Galois representations

All labels observed (8)

Label	Occurrences
Galois representations canonical	19
Artin representation	2
Galois representation	2
Artin representations	1
Galois representation theory	1
Tate modules	1
Weil–Deligne representations	1
de Rham 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.
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 ⓘ linked to: p-adic Hodge theory 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 ⓘ automorphic forms ⓘ fundamental groups of schemes ⓘ modular forms ⓘ motivic Galois groups ⓘ étale cohomology ⓘ
studiedBy	Andrew Wiles ⓘ Gerd Faltings ⓘ Jean-Pierre Serre ⓘ Pierre Deligne ⓘ Robert Langlands ⓘ
typicalCodomain	GL_n(C) ⓘ GL_n(Q_l) ⓘ 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 ⓘ
usedIn	Iwasawa theory ⓘ Langlands program ⓘ arithmetic of elliptic curves ⓘ p-adic Hodge theory ⓘ proofs of modularity theorems ⓘ study of L-functions ⓘ study of motives ⓘ

How these facts were elicited

Referenced by (28)

Full triples — surface form annotated when it differs from this entity's canonical label.

Fermat's Last Theorem → proofUses → Galois representations ⓘ

Emil Artin → notableWork → Artin representation ⓘ

linked to: Galois representations

Hasse–Weil zeta function → relatedTo → Galois representations ⓘ

Hasse–Weil zeta function → relatedTo → Weil–Deligne representations ⓘ

linked to: Galois representations

Chebotarev density theorem → isToolFor → Galois representations ⓘ

Weil cohomology → relatedTo → Tate modules ⓘ

linked to: Galois representations

Évariste Galois → influenced → Galois representations ⓘ

subject linked to: Galois

Ramanujan tau function → relatedTo → Galois representations ⓘ

Ono’s partition congruences → relatedTo → Galois representations ⓘ

p-adic numbers → usedIn → Galois representations ⓘ

algebraic number theory → fieldOfStudy → Galois representations ⓘ

Artin’s conjecture on L-functions → concerns → Galois representations ⓘ

p-adic Hodge theory → usesConcept → Galois representations ⓘ

p-adic Hodge theory → studiesProperty → de Rham representations ⓘ

linked to: Galois representations

Diophantine equations → relatedTo → Galois representations ⓘ

Tate Conjecture → concerns → Galois representations ⓘ

Langlands program → coreConcept → Galois representation ⓘ

linked to: Galois representations

Artin conductor → usedIn → Galois representation theory ⓘ

linked to: Galois representations

Artin conductor → appliesTo → Artin representations ⓘ

linked to: Galois representations

Frobenius element → usedIn → Galois representations ⓘ

Frobenius conjugacy class → definedInContextOf → Galois representations ⓘ

Galois cohomology → relatedTo → Galois representations ⓘ

Frobenius endomorphism → relatedTo → Galois representations ⓘ

Langlands dual group → relatedTo → Galois representations ⓘ

Chandrashekhar Khare → researchInterest → Galois representations ⓘ

Artin L-functions → associatedWith → Artin representation ⓘ

linked to: Galois representations

Frey curve → relatedConcept → Galois representation ⓘ

linked to: Galois representations

Ribet's theorem → usesConcept → Galois representations ⓘ

All labels observed (8)

How this entity was disambiguated Show

Statements (49)

How these facts were elicited Show

Referenced by (28)

How this entity was disambiguated

How these facts were elicited