Frey elliptic curve

Let a^p + b^p = c^p be a solution to Fermat's last theorem. Then the corresponding Frey curve is y^2 = x(x - a^p)(x + b^p). Ribet (1990a) showed that such curves cannot be modular, so if the Taniyama-Shimura conjecture were true, Frey curves couldn't exist and Fermat's last theorem would follow with b even and a congruent -1 (mod 4). Frey curves are semistable. Invariants include the elliptic discriminant Δ = a^(2p) b^(2p) c^(2p).

elliptic curve | Fermat's last theorem | Taniyama-Shimura conjecture