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.
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.