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=4
1
/
3
.(
Hint:
what
x
satisfy
φ
(
x
)=
x
?) Con-
clude that none of
C
2
,C
3
,C
4
is an eigenspace for
φ
.
(g) Let
E
1
be an elliptic curve with
j
= 3 that is 3-isogenous to
E
(it
exists by Theorem 12.19). Show that there does not exist
ν
(f) Show that
φ
(4
1
/
3
)
Z
such that
φP
=
νP
for all
P
in the kernel of the isogeny. This
shows that the restriction
j
= 0 is needed in Proposition 12.20.
∈
12.12 Let
E
1
,E
2
be elliptic curves defined over
F
q
and suppose there is an
isogeny
α
:
E
1
→ E
2
of degree
N
defined over
F
q
.
(a) Let
be a prime such that
qN
and let
n ≥
1. Show that
α
gives
an isomorphism
E
1
[
n
]
E
2
[
n
].
(b) Use Proposition 4.11 to show that #
E
1
(
F
q
)=#
E
2
(
F
q
).
12.13 Let
f
:
C
/L
1
→
C
/L
2
be a continuous map. This yields a continuous
map
f
:
C
→
C
/L
2
such that
f
(
z
)=
f
(
z
mod
L
1
). Let
f
(0) =
z
0
.Let
z
1
∈
C
and choose a path
γ
(
t
)
,
0
≤ t ≤
1, from 0 to
z
1
.
(a) Let 0
≤ t
1
≤
1. Show that there exists a complex-valued continuous
function
h
(
t
) defined in a small interval containing
t
1
,say(
t
1
−
, t
1
+
)
∩
[0
,
1] for some
, such that
h
(
t
)mod
L
2
=
f
(
γ
(
t
)). (
Hint:
Represent
C
/L
2
using a translated fundamental parallelogram that
contains
f
(
γ
(
t
1
)) in its interior.)
(b) As
t
1
runs through [0
,
1], the small intervals in part (a) give a
covering of the interval [0
,
1]. Since [0
,
1] is compact, there is a
finite set of values
t
(1)
1
< ···<t
(
n
1
whose intervals
I
1
,...,I
n
cover
all of [0
,
1]. Suppose that for some
t
0
with 0
≤ t
0
<
1, we have
a complex-valued continuous function
g
(
t
)on[0
,t
0
] such that
g
(
t
)
mod
L
2
=
f
(
γ
(
t
)). Show that if [0
,t
0
]
∩ I
j
is nonempty, and if
h
(
t
)
is the function on
I
j
constructed in part (a), then there is an
∈ L
2
such that
g
(
t
)=
h
(
t
)
−
for all
t ∈
[0
,t
0
]
∩ I
j
.(
Hint:
g
(
t
)
− h
(
t
)is
continuous and
L
2
is discrete.)
(c) Show that there exists a continuous function
g
:[0
,
1]
→
C
such
that
g
(
t
)mod
L
2
=
f
(
γ
(
t
)) for all
t ∈
[0
,
1].
(d) Define
f
(
z
1
)=
g
(1), where
z
1
and
g
are as above. Show that this
definition is independent of the choice of path
γ
.(
Hint:
Deform
one path into another continuously. The value of
g
(1) can change
only by a lattice point.)
(e) Show that the construction of
f
yields a continuous function
f
:
C
such that
f
(
z
mod
L
1
)=
f
(
z
)mod
L
2
for all
z
C
→
∈
C
.
12.14 Consider the elliptic curves
E
1
,
E
2
in Example 12.5. Use Velu's formulas
(Section 12.3) to compute the equations of
E
1
and
E
2
. Decide which
has
j
=
−
1andwhichhas
j
=
−
3.
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