30 2. Elliptic curves
Theorem 2.4.5 (weak Mordell-Weil). E(Q)/mE(Q) is a finite group
for all m 2.
We will discuss the proof of a special case of the weak Mordell-
Weil theorem in Section 2.9 (see Corollary 2.9.7).
It follows from the Mordell-Weil theorem and the general struc-
ture theory of finitely generated abelian groups that
E(Q)

=
E(Q)torsion
ZRE
. (2.4)
In other words, E(Q) is isomorphic to the direct sum of two abelian
groups (notice however that this decomposition is not canonical!).
The first summand is a finite group formed by all torsion elements,
i.e., those points on E of finite order:
E(Q)torsion = {P E(Q) : there is n N such that nP = O}.
The second summand of Eq. (2.4), sometimes called the free part, is
ZRE , i.e., RE copies of Z for some integer RE 0. It is generated
by RE points of E(Q) of infinite order (i.e., P E(Q) such that
nP = O for all non-zero n Z). The number RE is called the rank
of the elliptic curve E/Q. Notice, however, that the set
F = {P E(Q) : P is of infinite order} {O}
is not a subgroup of E(Q) if the torsion subgroup is non-trivial. For
instance, if T is a torsion point and P is of infinite order, then P and
P + T belong to F but T = (P + T ) P does not belong to F . This
fact makes the isomorphism of Eq. (2.4) not canonical because the
subgroup of E(Q) isomorphic to
ZRE
cannot be chosen, in general,
in a unique way.
Example 2.4.6. The following are some examples of elliptic curves
and their Mordell-Weil groups:
(1) The curve E1/Q : y2 = x3 + 6 has no rational points, other
than the point at infinity O. Therefore, there are no torsion
points (other than O) and no points of infinite order. In
particular, the rank is 0, and E1(Q) = {O}.
(2) The curve E2/Q :
y2
=
x3
+ 1 has only 6 rational points.
As we saw in Example 2.4.2, the point P = (2,3) has exact
order 6. Therefore E2(Q)

= Z/6Z is an isomorphism of
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