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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All labels observed (2)

Label	Occurrences
Tate–Shafarevich group	2
Shafarevich group of a torus canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T10063340 — 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: Shafarevich group of a torus
Context triple: [Hasse norm theorem, relatedTo, Shafarevich group of a torus]

A. Algebraic Groups and Class Fields
"Algebraic Groups and Class Fields" is a influential mathematical monograph that develops the deep connections between algebraic group theory and class field theory within number theory and arithmetic geometry.
B. Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
C. Adeles and Algebraic Groups
"Adeles and Algebraic Groups" is a foundational mathematical work by André Weil that develops the theory of adeles and its deep connections with algebraic groups and number theory.
D. Hasse–Weil bound for abelian varieties
The Hasse–Weil bound for abelian varieties is a fundamental result in arithmetic geometry that gives sharp estimates for the number of rational points on abelian varieties over finite fields in terms of their dimension and the field size.
E. Brauer–Manin obstruction
The Brauer–Manin obstruction is an arithmetic-geometric mechanism using the Brauer group and adelic points to explain failures of the Hasse principle and weak approximation for rational points on varieties.
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: Shafarevich group of a torus
Target entity description: 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.

A. Algebraic Groups and Class Fields
"Algebraic Groups and Class Fields" is a influential mathematical monograph that develops the deep connections between algebraic group theory and class field theory within number theory and arithmetic geometry.
B. Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
C. Adeles and Algebraic Groups
"Adeles and Algebraic Groups" is a foundational mathematical work by André Weil that develops the theory of adeles and its deep connections with algebraic groups and number theory.
D. Hasse–Weil bound for abelian varieties
The Hasse–Weil bound for abelian varieties is a fundamental result in arithmetic geometry that gives sharp estimates for the number of rational points on abelian varieties over finite fields in terms of their dimension and the field size.
E. Brauer–Manin obstruction
The Brauer–Manin obstruction is an arithmetic-geometric mechanism using the Brauer group and adelic points to explain failures of the Hasse principle and weak approximation for rational points on varieties.
F. None of above. chosen

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 ⓘ

How these facts were elicited

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 ⓘ