Frey curve
In mathematics, a Frey curve or Frey–Hellegouarch curve is the elliptic curve
The curve is named after Gerhard Frey and (sometimes) Yves Hellegouarch .
History[edit]
Yves Hellegouarch (1975) came up with the idea of associating solutions of Fermat's equation with a completely different mathematical object: an elliptic curve.[1] If ℓ is an odd prime and a, b, and c are positive integers such that
Gerhard Frey (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 Frey (1986) suggested that the Taniyama–Shimura–Weil conjecture implies Fermat's Last Theorem.[2] 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 implies Fermat's Last Theorem.[3]
Notes[edit]
References[edit]
- Frey, Gerhard (1986), "Links between stable elliptic curves and certain Diophantine equations", Annales Universitatis Saraviensis. Series Mathematicae, 1 (1): iv+40, ISSN 0933-8268, MR 0853387
- Frey, Gerhard (1982), "Rationale Punkte auf Fermatkurven und getwisteten Modulkurven", J. reine angew. Math., 331: 185–191
- Hellegouarch, Yves (1975), "Points d'ordre 2ph sur les courbes elliptiques" (PDF), Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica, 26 (3): 253–263, ISSN 0065-1036, MR 0379507
- Hellegouarch, Yves (2000), "Rectificatif à l'article de H. Darmon intitulé : "La Conjecture de Shimura-Taniyama-Weil est enfin démontré"", Gazette des Mathématiciens, 83, ISSN 0224-8999, archived from the original on 2012-02-04, retrieved 2012-01-02
- Hellegouarch, Yves (2002), Invitation to the mathematics of Fermat–Wiles, Boston, MA: Academic Press, ISBN 978-0-12-339251-0, MR 1475927
- Ribet, Kenneth A. (1990), "On modular representations of Gal(Q/Q) arising from modular forms", Inventiones Mathematicae, 100 (2): 431–476, Bibcode:1990InMat.100..431R, doi:10.1007/BF01231195, hdl:10338.dmlcz/147454, ISSN 0020-9910, MR 1047143, S2CID 120614740