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(b) Show that
g
(
x, y
)=
y
4
/
(
x
2
+1)
3
has no poles in
E
(
Q
) but does
have zeros in
E
(
Q
).
(c) Find the divisors of
f
and
g
(over
Q
).
11.2 Let
E
be an elliptic curve over a field
K
and let
m, n
be positive integers
that are not divisible by the characteristic of
K
.Let
S ∈ E
[
mn
]and
T ∈ E
[
n
]. Show that
e
mn
(
S, T
)=
e
n
(
mS, T
)
.
11.3 Suppose
f
is a function on an algebraic curve
C
such that div(
f
)=
[
P
]
−
[
Q
]forpoints
P
and
Q
. Show that
f
gives a bijection of
C
with
P
1
.
11.4 Show that part (3) of Proposition 11.1 follows from part (2). (
Hint:
Let
P
0
be any point and look at the function
f
−
f
(
P
0
).)
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