Cryptography Reference
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an injection
(
Q
×
/
Q
×
2
)
(
Q
×
/
Q
×
2
)
(
Q
×
/
Q
×
2
)
.
E
(
Q
)
/
2
E
(
Q
)
→
⊕
⊕
Proposition 8.13 says that if (
a, b, c
)(where
a, b, c
are chosen to be squarefree
integers) is in the image of
φ
,then
a, b, c
are products of primes in the set
S
of Proposition 8.13. Since
S
is finite, there are only finitely many such
a, b, c
mod squares. Therefore, the image of
φ
is finite. This proves the theorem.
REMARK 8.16
(for those who know some algebraic number theory) Let
K/
Q
be a finite extension. The theorem can be extended to say that if
E
is an elliptic curve over
K
then
E
(
K
)
/
2
E
(
K
) is finite. If we assume that
x
3
+
Ax
+
B
=(
x
−
e
1
)(
x
−
e
2
)(
x
−
e
3
) with all
e
i
∈
K
, then the proof is the
same except that the image of
φ
is contained in
(
K
×
/K
×
2
)
⊕
(
K
×
/K
×
2
)
⊕
(
K
×
/K
×
2
)
.
Let
O
K
be the ring of algebraic integers of
K
. To make things simpler, we
invert some elements in order to obtain a unique factorization domain. Take
a nonzero element from an integral ideal in each ideal class of
O
K
and let
M
be the multiplicative subset generated by these elements. Then
M
−
1
O
K
is a
principal ideal domain, hence a unique factorization domain. The analogue of
Proposition 8.13 says that the primes of
M
−
1
O
K
dividing
a, b, c
also divide
(
e
1
− e
2
)(
e
1
− e
3
)(
e
2
− e
3
). Let
S ⊂ M
−
1
O
K
be the set of prime divisors of
(
e
1
− e
2
)(
e
1
− e
3
)(
e
2
− e
3
). Then the image of
φ
is contained in the group
generated by
S
and the units of
M
−
1
O
K
. Since the class number of
K
is
finite,
M
is finitely generated. A generalization of the Dirichlet unit theorem
(often called the
S
-unit theorem) says that the units of
M
−
1
O
K
are a finitely
generated group. Therefore, the image of
φ
is a finitely generated abelian
group of exponent 2, hence is finite. This proves that
E
(
K
)
/
2
E
(
K
) is finite.
8.3 Heights and the Mordell-Weil Theorem
The purpose of this section is to change the weak Mordell-Weil theorem
into the Mordell-Weil theorem. This result was proved by Mordell in 1922 for
elliptic curves defined over
Q
. It was greatly generalized in 1928 by Weil in
his thesis, where he proved the result not only for elliptic curves over number
fields (that is, finite extensions of
Q
) but also for abelian varieties (higher-
dimensional analogues of elliptic curves).
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