In previous chapters we have seen how to solve the problem of finding all integral points on an elliptic curve. This is only one of the two fundamental diophantine questions which one can ask about an elliptic curve. In this chapter we look at the other question: What is the structure of the set of rational points on an elliptic curve? We shall give a sketch of the proof that the set of rational solutions to an elliptic curve forms a finitely generated abelian group, known as the Mordell–Weil group. There is no known effective proof of this result; we shall, however, outline a possible algorithmic proof which works fine in practice most of the time.

We first consider the basic theory of elliptic curves, which we shall just skim over. Those of you who have not met any of this before should perhaps consult any one of the excellent textbooks in the area such as [107], [172], [101], [25], [105], [176] or [40]. You could also, perhaps, consult the survey articles [22] and [194]. We shall only consider those parts of the theory of elliptic curves which we need in this book.

After outlining the basic theory we shall outline two ‘algorithms’ for determining E(ℚ)/2E(ℚ). These two methods should not really be called algorithms as they are not guaranteed to work.