Cryptography Reference
In-Depth Information
Chapter 8
Elliptic Curves over Q
As we saw in Chapter 1, elliptic curves over
Q
represent an interesting class of
Diophantine equations. In the present chapter, we study the group structure
of the set of rational points of an elliptic curve
E
defined over
Q
. First, we
show how the torsion points can be found quite easily. Then we prove the
Mordell-Weil theorem, which says that
E
(
Q
) is a finitely generated abelian
group. As we'll see in Section 8.6, the method of proof has its origins in
Fermat's method of infinite descent. Finally, we reinterpret the descent calcu-
lations in terms of Galois cohomology and define the Shafarevich-Tate group.
8.1 The Torsion Subgroup. The Lutz-Nagell The-
orem
The torsion subgroup of
E
(
Q
) is easy to calculate. In this section we'll give
examples of how this can be done. The crucial step is the following theorem,
which was used in Chapter 5 to study anomalous curves. For convenience, we
repeat some of the notation introduced there.
Let
a/b
= 0 be a rational number, where
a, b
are relatively prime integers.
Write
a/b
=
p
r
a
1
/b
1
with
p
a
1
b
1
. Define the
p
-adic valuation
to be
v
p
(
a/b
)=
r.
For example,
v
2
(7
/
40) =
−
3,
v
5
(50
/
3) = 2, and
v
7
(1
/
2) = 0. Define
v
p
(0) =
+
(so
v
p
(0)
>n
for every integer
n
).
Let
E
be an elliptic curve over
Z
given by
y
2
=
x
3
+
Ax
+
B
.Let
r ≥
1be
an integer. Define
∞
E
r
=
{
(
x, y
)
∈ E
(
Q
)
| v
p
(
x
)
≤−
2
r,
v
p
(
y
)
≤−
3
r} ∪ {∞}.
These are the points such that
x
has at least
p
2
r
in its denominator and
y
has at least
p
3
r
in its denominator. These should be thought of as the points
that are close to
∞
mod powers of
p
(that is,
p
-adically close to
∞
).
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