Ribet's theorem

E925498

mathematical theorem theorem in number theory

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.

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All labels observed (4)

Label	Occurrences
Ribet's theorem canonical	2
Ribet’s theorem	2
Ribet’s level-lowering theorem	1
level-lowering theorem	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]

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
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 ⓘ level-lowering theorem ⓘ linked to: Ribet's theorem ε-conjecture ⓘ
appliesTo	2-dimensional Galois representations of the absolute Galois group of the rationals ⓘ modular forms of weight 2 and higher ⓘ
author	Kenneth A. Ribet ⓘ linked to: Ken Ribet
buildsOn	work of Barry Mazur ⓘ work of Goro Shimura ⓘ work of Jean-Pierre Serre ⓘ linked to: Serre’s cohomological methods in algebraic geometry 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 ⓘ linked to: United States of America
field	number theory ⓘ
historicalImportance	key step linking Frey curve to modularity conjecture ⓘ
implies	epsilon conjecture of Serre ⓘ linked to: Serre’s conjecture on Galois representations
influenced	Andrew Wiles's proof strategy ⓘ Taylor–Wiles method ⓘ
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 ⓘ linked to: Ken Ribet
playsRoleIn	proof of Fermat's Last Theorem ⓘ
predecessor	Taniyama–Shimura conjecture ⓘ linked to: Taniyama–Shimura–Weil conjecture
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 ⓘ
relatedTo	Fermat's Last Theorem ⓘ Frey curve ⓘ Serre's modularity conjecture ⓘ linked to: Serre’s conjecture on Galois representations modularity theorem ⓘ linked to: modularity theorem for elliptic curves over Q
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 ⓘ Serre's conjecture ⓘ linked to: Serre’s conjecture on Galois representations congruences between modular forms ⓘ level lowering ⓘ modular elliptic curves ⓘ modular forms ⓘ
yearProved	1986 ⓘ

How these facts were elicited

Referenced by (6)

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

Ken Ribet → knownFor → Ribet's theorem ⓘ

Taniyama–Shimura–Weil conjecture → relatedTo → Ribet’s theorem ⓘ

linked to: Ribet's theorem

Frey curve → playedRoleIn → Ribet’s theorem ⓘ

linked to: Ribet's theorem

Frey curve → inspiredResult → Ribet’s level-lowering theorem ⓘ

linked to: Ribet's theorem

Frey curve construction → relatedTo → Ribet's theorem ⓘ

Ribet's theorem → alsoKnownAs → level-lowering theorem ⓘ

linked to: Ribet's theorem

All labels observed (4)

How this entity was disambiguated Show

Statements (50)

How these facts were elicited Show

Referenced by (6)

How this entity was disambiguated

How these facts were elicited