Taniyama–Shimura–Weil conjecture
E530307
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Taniyama–Shimura–Weil conjecture canonical | 3 |
| Modularity theorem | 1 |
| modularity theorem | 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.
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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
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Taniyama–Shimura–Weil conjecture Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Modularity theorem
this entity surface form:
modularity theorem