Cryptography Reference
In-Depth Information
The Jacobian of a curve satisfies the following u
ni
versal property. Let
φ
:
C
→
A
be a
morphism where
A
is an Abelian variety. Let
P
0
∈
C
(
k
) be such that
φ
(
P
0
)
=
0 and consider
the Abel-Jacobi map
ψ
:
C
→
J
C
(corresponding to
P
→
(
P
)
−
(
P
0
)). Then there is a
homomorphism of Abelian varieties
φ
:
J
C
→
φ
◦
A
such that
φ
=
ψ
.Exercise
10.5.4
gives a special case of this universal property.
Exercise 10.5.4
Let
C
:
y
2
x
6
a
2
x
4
a
4
x
2
=
+
+
+
a
6
over
k
where char(
k
)
=
2, and let
(
x
2
,y
)
be
non-constant rational map
φ
:
C
φ
(
x,y
)
=
→
E
over
k
where
E
is an elliptic
curve. Let
P
0
∈
C
(
k
) be such that
φ
(
P
0
)
=
O
E
. Show that the composition
Pic
0
k
C
(
k
)
→
(
C
)
→
E
(
k
)
where the first map is the Abel-Jacobi map
P
→
(
P
)
−
(
P
0
) and the second map,
φ
∗
,is
just the original map
φ
.
There is a vast literature on split Jacobians and we are unable to give a full survey. We
refer to Sections 4, 5 and 6 of Kuhn [
321
] or Chapter 14 of Cassels and Flynn [
115
]for
further examples.
10.6 Elements of order
n
We now bound the size of the set of elements of order dividing
n
in the divisor class group
of a curve. As with many other results in this chapter, the best approach is via the theory
of Abelian varieties. We state Theorem
10.6.1
for general curves, but without proof. The
result is immediate for Abelian varieties over
g
/L
where
L
C
, as they are isomorphic to
C
1
n
L/L
.
g
/L
are given by the
n
2
g
points in
is a rank 2
g
lattice. The elements of order
n
in
C
Theorem 10.6.1
Let C be a curve of genus g over
k
and let n
∈ N
.If
char(
k
)
=
0
or
1
then
#Pic
0
k
p>
0
then
#Pic
0
k
k
=
=
n
2
g
.If
char(
k
=
=
p
e
gcd(
n,
char(
))
(
C
)[
n
]
)
(
C
)[
p
]
≤
≤
where
0
e
g.
Proof
See Theorem 4 of Section 7 of Mumford [
398
].
10.7 Hyperelliptic curves over finite fields
There are a finite number of points on a curve
C
of genus
g
over a finite field
F
q
. There are
also finitely many possible values for the Mumford representation of a reduced divisor on
a hyperelliptic curve over a finite field. Hence, the divisor class group Pic
0
F
q
(
C
) of a curve
over a finite field is a finite group. Since the affine part of a reduced divisor is a sum of
at most
g
points (possibly defined over a field extension of degree bounded by
g
) it is not
surprising that there is a connection between
and #Pic
0
{
#
C
(
F
q
i
):1
≤
i
≤
g
}
F
q
(
C
). Indeed,
#Pic
0
there is also a connection between
F
q
). The aim of this
section is to describe these connections. We also give some important bounds on these
{
F
q
i
(
C
):1
≤
i
≤
g
}
and #
C
(
