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.

Try in SPARQL Jump to: Surface forms Statements Referenced by

Observed surface forms (1)

Surface form	Occurrences
Mordell–Weil group	1

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 NERFINISHED ⓘ
field	number theory ⓘ
generalizes	Mordell's theorem on rational points on elliptic curves over number fields NERFINISHED ⓘ
generalizesFrom	elliptic curves ⓘ
generalizesTo	abelian varieties ⓘ
hasKeyStep	weak Mordell–Weil theorem plus height descent NERFINISHED ⓘ
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 NERFINISHED ⓘ descent ⓘ finitely generated abelian group ⓘ height function ⓘ rank of an abelian variety ⓘ torsion subgroup ⓘ weak Mordell–Weil theorem NERFINISHED ⓘ
namedAfter	André Weil NERFINISHED ⓘ Louis Mordell NERFINISHED ⓘ
originallyProvedBy	Louis Mordell NERFINISHED ⓘ
relatedTo	Birch and Swinnerton-Dyer conjecture NERFINISHED ⓘ Faltings's theorem NERFINISHED ⓘ Néron–Tate height NERFINISHED ⓘ Shafarevich–Tate group NERFINISHED ⓘ
standardReference	André Weil "Variétés abéliennes et courbes algébriques" NERFINISHED ⓘ J. H. Silverman "The Arithmetic of Elliptic Curves" NERFINISHED ⓘ Serge Lang "Elliptic Curves: Diophantine Analysis" NERFINISHED ⓘ
statesThat	the group of rational points on an abelian variety over a number field is finitely generated ⓘ
subfield	Diophantine geometry NERFINISHED ⓘ 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 ⓘ

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Louis Mordell → knownFor → Mordell–Weil theorem ⓘ

Birch and Swinnerton-Dyer Conjecture → relatesConcept → Mordell–Weil theorem ⓘ

this entity surface form: Mordell–Weil group

Diophantine geometry → relatedTo → Mordell–Weil theorem ⓘ