Cryptography Reference
In-Depth Information
It remains to prove that this is equal to the norm of
f
evaluated at
Q
and we sketch
this when the extension is Galois
a
nd cyclic (the general case is simple linear algebra).
The elements
σ
(
C
1
)
/φ
∗
(
(
C
2
))) permute the m
P
i
and the ramification indices
e
φ
(
P
i
) are all equal. Since
c
i
∈ k ⊂
∈
Gal(
k
k
φ
∗
(
k
(
C
2
)) we have
f
≡
c
i
(mod m
P
i
) if and only if
σ
(
f
)
≡
c
i
(mod
σ
(m
P
i
)). Hence
c
e
φ
(
P
i
)
i
N
k
(
C
1
)
/φ
∗
(
k
(
C
2
))
(
f
)
=
σ
(
f
)
≡
(mod m
P
1
)
i
σ
∈
Gal(
k
(
C
1
)
/φ
∗
(
k
(
C
2
)))
φ
∗
(
(
C
2
)) this congruence holds modulo
φ
∗
(m
Q
). The result
and since N
k
(
C
1
)
/φ
∗
(
k
(
C
2
))
(
f
)
∈
k
follows.
We now give an important application of Theorem
8.3.8
, already stated as Theo-
rem
7.7.11
.
(
C
)
∗
. Then f has only finitely many
zeroes and poles (i.e.,
div(
f
)
is a divisor) and
deg(div(
f
))
Theorem 8.3.14
Let C be a curve over
k
and let f
∈ k
=
0
.
Proof
Let
D
=
(0 : 1)
−
(1 : 0) on
P
1
. Interpreting
f
as a rational map
f
:
C
→ P
1
as in
f
∗
(
D
) and, by part 1 of Theorem
8.3.8
,deg(
f
∗
(
D
))
Lemma
8.1.1
we have div(
f
)
=
=
deg(
f
)deg(
D
)
=
0. One also deduces that
f
has, counting with multiplicity, deg(
f
) poles
and zeroes.
Exercise 8.3.15
Let
φ
:
C
1
→
k
∈
C
2
be a rational map over
. Show that if
D
Div
k
(
C
1
)
∈
Div
k
(
C
2
)) then
φ
∗
(
D
) (respectively,
φ
∗
(
D
)) is defined over
k
(respectively,
D
.
8.4 Riemann-Roch spaces
=
P
n
P
(
P
) be a divisor on
C
.The
Definition 8.4.1
Let
C
be a curve over
k
and let
D
Riemann-Roch space
of
D
is
(
C
)
∗
:
v
P
(
f
)
L
k
(
D
)
={
f
∈ k
≥−
n
P
for all
P
∈
C
(
k
)
}∪{
0
}
.
We denote
L
k
(
D
)by
L
(
D
).
Lemma 8.4.2
Let C be a curve over
k
and let D be a divisor on C. Then:
1.
L
k
(
D
)
is a
k
-vector space.
D
implies
⊆
L
k
(
D
)
.
2. D
≤
L
k
(
D
)
3.
L
k
(0)
= k
,
L
k
(
D
)
={
0
}
if
deg(
D
)
<
0
.
1
and if D
≥
L
k
(
D
)
/
4. Let P
0
∈
C
(
k
)
. Then
dim
k
(
L
k
(
D
+
P
0
)
/
L
k
(
D
))
≤
D then
dim
k
(
deg(
D
)
L
k
(
D
))
≤
−
deg(
D
)
.
5.
L
k
(
D
)
is finite dimensional and if D
=
D
+
−
D
−
, where D
+
,D
−
are effective, then
dim
k
L
k
(
D
)
≤
deg(
D
+
)
+
1
.
6. If D
=
(
C
)
∗
then
L
k
(
D
)
are isomorphic as
D
+
div(
f
)
for some f
∈ k
L
k
(
D
)
and
k
-vector spaces.
