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E be a non-zero isogeny of elliptic curves over
Theorem 9.6.27
Let φ
:
E
k
.
deg(
φ
)
. Then
deg(
φ
)
d, φ
◦
φ
[
d
]
on E.
1. Let d
=
=
◦
φ
=
[
d
]
on E and φ
=
2. Let ψ
:
E
E
3
be an isogeny. Then ψ
=
φ
◦
ψ.
→
◦
φ
→
E be an isogeny. Then φ
=
φ
+
ψ.
3. Let ψ
:
E
+
ψ
4. φ
=
φ.
Proof
See Theorem III.6.2 of [
505
].
. Then
[
m
]
Corollary 9.6.28
Let E be an elliptic curve over
k
and let m
∈ Z
=
[
m
]
and
deg([
m
])
=
m
2
.
Proof
The first claim follows by induction from part 3 of Theorem
9.6.27
. The second
claim follows from part 1 of Theorem
9.6.27
and since [1]
=
[1]: write
d
=
deg([
m
]) and
=
[
m
][
m
]
[
m
2
]; since End
k
(
E
) is torsion-free (Lemma
9.6.11
) it follows that
use [
d
]
=
m
2
.
d
=
An important consequence of Corollary
9.6.28
is that it determines the possible group
structures of elliptic curves over finite fields. We return to this topic in Theorem
9.8.1
.
We end this section with another example of an isogeny.
Exercise 9.6.29
Let
k
be a field such that char(
k
)
=
2
,
3. Let
E
be an elliptic curve with a
subgroup of order 3 defined over
k
. Show that, after a suitable change of variable, one has
v
) and
v
2
a point
P
=
(0
,v
) such that [2]
P
=
(0
,
−
∈ k
. Show that
E
is
k
-isomorphic to a
curve of the form
a
6
a
2
x
a
6
2
.
y
2
x
3
1
=
+
+
Show that there is a
k
-isomorphism to a curve of the form
Y
2
X
3
1)
2
=
+
A
(
X
+
0
,
2
4
.
where
A
=
Exercise 9.6.30
(Doche, Icart and Kohel [161]) Let
k
be a field such that char(
k
)
=
2
,
3.
0
,
4
.
Consider the elliptic curve
E
:
y
2
x
3
1)
2
Let
u
∈ k
be such that
u
=
=
+
3
u
(
x
+
as
in Exercise
9.6
.2
9
. Then (0
,
√
3
u
) has order 3 and
G
±
√
3
u
)
={
O
E
,
(0
,
}
is a
k
-rational
subgroup of
E
(
k
). Show that
x
3
,y
1
4
ux
2
+
+
12
u
(
x
+
1)
12
u
x
+
2
φ
(
x,y
)
=
−
x
2
x
3
is an isogeny from
E
to
E
:
Y
2
X
3
9)
2
with ker(
φ
)
=
−
u
(3
X
−
4
u
+
=
G
. Determine the
dual isogeny to
φ
and show that
φ
◦
φ
=
[3].
