Cryptography Reference
In-Depth Information
class group of a curve is isomorphic to the Picard group of a curve (even though the Picard
group is usually defined differently, in terms of line bundles).
is Pic
0
k
Definition 7.8.1
The (degree zero)
divisor class group
of a curve
C
over
k
(
C
)
=
Div
0
k
(
C
)
/
Prin
k
(
C
).
We call two divisors
D
1
,D
2
∈
Div
0
k
(
C
)
linearly equivalent
and write
D
1
≡
D
2
if
D
1
−
Div
0
k
D
2
∈
Prin
k
(
C
). The equivalence cl
ass
(called a
divisor class
) of a divisor
D
∈
(
C
)
under linear equivalence is denoted
D
.
Example 7.8.2
By Lemma
7.7.8
,Pic
0
k
1
)
(
P
={
0
}
.
k
∈ k
=
∈
k
k
Theorem 7.8.3
Let C be a curve over
and let f
(
C
)
.Ifσ
(
f
)
f
f
or all σ
Gal(
/
)
∈ k
k
=
∈ k
∈ k
then f
(
C
)
.If
div(
f
)
is defined over
then f
ch for some c
and h
(
C
)
.
Proof
The first claim follows from Remark
5.4.14
(also see Remark
8.4.8
of Section
8.4
).
For the second statement let div(
f
) be defined over
=
div(
σ
(
f
)) where the second equality foll
o
ws from part 4 of Lemm
a
7.4.14
.
C
orollary
7.7.13
implies
σ
(
f
)
k
. Then div(
f
)
=
σ
(div(
f
))
∗
. The function
c
:Gal(
∗
is a
1
-cocycle
=
c
(
σ
)
f
for some
c
(
σ
)
∈ k
k
/
k
)
→ k
∗
is con-
(the fact that
c
(
στ
)
=
σ
(
c
(
τ
))
c
(
σ
) is immediate, the fact that
c
:Gal(
k
/
k
)
→ k
tinuous als
o
follows). Hence, Theorem
A.7.2
(Hi
lb
ert 90) implies that
c
(
σ
)
=
σ
(
γ
)
/γ
for
∗
. In other words, taking
h
some
γ
∈ k
=
f/γ
∈ k
(
C
), we have
σ
(
h
)
=
σ
(
f
)
/σ
(
γ
)
=
f/γ
=
h.
By the first part of the theorem
h
∈ k
(
C
).
Theorem
7.8.3
has the following important corollary, namely that Pic
0
k
(
C
) is a subgroup
of Pic
0
k
/
k
(
C
) for every extension
k
.
k
/
Corollary 7.8.4
Let C be a curve over
k
and let
k
be an algebraic extension. Then
Pic
0
k
(
C
)
injects into
Pic
0
k
(
C
)
.
Pic
0
k
(
C
) becomes trivial in Pic
0
Proof
Suppose a divisor class
D
∈
k
(
C
). Then there is
∈ k
(
C
)
∗
.But
some divisor
D
on
C
defined over
k
such that
D
=
div(
f
)forsome
f
Theorem
7.8.3
implies
D
=
div(
h
)forsome
h
∈ k
(
C
) and so the divisor class is trivial in
Pic
0
k
(
C
).
Corollary 7.8.5
Let
k
be a finite field. Let C be a curve over
k
. Define
Pic
0
k
(
C
)
Gal(
k
/
k
)
Pic
0
k
={
D
∈
(
C
):
σ
(
D
)
=
D for all σ
∈
Gal(
k
/
k
)
}
.
Then
Pic
0
k
Pic
0
k
(
C
)
Gal(
k
/
k
)
=
(
C
)
.
Proof
(Sketch) Let
G
=
Gal(
k
/
k
). Take Galois cohomology of the exact sequence
∗
(
C
)
∗
→
1
→ k
→ k
Prin
(
C
)
→
0
k
