Cryptography Reference
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not always the case. Let
E
be given by
y
2
=
x
3
+18
x
+ 72. Then
4
A
3
+27
B
2
= 163296 = 2
5
·
3
6
·
7
.
The Lutz-Nagell theorem would require us to check all
y
with
y
2
|
163296, which
amounts to checking all
y|
108 = 2
2
·
3
3
. Instead, the reduction mod 5 has 5
points and the reduction mod
11
has 8 points. It follows that the torsion
subgroup of
E
(
Q
) is trivial.
Finally, we mention a deep result of Mazur, which we will not prove (see
[77]).
THEOREM 8.11
Let
E
be an elliptic curve defined over
Q
.Thenthe torsion subgroup of
E
(
Q
)
isoneofthe follow ing:
Z
n
with
1
10
or
n
=12
,
Z
2
⊕
Z
2
n
with
1
≤ n ≤
4
.
≤
n
≤
REMARK 8.12
For each of the groups in the theorem, there are infinitely
many elliptic curves
E
(with distinct
j
-invariants) having that group as the
torsion subgroup of
E
(
Q
). See Exercise 8.1 for examples of each possibility.
8.2 Descent and the Weak Mordell-Weil Theo-
rem
We start with an example that has its origins in the work of Fermat (see
Section 8.6).
Example 8.5
Let's look at rational points on the curve
E
given by
y
2
=
x
(
x
−
2)(
x
+2)
.
If
y
=0,wehave
x
=0
,
±
2. Therefore, assume
y
= 0. Since the product of
x
,
x
2, and
x
+ 2 is a square, intuition suggests that each of these factors
should, in some sense, be close to being a square. Write
−
x
=
au
2
2=
bv
2
x
+2=
cw
2
x
−
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