Fermat's Last Theorem - Connection With Elliptic Curves

Connection With Elliptic Curves

The ultimately successful strategy for proving Fermat's Last Theorem was by proving the modularity theorem. The strategy was first described by Gerhard Frey in 1984. Frey noted that if Fermat's equation had a solution (a, b, c) for exponent p > 2, the corresponding elliptic curve

y2 = x (x − ap)(x + bp)

would have such unusual properties that the curve would likely violate the modularity theorem. This theorem, first conjectured in the mid-1950s and gradually refined through the 1960s, states that every elliptic curve is modular, meaning that it can be associated with a unique modular form.

Following this strategy, the proof of Fermat's Last Theorem required two steps. First, it was necessary to show that Frey's intuition was correct: that the above elliptic curve, if it exists, is always non-modular. Frey did not succeed in proving this rigorously; the missing piece was identified by Jean-Pierre Serre. This missing piece, the so-called "epsilon conjecture", was proven by Ken Ribet in 1986. Second, it was necessary to prove a special case of the modularity theorem. This special case (for semistable elliptic curves) was proven by Andrew Wiles in 1995.

Thus, the epsilon conjecture showed that any solution to Fermat's equation could be used to generate a non-modular semistable elliptic curve, whereas Wiles' proof showed that all such elliptic curves must be modular. This contradiction implies that there can be no solutions to Fermat's equation, thus proving Fermat's Last Theorem.