Cryptography Reference
In-Depth Information
Exercise 7.4.6
Show that
v
P
(
f
) doe
s
not depend on the underlying field. In other words,
if
k
is an algebraic extension of
∈
m
P,
k
(
C
)
m
k
in
k
then
v
P
(
f
)
=
max
{
m
∈ Z
≥
0
:
f
}
.
Lemma 7.4.7
Let C be a curve over
k
and P
∈
C
(
k
)
. Let t
P
∈
O
P,
k
(
C
)
be any uni-
∈ Z
≥
0
:
f/t
P
∈
formiser at P. Let f
∈
O
P,
k
(
C
)
be such that f
=
0
. Thenv
P
(
f
)
=
max
{
m
t
v
P
(
f
P
u for some u
∈
O
P,
k
(
C
)
∗
.
O
P,
k
(
C
)
}
and f
=
Exercise 7.4.8
Prove Lemma
7.4.7
.
Writing a function
f
as
t
v
P
(
f
)
P
(
C
)
∗
is analogous to writing a polyno-
u
for some
u
∈
O
P,
k
a
)
m
G
(
x
) where
G
(
x
)
mial
F
(
x
)
0.
Hopefully, the reader is convinced that this is a powerful tool. For example, it enables a sim-
ple proof of Exercise
7.4.9
. Further, one can represent a function
f
as a formal power series
n
=
v
P
(
f
)
a
n
t
P
where
a
n
∈ k
∈ k
[
x
]intheform
F
(
x
)
=
(
x
−
∈ k
[
x
] satisfies
G
(
a
)
=
; see Exercises 2-30 to 2-32 of Fulton [
199
]. Such expansions
will used in Chapters
25
and
26
, but we do not develop the theory rigorously.
Exercise 7.4.9
Let
C
be a curve over
k
and
P
∈
C
(
k
). Let
f,h
∈
O
P,
k
(
C
) be such that
f,h
=
0. Show that
v
P
(
fh
)
=
v
P
(
f
)
+
v
P
(
h
).
Lemma 7.4.10
Let C be a curve over
k
,letP
∈
C
(
k
)
and let f
∈ k
(
C
)
. Then f can be
written as f
1
/f
2
where f
1
,f
2
∈
O
P,
k
(
C
)
.
Proof
Without
l
oss of generality,
C
is affine. By definition,
f
=
f
1
/f
2
where
f
1
,f
2
∈ k
[
C
].
Since
k
[
C
]
⊂ k
[
C
]
⊂
O
P,
k
(
C
) the result follows.
Definition 7.4.11
Let
C
be a curve over
k
and let
f
∈ k
(
C
). A point
P
∈
C
(
k
)iscalled
∈
O
P,
k
(
C
). If
f
=
f
1
/f
2
∈ k
(
C
) where
f
1
,f
2
∈
O
P,
k
(
C
) then define
a
pole
of
f
if
f
=
−
v
P
(
f
)
v
P
(
f
1
)
v
P
(
f
2
).
Exercise 7.4.12
Show that if
P
∈
C
(
k
) is a pole of
f
∈ k
(
C
) then
v
P
(
f
)
<
0 and
P
is a
zero of 1
/f
.
Lemma 7.4.13
For every function f
∈ k
(
C
)
the order v
P
(
f
)
of f at P is independent of
the choice of representative of f .
We now give some properties of
v
P
(
f
).
Lemma 7.4.14
Let P
∈
C
(
k
)
. Then v
P
is a discrete valuation on
k
(
C
)
. Furthermore, the
following properties hold.
∗
then v
P
(
f
)
∈ k
=
1. If f
0
.
2. If c
∈ k
and if v
P
(
f
)
<
0
then v
P
(
f
+
c
)
=
v
P
(
f
)
.
(
C
)
∗
3. If f
1
,f
2
∈ k
are such that v
P
(
f
1
)
=
v
P
(
f
2
)
then v
P
(
f
1
+
f
2
)
=
min
{
v
P
(
f
1
)
,
.
4. Suppose C is defined over
v
P
(
f
2
)
}
k
and let P
∈
C
(
k
)
. Let σ
∈
Gal(
k
/
k
)
. Then v
P
(
f
)
=
v
σ
(
P
)
(
σ
(
f
))
.
