Frey curve

Template:Short description In mathematics, a Frey curve or Frey–Hellegouarch curve is the elliptic curve $$y^2=x(x-\alpha )(x+\beta )$$ associated with an ABC triple $\alpha +\beta =\gamma$. This relates properties of solutions of equations to elliptic curves. This curve was popularized in its application to Fermat’s Last Theorem where one investigates a (hypothetical) solution of Fermat's equation

$a^{\ell }+b^{\ell }=c^{\ell }.$

The curve is named after Gerhard Frey and (sometimes) Template:Ill.

History

Template:Harvs came up with the idea of associating solutions $(a,b,c)$ of Fermat's equation with a completely different mathematical object: an elliptic curve.Template:Sfnp If ℓ is an odd prime and a, b, and c are positive integers such that $$a^{\ell }+b^{\ell }=c^{\ell },$$ then a corresponding Frey curve is an algebraic curve given by the equation $$y^2=x(x-a^{\ell })(x+b^{\ell }),$$ or, equivalently $$y^2=x(x-a^{\ell })(x-c^{\ell }).$$ This is a nonsingular algebraic curve of genus one defined over Q, and its projective completion is an elliptic curve over Q.

Template:Harvs 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 Template:Harvtxt suggested that the Taniyama–Shimura–Weil conjecture implies Fermat's Last Theorem.Template:Sfnmp 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, Template:Harvtxt proved the epsilon conjecture, thereby proving that the Taniyama–Shimura–Weil conjecture implies Fermat's Last Theorem.Template:Sfnp

References

• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation