Shimura varieties

E839486

algebraic variety higher-dimensional algebraic variety object of arithmetic geometry

Shimura varieties are higher-dimensional algebraic varieties that generalize modular curves and play a central role in the Langlands program by connecting number theory, automorphic forms, and arithmetic geometry.

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Observed surface forms (1)

Surface form	Occurrences
Shimura curve	1

Statements (49)

Predicate	Object
instanceOf	algebraic variety ⓘ higher-dimensional algebraic variety ⓘ object of arithmetic geometry ⓘ
appearsIn	André–Oort conjecture NERFINISHED ⓘ Langlands correspondence NERFINISHED ⓘ Mumford–Tate conjecture NERFINISHED ⓘ Zilber–Pink conjecture NERFINISHED ⓘ
definedFrom	Shimura datum NERFINISHED ⓘ conjugacy class of homomorphisms from Deligne torus ⓘ reductive algebraic group over Q ⓘ
generalizes	modular curves ⓘ
hasApplication	study of Hodge structures ⓘ study of periods and motives ⓘ study of rational points ⓘ
hasComponent	Shimura variety of Hodge type NERFINISHED ⓘ Shimura variety of PEL type ⓘ Shimura variety of abelian type NERFINISHED ⓘ connected Shimura variety ⓘ
hasProperty	admit minimal (Baily–Borel) compactifications ⓘ admit toroidal compactifications ⓘ admits canonical models over reflex field ⓘ can have compactifications ⓘ defined over a number field called reflex field ⓘ often quasi-projective ⓘ
hasStructure	Hecke correspondences NERFINISHED ⓘ Shimura datum NERFINISHED ⓘ canonical model over a number field ⓘ complex analytic uniformization ⓘ special points ⓘ special subvarieties ⓘ
namedAfter	Goro Shimura NERFINISHED ⓘ
relatedTo	Galois representations ⓘ L-functions NERFINISHED ⓘ Langlands program NERFINISHED ⓘ arithmetic geometry ⓘ automorphic forms ⓘ number theory ⓘ representation theory ⓘ
specialCaseOf	Hilbert modular varieties NERFINISHED ⓘ Siegel modular varieties NERFINISHED ⓘ moduli space of abelian varieties with additional structure ⓘ unitary Shimura varieties NERFINISHED ⓘ
studiedBy	Goro Shimura NERFINISHED ⓘ Pierre Deligne NERFINISHED ⓘ Robert Langlands NERFINISHED ⓘ
usedIn	construction of Galois representations from automorphic forms ⓘ formulation of reciprocity laws ⓘ proofs of modularity lifting theorems ⓘ study of motives ⓘ

Referenced by (2)

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

Hilbert’s twelfth problem → relatedTo → Shimura varieties ⓘ

Teichmüller curve → relatedTo → Shimura varieties ⓘ

this entity surface form: Shimura curve