modularity conjecture
E921625
The modularity conjecture is a central statement in number theory asserting that every elliptic curve over the rational numbers corresponds to a modular form, a result whose proof underpins the modern proof of Fermat’s Last Theorem.
All labels observed (1)
| Label | Occurrences |
|---|---|
| modularity conjecture canonical | 3 |
How this entity was disambiguated
This entity first appeared as the object of triple T11365714 — 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: modularity conjecture Context triple: [Gerhard Frey, contributedTo, modularity conjecture]
-
A.
reciprocity conjecture
The reciprocity conjecture is a far-reaching set of ideas in number theory and representation theory that generalizes classical reciprocity laws by relating Galois groups to automorphic forms within the Langlands program.
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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.
Monstrous Moonshine conjecture
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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D.
Fontaine–Mazur conjecture
The Fontaine–Mazur conjecture is a central open problem in number theory that predicts which p-adic Galois representations of number fields arise from geometry or from automorphic forms.
-
E.
Tate Conjecture
The Tate Conjecture is a major open problem in arithmetic geometry that predicts a deep connection between algebraic cycles on varieties over finite fields and their Galois-invariant étale cohomology classes.
- 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: modularity conjecture Target entity description: The modularity conjecture is a central statement in number theory asserting that every elliptic curve over the rational numbers corresponds to a modular form, a result whose proof underpins the modern proof of Fermat’s Last Theorem.
-
A.
reciprocity conjecture
The reciprocity conjecture is a far-reaching set of ideas in number theory and representation theory that generalizes classical reciprocity laws by relating Galois groups to automorphic forms within the Langlands program.
-
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.
Monstrous Moonshine conjecture
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.
-
D.
Fontaine–Mazur conjecture
The Fontaine–Mazur conjecture is a central open problem in number theory that predicts which p-adic Galois representations of number fields arise from geometry or from automorphic forms.
-
E.
Tate Conjecture
The Tate Conjecture is a major open problem in arithmetic geometry that predicts a deep connection between algebraic cycles on varieties over finite fields and their Galois-invariant étale cohomology classes.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical conjecture
ⓘ
statement in number theory ⓘ |
| alsoKnownAs |
Taniyama–Shimura conjecture
ⓘ
linked to:
Taniyama–Shimura–Weil conjecture
Taniyama–Shimura–Weil conjecture ⓘ modularity theorem for elliptic curves over Q ⓘ |
| appliesTo | elliptic curves defined over Q ⓘ |
| asserts |
every elliptic curve over Q corresponds to a modular form
ⓘ
every elliptic curve over the rational numbers is modular ⓘ |
| concerns |
elliptic curves over the rational numbers
ⓘ
modular forms ⓘ |
| connectedTo |
Hasse–Weil L-function of an elliptic curve
ⓘ
cusp forms of weight 2 and level N ⓘ |
| equivalentFormulationInvolves | equality of L-functions of elliptic curves and modular forms ⓘ |
| field | number theory ⓘ |
| hasConsequence |
classification of elliptic curves over Q via modular forms
ⓘ
connections between arithmetic geometry and automorphic forms ⓘ |
| historicallyFormulatedBy |
André Weil
ⓘ
Goro Shimura ⓘ Yutaka Taniyama ⓘ |
| implies |
Fermat’s Last Theorem
ⓘ
linked to:
Fermat's Last Theorem
|
| influenced |
development of the Langlands correspondence for GL(2)
ⓘ
modern research in arithmetic geometry ⓘ |
| involvesObject |
congruence subgroups of SL(2,Z)
ⓘ
rational points on elliptic curves ⓘ weight 2 modular forms ⓘ |
| isGeneralizedBy | modularity conjectures for higher-dimensional abelian varieties ⓘ |
| isSpecialCaseOf |
Langlands reciprocity conjectures
ⓘ
linked to:
Langlands program
|
| originallyConjecturedInDecade | 1950s ⓘ |
| partiallyProvedBy |
Andrew Wiles
ⓘ
Richard Taylor ⓘ |
| proofCompletedInYear | 2001 ⓘ |
| provedBy |
Brian Conrad
ⓘ
Christophe Breuil ⓘ Fred Diamond ⓘ Richard Taylor ⓘ |
| provedUsing |
Galois deformation theory
ⓘ
Iwasawa theory techniques ⓘ R=T theorems ⓘ modularity lifting theorems ⓘ properties of Hecke algebras ⓘ |
| relatedTo |
Langlands program
ⓘ
Shimura–Taniyama–Weil conjecture ⓘ
linked to:
Taniyama–Shimura–Weil conjecture
|
| relatesConcept |
Galois representations
ⓘ
L-functions ⓘ elliptic curves ⓘ modular curves ⓘ |
| status | proved ⓘ |
| type | modularity theorem ⓘ |
| usedInProofOf |
Fermat’s Last Theorem
ⓘ
linked to:
Fermat's Last Theorem
|
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: modularity conjecture Description of subject: The modularity conjecture is a central statement in number theory asserting that every elliptic curve over the rational numbers corresponds to a modular form, a result whose proof underpins the modern proof of Fermat’s Last Theorem.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.