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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