Langlands program

E753154

mathematical theory research program

The Langlands program is a far-reaching web of conjectures and theories in number theory and representation theory that seeks deep connections between Galois groups and automorphic forms, unifying many areas of modern mathematics.

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Observed surface forms (8)

Surface form	Occurrences
local Langlands correspondence	2
Langlands conjectures	1
Langlands correspondence	1
functoriality conjecture	1
geometric Langlands correspondence	1
geometric Langlands program	1
geometric Langlands theory	1
local Langlands program (via theta correspondence)	1

Statements (50)

Predicate	Object
instanceOf	mathematical theory ⓘ research program ⓘ
aimsTo	generalize class field theory ⓘ relate Galois groups to automorphic forms ⓘ unify number theory and representation theory ⓘ
basedOn	Langlands correspondence NERFINISHED ⓘ
conjecturedBy	Robert Langlands NERFINISHED ⓘ
coreConcept	Galois representation NERFINISHED ⓘ L-function NERFINISHED ⓘ automorphic representation ⓘ functoriality ⓘ motives ⓘ
field	algebraic geometry ⓘ arithmetic geometry ⓘ harmonic analysis ⓘ number theory ⓘ representation theory ⓘ
hasApproach	Shimura varieties NERFINISHED ⓘ endoscopy theory ⓘ geometric methods via perverse sheaves and D-modules ⓘ p-adic Hodge theory ⓘ theta correspondence ⓘ trace formula ⓘ
hasPart	Langlands functoriality conjecture NERFINISHED ⓘ Ramanujan–Petersson conjecture (automorphic form version) NERFINISHED ⓘ geometric Langlands program NERFINISHED ⓘ global Langlands correspondence NERFINISHED ⓘ local Langlands correspondence NERFINISHED ⓘ reciprocity conjecture NERFINISHED ⓘ
inception	1967 ⓘ
influenced	development of geometric representation theory ⓘ modularity theorem NERFINISHED ⓘ proof of Fermat's Last Theorem ⓘ theory of motives ⓘ
influencedBy	Taniyama–Shimura conjecture NERFINISHED ⓘ class field theory NERFINISHED ⓘ
namedAfter	Robert Langlands NERFINISHED ⓘ
notableResult	Jacquet–Langlands correspondence NERFINISHED ⓘ Langlands correspondence for function fields (Drinfeld and Lafforgue) NERFINISHED ⓘ Langlands–Tunnell theorem NERFINISHED ⓘ base change for GL(2) ⓘ global Langlands correspondence for GL(2) over number fields (partial) ⓘ local Langlands correspondence for GL(n) ⓘ proof of Sato–Tate conjecture in many cases ⓘ
openProblem	full global Langlands correspondence for number fields ⓘ general functoriality for all reductive groups ⓘ
relates	Hecke eigenforms NERFINISHED ⓘ adelic groups ⓘ automorphic representations of reductive groups over global fields ⓘ n-dimensional Galois representations ⓘ

Referenced by (25)

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

Joseph Bernstein → areaOfInfluence → Langlands program ⓘ

Euler products for automorphic L-functions → builtFrom → Langlands program ⓘ

this entity surface form: local Langlands correspondence

Vladimir Drinfeld → contributedTo → Langlands program ⓘ

this entity surface form: geometric Langlands program

Euler products for automorphic L-functions → field → Langlands program ⓘ

Weil group → fieldOfStudy → Langlands program ⓘ

Alexander Beilinson → influenced → Langlands program ⓘ

this entity surface form: geometric Langlands theory

Weil representation → isFundamentalIn → Langlands program ⓘ

this entity surface form: local Langlands program (via theta correspondence)

Serre’s conjecture on Galois representations → isRelatedTo → Langlands program ⓘ

Chebotarev density theorem → isToolFor → Langlands program ⓘ

Robert Langlands → knownFor → Langlands program ⓘ

this entity surface form: Langlands conjectures

Robert Langlands → knownFor → Langlands program ⓘ

this entity surface form: functoriality conjecture

Vladimir Drinfeld → knownFor → Langlands program ⓘ

this entity surface form: geometric Langlands correspondence

Robert Langlands → notableIdea → Langlands program ⓘ

this entity surface form: Langlands correspondence

Hilbert’s twelfth problem → relatedTo → Langlands program ⓘ

Iwasawa theory → relatedTo → Langlands program ⓘ

Ramanujan–Petersson conjecture → relatedTo → Langlands program ⓘ

Representation Theory and Automorphic Functions → relatedTo → Langlands program ⓘ

Deligne–Lusztig theory → relatesTo → Langlands program ⓘ

Hasse–Weil zeta function → studiedIn → Langlands program ⓘ

L-functions → usedIn → Langlands program ⓘ

subject surface form: L-function

Weil group → usedIn → Langlands program ⓘ

this entity surface form: local Langlands correspondence

étale cohomology → usedIn → Langlands program ⓘ