Frey curve: Difference between revisions

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In mathematics, a '''Frey curve''' or '''Frey–Hellegouarch''' curve is the [[elliptic curve]]
::<math>y^2 = x(x - a^\ell)(x + b^\ell)\ </math>
associated with a solution of Fermat's equation
This is a nonsingular algebraic curve of genus one defined over '''Q''', and its [[projective completion]] is an elliptic curve over '''Q'''.
In 1982 [[Gerhard Frey]] called attention to the unusual properties of the same curve as Hellegouarch, now 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. 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.