Mordell–Weil theorem

For an abelian variety over a number field, the group of rational points is finitely generated. Its structure is a finite torsion subgroup together with a free abelian group of finite rank.

Abelian variety: A complete algebraic group variety, generalizing elliptic curves to higher dimensions. The theorem applies to rational points on every abelian variety over a number field.
Number field: A finite extension of the rational numbers. The theorem’s base field must have finite degree over the rationals.
Louis Mordell: A British mathematician whose work on Diophantine equations led to Mordell’s theorem. His 1922 theorem established finite generation for rational points on elliptic curves over the rationals.
Rank of an elliptic curve: The rank of the free abelian part of an elliptic curve’s rational-point group. For elliptic curves, this measures the number of independent nontorsion rational points.
Néron–Tate height: A quadratic height function on points of an abelian variety, built from arithmetic height and the group law. Height bounds control which points can have bounded arithmetic complexity.
Rational point (algebraic geometry): A point on an algebraic variety whose coordinates lie in the specified base field. The theorem concerns points whose coordinates lie in the number field.
André Weil: A French mathematician who proved a broad finite-generation theorem for abelian varieties over number fields. Weil extended Mordell’s result to its general form in 1928.
Torsion subgroup of an elliptic curve: The finite subgroup of rational points on an elliptic curve that have finite order. The theorem guarantees this subgroup is finite over number fields.
Weak Mordell–Weil theorem: A theorem that the quotient of an abelian variety’s rational points by multiplication by an integer is finite. It supplies the finite-quotient step that combines with height reduction to prove finite generation.
Finitely generated abelian group: An abelian group generated by a finite set of elements. This is the precise algebraic structure asserted for the rational points.

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