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 The curve is named after Gerhard Frey and (sometimes) Yves Hellegouarch [fr; de].
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
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.
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.