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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Frey curve construction canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11365711 — 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: Frey curve construction Context triple: [Gerhard Frey, knownFor, Frey curve construction]
-
A.
Peano curve
The Peano curve is a space-filling fractal curve that continuously maps a one-dimensional interval onto a two-dimensional area, demonstrating that a line can completely fill a square.
-
B.
Random Curves
Random Curves is a mathematics book by Neal Koblitz that explores probabilistic and heuristic methods in number theory and algebraic geometry, particularly in relation to elliptic curves and cryptographic applications.
-
C.
Bezier curves
Bézier curves are mathematically defined parametric curves widely used in computer graphics and digital design to model smooth, scalable shapes and paths.
-
D.
Fermat curve
A Fermat curve is an algebraic curve defined by an equation of the form \(x^n + y^n = 1\), studied in number theory and algebraic geometry for its rich arithmetic and geometric properties.
-
E.
Page curve
The Page curve is a theoretical prediction in black hole physics that describes how the entanglement entropy of Hawking radiation should rise and then fall over time if black hole evaporation is ultimately unitary.
- 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: Frey curve construction Target entity description: 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.
-
A.
Peano curve
The Peano curve is a space-filling fractal curve that continuously maps a one-dimensional interval onto a two-dimensional area, demonstrating that a line can completely fill a square.
-
B.
Random Curves
Random Curves is a mathematics book by Neal Koblitz that explores probabilistic and heuristic methods in number theory and algebraic geometry, particularly in relation to elliptic curves and cryptographic applications.
-
C.
Bezier curves
Bézier curves are mathematically defined parametric curves widely used in computer graphics and digital design to model smooth, scalable shapes and paths.
-
D.
Fermat curve
A Fermat curve is an algebraic curve defined by an equation of the form \(x^n + y^n = 1\), studied in number theory and algebraic geometry for its rich arithmetic and geometric properties.
-
E.
Page curve
The Page curve is a theoretical prediction in black hole physics that describes how the entanglement entropy of Hawking radiation should rise and then fall over time if black hole evaporation is ultimately unitary.
- F. None of above. chosen
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
ⓘ
linked to:
Serre’s conjecture on Galois representations
epsilon conjecture of Serre ⓘ
linked to:
Serre’s conjecture on Galois representations
|
| 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 ⓘ |
| methodType | reductio ad absurdum technique ⓘ |
| namedAfter | Gerhard Frey ⓘ |
| 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 ⓘ Taniyama–Shimura–Weil conjecture ⓘ 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
ⓘ
Gerhard Frey ⓘ Jean-Pierre Serre ⓘ Ken Ribet ⓘ |
| usedInProofOf | Fermat's Last Theorem ⓘ |
| usesObject |
Fermat equation x^n + y^n = z^n
ⓘ
elliptic curve ⓘ |
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: Frey curve construction Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.