Mordell-Weil rank (reviewed)

The Mordell-Weil theorem states that the set of rational points on an abelian variety over a number field forms a finitely generated abelian group, hence isomorphic to a group of the form $T \oplus \Z^r$, where $T$ is a finite torsion group. The integer $r\ge 0$ is the Mordell-Weil rank of the abelian variety.