Mordell–Weil theorem
E641515
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Mordell–Weil theorem canonical | 2 |
| Mordell–Weil group | 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]
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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.
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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.
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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.
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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 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 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Mordell–Weil theorem Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Mordell–Weil group