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 calculations in terms of Galois cohomology and define the Shafarevich-Tate group.