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