Mordell–Weil theorem

In mathematics, the Mordell–Weil theorem states that for an abelian variety ${\displaystyle A}$ over a number field ${\displaystyle K}$, the group ${\displaystyle A(K)}$ of K-rational points of ${\displaystyle A}$ is a finitely-generated abelian group, called the Mordell–Weil group. The case with ${\displaystyle A}$ an elliptic curve ${\displaystyle E}$ and ${\displaystyle K}$ the field of rational numbers is Mordell's theorem, answering a question apparently posed by Henri Poincaré around 1901; it was proved by Louis Mordell in 1922. It is a foundational theorem of Diophantine geometry and the arithmetic of abelian varieties.