Frey curve construction
E921624
The Frey curve construction is a method in number theory that associates an elliptic curve to a putative solution of Fermat’s Last Theorem, playing a key role in the proof by linking the theorem to modularity.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
concept in number theory
ⓘ
mathematical construction ⓘ |
| aim | show incompatibility between a Fermat counterexample and modularity theorem ⓘ |
| argumentStyle | contradiction via modularity ⓘ |
| assumes | existence of a nontrivial solution to Fermat's equation ⓘ |
| coreIdea | associate an elliptic curve to a putative solution of Fermat's equation ⓘ |
| countryOfOrigin | Germany ⓘ |
| field |
Diophantine equations
ⓘ
arithmetic geometry ⓘ number theory ⓘ |
| historicalRole |
linked Fermat's Last Theorem to modularity of elliptic curves
ⓘ
motivated Ribet's proof of the epsilon conjecture ⓘ provided strategy to derive contradiction from a hypothetical Fermat counterexample ⓘ |
| input | putative nontrivial integer solution of Fermat's equation ⓘ |
| inspiredConjecture |
Serre's modularity conjecture
NERFINISHED
ⓘ
epsilon conjecture of Serre NERFINISHED ⓘ |
| involves |
conductor of an elliptic curve
ⓘ
discriminant of an elliptic curve ⓘ modular forms of weight 2 ⓘ |
| keyProperty |
expected non-modularity under existence of a Fermat counterexample
ⓘ
special behavior of Galois representations attached to the curve ⓘ unusual conductor of the associated elliptic curve ⓘ |
| logicalRole | reduces Fermat's Last Theorem to a modularity statement for elliptic curves ⓘ |
| mainApplication | Fermat's Last Theorem NERFINISHED ⓘ |
| methodType | reductio ad absurdum technique ⓘ |
| namedAfter | Gerhard Frey NERFINISHED ⓘ |
| notableFeature | simple definition but deep arithmetic consequences ⓘ |
| output |
elliptic curve over the rational numbers
ⓘ
semistable elliptic curve ⓘ |
| propertyTested | modularity of the associated elliptic curve ⓘ |
| relatedTo |
Galois representations of elliptic curves
ⓘ
Ribet's theorem NERFINISHED ⓘ Taniyama–Shimura–Weil conjecture NERFINISHED ⓘ level-lowering theorems ⓘ modularity theorem ⓘ |
| studiedIn |
research on Diophantine equations
ⓘ
research on elliptic curves ⓘ research on modular forms ⓘ |
| timePeriod | late 20th century ⓘ |
| typicalExponent | integer n > 2 ⓘ |
| usedBy |
Andrew Wiles
NERFINISHED
ⓘ
Gerhard Frey NERFINISHED ⓘ Jean-Pierre Serre NERFINISHED ⓘ Ken Ribet NERFINISHED ⓘ |
| usedInProofOf | Fermat's Last Theorem NERFINISHED ⓘ |
| usesObject |
Fermat equation x^n + y^n = z^n
ⓘ
elliptic curve ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.