Wiles's Proof of Fermat's Last Theorem - Progress of The Previous Decades

Progress of The Previous Decades

Fermat's Last Theorem states that no three positive integers a, b and c can satisfy the equation

if n is an integer greater than two.

In the 1950s and 1960s a connection between elliptic curves and modular forms was conjectured by the Japanese mathematician Goro Shimura based on ideas posed by Yutaka Taniyama. In the West it became well known through a 1967 paper by André Weil. With Weil giving conceptual evidence for it, it is sometimes called the Taniyama–Shimura–Weil conjecture. It states that every rational elliptic curve is modular.

On a separate branch of development, in the late 1960s, Yves Hellegouarch came up with the idea of associating solutions (a,b,c) of Fermat's equation with a completely different mathematical object: an elliptic curve. The curve consists of all points in the plane whose coordinates (x, y) satisfy the relation

Such an elliptic curve would enjoy very special properties, which are due to the appearance of high powers of integers in its equation and the fact that an + bn = cn is a nth power as well.

In 1982–1985, 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. Again, the conjecture says that each elliptic curve with rational coefficients can be constructed in an entirely different way, not by giving its equation but by using modular functions to parametrize coordinates x and y of the points on it. Thus, according to the conjecture, any elliptic curve over Q would have to be a modular elliptic curve, yet if a solution to Fermat's equation with non-zero a, b, c and n greater than 2 existed, the corresponding curve would not be modular, resulting in a contradiction. As such, a proof or disproof of either of Fermat's Last Theorem or the Taniyama–Shimura-Weil conjecture would simultaneously prove or disprove the other.

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 of his proposal; the missing part became known as the epsilon conjecture or ε-conjecture (now known as Ribet's theorem). Serre's main interest was in an even more ambitious conjecture, Serre's conjecture on modular Galois representations, which would imply the Taniyama–Shimura conjecture. Although in the preceding twenty or thirty years much evidence had been accumulated to form conjectures about elliptic curves, the main reason to believe that these various conjectures were true lay not in the numerical confirmations, but in a remarkably coherent and attractive mathematical picture that they presented. Equally it could happen that one or more of these conjectures were actually untrue.

Following this strategy, a 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 (now known as a Frey curve), if it exists, is always non-modular. Frey did not quite succeed in proving this rigorously; the missing piece (the so-called "epsilon conjecture", now known as Ribet's theorem) was noticed by Jean-Pierre Serre and proven in 1986 by Ken Ribet. Second, it was necessary to prove the modularity theorem—or at least to prove it for the sub-class of cases (known as semistable elliptic curves) which included Frey's equation.

• Ribet's theorem—if proven—would show that any solution to Fermat's equation could be used to generate a semistable elliptic curve that was not modular;
• The modularity theorem—if proven for Frey's equation—would show that all such elliptic curves must be modular.
• The contradiction implies that no solutions can exist to Fermat's equation, thus proving Fermat's Last Theorem.

In the summer of 1986, Ken Ribet succeeded in proving the epsilon conjecture. (His article was published in 1990.) He demonstrated that, just as Frey had anticipated, a special case of the Taniyama–Shimura conjecture (still unproven at the time), together with the now proven epsilon conjecture, implies Fermat's Last Theorem. Thus, if the Taniyama–Shimura conjecture is true for semistable elliptic curves, then Fermat's Last Theorem would be true. However this theoretical approach was widely considered unattainable, since the Taniyama-Shimura conjecture was itself widely seen as completely inaccessible to proof with current knowledge. For example, Wiles' ex-supervisor John Coates states that it seemed "impossible to actually prove", and Ken Ribet considered himself "one of the vast majority of people who believed was completely inaccessible".

Hearing of the 1986 proof of the epsilon conjecture, Wiles decided to begin researching exclusively towards a proof of the Taniyama-Shimura conjecture. Ribet later commented that "Andrew Wiles was probably one of the few people on earth who had the audacity to dream that you can actually go and prove ."