reciprocity conjecture

E870216

conjecture in number theory conjecture in representation theory mathematical conjecture

The reciprocity conjecture is a far-reaching set of ideas in number theory and representation theory that generalizes classical reciprocity laws by relating Galois groups to automorphic forms within the Langlands program.

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Statements (44)

Predicate	Object
instanceOf	conjecture in number theory ⓘ conjecture in representation theory ⓘ mathematical conjecture ⓘ
aimsToDescribe	relationship between arithmetic of fields and harmonic analysis on groups ⓘ
associatedWith	Langlands correspondence NERFINISHED ⓘ Robert Langlands NERFINISHED ⓘ
basedOn	classical reciprocity laws ⓘ
concerns	L-functions NERFINISHED ⓘ automorphic L-functions ⓘ motivic Galois representations ⓘ
coreIdea	correspondence between Galois representations and automorphic representations ⓘ generalization of abelian class field theory ⓘ
field	Galois theory NERFINISHED ⓘ arithmetic geometry ⓘ automorphic forms ⓘ number theory ⓘ representation theory ⓘ
generalizes	higher reciprocity laws ⓘ quadratic reciprocity law ⓘ
hasAspect	global reciprocity ⓘ local reciprocity ⓘ
historicalRoot	Artin reciprocity law NERFINISHED ⓘ Hilbert reciprocity law NERFINISHED ⓘ Kronecker–Weber theorem NERFINISHED ⓘ
implies	class field theory in the abelian case ⓘ
influences	arithmetic geometry ⓘ modern algebraic number theory ⓘ theory of automorphic forms ⓘ
motivation	unify reciprocity phenomena in number theory ⓘ
partOf	Langlands program NERFINISHED ⓘ
relatedTo	Taniyama–Shimura–Weil conjecture NERFINISHED ⓘ functoriality conjecture NERFINISHED ⓘ modularity of elliptic curves ⓘ
relates	Galois groups NERFINISHED ⓘ automorphic forms ⓘ automorphic representations ⓘ
scope	far-reaching generalization of reciprocity laws ⓘ
status	open problem ⓘ
typicalDomain	global fields ⓘ local fields ⓘ number fields ⓘ
usesConcept	Galois representations into L-groups ⓘ automorphic representations of adelic groups ⓘ reductive algebraic groups ⓘ

Referenced by (1)

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

Robert Langlands → knownFor → reciprocity conjecture ⓘ