Galois representations
E534413
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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Galois representations canonical | 5 |
| Artin representation | 1 |
| Tate modules | 1 |
| Weil–Deligne representations | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5570528 — 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: Galois representations Context triple: [Fermat's Last Theorem, proofUses, Galois representations]
-
A.
Serre’s conjecture on Galois representations
Serre’s conjecture on Galois representations is a landmark statement in number theory that predicts which two-dimensional mod p Galois representations of the absolute Galois group of the rationals arise from modular forms.
-
B.
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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C.
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.
-
D.
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.
-
E.
Cohomologie Galoisienne
Cohomologie Galoisienne is a foundational monograph by Jean-Pierre Serre that systematically develops Galois cohomology and its deep applications in number theory and algebraic geometry.
- 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: Galois representations Target entity description: 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.
-
A.
Serre’s conjecture on Galois representations
Serre’s conjecture on Galois representations is a landmark statement in number theory that predicts which two-dimensional mod p Galois representations of the absolute Galois group of the rationals arise from modular forms.
-
B.
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.
-
C.
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.
-
D.
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.
-
E.
Cohomologie Galoisienne
Cohomologie Galoisienne is a foundational monograph by Jean-Pierre Serre that systematically develops Galois cohomology and its deep applications in number theory and algebraic geometry.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
representation of a group ⓘ |
| centralIn |
modularity theorem for elliptic curves
ⓘ
proof of Fermat's Last Theorem ⓘ |
| codomain | general linear group over a field ⓘ |
| definedAs | group homomorphisms from Galois groups to matrix groups ⓘ |
| domain | Galois group of a field extension ⓘ |
| encodes |
arithmetic information about field extensions
ⓘ
information about algebraic numbers ⓘ information about algebraic varieties ⓘ information about modular forms ⓘ |
| field |
algebraic geometry
ⓘ
arithmetic geometry ⓘ number theory ⓘ representation theory ⓘ |
| formalism | continuous homomorphisms with respect to profinite topology ⓘ |
| hasType |
Artin representation
ⓘ
Hodge–Tate representation NERFINISHED ⓘ crystalline representation ⓘ de Rham representation ⓘ geometric Galois representation ⓘ l-adic Galois representation ⓘ p-adic Galois representation ⓘ |
| oftenAssumed | continuous with respect to l-adic topology ⓘ |
| relatedTo |
Tate modules of abelian varieties
ⓘ
Weil–Deligne representations NERFINISHED ⓘ automorphic forms ⓘ fundamental groups of schemes ⓘ modular forms ⓘ motivic Galois groups NERFINISHED ⓘ étale cohomology ⓘ |
| studiedBy |
Andrew Wiles
NERFINISHED
ⓘ
Gerd Faltings NERFINISHED ⓘ Jean-Pierre Serre NERFINISHED ⓘ Pierre Deligne NERFINISHED ⓘ Robert Langlands NERFINISHED ⓘ |
| typicalCodomain |
GL_n(C)
ⓘ
GL_n(Q_l) NERFINISHED ⓘ GL_n(Z_l) ⓘ |
| typicalDomain |
absolute Galois group of a local field
ⓘ
absolute Galois group of a number field ⓘ absolute Galois group of the rational numbers NERFINISHED ⓘ |
| usedIn |
Iwasawa theory
NERFINISHED
ⓘ
Langlands program NERFINISHED ⓘ arithmetic of elliptic curves ⓘ p-adic Hodge theory ⓘ proofs of modularity theorems ⓘ study of L-functions ⓘ study of motives ⓘ |
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: Galois representations Description of subject: 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.
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Artin representation
this entity surface form:
Weil–Deligne representations
this entity surface form:
Tate modules
subject surface form:
Évariste Galois