Cryptography Reference
In-Depth Information
We claim that
G
=
E
(
Q
). Suppose not. Let
P
E
(
Q
) be an element not
in
G
. Since, for a point
P
, there are only finitely many points of height less
than
P
, we may change
P
to one of these, if necessary, and assume
P
has the
smallest height among points not in
G
. We may write
∈
P − R
i
=2
P
1
for some
i
and some
P
1
. By Theorem 8.18,
4
h
(
P
1
)=
h
(2
P
1
)=
h
(
P − R
i
)
=2
h
(
P
)+2
h
(
R
i
)
−
h
(
P
+
R
i
)
≤
2
h
(
P
)+2
c
+0
<
2
h
(
P
)+2
h
(
P
)=4
h
(
P
)
(since
c<h
(
P
), because
P
=
Q
j
). Therefore,
h
(
P
1
)
< h
(
P
)
.
Since
P
had the smallest height for points not in
G
,wemusthave
P
1
∈ G
.
Therefore,
P
=
R
i
+2
P
1
∈ G.
This contradiction proves that
E
(
Q
)=
G
. This completes the proof of the
Mordell-Weil theorem.
It remains to prove Theorem 8.18. The key step is the following.
PROPOSITION 8.19
Thereexistsaconstant
c
1
su ch that
|
h
(
P
+
Q
)+
h
(
P
−
Q
)
−
2
h
(
P
)
−
2
h
(
Q
)
|≤
c
1
for all
P, Q ∈ E
(
Q
)
.
The proof is rather technical, so we postpone it in order to complete the
proof of Theorem 8.18.
P roof of T heorem 8.18:
Proofofparts (1) and (2):
Letting
Q
=
P
in Proposition 8.19, we obtain
|
h
(2
P
)
−
4
h
(
P
)
|≤
c
1
(8.4)
for all
P
. Define
h
(
P
)=
1
2
1
4
n
h
(2
n
P
)
.
We need to prove the limit exists. We have
lim
n
→∞
4
n
h
(2
n
P
)=
h
(
P
)+
∞
1
1
4
j
(
h
(2
j
P
)
4
h
(2
j−
1
P
))
.
lim
n
−
(8.5)
→∞
j
=1
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