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| In mathematics, a '''Frey curve''' or '''Frey–Hellegouarch''' curve is the [[elliptic curve]]
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| ::<math>y^2 = x(x - a^\ell)(x + b^\ell)\ </math>
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| associated with a solution of Fermat's equation
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| :<math>a^\ell + b^\ell = c^\ell.\ </math>
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| ==History==
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| {{harvs|first=Yves|last=Hellegouarch|year=1975|txt}} came up with the idea of associating solutions (a,b,c) of Fermat's equation with a completely different mathematical object: an elliptic curve.
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| If ℓ is an odd prime and ''a'', ''b'', and ''c'' are positive integers such that
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| :<math>a^\ell + b^\ell = c^\ell,\ </math>
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| then a corresponding Frey curve is an algebraic curve given by the equation
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| :<math>y^2 = x(x - a^\ell)(x + b^\ell)\ </math>
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| or, equivalently
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| :<math>y^2 = x(x - a^\ell)(x - c^\ell).\ </math>
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| This is a nonsingular algebraic curve of genus one defined over '''Q''', and its [[projective completion]] is an elliptic curve over '''Q'''.
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| {{harvs|first=Gerhard|last=Frey|authorlink=Gerhard Frey|year=1982}} called attention to the unusual properties of the same curve as Hellegouarch, which became called a Frey curve. This provided a bridge between Fermat and Taniyama by showing that a counterexample to [[Fermat's Last Theorem]] would create such a curve that would not be modular. The conjecture attracted considerable interest when {{harvtxt|Frey|1986}} suggested that the [[Taniyama–Shimura–Weil conjecture]] implies Fermat's Last Theorem. However, his argument was not complete. In 1985, Jean-Pierre Serre proposed that a Frey curve could not be modular and provided a partial proof of this. This showed that a proof of the semistable case of the Taniyama-Shimura conjecture would imply Fermat's Last Theorem. Serre did not provide a complete proof and what was missing became known as the [[epsilon conjecture]] or ε-conjecture. In the summer of 1986, Ribet (1990) proved the epsilon conjecture, thereby proving that the Taniyama–Shimura–Weil conjecture implied Fermat's Last Theorem.
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| ==References==
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| *{{Citation | last1=Frey | first1=Gerhard | title=Links between stable elliptic curves and certain Diophantine equations | mr=853387 | year=1986 | journal=Annales Universitatis Saraviensis. Series Mathematicae | issn=0933-8268 | volume=1 | issue=1 | pages=iv+40}}
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| *{{Citation | last1=Frey | first1=Gerhard | title=Rationale Punkte auf Fermatkurven und getwisteten Modulkurven| year=1982 | journal=J.reine u.angew.Math | volume=331 | pages=185–191}}
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| *{{Citation | last1=Hellegouarch | first1=Yves | title=Rectificatif à l'article de H. Darmon intitulé : "La Conjecture de Shimura-Taniyama-Weil est enfin démontré" | url=http://www.math.unicaen.fr/~nitaj/hellegouarch.html | year=2000 | journal=Gazette des Mathématiciens | issn=0224-8999 | volume=83}}
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| *{{Citation | last1=Hellegouarch | first1=Yves | title=Points d'ordre 2p<sup>h</sup> sur les courbes elliptiques | url=http://matwbn.icm.edu.pl/tresc.php?wyd=6&tom=26 | mr=0379507 | year=1974 | journal=Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica | issn=0065-1036 | volume=26 | issue=3 | pages=253–263}}
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| *{{Citation | last1=Hellegouarch | first1=Yves | title=Invitation to the mathematics of Fermat-Wiles | publisher=[[Academic Press]] | location=Boston, MA | isbn=978-0-12-339251-0 | mr=1475927 | year=2002}}
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| [[Category:Number theory]]
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