Cassels–Tate pairing

E654586

bilinear pairing construction in arithmetic geometry mathematical concept

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.

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Observed surface forms (1)

Surface form	Occurrences
Tate–Shafarevich group	1

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 ⓘ

Referenced by (2)

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

J. W. S. Cassels → notableConcept → Cassels–Tate pairing ⓘ

Birch and Swinnerton-Dyer Conjecture → relatesConcept → Cassels–Tate pairing ⓘ

this entity surface form: Tate–Shafarevich group