This book will describe the recent proof of Fermat's Last The- orem by Andrew Wiles, aided by Richard Taylor, for graduate students and faculty with a. Annals of Mathematics, (), Pierre de Fermat. Andrew John Wiles. Modular elliptic curves and. Fermat's Last Theorem. By Andrew John Wiles *. Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a "The Proof of Fermat's Last Theorem by R. Taylor and A. Wiles" (PDF). Notices of the AMS. 42 (7): – ISSN Zbl Frey.
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Fermat's Last Theorem states that the equation xn + yn = zn, xyz = Prehistory: The only case of Fermat's Last Theorem for which Fermat actu-. PDF | On Mar 1, , Shailesh A. Shirali and others published The story of Fermat's Last Theorem. PDF | 15+ minutes read | On Feb 18, , David Cole and others published A Simpler Proof of Fermat's Last Theorem.
The links between these three steps were all well-understood in By that year, most number theorists were convinced — though proof would have to wait — that every Galois representation could be assigned, again by a precise rule, D a modular form, which is a kind of two-dimensional generalization of the familiar sine and cosine functions from trigonometry.
But there are no such forms.
Table of contents
Therefore there is no modular form D , no Galois representation C , no equation B , and no solution A. The only thing left to do was to establish the missing link between C and D , which mathematicians call the modularity conjecture.
Twenty years after Yutaka Taniyama and Goro Shimura, in the s, first intimated the link between B and D , via C , mathematicians had grown convinced that this must be right. The connection was simply too good not to be true. But the modularity conjecture itself looked completely out of reach.
Objects of type C and D were just too different. I will not try to untangle this ambiguity.
But if what the logicians had in mind was to formally verify the published proof of the relation between C and D , then they were setting their sights too low. For one thing, Wiles only proved a bit more than enough of the modularity conjecture to complete the A to E deduction. His work was extended to a full proof of the modularity theorem over the following 6 years by others, who built on Wiles's work.
During 21—23 June Wiles announced and presented his proof of the Taniyama—Shimura conjecture for semi-stable 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.
After the announcement, Nick Katz was appointed as one of the referees to review Wiles's manuscript. In the course of his review, he asked Wiles a series of clarifying questions that led Wiles to recognise that the proof contained a gap.
There was an error in one critical portion of the proof which gave a bound for the order of a particular group: The error would not have rendered his work worthless — each part of Wiles's work was highly significant and innovative by itself, as were the many developments and techniques he had created in the course of his work, and only one part was affected.
Wiles's proof of Fermat's Last Theorem
Wiles spent almost a year trying to repair his proof, initially by himself and then in collaboration with his former student Richard Taylor , without success. 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. But instead of being fixed, the problem, which had originally seemed minor, now seemed very significant, far more serious, and less easy to resolve.
Wiles states that on the morning of 19 September , 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. He states that he was having a final look to try and understand the fundamental reasons why his approach could not be made to work, when he had a sudden insight that the specific reason why the Kolyvagin—Flach approach would not work directly, also meant that his original attempts using Iwasawa theory could be made to work if he strengthened it using his experience gained from the Kolyvagin—Flach approach since then.
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: On 6 October Wiles asked three colleagues including Faltings to review his new proof,  and on 24 October Wiles submitted two manuscripts, "Modular elliptic curves and Fermat's Last Theorem"  and "Ring theoretic properties of certain Hecke algebras",  the second of which Wiles had written with Taylor and proved that certain conditions were met which were needed to justify the corrected step in the main paper.
The two papers were vetted and finally published as the entirety of the May issue of the Annals of Mathematics. The new proof was widely analysed, and became accepted as likely correct in its major components. Fermat claimed to " As noted above, Wiles proved the Taniyama—Shimura—Weil conjecture for the special case of semistable elliptic curves, rather than for all elliptic curves. In , Dutch computer scientist Jan Bergstra posed the problem of formalizing Wiles' proof in such a way that it could be verified by computer.
Wiles used proof by contradiction , in which one assumes the opposite of what is to be proved, and show if that were true, it would create a contradiction. The contradiction shows that the assumption must have been incorrect. The proof falls roughly in two parts. In the first part, Wiles proves a general result about " lifts ", known as the "modularity lifting theorem". This first part allows him to prove results about elliptic curves by converting them to problems about Galois representations of elliptic curves.
He then uses this result to prove that all semi-stable curves are modular, by proving that the Galois representations of these curves are modular, instead. We will set up our proof by initially seeing what happens if Fermat's Last Theorem is incorrect, and showing hopefully that this would always lead to a contradiction.
Wiles aims first of all to prove a result about these representations, that he will use later: Proving this is helpful in two ways: Together, these allow us to work with representations of curves rather than directly with elliptic curves themselves. Our original goal will have been transformed into proving the modularity of geometric Galois representations of semi-stable elliptic curves, instead.
Wiles described this realization as a "key breakthrough". 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 is Wiles' lifting theorem or modularity lifting theorem , a major and revolutionary accomplishment at the time. So we can try to prove all of our elliptic curves are modular by using one prime number as p - but if we do not succeed in proving this for all elliptic curves, perhaps we can prove the rest by choosing different prime numbers as 'p' for the difficult cases.
The proof must cover the Galois representations of all semi-stable elliptic curves E , but for each individual curve, we only need to prove it is modular using one prime number p. From above, it does not matter which prime is chosen for the representations.
We can use any one prime number that is easiest. So the proof splits in two at this point. 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. Wiles denotes this matching or mapping that, more specifically, is a ring homomorphism:.
Wiles had the insight that in many cases this ring homomorphism could be a ring isomorphism Conjecture 2.
This is sometimes referred to as the "numerical criterion". Given this result, Fermat's Last Theorem is reduced to the statement that two groups have the same order.
Much of the text of the proof leads into topics and theorems related to ring theory and commutation theory. In treating deformations, Wiles defined four cases, with the flat deformation case requiring more effort to prove and treated in a separate article in the same volume entitled "Ring-theoretic properties of certain Hecke algebras".
Gerd Faltings , in his bulletin, gives the following commutative diagram p.
Wiles's proof of Fermat's Last Theorem
In order to perform this matching, Wiles had to create a class number formula CNF. He first attempted to use horizontal Iwasawa theory but that part of his work had an unresolved issue such that he could not create a CNF.
At the end of the summer of , he learned about an Euler system recently developed by Victor Kolyvagin and Matthias Flach that seemed "tailor made" for the inductive part of his proof, which could be used to create a CNF, and so Wiles set his Iwasawa work aside and began working to extend Kolyvagin and Flach's work instead, in order to create the CNF his proof would require. Since his work relied extensively on using the Kolyvagin—Flach approach, which was new to mathematics and to Wiles, and which he had also extended, in January he asked his Princeton colleague, Nick Katz , to help him review his work for subtle errors.
Their conclusion at the time was that the techniques Wiles used seemed to work correctly. Wiles' 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.
Less obvious is that given a modular form of a certain special type, a Hecke eigenform with eigenvalues in Q , one also gets a homomorphism from the absolute Galois group. This goes back to Eichler and Shimura. The resulting representation is not usually 2-dimensional, but the Hecke operators cut out a 2-dimensional piece. It is easy to demonstrate that these representations come from some elliptic curve but the converse is the difficult part to prove.
An essential point is to impose a sufficient set of conditions on the Galois representation; otherwise, there will be too many lifts and most will not be modular. These conditions should be satisfied for the representations coming from modular forms and those coming from elliptic curves.
In his page article published in , Wiles divides the subject matter up into the following chapters preceded here by page numbers:.
Gerd Faltings subsequently provided some simplifications to the proof, primarily in switching from geometric constructions to rather simpler algebraic ones. Wiles's paper is over pages long and often uses the specialised symbols and notations of group theory , algebraic geometry , commutative algebra , and Galois theory. Remarks on Fermat's last theorem, Austral. Wiles A.
Annals of Mathematics 3 : pp. Doi: Heath-Brown D. The first case of Fermat's last theorem. Intelligencer7 4 , pp Barlow P. An Elementary Investigation of Theory of Numbers.
Paul's Church-Yard, London: J. Gautschi, W. SIAM Review 50 1 : pp On Euler's hypothetical proof. Mathematical Notes 82 Del Centina, A. Archive for History of Exact Sciences September Carmichael, R.
The Theory of numbers and Diophantine Analysis. Dover N. Legendre, A. Institut France 6: Paris 9. Pures Appl. Ribet K. Galois representation and modular forms. Bulletin AMS 32, pp Ribet, K. Toulouse Math. Serge L. Some History of the Shimura-Taniyama Conjecture.November International Press. From Ribet's Theorem and the Frey Curve, any 4 numbers able to be used to disprove Fermat's Last Theorem could also be used to make a semistable elliptic curve "Frey's curve" that could never be modular; But if the Taniyama—Shimura—Weil conjecture were also true for semistable elliptic curves, then by definition every Frey's curve that existed must be modular.
But number theorists were not the only ones electrified by this story. Wiles denotes this matching or mapping that, more specifically, is a ring homomorphism:. If the UTF is false, then there would be an elliptic curve such that can't be associated with any modular form, and therefore the Taniyama-Shimura conjecture would be false.
Serre's main interest was in an even more ambitious conjecture, Serre's conjecture on modular Galois representations , which would imply the Taniyama—Shimura—Weil conjecture.
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