Cryptography Reference
In-Depth Information
There is a natural map
φ
∗
on divisors that is called the
pushforward
(it is called the
divisor-norm map
in Section VII.7 of Lorenzini [
355
]).
Definition 8.3.4
Let
φ
:
C
1
→
C
2
be a non-constant morphism of curves. Define the
pushforward
φ
∗
:Div
k
(
C
1
)
→
Div
(
C
2
)
k
by
φ
∗
(
P
)
=
φ
(
P
) and extend to the whole of Div
(
C
1
) by linearity.
k
(
C
2
) that corresponds (in the sense of property
4 of Theorem
8.3.8
) to the pushforward. This is achieved using the norm map with respect
to the extension
It remains to find a map from
k
(
C
1
)to
k
(
C
1
)
/φ
∗
(
k
k
(
C
2
)). As we will show in Lemma
8.3.13
, this norm satisfies,
=
φ
(
P
)
=
Q
f
(
P
)
e
φ
(
P
)
.
for
f
∈ k
(
C
1
) and
Q
∈
C
2
(
k
), N
k
(
C
1
)
/φ
∗
(
k
(
C
2
))
(
f
)(
Q
)
Definition 8.3.5
Let
C
1
,C
2
be curves over
C
2
be a non-constant rational
map. Let
N
k
(
C
1
)
/φ
∗
k
(
C
2
)
be the usual norm map in field theory (see Section
A.6
). Define
k
and let
φ
:
C
1
→
k
→ k
φ
∗
:
(
C
1
)
(
C
2
)
(
φ
∗
)
−
1
(N
k
(
C
1
)
/φ
∗
(
k
(
C
2
))
(
f
)).
by
φ
∗
(
f
)
=
φ
∗
(
Note that the definition of
φ
∗
(
f
) makes sense since N
k
(
C
1
)
/φ
∗
k
(
C
2
)
(
f
)
∈
k
(
C
2
)) and
so is of the form
h
◦
φ
for some
h
∈ k
(
C
2
). So
φ
∗
(
f
)
=
h
.
1
x
2
. Then
Example 8.3.6
Let
C
1
=
C
2
= A
and
φ
:
C
1
→
C
2
be given by
φ
(
x
)
=
φ
∗
(
(
x
2
) and
φ
∗
(
x
2
/
(
x
k
(
C
2
))
= k
k
(
C
1
)
=
k
(
C
2
))(
x
). Let
f
(
x
)
=
−
1). Then
x
2
x
)
2
x
4
(
−
N
k
(
C
1
)
/φ
∗
k
(
C
2
)
(
f
)
=
f
(
x
)
f
(
−
x
)
=
1)
=
1
,
(
x
−
1)
(
−
x
−
−
x
2
+
X
2
/
(1
which is
h
◦
φ
for
h
(
X
)
=
−
X
). Hence,
φ
∗
(
f
(
x
))
=−
f
(
x
).
V
(
y
2
x
2
2
,
C
2
= A
1
Exercise 8.3.7
Let
C
1
=
=
+
1)
⊆ A
and let
φ
:
C
1
→
C
2
be given
by
φ
(
x,y
)
=
x
.Let
f
(
x,y
)
=
x/y
. Show that
−
x
2
N
k
(
C
1
)
/φ
∗
k
(
C
2
)
(
f
)
=
x
2
+
1
X
2
/
(
X
2
and so
φ
∗
(
f
)
=
h
(
X
) where
h
(
X
)
=−
+
1).
We now state the main properties of the pullback and pushforward.
Theorem 8.3.8
Let φ
:
C
1
→
C
2
be a non-constant morphism of curves over
k
. Then:
1.
deg(
φ
∗
(
D
))
=
deg(
φ
)deg(
D
)
for all D
∈
Div(
C
2
)
.
2. φ
∗
(div(
f
))
div(
φ
∗
f
)
for all f
(
C
2
)
∗
.
=
∈ k
3.
deg(
φ
∗
(
D
))
=
deg(
D
)
for all D
∈
Div(
C
1
)
.
(
C
1
)
∗
.
4. φ
∗
(div(
f
))
=
div(
φ
∗
(
f
))
for all f
∈ k
5. φ
∗
(
φ
∗
(
D
))
=
deg(
φ
)
D for D
∈
Div(
C
2
)
.