Relation of Fermat's Last Theorem and elliptic curves

Suppose that A^n+B^n=C^n is a counterexample to Fermat's Last Theorem, that is A,B,C are nonzero integers and n is an integer at least 2. Consider the elliptic curve E

y^2=x(x-A^n)(x+B^n)

which was introduced by G. Frey. The crucial fact is that the discriminant of the cubic on the right (the square of the product of differences of the roots) is -(ABC)^n which is a nth power. When n is divisible by an odd prime this places strong conditions on the arithmetic of the curve E. Frey suggested that the conjectured viewpoint (at the time) about elliptic curves implied that such curves could not exist, that is there are no counterexamples to Fermat's Last Theorem.