Cryptography Reference
In-Depth Information
U
) in the integral closure of
φ
∗
(
(for
Q
(
C
1
) yields the results. Details of this
approach are given in Section VII.5 of Lorenzini [
355
], Section I.4 of Serre [
488
] (especially
Propositions I.10 and I.11), Chapter 1 of Lang [
327
] and Chapter XII of Lang [
329
]. An
analogous ring-theoretic formulation is used in Proposition II.6.9 of Hartshorne [
252
]. A
different method is to study extensions of valuations directly
, f
or example see Section III.1
of Stichtenoth [
529
]. Note that, since we consider points over
∈
k
[
U
]) in
k
, the notion of residue degree
does not arise, which simplifies the presentation compared with many texts.
k
Definition 8.2.1
Let
F
2
be a field of transcendence degree 1 over
.Let
F
1
/F
2
be a finite
extension. Let
v
be a discrete valuation on
F
2
. A valuation
v
on
F
1
is an
extension
of
v
(or,
v
is the
restriction
of
v
)if
k
F
2
:
v
(
f
)
. We write
v
|
{
f
∈
F
2
:
v
(
f
)
≥
0
}={
f
∈
≥
0
}
v
if this is the case.
Note that if
v
is an extension of
v
as above then one does not necessarily have
v
(
f
)
=
F
2
(indeed, we will see later that
v
(
f
)
v
(
f
) for all
f
∈
=
ev
(
f
)forsome
e
∈ N
).
φ
∗
(
(
C
1
). We
now explain the relation between extensions of valuations from
F
2
to
F
1
and preimages of
points under
φ
.
Let
φ
:
C
1
→
C
2
be a morphism of curves and let
F
2
=
k
(
C
2
)) and
F
1
= k
Lemma 8.2.2
Let φ
:
C
1
→
k
C
2
be a non-constant morphism of
c
urves over
(t
hi
s is short-
k
∈
k
∈
k
hand for C
1
,C
2
and φ all being defined over
). Let P
C
1
(
)
and Q
C
2
(
)
. Denote
by v the valuation on φ
∗
(
k
(
C
2
))
⊆ k
(
C
1
)
defined by v
(
φ
∗
(
f
))
=
v
Q
(
f
)
for f
∈ k
(
C
2
)
.If
φ
(
P
)
=
Q then v
P
is an extension of v.
Q
we have
φ
∗
(
f
)
Proof
Let
f
∈ k
(
C
2
). Since
φ
(
P
)
=
=
f
◦
φ
regular at
P
if and only if
f
is regular at
Q
. Hence,
v
P
(
φ
∗
(
f
))
≥
0 if and only if
v
Q
(
f
)
≥
0. It follows that
v
P
|
v
.
Le
m
ma 8.2.3
Let the notation be as in Lemm
a
8.2.2
. In particular, P
∈
C
1
(
k
)
,
Q
∈
(
C
1
)
andv is the valuation onφ
∗
(
C
2
(
k
)
,v
P
is the correspo
n
ding valuation onF
1
= k
k
(
C
2
))
corresponding to v
Q
on
k
(
C
2
)
. Then v
P
|
v implies φ
(
P
)
=
Q.
In other words, Lemmas
8.2.2
and
8.2.3
show that
φ
(
P
)
=
Q
if and only if the maximal
(
C
1
) contains
φ
∗
(m
Q
) where m
Q
is the maximal ideal in
ideal m
P
in
(
C
2
).
This is the connection between the behaviour of points under morphisms and the splitting
of ideals in Dedekind domains.
We already know that a non-constant morphism of curves is dominant, but the next result
makes the even stronger statement that a morphism is surjective.
O
P
⊆ k
O
Q
⊆ k
k
Theorem 8.2.4
Let C
1
and C
2
be curves over
(in particular, they are projective and
non-singular). Let
φ
:
C
1
→
k
C
2
be a non-constant morphism of curves over
. Then φ is
surjective from C
1
(
k
)
to C
2
(
k
)
.
Proof
Proposition VII.5.7 of Lorenzini [
355
].
