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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All labels observed (6)

Label	Occurrences
Taniyama–Shimura–Weil conjecture canonical	9
Taniyama–Shimura conjecture	6
modularity theorem	4
Shimura–Taniyama conjecture	2
Modularity theorem	1
Shimura–Taniyama–Weil conjecture	1

How this entity was disambiguated

This entity first appeared as the object of triple T5570531 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Taniyama–Shimura–Weil conjecture
Context triple: [Fermat's Last Theorem, relatedConjecture, Taniyama–Shimura–Weil conjecture]

A. Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
B. Ramanujan–Petersson conjecture
The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
C. Faltings' theorem
Faltings' theorem is a landmark result in arithmetic geometry that proves every algebraic curve of genus greater than one over a number field has only finitely many rational points.
D. Fermat's Last Theorem
Fermat's Last Theorem is a famous statement in number theory asserting that there are no whole-number solutions to the equation xⁿ + yⁿ = zⁿ for integers n greater than 2, a problem that remained unsolved for over three centuries until it was proved by Andrew Wiles in the 1990s.
E. Hodge Conjecture
The Hodge Conjecture is a major unsolved problem in algebraic geometry that predicts which cohomology classes on a non-singular projective complex variety arise from algebraic subvarieties.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Taniyama–Shimura–Weil conjecture
Target entity description: 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.

A. Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
B. Ramanujan–Petersson conjecture
The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
C. Faltings' theorem
Faltings' theorem is a landmark result in arithmetic geometry that proves every algebraic curve of genus greater than one over a number field has only finitely many rational points.
D. Fermat's Last Theorem
Fermat's Last Theorem is a famous statement in number theory asserting that there are no whole-number solutions to the equation xⁿ + yⁿ = zⁿ for integers n greater than 2, a problem that remained unsolved for over three centuries until it was proved by Andrew Wiles in the 1990s.
E. Hodge Conjecture
The Hodge Conjecture is a major unsolved problem in algebraic geometry that predicts which cohomology classes on a non-singular projective complex variety arise from algebraic subvarieties.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	mathematical conjecture ⓘ precursor of the modularity theorem ⓘ statement in number theory ⓘ
alsoKnownAs	Shimura–Taniyama conjecture ⓘ linked to: Taniyama–Shimura–Weil conjecture Taniyama–Shimura conjecture ⓘ linked to: Taniyama–Shimura–Weil conjecture modularity conjecture ⓘ modularity theorem for elliptic curves over Q ⓘ
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 ⓘ Goro Shimura ⓘ Yutaka Taniyama ⓘ
centralTo	Langlands program for GL(2) over Q ⓘ linked to: Langlands program
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 ⓘ linked to: Fermat's Last Theorem 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 ⓘ linked to: Taniyama–Shimura–Weil conjecture
provedInFullBy	Brian Conrad ⓘ Christophe Breuil ⓘ Fred Diamond ⓘ Richard Taylor ⓘ
provedInSpecialCaseBy	Andrew Wiles ⓘ Richard Taylor ⓘ
relatedConjecture	Langlands reciprocity conjecture ⓘ
relatedTo	Ribet’s theorem ⓘ linked to: Ribet's theorem Serre’s conjecture ⓘ linked to: Serre’s conjecture on Galois representations
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 ⓘ linked to: Fermat's Last Theorem

How these facts were elicited

Referenced by (23)

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

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

Birch and Swinnerton-Dyer Conjecture → relatedTo → Modularity theorem ⓘ

linked to: Taniyama–Shimura–Weil conjecture

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

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

Serre’s conjecture on Galois representations → isRelatedTo → modularity theorem ⓘ

linked to: Taniyama–Shimura–Weil conjecture

Goro Shimura → knownFor → Shimura–Taniyama conjecture ⓘ

linked to: Taniyama–Shimura–Weil conjecture

Hecke operators → appearIn → modularity theorem ⓘ

linked to: Taniyama–Shimura–Weil conjecture

Hecke theory → relatedTo → modularity theorem ⓘ

linked to: Taniyama–Shimura–Weil conjecture

Taniyama–Shimura–Weil conjecture → alsoKnownAs → Taniyama–Shimura conjecture ⓘ

linked to: Taniyama–Shimura–Weil conjecture

Taniyama–Shimura–Weil conjecture → alsoKnownAs → Shimura–Taniyama conjecture ⓘ

linked to: Taniyama–Shimura–Weil conjecture

Taniyama–Shimura–Weil conjecture → nowKnownAs → modularity theorem ⓘ

linked to: Taniyama–Shimura–Weil conjecture

Artin’s conjecture on L-functions → relatedTo → Taniyama–Shimura conjecture ⓘ

linked to: Taniyama–Shimura–Weil conjecture

Fermat's Enigma → describes → Taniyama–Shimura conjecture ⓘ

linked to: Taniyama–Shimura–Weil conjecture

Langlands program → influencedBy → Taniyama–Shimura conjecture ⓘ

linked to: Taniyama–Shimura–Weil conjecture

Introduction to Elliptic Curves and Modular Forms → relatedTo → Taniyama–Shimura–Weil conjecture ⓘ

reciprocity conjecture → relatedTo → Taniyama–Shimura–Weil conjecture ⓘ

Fontaine–Mazur conjecture → relatedTo → Taniyama–Shimura–Weil conjecture ⓘ

Frey curve → associatedWith → Taniyama–Shimura–Weil conjecture ⓘ

Frey curve construction → relatedTo → Taniyama–Shimura–Weil conjecture ⓘ

modularity conjecture → alsoKnownAs → Taniyama–Shimura conjecture ⓘ

linked to: Taniyama–Shimura–Weil conjecture

modularity conjecture → alsoKnownAs → Taniyama–Shimura–Weil conjecture ⓘ

modularity conjecture → relatedTo → Shimura–Taniyama–Weil conjecture ⓘ

linked to: Taniyama–Shimura–Weil conjecture

Ribet's theorem → predecessor → Taniyama–Shimura conjecture ⓘ

linked to: Taniyama–Shimura–Weil conjecture

All labels observed (6)

How this entity was disambiguated Show

Statements (47)

How these facts were elicited Show

Referenced by (23)

How this entity was disambiguated

How these facts were elicited