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 | 1 |
| 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
NERFINISHED
ⓘ
abelian varieties ⓘ elliptic curves ⓘ |
| assumes | global field structure of a number field ⓘ |
| codomain | Q/Z ⓘ |
| constructionUses |
Galois cohomology
NERFINISHED
ⓘ
Weil pairing NERFINISHED ⓘ 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
NERFINISHED
ⓘ
Tate–Shafarevich group of an abelian variety NERFINISHED ⓘ Tate–Shafarevich group of an abelian variety over a number field NERFINISHED ⓘ |
| domain | Tate–Shafarevich group × Tate–Shafarevich group NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
John W. S. Cassels NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
Poitou–Tate duality NERFINISHED ⓘ Weil–Châtelet group NERFINISHED ⓘ |
| type | global duality pairing ⓘ |
| usedIn |
Birch and Swinnerton-Dyer conjecture
NERFINISHED
ⓘ
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
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
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 (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Tate–Shafarevich group