Cryptography Reference
In-Depth Information
Despite not discussing isogenies in full generality, it is possible to discuss isogenies that
arise from maps between curves purely in terms of divisor class groups. We now give some
examples, but first introduce a natural notation.
Pic
0
k
Definition 10.5.1
Let
C
be a curve over a field
k
and let
n
∈ N
.For
D
∈
(
C
) define
[
n
]
D
=
D
+···+
D
(
n
times)
.
Indeed, we usually assume that [
n
]
D
is a reduced divisor representing the divisor class
nD
.
Define
Pic
0
k
Pic
0
k
(
C
)[
n
]
={
D
∈
(
C
):[
n
]
D
=
0
}
.
Recall from Corollary
8.3.10
that if
φ
:
C
1
→
C
2
is a non-constant rational map (and
hence a non-constant morphism) over
k
between two curves then there are corresponding
group homomorphisms
φ
∗
:Pic
0
k
Pic
0
k
(
C
1
) and
φ
∗
:Pic
0
k
Pic
0
k
(
C
2
)
→
(
C
1
)
→
(
C
2
). Further-
more, by part 5 of Theorem
8.3.8
we have
φ
∗
φ
∗
(
D
)
[deg(
φ
)]
D
on Pic
0
k
=
(
C
2
).
In the special case of a non-constant rational map
φ
:
C
→
E
over
k
where
E
is an
Pic
0
k
elliptic curve we can compose with the Abel-Jacobi map
E
→
(
E
) of Theorem
7.9.8
O
E
) to obtain group homomorphisms that we call
φ
∗
:
E
Pic
0
k
given by
P
→
(
P
)
−
(
→
(
C
)
and
φ
∗
:Pic
0
(
C
)
→
E
.
k
Exercise 10.5.2
Let
φ
:
C
→
E
be a non-constant rational map over
k
where
E
is an elliptic
.Let
φ
∗
:
E
Pic
0
k
(
C
) and
φ
∗
:Pic
0
k
curve over
k
→
(
C
)
→
E
be th
e g
roup homomorphisms
as above. Show that
φ
∗
is surjective as a map from Pic
0
k
(
C
)to
E
(
k
) and that the kernel of
φ
∗
is contained in
E
[deg(
φ
)].
If
C
is a curve of genus 2 and there are two non-constant rational maps
φ
i
:
C
→
E
i
over
k
for elliptic c
ur
ves
E
1
,
E
2
then one naturally has a group homomorphism
φ
1
,
∗
×
φ
2
,
∗
:
Pic
0
k
ker(
φ
2
,
∗
) is finite then it follows from the theory
of Abelian varieties that the Jacobian variety
J
C
is isogenous to the product
E
1
×
(
C
)
→
E
1
(
k
)
×
E
2
(
k
). If ker(
φ
1
,
∗
)
∩
E
2
of the
elliptic curves and one says that
J
C
is a
split Jacobian
.
Example 10.5.3
Let
C
:
y
2
x
6
2
x
2
=
+
+
1 be a genus 2 curve over
F
11
. Consider the
rational maps
E
1
:
Y
2
X
3
φ
1
:
C
→
=
+
2
X
+
1
(
x
2
,y
) and
given by
φ
1
(
x,y
)
=
E
2
:
Y
2
X
3
2
X
2
φ
2
:
C
→
=
+
+
1
(1
/x
2
,y/x
3
). The two elliptic curves
E
1
and
E
2
are neither isomorphic
or isogenous. One has #
E
1
(
given by
φ
2
(
x,y
)
=
14 and #Pic
0
F
11
)
=
16, #
E
2
(
F
11
)
=
F
11
(
C
)
=
14
·
16.
It can be shown (this is not trivial) that ker(
φ
1
,
∗
)
∩
ker(
φ
2
,
∗
) is finite. Further, since
deg(
φ
1
)
=
deg(
φ
2
)
=
2 it can be shown that the kernel of
φ
1
,
∗
×
φ
2
,
∗
is contained in
Pic
0
k
(
C
)[2].
