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 $A(K)$.

As proved by Weil, $A(K)$ is a finitely generated. This implies that the abelian group $A(K)$ has a unique decomposition of the form \[ A(K)\simeq \Z^r \oplus T, \] where the finite group $T$ is the torsion subgroup $T$, and the nonnegative integer $r$ is the Mordell-Weil rank of $A$.

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