Cryptography Reference
In-Depth Information
φ
)
∗
=
6. If ψ
:
C
2
→
C
3
is another non-constant rational map of curves over
k
then
(
ψ
◦
φ
∗
◦
ψ
∗
and
(
ψ
◦
φ
)
∗
=
ψ
∗
◦
φ
∗
.
Proof
Most of the claims follow from earlier results such as Lemma
8.2.6
, Theorem
8.2.9
and Lemma
8.2.12
; also see Proposition II.3.6 of Silverman [
505
]. Claim 4 is harder; see
Proposition VII.7.8 and Lemma VII.7.16 of Lorenzini [
355
].
Exercise 8.3.9
Give all the details in the proof of Theorem
8.3.8
.
Corollary 8.3.10
Let φ
:
C
1
→
C
2
be a non-constant morphism of curves over
k
. Then the
induced maps φ
∗
:Pic
0
k
Pic
0
k
(
C
1
)
and φ
∗
:Pic
0
Pic
0
k
→
→
(
C
2
)
(
C
1
)
(
C
2
)
on divisor class
k
groups are well-defined group homomorphisms.
Proof
The maps
φ
∗
and
φ
∗
are well-defined on divisor classes by parts 2 and 4 of
Theorem
8.3.8
. The homomorphic property follows from the linearity of the definitions.
Exercise 8.3.11
Show that if
φ
:
C
1
→
k
C
2
is an isomorphism of curves over
then
(
C
1
)
=
Pic
0
k
Pic
k
(
C
2
) (isomorphic as groups). Give an example to show that the converse is
not true.
A further corollary of this result is that a rational map
φ
:
E
1
→
E
2
between
elliptic curves such that
φ
(
O
E
1
)
=
O
E
2
is automatically a group homomorphism (see
Theorem
9.2.1
).
P
1
→ P
1
=
(
x
2
/z
2
=
−
Exercise 8.3.12
Let
φ
:
be defined by
φ
((
x
:
z
))
:1).Let
D
(
1:
1)
+
(0 : 1). Compute
φ
∗
(
D
),
φ
∗
(
D
),
φ
∗
φ
∗
(
D
) and
φ
∗
φ
∗
(
D
).
We now make an observation that was mentioned when we defined
φ
∗
on
(1 : 0)
−
k
(
C
1
).
Lemma 8.3.13
Let
φ
:
C
1
→
C
2
be a non-constant morphism of cur
v
es over
k
. Let f
∈
(
C
1
)
∗
and Q
k
∈
C
2
(
k
)
. Suppose that v
P
(
f
)
=
0
for all points P
∈
C
1
(
k
)
such that φ
(
P
)
=
Q. Then
f
(
P
)
e
φ
(
P
)
f
(
φ
∗
(
Q
))
.
N
k
(
C
1
)
/φ
∗
(
k
(
C
2
))
(
f
)(
Q
)
=
=
P
∈
C
1
(
k
=
Q
):
φ
(
P
)
Another formulation would be: f of conorm of Q equals norm of f at Q.
Proo
f
(Sketch) This uses similar ideas to the proof of part 4 of Theorem
8.3.8
.Wework
over
.
As always,
k
(
C
1
) is a finite extension of
φ
∗
(
φ
∗
(
k
k
(
C
2
)). Let
A
=
O
Q
(
C
2
)) and let
B
be
(
C
1
). Then
B
is
a Dedekind domain and the ideal
φ
∗
(m
Q
)
the integral closure of
A
in
k
splits as a product
i
m
e
φ
(
P
i
)
where
P
i
∈
C
1
(
k
) are distinct points such that
φ
(
P
i
)
=
Q
.
P
i
By assumption,
f
has no poles at
P
i
and so
f
∈
B
. Note that
f
(
P
i
)
=
c
i
∈ k
if and only
if
f
≡
c
i
(mod m
P
i
). Hence, the right-hand side is
c
e
φ
(
P
i
)
i
f
(
P
i
)
e
φ
(
P
i
)
(
f
(mod m
P
i
))
e
φ
(
P
i
)
.
=
=
i
i
i
