Shafarevich group of a torus
E839566
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Tate–Shafarevich group | 2 |
| Shafarevich group of a torus canonical | 1 |
| Shafarevich–Tate group | 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 ⓘ |
| 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 ⓘ |
| property |
functorial in morphisms of tori
ⓘ
torsion abelian group in many arithmetic situations ⓘ |
| relatedTo |
Brauer–Manin obstruction
ⓘ
Hasse principle ⓘ Poitou–Tate duality ⓘ Tate–Shafarevich group ⓘ 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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
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: Shafarevich group of a torus Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Shafarevich group of a torus
linked to: Shafarevich group of a torus
linked to: Shafarevich group of a torus