Langlands program
E753154
The Langlands program is a far-reaching web of conjectures and theories in number theory and representation theory that seeks deep connections between Galois groups and automorphic forms, unifying many areas of modern mathematics.
All labels observed (9)
How this entity was disambiguated
This entity first appeared as the object of triple T8733479 — 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: Langlands program Context triple: [Hasse–Weil zeta function, studiedIn, Langlands program]
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A.
Galois representations
Galois representations are homomorphisms from Galois groups of field extensions into matrix groups that encode deep arithmetic information and link number theory with algebraic geometry and modular forms.
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B.
Representation Theory and Automorphic Functions
"Representation Theory and Automorphic Functions" is a seminal mathematical work by Israel Gelfand that develops the connections between representation theory of groups and the theory of automorphic forms, with deep applications in number theory and harmonic analysis.
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C.
Weil conjectures
The Weil conjectures are a set of deep statements about the zeta functions of algebraic varieties over finite fields that guided the development of modern algebraic geometry and were ultimately proved using étale cohomology.
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D.
Iwasawa theory
Iwasawa theory is a branch of number theory that studies the growth of arithmetic invariants in infinite towers of number fields, particularly using p-adic methods.
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E.
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.
- 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: Langlands program Target entity description: The Langlands program is a far-reaching web of conjectures and theories in number theory and representation theory that seeks deep connections between Galois groups and automorphic forms, unifying many areas of modern mathematics.
-
A.
Galois representations
Galois representations are homomorphisms from Galois groups of field extensions into matrix groups that encode deep arithmetic information and link number theory with algebraic geometry and modular forms.
-
B.
Representation Theory and Automorphic Functions
"Representation Theory and Automorphic Functions" is a seminal mathematical work by Israel Gelfand that develops the connections between representation theory of groups and the theory of automorphic forms, with deep applications in number theory and harmonic analysis.
-
C.
Weil conjectures
The Weil conjectures are a set of deep statements about the zeta functions of algebraic varieties over finite fields that guided the development of modern algebraic geometry and were ultimately proved using étale cohomology.
-
D.
Iwasawa theory
Iwasawa theory is a branch of number theory that studies the growth of arithmetic invariants in infinite towers of number fields, particularly using p-adic methods.
-
E.
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.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theory
ⓘ
research program ⓘ |
| aimsTo |
generalize class field theory
ⓘ
relate Galois groups to automorphic forms ⓘ unify number theory and representation theory ⓘ |
| basedOn | Langlands correspondence NERFINISHED ⓘ |
| conjecturedBy | Robert Langlands NERFINISHED ⓘ |
| coreConcept |
Galois representation
NERFINISHED
ⓘ
L-function NERFINISHED ⓘ automorphic representation ⓘ functoriality ⓘ motives ⓘ |
| field |
algebraic geometry
ⓘ
arithmetic geometry ⓘ harmonic analysis ⓘ number theory ⓘ representation theory ⓘ |
| hasApproach |
Shimura varieties
NERFINISHED
ⓘ
endoscopy theory ⓘ geometric methods via perverse sheaves and D-modules ⓘ p-adic Hodge theory ⓘ theta correspondence ⓘ trace formula ⓘ |
| hasPart |
Langlands functoriality conjecture
NERFINISHED
ⓘ
Ramanujan–Petersson conjecture (automorphic form version) NERFINISHED ⓘ geometric Langlands program NERFINISHED ⓘ global Langlands correspondence NERFINISHED ⓘ local Langlands correspondence NERFINISHED ⓘ reciprocity conjecture NERFINISHED ⓘ |
| inception | 1967 ⓘ |
| influenced |
development of geometric representation theory
ⓘ
modularity theorem NERFINISHED ⓘ proof of Fermat's Last Theorem ⓘ theory of motives ⓘ |
| influencedBy |
Taniyama–Shimura conjecture
NERFINISHED
ⓘ
class field theory NERFINISHED ⓘ |
| namedAfter | Robert Langlands NERFINISHED ⓘ |
| notableResult |
Jacquet–Langlands correspondence
NERFINISHED
ⓘ
Langlands correspondence for function fields (Drinfeld and Lafforgue) NERFINISHED ⓘ Langlands–Tunnell theorem NERFINISHED ⓘ base change for GL(2) ⓘ global Langlands correspondence for GL(2) over number fields (partial) ⓘ local Langlands correspondence for GL(n) ⓘ proof of Sato–Tate conjecture in many cases ⓘ |
| openProblem |
full global Langlands correspondence for number fields
ⓘ
general functoriality for all reductive groups ⓘ |
| relates |
Hecke eigenforms
NERFINISHED
ⓘ
adelic groups ⓘ automorphic representations of reductive groups over global fields ⓘ n-dimensional Galois representations ⓘ |
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: Langlands program Description of subject: The Langlands program is a far-reaching web of conjectures and theories in number theory and representation theory that seeks deep connections between Galois groups and automorphic forms, unifying many areas of modern mathematics.
Referenced by (25)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
local Langlands correspondence
this entity surface form:
local Langlands program (via theta correspondence)
this entity surface form:
Langlands conjectures
this entity surface form:
functoriality conjecture
this entity surface form:
Langlands correspondence
this entity surface form:
geometric Langlands theory
this entity surface form:
geometric Langlands correspondence
this entity surface form:
geometric Langlands program
this entity surface form:
local Langlands correspondence
subject surface form:
L-function