Frey curve

{{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.{{sfnp|Hellegouarch|1975}} If ℓ is an odd prime and a, b, and c are positive integers such that

$$a^\backslash ell\; +\; b^\backslash ell\; =\; c^\backslash ell,$$

then a corresponding Frey curve is an algebraic curve given by the equation

$$y^2\; =\; x(x\; -\; a^\backslash ell)(x\; +\; b^\backslash ell),$$

or, equivalently

$$y^2\; =\; x(x\; -\; a^\backslash ell)(x\; -\; c^\backslash ell).$$

This is a nonsingular algebraic curve of genus one defined over Q, and its projective completion is an elliptic curve over Q.

{{harvs|first=Gerhard|last=Frey|authorlink=Gerhard Frey|year=1982|txt}} 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.{{sfnmp|1a1=Frey|1y=1982|2a1=Frey|2y=1986}} 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, {{harvtxt|Ribet|1990}} proved the epsilon conjecture, thereby proving that the Taniyama–Shimura–Weil conjecture implies Fermat's Last Theorem.{{sfnp|Ribet|1990}}

{{reflist}}

• {{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}}
• {{Citation | last1=Frey | first1=Gerhard | title=Rationale Punkte auf Fermatkurven und getwisteten Modulkurven| year=1982 | journal=J. reine angew. Math. | volume=331 | pages=185–191}}
• {{Citation | last1=Hellegouarch | first1=Yves | title=Points d'ordre 2p^h sur les courbes elliptiques | url=http://matwbn.icm.edu.pl/ksiazki/aa/aa26/aa2636.pdf| mr=0379507 | year=1975 | journal=Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica | issn=0065-1036 | volume=26 | issue=3 | pages=253–263}}
• {{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 | access-date=2012-01-02 | archive-url=https://web.archive.org/web/20120204094533/http://www.math.unicaen.fr/~nitaj/hellegouarch.html | archive-date=2012-02-04 | url-status=dead }}
• {{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}}
• {{Citation | last1=Ribet | first1=Kenneth A. | title=On modular representations of Gal({{overline|Q}}/Q) arising from modular forms | doi=10.1007/BF01231195 | mr=1047143 | year=1990 | journal=Inventiones Mathematicae | issn=0020-9910 | volume=100 | issue=2 | pages=431–476| bibcode=1990InMat.100..431R | hdl=10338.dmlcz/147454 | s2cid=120614740 | hdl-access=free }}

Category:Number theory