Hasse invariant
E207312
The Hasse invariant is an arithmetic invariant in number theory and algebraic geometry that classifies structures such as quadratic forms or elliptic curves over local and global fields, playing a key role in local-global principles.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Hasse invariant canonical | 1 |
| Hilbert symbol | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1862414 — 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: Hasse invariant Context triple: [Helmut Hasse, notableWork, Hasse invariant]
-
A.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
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B.
Deuring reduction theorem
The Deuring reduction theorem is a result in number theory that relates the reduction of elliptic curves with complex multiplication modulo primes to the theory of quaternion algebras and endomorphism rings.
-
C.
Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
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D.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
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E.
Levine-Fricke Field
Levine-Fricke Field is the home softball stadium of the University of California, Berkeley Golden Bears, located on the university’s campus in Berkeley, California.
- 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: Hasse invariant Target entity description: The Hasse invariant is an arithmetic invariant in number theory and algebraic geometry that classifies structures such as quadratic forms or elliptic curves over local and global fields, playing a key role in local-global principles.
-
A.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
-
B.
Deuring reduction theorem
The Deuring reduction theorem is a result in number theory that relates the reduction of elliptic curves with complex multiplication modulo primes to the theory of quaternion algebras and endomorphism rings.
-
C.
Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
-
D.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
-
E.
Levine-Fricke Field
Levine-Fricke Field is the home softball stadium of the University of California, Berkeley Golden Bears, located on the university’s campus in Berkeley, California.
- F. None of above. chosen
Statements (51)
| Predicate | Object |
|---|---|
| instanceOf |
arithmetic invariant
ⓘ
mathematical invariant ⓘ notion in algebraic geometry ⓘ notion in number theory ⓘ |
| appearsIn |
classification of quaternion algebras
ⓘ
statement of the Hasse–Minkowski theorem ⓘ theory of p-divisible groups and Dieudonné modules ⓘ theory of supersingular elliptic curves ⓘ |
| appliesTo |
Brauer group elements
ⓘ
central simple algebras ⓘ elliptic curves over finite fields ⓘ elliptic curves over local fields ⓘ modular forms ⓘ p-divisible groups ⓘ quadratic forms over global fields ⓘ quadratic forms over local fields ⓘ |
| context |
Galois cohomology
ⓘ
algebraic geometry over finite fields ⓘ arithmetic of elliptic curves ⓘ arithmetic of quadratic forms ⓘ modular curves ⓘ |
| hasDomain |
finite fields
ⓘ
function fields ⓘ global fields ⓘ local fields ⓘ number fields ⓘ p-adic fields ⓘ |
| namedAfter | Helmut Hasse ⓘ |
| relatedTo |
Brauer group
ⓘ
Hasse principle ⓘ Hilbert symbol ⓘ Witt group of quadratic forms ⓘ discriminant of a quadratic form ⓘ global invariants ⓘ local invariants ⓘ |
| roleIn |
classification of central simple algebras over global fields
ⓘ
classification of central simple algebras over local fields ⓘ classification of quadratic forms over number fields ⓘ description of the Brauer group of a global field ⓘ description of the Brauer group of a local field ⓘ |
| usedIn |
Hasse principle
ⓘ
surface form:
Hasse–Minkowski theorem
classification of elliptic curves ⓘ classification of quadratic forms ⓘ local class field theory ⓘ local-global principles ⓘ |
| usedToDistinguish |
isomorphism classes of central simple algebras
ⓘ
isomorphism classes of quadratic forms ⓘ ordinary and supersingular elliptic curves in characteristic p ⓘ |
| valueType |
element of {+1,-1} for certain quadratic forms
ⓘ
integer modulo p-1 for some p-adic contexts ⓘ rational number modulo 1 for central simple algebras ⓘ |
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: Hasse invariant Description of subject: The Hasse invariant is an arithmetic invariant in number theory and algebraic geometry that classifies structures such as quadratic forms or elliptic curves over local and global fields, playing a key role in local-global principles.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Hilbert symbol