Cryptography Reference
In-Depth Information
Proof
Proposition 8.3 of Fulton [
199
].
=
P
∈
C
(
k
)
n
P
(
P
) be a divisor on
C
. Explain why
(
C
)
∗
:
Exercise 8.4.3
Let
D
{
f
∈ k
v
P
(
f
)
=
n
P
for all
P
∈
C
(
k
)
}∪{
0
}
is not usually a
k
-vector space.
Definition 8.4.4
Let
C
be a curve over
k
and let
D
be a divisor on
C
. Define
=
dim
k
L
k
(
D
)
.
k
(
D
)
Write
(
D
)
=
k
(
D
).
Exercise 8.4.5
Show that
k
(0)
=
1 and, for
f
∈ k
(
C
),
k
(div(
f
))
=
1.
Theorem 8.4.6
(Riemann's theorem) Let C be a curve over
(in particular, non-singular
and projec
ti
ve). Then there exists a unique minimal integer g such that, for all divisors D
on C over
k
k
≥
+
−
k
(
D
)
deg(
D
)
1
g.
Proof
See Proposition I.4.14 of Stichtenoth [
529
], Section 8.3 (page 196) of Fulton [
199
]
or Theorem 2.3 of Moreno [
395
].
Definition 8.4.7
The number
g
in Theorem
8.4.6
is called the
genus
of
C
.
Note that the genus is independent of the model of the curve
C
and so one can associate
the genus with the function field or birational equivalence class of the curve.
∈ k
Remark 8.4.8
We give an alternative j
us
tification for Remark
5.4.14
. Suppose
f
(
C
)
=
∈
k
k
=
is such that
σ
(
f
)
f
for all
σ
Gal(
/
). Write
D
div(
f
). Note that
D
is defined
over
(
D
), which has dimension 1 by Exercise
8.4.5
. Now, performing the
Brill-Noether proof of Riemann's theorem (e.g., see Section 8.5 of Fulton [
199
]), one can
show that
k
. Then
f
∈
L
k
L
k
(
D
) contains a function
h
∈ k
(
C
). It follows that div(
h
)
=
D
and that
f
=
ch
for some
c
∈ k
. Hence, Theorem
7.8.3
is proved.
8.5 Derivations and differentials
Differentials arise in differential geometry: a manifold is described by open patches homeo-
morphic to
n
for complex manifolds) with coordinate functions
x
1
,...,x
n
and the
differentials
dx
i
arise naturally. It turns out that differentials can be described in a purely
formal way (i.e., without reference to limits).
When working over general fields (such as finite fields) it no longer makes sense
to consider differentiation as a process defined by limits. But the formal description of
differentials makes sense and the concept turns out to be useful.
We first explain how to generalise partial differentiation to functions on curves. We can
then define differentials. Throughout this section, if
F
(
x,y
) is a polynomial or rational
function then
∂F/∂x
denotes standard undergraduate partial differentiation.
R
n
(or
C
