Ribet's theorem
E925498
Ribet's theorem is a result in number theory that linked certain modular forms to Galois representations and played a crucial role in the proof of Fermat's Last Theorem.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Ribet's theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11432817 — 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: Ribet's theorem Context triple: [Ken Ribet, knownFor, Ribet's theorem]
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A.
Taniyama–Shimura–Weil conjecture
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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B.
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.
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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.
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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.
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E.
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.
- 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: Ribet's theorem Target entity description: Ribet's theorem is a result in number theory that linked certain modular forms to Galois representations and played a crucial role in the proof of Fermat's Last Theorem.
-
A.
Taniyama–Shimura–Weil conjecture
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.
-
B.
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.
-
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.
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.
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.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in number theory ⓘ |
| alsoKnownAs |
epsilon conjecture
NERFINISHED
ⓘ
level-lowering theorem NERFINISHED ⓘ ε-conjecture NERFINISHED ⓘ |
| appliesTo |
2-dimensional Galois representations of the absolute Galois group of the rationals
ⓘ
modular forms of weight 2 and higher ⓘ |
| author | Kenneth A. Ribet NERFINISHED ⓘ |
| buildsOn |
work of Barry Mazur
ⓘ
work of Goro Shimura ⓘ work of Jean-Pierre Serre NERFINISHED ⓘ work of Yutaka Taniyama ⓘ |
| concerns |
level-lowering for modular Galois representations
ⓘ
relationship between level of modular forms and conductor of Galois representations ⓘ |
| consequence | non-existence of non-trivial solutions to Fermat's equation follows from modularity of semistable elliptic curves ⓘ |
| countryOfOrigin |
United States of America
ⓘ
surface form:
United States
|
| field | number theory ⓘ |
| historicalImportance | key step linking Frey curve to modularity conjecture ⓘ |
| implies | epsilon conjecture of Serre NERFINISHED ⓘ |
| influenced |
Andrew Wiles's proof strategy
ⓘ
Taylor–Wiles method NERFINISHED ⓘ |
| involves |
conductor of an elliptic curve
ⓘ
level of a modular form ⓘ odd irreducible Galois representations ⓘ residual Galois representations modulo a prime ⓘ |
| language | English ⓘ |
| mathematicsSubjectClassification |
11F11
ⓘ
11F80 ⓘ |
| namedAfter | Kenneth A. Ribet NERFINISHED ⓘ |
| playsRoleIn | proof of Fermat's Last Theorem ⓘ |
| predecessor | Taniyama–Shimura conjecture NERFINISHED ⓘ |
| provedUsing |
Galois representations attached to modular forms
ⓘ
Mazur's results on rational isogenies of prime degree ⓘ congruences between cusp forms ⓘ properties of modular forms of weight 2 ⓘ |
| publicationVenue | Inventiones Mathematicae NERFINISHED ⓘ |
| relatedTo |
Fermat's Last Theorem
NERFINISHED
ⓘ
Frey curve NERFINISHED ⓘ Serre's modularity conjecture NERFINISHED ⓘ modularity theorem NERFINISHED ⓘ |
| states | if a certain semistable elliptic curve associated to a Frey curve were modular, then it would contradict properties of modular forms of specific level ⓘ |
| subfield |
algebraic number theory
ⓘ
arithmetic geometry ⓘ |
| usesConcept |
Galois representations
NERFINISHED
ⓘ
Serre's conjecture NERFINISHED ⓘ congruences between modular forms ⓘ level lowering ⓘ modular elliptic curves ⓘ modular forms ⓘ |
| yearProved | 1986 ⓘ |
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: Ribet's theorem Description of subject: Ribet's theorem is a result in number theory that linked certain modular forms to Galois representations and played a crucial role in the proof of Fermat's Last Theorem.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.