Frey curve
E921623
The Frey curve is an elliptic curve construction introduced by Gerhard Frey that played a pivotal role in linking Fermat’s Last Theorem to the modularity conjecture, ultimately contributing to its proof.
Statements (30)
| Predicate | Object |
|---|---|
| instanceOf |
elliptic curve
ⓘ
mathematical object ⓘ |
| appearsIn | proof strategy for Fermat’s Last Theorem via modularity lifting ⓘ |
| associatedWith |
Fermat’s Last Theorem
NERFINISHED
ⓘ
Taniyama–Shimura–Weil conjecture NERFINISHED ⓘ modularity conjecture NERFINISHED ⓘ |
| centralIdea | derive contradiction between non-modularity and modularity theorem ⓘ |
| consideredBy |
Jean-Pierre Serre
NERFINISHED
ⓘ
Ken Ribet NERFINISHED ⓘ |
| constructedFrom | putative solution of Fermat’s equation a^p + b^p = c^p with p > 2 ⓘ |
| constructionType | elliptic curve construction ⓘ |
| definedOver | rational numbers ⓘ |
| field |
arithmetic geometry
ⓘ
number theory ⓘ |
| hasProperty |
conductor depending on the Fermat triple
ⓘ
semistable ⓘ would be non-modular if a non-trivial Fermat solution existed ⓘ |
| inspiredResult | Ribet’s level-lowering theorem NERFINISHED ⓘ |
| introducedBy | Gerhard Frey NERFINISHED ⓘ |
| introducedInDecade | 1980s ⓘ |
| motivated | link between Fermat’s Last Theorem and modularity conjecture ⓘ |
| namedAfter | Gerhard Frey NERFINISHED ⓘ |
| playedRoleIn |
Ribet’s theorem
NERFINISHED
ⓘ
Wiles’s proof of Fermat’s Last Theorem ⓘ |
| relatedConcept |
Galois representation
NERFINISHED
ⓘ
epsilon conjecture of Serre NERFINISHED ⓘ modular elliptic curve ⓘ |
| studiedIn | Diophantine equations ⓘ |
| typicalEquationForm | y^2 = x(x - a^p)(x + b^p) ⓘ |
| usedInProofOf | Fermat’s Last Theorem NERFINISHED ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.