Mordell-Weil group (reviewed)

For an elliptic curve $E$ defined over a field $K$, the Mordell-Weil group of $E/K$ is the group $E(K)$ of $K$-rational points of $E$. It is a finitely-generated Abelian group.

This is a special case of the Mordell-Weil group of an abelian variety.

The Mordell-Weil Theorem, first proved by Mordell for elliptic curves defined over $\Q$ and later generalized by Weil to abelian varieties $A$ over general number fields $K$, states that, if $K$ is a number field, then $A(K)$ is a finitely generated abelian group. Its rank is called the Mordell-Weil rank of $A$ over $K$.

The Mordell-Weil theorem implies in particular that the torsion subgroup $E(K)_{\mathrm{tor}}$ of $E(K)$ is finite, and thus that the torsion order of $E$, one of the BSD invariants, is finite.