Monstrous Moonshine conjecture

E656687

The Monstrous Moonshine conjecture is a famous result in mathematics that reveals a deep and unexpected connection between the Monster finite simple group and modular functions in number theory.

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Predicate Object
instanceOf mathematical conjecture
result in number theory
alsoKnownAs Moonshine conjecture
associatedWith Monster group
largest sporadic simple group
sporadic simple groups
concerns relationship between Monster group representations and modular function coefficients
unexpected connection between finite simple groups and modular functions
connectedTo conformal field theory
genus zero property of modular curves
string theory
coreIdea Fourier coefficients of the modular j-invariant are linearly related to sums of dimensions of Monster group representations
field finite group theory
group theory
modular forms
number theory
representation theory
vertex operator algebras
formulatedInPublication Monstrous Moonshine (Conway–Norton paper)
linked to: monstrous moonshine
formulatedInYear 1979
historicalNote name refers to the mysterious and surprising nature of the connection (moonshine) and the Monster group (monstrous)
implies McKay–Thompson series are Hauptmoduln for genus zero groups
existence of a graded infinite-dimensional Monster module
inspired Mathieu moonshine
umbral moonshine
involves McKay–Thompson series
graded representation of the Monster group
modular functions for genus zero groups
modular j-invariant
vertex operator algebra structure
keyObject Monster vertex operator algebra V^natural
mathematicsSubjectClassification 11Fxx
20C34
proofCompletedBy Borcherds proof of the Moonshine conjecture
proofPublishedInYear 1992
provedBy Richard E. Borcherds
recognizedBy Fields Medal for Richard Borcherds in 1998
relates Fourier coefficients of modular functions
Monster group
dimensions of irreducible representations of the Monster group
modular forms of weight 0
modular functions
statedBy John H. Conway
Simon P. Norton
status proved
usesInProof automorphic forms
generalized Kac–Moody algebras
vertex operator algebras

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Simon P. Norton knownFor Monstrous Moonshine conjecture
Conway–Norton collaboration resultedIn Conway–Norton conjectures on monstrous moonshine
linked to: Monstrous Moonshine conjecture
M isCentralObjectIn monstrous moonshine conjectures
linked to: Monstrous Moonshine conjecture
Monstrous Moonshine conjecture implies McKay–Thompson series are Hauptmoduln for genus zero groups
linked to: Monstrous Moonshine conjecture
Monstrous Moonshine conjecture alsoKnownAs Moonshine conjecture
linked to: Monstrous Moonshine conjecture