Weil–Deligne group
E860116
The Weil–Deligne group is an extension of the Weil group by a copy of the additive group that encodes both arithmetic and monodromy data, playing a central role in the local Langlands correspondence and the study of l-adic Galois representations.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Weil–Deligne group canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10389109 — 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: Weil–Deligne group Context triple: [Weil group, hasVariant, Weil–Deligne group]
-
A.
Weil group
The Weil group is an extension of the absolute Galois group introduced by André Weil to refine class field theory and play a central role in the formulation of the local and global Langlands correspondences.
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B.
Deligne–Lusztig theory
Deligne–Lusztig theory is a framework in algebraic geometry and representation theory that constructs and studies representations of finite groups of Lie type using varieties defined over finite fields.
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C.
Weil cohomology
Weil cohomology is a type of cohomology theory for algebraic varieties that satisfies specific axioms enabling the proof of the Weil conjectures and the development of modern algebraic geometry.
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D.
Frobenius element
The Frobenius element is a distinguished element in a Galois group associated to an unramified prime, encoding how that prime splits in a field extension and playing a central role in algebraic number theory and arithmetic geometry.
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E.
Brauer group
The Brauer group is an algebraic structure that classifies equivalence classes of central simple algebras over a field (or more general schemes), playing a key role in number theory, algebraic geometry, and cohomology.
- 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: Weil–Deligne group Target entity description: The Weil–Deligne group is an extension of the Weil group by a copy of the additive group that encodes both arithmetic and monodromy data, playing a central role in the local Langlands correspondence and the study of l-adic Galois representations.
-
A.
Weil group
The Weil group is an extension of the absolute Galois group introduced by André Weil to refine class field theory and play a central role in the formulation of the local and global Langlands correspondences.
-
B.
Deligne–Lusztig theory
Deligne–Lusztig theory is a framework in algebraic geometry and representation theory that constructs and studies representations of finite groups of Lie type using varieties defined over finite fields.
-
C.
Weil cohomology
Weil cohomology is a type of cohomology theory for algebraic varieties that satisfies specific axioms enabling the proof of the Weil conjectures and the development of modern algebraic geometry.
-
D.
Frobenius element
The Frobenius element is a distinguished element in a Galois group associated to an unramified prime, encoding how that prime splits in a field extension and playing a central role in algebraic number theory and arithmetic geometry.
-
E.
Brauer group
The Brauer group is an algebraic structure that classifies equivalence classes of central simple algebras over a field (or more general schemes), playing a key role in number theory, algebraic geometry, and cohomology.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
extension of groups
ⓘ
mathematical object ⓘ topological group ⓘ |
| appearsIn |
Deligne’s theory of local constants
ⓘ
Grothendieck’s theory of ℓ-adic representations ⓘ |
| associatedWith |
complex representation
ⓘ
non-archimedean local field ⓘ p-adic field ⓘ |
| captures |
inertia action
ⓘ
tame ramification ⓘ unipotent monodromy ⓘ wild ramification ⓘ |
| context |
local class field theory
ⓘ
non-abelian local class field theory ⓘ |
| definedOver | local field ⓘ |
| encodes |
arithmetic data
ⓘ
monodromy data ⓘ |
| fieldOfStudy |
Langlands program
NERFINISHED
ⓘ
arithmetic geometry ⓘ number theory ⓘ representation theory ⓘ |
| formalizedBy | Pierre Deligne NERFINISHED ⓘ |
| generalizationOf | Weil group NERFINISHED ⓘ |
| hasComponent |
Weil group
NERFINISHED
ⓘ
additive group of complex numbers ⓘ nilpotent operator ⓘ |
| hasParameterization | Langlands parameter via homomorphism into L-group ⓘ |
| hasRole | local Langlands group candidate ⓘ |
| hasStructure | semidirect product ⓘ |
| namedAfter |
André Weil
NERFINISHED
ⓘ
Pierre Deligne NERFINISHED ⓘ |
| relatedTo |
Frobenius element
NERFINISHED
ⓘ
Weil group NERFINISHED ⓘ Weil–Deligne parameter NERFINISHED ⓘ Weil–Deligne representation NERFINISHED ⓘ absolute Galois group ⓘ local Langlands correspondence NERFINISHED ⓘ monodromy operator ⓘ ℓ-adic Galois representation ⓘ |
| usedIn |
classification of smooth representations of p-adic groups
ⓘ
local Langlands parameterization NERFINISHED ⓘ study of local L-functions ⓘ study of local ε-factors ⓘ study of étale cohomology ⓘ study of ℓ-adic sheaves ⓘ |
| usedToDefine | Weil–Deligne representation of a local field NERFINISHED ⓘ |
| usedToRelate | Galois representations and automorphic 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: Weil–Deligne group Description of subject: The Weil–Deligne group is an extension of the Weil group by a copy of the additive group that encodes both arithmetic and monodromy data, playing a central role in the local Langlands correspondence and the study of l-adic Galois representations.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.