Cryptography Reference
In-Depth Information
Having shown that every
v
P
is a discrete valuation on
k
(
C
), it is natural to ask whether
every discrete valuation on
). To make this true over
fields that are not algebraically closed requires a more general notion of
a
point of
C
defined over
k
(
C
)is
v
P
for some point
P
∈
C
(
k
k
. Instead of doing this, we conti
nu
e to work with points over
k
and show in
Theorem
7.5.2
that every discrete valuation on
k
(
C
)is
v
P
for some
P
∈
C
(
k
). But first we
give some examples.
Example 7.4.15
Let
E
:
y
2
=
x
(
x
−
1)(
x
+
1) over
k
and let
P
=
(1
,
0)
∈
E
(
k
). We deter-
mine
v
P
(
x
)
,v
P
(
x
−
1)
,v
P
(
y
) and
v
P
(
x
+
y
−
1).
=
=
=
First,
x
(
P
)
1so
v
P
(
x
)
0. For the rest, since
P
ι
(
P
) we take the uniformiser to
be
t
P
=
=
y
. Hence,
v
P
(
y
)
1. Since
y
2
/
(
x
(
x
x
−
1
=
+
1))
and 1
/
(
x
(
x
+
1))
∈
O
P
we have
v
P
(
x
−
1)
=
2.
Finally,
f
(
x,y
)
=
x
+
y
−
1
=
y
+
(
x
−
1)
so
v
P
(
f
(
x,y
))
=
min
{
v
P
(
y
)
,v
P
(
x
−
1)
}=
min
{
1
,
2
}=
1. One can see this directly by writing
f
(
x,y
)
=
y
(1
+
y/x
(
x
+
1)).
Lemma 7.4.16
Let E be an elliptic curve. Then v
O
E
(
x
)
=−
2
and v
O
E
(
y
)
=−
3
.
Proof
We consider the projective equation, so that the functions become
x/z
and
y/z
then
set
y
=
1 so that we are considering
x/z
and 1
/z
on
a
3
z
2
x
3
a
2
x
2
z
a
4
xz
2
a
6
z
3
.
z
+
a
1
xz
+
=
+
+
+
(
x
3
) and so
v
O
E
(
x
)
As in Example
7.3.4
,wehave
z
∈
=
1
,v
O
E
(
z
)
=
3. This implies
v
O
E
(1
/z
)
=−
3 and
v
O
E
(
x/z
)
=−
2 as claimed.
7.5 Valuations and points on curves
L
et
C
be a curve over
k
and
P
∈
C
(
k
). We have shown that
v
P
(
f
) is a discrete valuation on
k
(
C
). The aim
o
f this section is to show (using the weak
N
ullstellensatz) that every discrete
valuation
v
on
k
(
C
)arisesas
v
P
for some point
P
∈
C
(
k
).
Lemma 7.5.1
Let C be a curve over
k
and let v be a discrete valua
ti
on on
k
(
C
)
. Write
n
R
v
,
m
v
for the corresponding valuation ring and maximal ideal (over
k
). S
up
pose C
⊂ P
[
ϕ
−
i
(
C
)]
is a
with coordinates
(
x
0
:
···
:
x
n
)
. Then there exists some
0
≤
i
≤
n such that
k
subring of R
v
(where ϕ
−
i
is as in Definition
5.2.19
).
Theorem 7.5.2
Let C b
e
a curve over
k
and let v be a discrete valuation on
k
(
C
)
. Then
v
=
v
P
for some P
∈
C
(
k
)
.
Proof
See
Corollary
I.6.6
of
Hartshorne
[252]
or
Theorem
VI.9.1
of
Lorenzini
[355].
