraic curve in a natural way. This takes some work.
Example 15.1
When
K
=
C
, we can see this one-to-one correspondence as follows.
An
elliptic curve can be represented as
E
τ
=
C
/
(
Z
τ
+
Z
)
,
with
τ
∈H
, the upper half plane. The set
C
τ
=
0
,
N
, ...,
N
−
1
1
N
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