Mordell–Weil theorem

E641515

mathematical theorem

The Mordell–Weil theorem is a fundamental result in number theory stating that the group of rational points on an abelian variety (in particular, an elliptic curve) over a number field is finitely generated.

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All labels observed (6)

Label	Occurrences
Mordell–Weil theorem canonical	7
Mordell's theorem on rational points on elliptic curves over number fields	1
Mordell–Weil group	1
Mordell’s theorem	1
weak Mordell–Weil theorem	1
weak Mordell–Weil theorem plus height descent	1

How this entity was disambiguated

This entity first appeared as the object of triple T7078801 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Mordell–Weil theorem
Context triple: [Louis Mordell, knownFor, Mordell–Weil theorem]

A. Faltings' theorem
Faltings' theorem is a landmark result in arithmetic geometry that proves every algebraic curve of genus greater than one over a number field has only finitely many rational points.
B. Mordell curve
A Mordell curve is an elliptic curve of the form \(y^2 = x^3 + k\) over a field, central to number theory and Diophantine geometry.
C. Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
D. Hermite–Minkowski theorem
The Hermite–Minkowski theorem is a fundamental result in algebraic number theory that gives a finiteness bound on the number of number fields of a given degree and discriminant.
E. Taniyama–Shimura–Weil conjecture
The Taniyama–Shimura–Weil conjecture, now the modularity theorem, asserts that every elliptic curve over the rational numbers is modular and played a central role in the proof of Fermat’s Last Theorem.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Mordell–Weil theorem
Target entity description: The Mordell–Weil theorem is a fundamental result in number theory stating that the group of rational points on an abelian variety (in particular, an elliptic curve) over a number field is finitely generated.

A. Faltings' theorem
Faltings' theorem is a landmark result in arithmetic geometry that proves every algebraic curve of genus greater than one over a number field has only finitely many rational points.
B. Mordell curve
A Mordell curve is an elliptic curve of the form \(y^2 = x^3 + k\) over a field, central to number theory and Diophantine geometry.
C. Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
D. Hermite–Minkowski theorem
The Hermite–Minkowski theorem is a fundamental result in algebraic number theory that gives a finiteness bound on the number of number fields of a given degree and discriminant.
E. Taniyama–Shimura–Weil conjecture
The Taniyama–Shimura–Weil conjecture, now the modularity theorem, asserts that every elliptic curve over the rational numbers is modular and played a central role in the proof of Fermat’s Last Theorem.
F. None of above. chosen

Statements (45)

Predicate	Object
instanceOf	mathematical theorem ⓘ
appearsIn	textbooks on arithmetic geometry ⓘ textbooks on elliptic curves ⓘ
classification	gives structure theorem for rational points on abelian varieties over number fields ⓘ
concerns	abelian varieties ⓘ elliptic curves ⓘ number fields ⓘ rational points ⓘ
describes	structure of rational points as a finitely generated abelian group ⓘ
extendedBy	André Weil ⓘ
field	number theory ⓘ
generalizes	Mordell's theorem on rational points on elliptic curves over number fields ⓘ linked to: Mordell–Weil theorem
generalizesFrom	elliptic curves ⓘ
generalizesTo	abelian varieties ⓘ
hasKeyStep	weak Mordell–Weil theorem plus height descent ⓘ linked to: Mordell–Weil theorem
holdsOver	number fields ⓘ
implies	finiteness of generators for rational points on an elliptic curve over a number field ⓘ the group of rational points is isomorphic to a finite torsion subgroup plus a free abelian group of finite rank ⓘ the group of rational points on an elliptic curve over a number field is finitely generated ⓘ
involvesConcept	Mordell–Weil group ⓘ descent ⓘ finitely generated abelian group ⓘ height function ⓘ rank of an abelian variety ⓘ torsion subgroup ⓘ weak Mordell–Weil theorem ⓘ linked to: Mordell–Weil theorem
namedAfter	André Weil ⓘ Louis Mordell ⓘ
originallyProvedBy	Louis Mordell ⓘ
relatedTo	Birch and Swinnerton-Dyer conjecture ⓘ linked to: Birch and Swinnerton-Dyer Conjecture Faltings's theorem ⓘ linked to: Faltings' theorem Néron–Tate height ⓘ Shafarevich–Tate group ⓘ linked to: Shafarevich group of a torus
standardReference	André Weil "Variétés abéliennes et courbes algébriques" ⓘ linked to: Sur les courbes algébriques et les variétés qui s’en déduisent J. H. Silverman "The Arithmetic of Elliptic Curves" ⓘ linked to: Arithmetic of Elliptic Curves Serge Lang "Elliptic Curves: Diophantine Analysis" ⓘ linked to: Elliptic Curves: Diophantine Analysis
statesThat	the group of rational points on an abelian variety over a number field is finitely generated ⓘ
subfield	Diophantine geometry ⓘ arithmetic geometry ⓘ
topic	Diophantine equations ⓘ rational points on varieties ⓘ
usedIn	proofs of finiteness results for Diophantine equations ⓘ study of abelian varieties over global fields ⓘ study of elliptic curves over number fields ⓘ
yearOfOriginalProof	1922 ⓘ

How these facts were elicited