reciprocity conjecture
E870216
The reciprocity conjecture is a far-reaching set of ideas in number theory and representation theory that generalizes classical reciprocity laws by relating Galois groups to automorphic forms within the Langlands program.
All labels observed (1)
| Label | Occurrences |
|---|---|
| reciprocity conjecture canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T10550215 — 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: reciprocity conjecture Context triple: [Robert Langlands, knownFor, reciprocity conjecture]
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A.
Tate Conjecture
The Tate Conjecture is a major open problem in arithmetic geometry that predicts a deep connection between algebraic cycles on varieties over finite fields and their Galois-invariant étale cohomology classes.
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B.
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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C.
Ramanujan–Petersson conjecture
The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
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D.
quadratic reciprocity law
The quadratic reciprocity law is a fundamental theorem in number theory that characterizes when a quadratic equation modulo one odd prime has solutions in terms of solvability modulo another, revealing a deep symmetry between primes.
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E.
Taniyama–Shimura–Weil conjecture
The Taniyama–Shimura–Weil conjecture, now the modularity theorem, asserts that every elliptic curve over the rational numbers is modular and played a central role in the proof of Fermat’s Last Theorem.
- 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: reciprocity conjecture Target entity description: The reciprocity conjecture is a far-reaching set of ideas in number theory and representation theory that generalizes classical reciprocity laws by relating Galois groups to automorphic forms within the Langlands program.
-
A.
Tate Conjecture
The Tate Conjecture is a major open problem in arithmetic geometry that predicts a deep connection between algebraic cycles on varieties over finite fields and their Galois-invariant étale cohomology classes.
-
B.
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.
-
C.
Ramanujan–Petersson conjecture
The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
-
D.
quadratic reciprocity law
The quadratic reciprocity law is a fundamental theorem in number theory that characterizes when a quadratic equation modulo one odd prime has solutions in terms of solvability modulo another, revealing a deep symmetry between primes.
-
E.
Taniyama–Shimura–Weil conjecture
The Taniyama–Shimura–Weil conjecture, now the modularity theorem, asserts that every elliptic curve over the rational numbers is modular and played a central role in the proof of Fermat’s Last Theorem.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
conjecture in number theory
ⓘ
conjecture in representation theory ⓘ mathematical conjecture ⓘ |
| aimsToDescribe | relationship between arithmetic of fields and harmonic analysis on groups ⓘ |
| associatedWith |
Langlands correspondence
ⓘ
linked to:
local Langlands correspondence
Robert Langlands ⓘ |
| basedOn | classical reciprocity laws ⓘ |
| concerns |
L-functions
ⓘ
automorphic L-functions ⓘ motivic Galois representations ⓘ |
| coreIdea |
correspondence between Galois representations and automorphic representations
ⓘ
generalization of abelian class field theory ⓘ |
| field |
Galois theory
ⓘ
arithmetic geometry ⓘ automorphic forms ⓘ number theory ⓘ representation theory ⓘ |
| generalizes |
higher reciprocity laws
ⓘ
quadratic reciprocity law ⓘ |
| hasAspect |
global reciprocity
ⓘ
local reciprocity ⓘ |
| historicalRoot |
Artin reciprocity law
ⓘ
Hilbert reciprocity law ⓘ Kronecker–Weber theorem ⓘ |
| implies | class field theory in the abelian case ⓘ |
| influences |
arithmetic geometry
ⓘ
modern algebraic number theory ⓘ theory of automorphic forms ⓘ |
| motivation | unify reciprocity phenomena in number theory ⓘ |
| partOf | Langlands program ⓘ |
| relatedTo |
Taniyama–Shimura–Weil conjecture
ⓘ
functoriality conjecture ⓘ
linked to:
Langlands program
modularity of elliptic curves ⓘ |
| relates |
Galois groups
ⓘ
linked to:
Galois group
automorphic forms ⓘ automorphic representations ⓘ |
| scope | far-reaching generalization of reciprocity laws ⓘ |
| status | open problem ⓘ |
| typicalDomain |
global fields
ⓘ
local fields ⓘ number fields ⓘ |
| usesConcept |
Galois representations into L-groups
ⓘ
automorphic representations of adelic groups ⓘ reductive algebraic groups ⓘ |
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: reciprocity conjecture Description of subject: The reciprocity conjecture is a far-reaching set of ideas in number theory and representation theory that generalizes classical reciprocity laws by relating Galois groups to automorphic forms within the Langlands program.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.