Cryptography Reference
In-Depth Information
Definition 8.5.30
Any divisor div(
ω
) is called a
canonical divisor
.Theset
{
div(
ω
):
ω
∈
k
(
C
)
}
is the
canonical divisor class
.
1
.
Example 8.5.31
We determine the canonical class of
C
= P
Let
ω
=
dx
. Since
x
is a uniformiser at the point 0 we have
v
0
(
ω
)
=
v
0
(
dx/dx
)
=
0.
More generally, for
P
∈ k
we have (
x
−
P
) a uniformiser and
v
P
(
ω
)
=
v
P
(
dx/d
(
x
−
x
−
2
)
dx
so
v
∞
(
ω
)
P
))
=
v
P
(1)
=
0. Finally, a uniformiser at
∞
is
t
=
1
/x
and
dt
=
(
−
=
x
2
)
v
∞
(
−
=−
2. Hence, div(
ω
)
=−
2
∞
and the degree of div(
ω
)is-2.
Example 8.5.32
We determine the divisor of a differential on an elliptic curve
E
in
Weierstrass form. Rather than computing div(
dx
) it is easier to compute div(
ω
)for
dx
ω
=
a
3
.
2
y
+
a
1
x
+
Let
P
∈
E
(
k
). There are three cases, if
P
=
O
E
then one can take uniformiser
t
=
x/y
,if
P
=
(
x
P
,y
P
)
=
ι
(
P
) then take uniformiser (
y
−
y
P
) (and note that
v
P
(2
y
+
a
1
x
+
a
3
)
=
1
in this case) and otherwise take uniformiser (
x
−
x
P
) and note that
v
P
(2
y
+
a
1
x
+
a
3
)
=
0.
We deal with the general case first. Since
dx/d
(
x
−
x
P
)
=
∂x/∂
(
x
−
x
P
)
=
1 it follows
t
−
2
f
and
y
t
−
3
h
for some functions
that
v
P
(
ω
)
=
0. For the case
P
=
O
E
write
x
=
=
f,h
∈ k
(
E
) regular at
O
E
and with
f
(
O
E
)
,h
(
O
E
)
=
0. One can verify that
2
t
−
3
f
t
−
2
f
tf
ω
dt
=
−
+
−
2
f
+
a
3
=
2
t
−
3
h
+
a
1
t
−
2
f
+
2
h
+
a
1
tf
+
a
3
t
3
and so
v
O
E
(
ω
)
=
0. Finally, when
P
=
ι
(
P
) we must consider
dx
1
∂y/∂x
=
2
y
+
a
1
x
+
a
3
y
P
)
=
a
4
.
d
(
y
−
3
x
2
+
2
a
2
x
+
=
(1
/
(3
x
2
+
+
−
It follows that
ω
2
a
2
x
a
4
))
d
(
y
y
P
) and, since
P
is not a singular point,
3
x
P
+
0.
In other words, we have shown that div(
ω
)
2
a
2
x
P
+
a
4
=
0 and so
v
P
(
ω
)
=
=
0. One can verify that
div(
dx
)
=
(
P
1
)
+
(
P
2
)
+
(
P
3
)
−
3(
O
E
)
where
P
1
,P
2
,P
3
are the three non-trivial points of order 2 in
E
(
k
).
Exercise 8.5.33
Show that
dx
dy
a
3
=
2
y
+
a
1
x
+
3
x
2
+
2
a
2
x
+
a
4
−
a
1
y
on an elliptic curve.
Definition 8.5.34
Let
φ
:
C
1
→
C
2
be a non-constant morphism of curves over
k
. Define
the function
φ
∗
:
k
(
C
2
)
→
k
(
C
1
)by
φ
∗
(
fdx
)
φ
∗
(
f
)
d
(
φ
∗
(
x
))
.
=
