Wiles's proof of Fermat's Last Theorem
[1]: 203–205, 223, 226 Wiles first announced his proof on 23 June 1993 at a lecture in Cambridge entitled "Modular Forms, Elliptic Curves and Galois Representations".
One year later on 19 September 1994, in what he would call "the most important moment of [his] working life", Wiles stumbled upon a revelation that allowed him to correct the proof to the satisfaction of the mathematical community.
It also uses standard constructions of modern algebraic geometry such as the category of schemes, significant number theoretic ideas from Iwasawa theory, and other 20th-century techniques which were not available to Fermat.
When announcing that Wiles had won the Abel Prize, the Norwegian Academy of Science and Letters described his achievement as a "stunning proof".
Between its publication and Andrew Wiles's eventual solution over 350 years later, many mathematicians and amateurs attempted to prove this statement, either for all values of n > 2, or for specific cases.
Separately from anything related to Fermat's Last Theorem, in the 1950s and 1960s Japanese mathematician Goro Shimura, drawing on ideas posed by Yutaka Taniyama, conjectured that a connection might exist between elliptic curves and modular forms.
Around 50 years after first being proposed, the conjecture was finally proven and renamed the modularity theorem, largely as a result of Andrew Wiles's work described below.
On yet another separate branch of development, in the late 1960s, Yves Hellegouarch came up with the idea of associating hypothetical solutions (a, b, c) of Fermat's equation with a completely different mathematical object: an elliptic curve.
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 parametrise coordinates x and y of the points on it.
If the link identified by Frey could be proven, then in turn, it would mean that a disproof of Fermat's Last Theorem would disprove the Taniyama–Shimura–Weil conjecture, or by contraposition, a proof of the latter would prove the former as well.
However, despite the progress made by Serre and Ribet, this approach to Fermat was widely considered unusable as well, since almost all mathematicians saw the Taniyama–Shimura–Weil conjecture itself as completely inaccessible to proof with current knowledge.
[1]: 203–205, 223, 226 For example, Wiles's ex-supervisor John Coates stated that it seemed "impossible to actually prove",[1]: 226 and Ken Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible".
[1]: 223 Hearing of Ribet's 1986 proof of the epsilon conjecture, English mathematician Andrew Wiles, who had studied elliptic curves and had a childhood fascination with Fermat, decided to begin working in secret towards a proof of the Taniyama–Shimura–Weil conjecture, since it was now professionally justifiable,[11] as well as because of the enticing goal of proving such a long-standing problem.
During 21–23 June 1993, Wiles announced and presented his proof of the Taniyama–Shimura conjecture for semistable elliptic curves, and hence of Fermat's Last Theorem, over the course of three lectures delivered at the Isaac Newton Institute for Mathematical Sciences in Cambridge, England.
There was an error in one critical portion of the proof which gave a bound for the order of a particular group: the Euler system used to extend Kolyvagin and Flach's method was incomplete.
Mathematicians were beginning to pressure Wiles to disclose his work whether or not complete, so that the wider community could explore and use whatever he had managed to accomplish.
[16] Wiles states that on the morning of 19 September 1994, he was on the verge of giving up and was almost resigned to accepting that he had failed, and to publishing his work so that others could build on it and find the error.
Each was inadequate by itself, but fixing one approach with tools from the other would resolve the issue and produce a class number formula (CNF) valid for all cases that were not already proven by his refereed paper:[13][17] I was sitting at my desk examining the Kolyvagin–Flach method.
[20][21] Wiles's proof is very complex, and incorporates the work of so many other specialists that it was suggested in 1994 that only a small number of people were capable of fully understanding at that time all the details of what he had done.
Over the following years, Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor (sometimes abbreviated as "BCDT") carried the work further, ultimately proving the Taniyama–Shimura–Weil conjecture for all elliptic curves in a 2001 paper.
Our original goal will have been transformed into proving the modularity of geometric Galois representations of semistable elliptic curves, instead.
To show that a geometric Galois representation of an elliptic curve is a modular form, we need to find a normalized eigenform whose eigenvalues (which are also its Fourier series coefficients) satisfy a congruence relationship for all but a finite number of primes.
This proves: Wiles opted to attempt to match elliptic curves to a countable set of modular forms.
He found that this direct approach was not working, so he transformed the problem by instead matching the Galois representations of the elliptic curves to modular forms.
[25] By the spring of 1993, his work had covered all but a few families of elliptic curves, and in early 1993, Wiles was confident enough of his nearing success to let one trusted colleague into his secret.
[1]: 261–265 [26] Wiles's use of Kolyvagin–Flach would later be found to be the point of failure in the original proof submission, and he eventually had to revert to Iwasawa theory and a collaboration with Richard Taylor to fix it.
In May 1993, while reading a paper by Mazur, Wiles had the insight that the 3/5 switch would resolve the final issues and would then cover all elliptic curves.
representation has an image which is too small, one runs into trouble with the lifting argument, and in this case, there is a final trick which has since been studied in greater generality in the subsequent work on the Serre modularity conjecture.
[27] In his 108-page article published in 1995, Wiles divides the subject matter up into the following chapters (preceded here by page numbers): Gerd Faltings subsequently provided some simplifications to the 1995 proof, primarily in switching from geometric constructions to rather simpler algebraic ones.
[31] F. Q. Gouvêa's 1994 article "A Marvelous Proof", which reviews some of the required topics, won a Lester R. Ford award from the Mathematical Association of America.
