Mordell-Weil group of an abelian variety (reviewed)

The Mordell-Weil group of an abelian variety $A$ over a number field $K$ is its group of $K$-rational points $A(K)$.

Weil, building on Mordell's theorem for elliptic curves over $\Q$, proved that the abelian group $A(K)$ is finitely generated. Thus \[ A(K)\simeq \Z^r \oplus T, \] where $r$ is a nonnegative integer called the Mordell-Weil rank of $A$, and $T$ is a finite abelian group called the torsion subgroup.

The torsion subgroup $T$ is the product of at most $2g$ cyclic groups, where $g$ is the dimension of $A$.