Galois representations

E534413

mathematical concept representation of a group

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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Observed surface forms (3)

Surface form	Occurrences
Artin representation	1
Tate modules	1
Weil–Deligne representations	1

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 ⓘ

Referenced by (8)

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

Fermat's Last Theorem → proofUses → Galois representations ⓘ

Emil Artin → notableWork → Galois representations ⓘ

this entity surface form: Artin representation

Hasse–Weil zeta function → relatedTo → Galois representations ⓘ

this entity surface form: Weil–Deligne representations

Chebotarev density theorem → isToolFor → Galois representations ⓘ

Weil cohomology → relatedTo → Galois representations ⓘ

this entity surface form: Tate modules

Galois → influenced → Galois representations ⓘ

subject surface form: Évariste Galois

Ramanujan tau function → relatedTo → Galois representations ⓘ