Cassels–Tate pairing
E654586
The Cassels–Tate pairing is a bilinear pairing on the Tate–Shafarevich group of an abelian variety over a number field that plays a central role in arithmetic geometry and the study of rational points.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Cassels–Tate pairing canonical | 2 |
| Tate–Shafarevich group | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7304637 — 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: Cassels–Tate pairing Context triple: [J. W. S. Cassels, notableConcept, Cassels–Tate pairing]
-
A.
Weil pairing
The Weil pairing is a bilinear, alternating, non-degenerate pairing on the torsion points of an elliptic curve, fundamental in number theory and modern cryptography.
-
B.
Hasse bound for elliptic curves
The Hasse bound for elliptic curves is a fundamental result in number theory that gives tight limits on how far the number of points on an elliptic curve over a finite field can deviate from the size of the field plus one.
-
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.
Koblitz curves
Koblitz curves are a special class of elliptic curves defined over binary fields that enable particularly efficient and fast implementations of elliptic curve cryptography.
-
E.
Mordell–Weil 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.
- 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: Cassels–Tate pairing Target entity description: The Cassels–Tate pairing is a bilinear pairing on the Tate–Shafarevich group of an abelian variety over a number field that plays a central role in arithmetic geometry and the study of rational points.
-
A.
Weil pairing
The Weil pairing is a bilinear, alternating, non-degenerate pairing on the torsion points of an elliptic curve, fundamental in number theory and modern cryptography.
-
B.
Hasse bound for elliptic curves
The Hasse bound for elliptic curves is a fundamental result in number theory that gives tight limits on how far the number of points on an elliptic curve over a finite field can deviate from the size of the field plus one.
-
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.
Koblitz curves
Koblitz curves are a special class of elliptic curves defined over binary fields that enable particularly efficient and fast implementations of elliptic curve cryptography.
-
E.
Mordell–Weil 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.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
bilinear pairing
ⓘ
construction in arithmetic geometry ⓘ mathematical concept ⓘ |
| appearsIn | formulations of the Birch and Swinnerton-Dyer conjecture for abelian varieties ⓘ |
| associatedWith |
Tate–Shafarevich group of an elliptic curve
ⓘ
linked to:
Tate–Shafarevich group
abelian varieties ⓘ elliptic curves ⓘ |
| assumes | global field structure of a number field ⓘ |
| codomain | Q/Z ⓘ |
| constructionUses |
Galois cohomology
ⓘ
Weil pairing ⓘ cup product in cohomology ⓘ |
| context |
Mordell–Weil group and its Selmer groups
ⓘ
Selmer group of an abelian variety ⓘ |
| definedFor |
abelian variety over a number field
ⓘ
principally polarized abelian variety ⓘ |
| definedOn |
Tate–Shafarevich group
ⓘ
Tate–Shafarevich group of an abelian variety ⓘ Tate–Shafarevich group of an abelian variety over a number field ⓘ
linked to:
Tate–Shafarevich group
|
| domain |
Tate–Shafarevich group × Tate–Shafarevich group
ⓘ
linked to:
Tate–Shafarevich group
|
| field |
arithmetic geometry
ⓘ
number theory ⓘ |
| generalizes | pairing on the Tate–Shafarevich group of an elliptic curve defined by Cassels ⓘ |
| helpsDetermine | parity of the rank in some cases ⓘ |
| is |
alternating
ⓘ
bilinear ⓘ functorial in isogenies ⓘ skew-symmetric up to sign ⓘ |
| localComponents | pairings at each completion of the number field ⓘ |
| mathematicalDiscipline |
algebraic geometry
ⓘ
algebraic number theory ⓘ |
| namedAfter |
John Tate
ⓘ
John W. S. Cassels ⓘ
linked to:
John William Scott Cassels
|
| property |
conjecturally non-degenerate when the Tate–Shafarevich group is finite
ⓘ
its left and right kernels coincide with the maximal divisible subgroup of the Tate–Shafarevich group ⓘ non-degenerate modulo the maximal divisible subgroup ⓘ |
| relatedTo |
Néron–Tate height pairing
ⓘ
linked to:
Néron–Tate height
Poitou–Tate duality ⓘ Weil–Châtelet group ⓘ |
| type | global duality pairing ⓘ |
| usedIn |
Birch and Swinnerton-Dyer conjecture
ⓘ
linked to:
Birch and Swinnerton-Dyer Conjecture
analysis of the structure of the Tate–Shafarevich group ⓘ descent theory ⓘ obstruction theory for rational points ⓘ study of rational points on abelian varieties ⓘ study of rational points on elliptic curves ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Cassels–Tate pairing Description of subject: The Cassels–Tate pairing is a bilinear pairing on the Tate–Shafarevich group of an abelian variety over a number field that plays a central role in arithmetic geometry and the study of rational points.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Cassels–Tate pairing