Tamagawa numbers
E685697
arithmetic invariant
invariant of algebraic groups
invariant of elliptic curves
number theoretic invariant
Tamagawa numbers are arithmetic invariants attached to algebraic groups or elliptic curves that measure certain volume or local factor contributions in number theory, notably appearing in the Birch and Swinnerton-Dyer conjecture.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Tamagawa numbers canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7743537 — 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: Tamagawa numbers Context triple: [Birch and Swinnerton-Dyer Conjecture, relatesConcept, Tamagawa numbers]
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A.
Namba
Namba is a major commercial and entertainment district in Osaka, Japan, known for its bustling nightlife, shopping, and iconic neon-lit streets.
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B.
Namba Marui
Namba Marui is a major Marui department store and shopping complex located in Osaka’s bustling Namba district, known for its fashion, dining, and entertainment options.
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C.
Temwaiku
Temwaiku is a village and district within South Tarawa in Kiribati, known as one of the populated islets forming the country's capital area.
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D.
Namba Grand Kagetsu
Namba Grand Kagetsu is a famous comedy theater in Osaka, Japan, known as a central venue for Yoshimoto Kogyo’s manzai and variety performances.
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E.
Eightfold Way
The Eightfold Way is a classification scheme for hadrons in particle physics that organizes them into symmetry-based groups, laying the groundwork for the quark model.
- 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: Tamagawa numbers Target entity description: Tamagawa numbers are arithmetic invariants attached to algebraic groups or elliptic curves that measure certain volume or local factor contributions in number theory, notably appearing in the Birch and Swinnerton-Dyer conjecture.
-
A.
Namba
Namba is a major commercial and entertainment district in Osaka, Japan, known for its bustling nightlife, shopping, and iconic neon-lit streets.
-
B.
Namba Marui
Namba Marui is a major Marui department store and shopping complex located in Osaka’s bustling Namba district, known for its fashion, dining, and entertainment options.
-
C.
Temwaiku
Temwaiku is a village and district within South Tarawa in Kiribati, known as one of the populated islets forming the country's capital area.
-
D.
Namba Grand Kagetsu
Namba Grand Kagetsu is a famous comedy theater in Osaka, Japan, known as a central venue for Yoshimoto Kogyo’s manzai and variety performances.
-
E.
Eightfold Way
The Eightfold Way is a classification scheme for hadrons in particle physics that organizes them into symmetry-based groups, laying the groundwork for the quark model.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
arithmetic invariant
ⓘ
invariant of algebraic groups ⓘ invariant of elliptic curves ⓘ number theoretic invariant ⓘ |
| appearsAsFactorIn |
BSD formula denominator
ⓘ
volume computations for arithmetic quotients ⓘ |
| appearsIn |
Birch and Swinnerton-Dyer formula
NERFINISHED
ⓘ
formula for the leading term of the L-function of an elliptic curve at s = 1 ⓘ mass formulae for algebraic groups ⓘ |
| associatedWith |
algebraic group over a number field
ⓘ
elliptic curve over a number field ⓘ |
| conjecturallyRelatedTo |
finiteness of Tate–Shafarevich groups
ⓘ
rank of elliptic curves ⓘ |
| context |
Langlands program
NERFINISHED
ⓘ
arithmetic of reductive groups ⓘ automorphic forms ⓘ |
| definedOver | number fields ⓘ |
| definedUsing |
adelic quotient of an algebraic group
ⓘ
product of local measures ⓘ |
| describedAs | volume of adelic points modulo rational points ⓘ |
| field |
algebraic geometry
ⓘ
arithmetic geometry ⓘ number theory ⓘ |
| historicalDevelopment | introduced in the mid-20th century ⓘ |
| namedAfter | Toshio Tamagawa NERFINISHED ⓘ |
| property |
factorizes as a product of local contributions
ⓘ
invariant under isomorphism of algebraic groups over a number field ⓘ |
| relatedTo |
Galois cohomology
ⓘ
Haar measure ⓘ Tamagawa measure NERFINISHED ⓘ Weil’s adelic formalism NERFINISHED ⓘ adelic points ⓘ cohomology of algebraic groups ⓘ local factors of L-functions ⓘ rational points ⓘ |
| specialCase |
Tamagawa number of a semisimple algebraic group
NERFINISHED
ⓘ
Tamagawa number of a torus ⓘ Tamagawa number of an elliptic curve ⓘ |
| studiedBy |
André Weil
NERFINISHED
ⓘ
Goro Shimura NERFINISHED ⓘ John Tate NERFINISHED ⓘ Yutaka Taniyama NERFINISHED ⓘ |
| takesValuesIn | positive rational numbers ⓘ |
| usedIn |
Birch and Swinnerton-Dyer conjecture
NERFINISHED
ⓘ
Tamagawa measure theory NERFINISHED ⓘ Weil conjectures for algebraic groups NERFINISHED ⓘ arithmetic of elliptic curves ⓘ rational points on abelian varieties ⓘ study of L-functions ⓘ |
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: Tamagawa numbers Description of subject: Tamagawa numbers are arithmetic invariants attached to algebraic groups or elliptic curves that measure certain volume or local factor contributions in number theory, notably appearing in the Birch and Swinnerton-Dyer conjecture.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.