Taniyama–Shimura–Weil conjecture

E530307

mathematical conjecture precursor of the modularity theorem statement in number theory

The Taniyama–Shimura–Weil conjecture, now the modularity theorem, asserts that every elliptic curve over the rational numbers is modular and played a central role in the proof of Fermat’s Last Theorem.

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Observed surface forms (2)

Surface form	Occurrences
Modularity theorem	1
modularity theorem	1

Statements (47)

Predicate	Object
instanceOf	mathematical conjecture ⓘ precursor of the modularity theorem ⓘ statement in number theory ⓘ
alsoKnownAs	Shimura–Taniyama conjecture NERFINISHED ⓘ Taniyama–Shimura conjecture NERFINISHED ⓘ modularity conjecture NERFINISHED ⓘ modularity theorem for elliptic curves over Q NERFINISHED ⓘ
asserts	every elliptic curve over Q is a quotient of the Jacobian of a modular curve ⓘ every elliptic curve over Q is associated to a modular form of weight 2 ⓘ every elliptic curve over the rational numbers is modular ⓘ
associatedWith	André Weil NERFINISHED ⓘ Goro Shimura NERFINISHED ⓘ Yutaka Taniyama NERFINISHED ⓘ
centralTo	Langlands program for GL(2) over Q NERFINISHED ⓘ
codomainObjects	normalized newforms of weight 2 and level N ⓘ
concerns	elliptic curves over the rational numbers ⓘ modular forms ⓘ
domainOfDefinition	elliptic curves defined over Q ⓘ
field	number theory ⓘ
generalizationOf	modularity of semistable elliptic curves over Q ⓘ
hasConsequence	classification of elliptic curves over Q via modular forms ⓘ
historicalPeriod	20th century mathematics ⓘ
implies	Fermat’s Last Theorem NERFINISHED ⓘ modularity of all elliptic curves over Q ⓘ
involvesConcept	Galois representations ⓘ L-functions of elliptic curves ⓘ L-functions of modular forms ⓘ modular curves ⓘ
logicalStepFor	Ribet’s reduction of Fermat’s Last Theorem to modularity ⓘ
nowKnownAs	modularity theorem NERFINISHED ⓘ
provedInFullBy	Brian Conrad NERFINISHED ⓘ Christophe Breuil NERFINISHED ⓘ Fred Diamond NERFINISHED ⓘ Richard Taylor NERFINISHED ⓘ
provedInSpecialCaseBy	Andrew Wiles NERFINISHED ⓘ Richard Taylor NERFINISHED ⓘ
relatedConjecture	Langlands reciprocity conjecture ⓘ
relatedTo	Ribet’s theorem NERFINISHED ⓘ Serre’s conjecture NERFINISHED ⓘ
relates	cusp forms of weight 2 for congruence subgroups of SL(2,Z) ⓘ elliptic curves over Q ⓘ
statesCorrespondenceBetween	isogeny classes of elliptic curves over Q and newforms of weight 2 with rational Fourier coefficients ⓘ
status	proved ⓘ
subfield	algebraic number theory ⓘ arithmetic geometry ⓘ
typeOf	modularity statement ⓘ
usedInProofOf	Fermat’s Last Theorem NERFINISHED ⓘ

Referenced by (5)

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

Serre’s conjecture on Galois representations → isRelatedTo → Taniyama–Shimura–Weil conjecture ⓘ

this entity surface form: modularity theorem

Fermat's Last Theorem → relatedConjecture → Taniyama–Shimura–Weil conjecture ⓘ

Birch and Swinnerton-Dyer Conjecture → relatedTo → Taniyama–Shimura–Weil conjecture ⓘ

this entity surface form: Modularity theorem

Hasse–Weil zeta function → relatedTo → Taniyama–Shimura–Weil conjecture ⓘ