Shafarevich group of a torus

E839566

Galois cohomology group arithmetic invariant cohomological invariant

The Shafarevich group of a torus is an arithmetic invariant measuring the failure of local-global principles for principal homogeneous spaces under an algebraic torus over a global field.

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Observed surface forms (1)

Surface form	Occurrences
Tate–Shafarevich group	2

Statements (40)

Predicate	Object
instanceOf	Galois cohomology group ⓘ arithmetic invariant ⓘ cohomological invariant ⓘ
appearsIn	descent theory for tori ⓘ obstruction theory for rational points ⓘ study of norm one tori ⓘ
associatedWith	algebraic torus ⓘ principal homogeneous space ⓘ
captures	classes trivial in all local cohomology groups ⓘ
context	Galois cohomology ⓘ arithmetic geometry ⓘ number theory ⓘ
definedAs	kernel of the localization map from global to local Galois cohomology for a torus ⓘ
definedOver	global field ⓘ
dependsOn	choice of global field ⓘ isomorphism class of the torus ⓘ
generalizes	Tate–Shafarevich group of an abelian variety NERFINISHED ⓘ
involves	Galois cohomology group H^1 of a torus ⓘ local Galois cohomology groups at all places of a global field ⓘ
isSubsetOf	H^1 of the global Galois group with values in the torus ⓘ
mathematicalDomain	algebraic geometry ⓘ algebraic number theory ⓘ cohomology of groups ⓘ
measures	failure of local-global principle ⓘ obstruction to the Hasse principle ⓘ
namedAfter	Igor Shafarevich NERFINISHED ⓘ
property	functorial in morphisms of tori ⓘ torsion abelian group in many arithmetic situations ⓘ
relatedTo	Brauer–Manin obstruction NERFINISHED ⓘ Hasse principle NERFINISHED ⓘ Poitou–Tate duality NERFINISHED ⓘ Tate–Shafarevich group NERFINISHED ⓘ class field theory for tori ⓘ principal homogeneous spaces under tori ⓘ weak approximation ⓘ
usedIn	arithmetic of algebraic tori ⓘ classification of torsors under tori ⓘ study of rational points on varieties with torus actions ⓘ
usedToStudy	failure of weak approximation on tori ⓘ local-global principles for torsors under tori ⓘ

Referenced by (3)

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

Hasse norm theorem → relatedTo → Shafarevich group of a torus ⓘ

Cohomologie Galoisienne → topic → Shafarevich group of a torus ⓘ

this entity surface form: Tate–Shafarevich group

John Tate → notableWork → Shafarevich group of a torus ⓘ

this entity surface form: Tate–Shafarevich group