Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic blogger.comer with Ribet's theorem, it provides a proof for Fermat's Last blogger.com Fermat's Last Theorem and the modularity theorem were almost universally considered inaccessible to proof by contemporaneous mathematicians, meaning that they Oct 20, · Future Proof. Posted October 20, by Joshua M Brown. Future Proof Festival: September 11thth I have FREE registrations for financial advisors on a Oct 07, · Proof: Directed by John Madden. With Gwyneth Paltrow, Anthony Hopkins, Jake Gyllenhaal, Danny McCarthy. The daughter of a brilliant but mentally disturbed mathematician, recently deceased, tries to come to grips with her possible inheritance: his insanity. Complicating matters are one of her father's ex-students, who wants to search through his papers, and her estranged sister, who
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Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theoremit provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the modularity theorem were almost universally considered inaccessible to proof by contemporaneous mathematicians, meaning that they were believed to be impossible to prove using current knowledge, proof reading papers.
Wiles first announced his proof on 23 June at a lecture in Cambridge entitled "Modular Forms, Elliptic Curves and Galois Representations", proof reading papers. One year later on 19 Septemberin what he would call "the most important moment of [his] working life", Wiles stumbled upon a revelation that allowed him to correct the proof reading papers to the satisfaction of the mathematical community.
The corrected proof was published in Wiles's proof uses many techniques from algebraic geometry and number theoryand has many ramifications in these branches of mathematics, proof reading papers. It also uses standard constructions of modern algebraic geometry, such as the category of schemes and Iwasawa theoryand other 20th-century techniques which were not available to Fermat. Together, the two papers which contain the proof are pages long, [4] [5] and consumed over seven years of Wiles's research time.
John Coates described the proof as one proof reading papers the highest achievements of number theory, and Proof reading papers Conway called it "the proof of the [20th] century. For proving Fermat's Last Theorem, he was knightedand received other honours such as the Abel Prize.
When announcing that Wiles had won the Abel Prize, the Norwegian Academy of Science and Letters described his achievement as a "stunning proof", proof reading papers.
Fermat's Last Theoremformulated instates that no three distinct positive integers aband c can satisfy the equation, proof reading papers. Over time, this simple assertion became one of the most famous unproved claims in mathematics.
It spurred the development of entire new areas within number theory. Proofs were eventually found for all values of n up to around 4 million, first by hand, and later by computer. However, no general proof was found that would be valid for all possible values of nnor even a hint how such a proof could be undertaken.
Separately from anything related to Fermat's Last Theorem, in the s and s Japanese mathematician Goro Shimuraproof reading papers, drawing on ideas posed by Yutaka Taniyamaconjectured that a connection proof reading papers exist between proof reading papers curves and modular forms, proof reading papers. These were mathematical objects with no known connection between them. Taniyama and Shimura posed the question whether, unknown to mathematicians, the two kinds of object were actually identical mathematical objects, just seen in different ways.
They conjectured that every rational elliptic curve is also modular. This became known as the Taniyama—Shimura conjecture. In the West, this conjecture became well known through a paper by André Weilwho gave conceptual evidence for it; thus, it is sometimes called the Taniyama—Shimura—Weil conjecture.
By aroundmuch evidence had been accumulated to form conjectures about elliptic curves, and many papers had been written which examined the consequences if the conjecture was true, but the actual conjecture itself was unproven and generally considered inaccessible—meaning that mathematicians believed a proof of the conjecture was probably impossible using current knowledge.
For decades, the conjecture remained an important but unsolved problem in mathematics, proof reading papers. Around 50 years after first being proposed, proof reading papers, the conjecture was proof reading papers proven and renamed the modularity theoremlargely as a proof reading papers of Andrew Wiles's work described below.
On yet another separate branch of development, in the late s, Yves Hellegouarch came up with the idea of associating hypothetical solutions abc of Fermat's equation with a completely different mathematical object: an elliptic curve. In —, Gerhard Frey called attention to the unusual properties of this same curve, proof reading papers, now called a Frey curve.
He showed that it was likely that the curve could link Fermat and Taniyama, since any counterexample to Fermat's Last Theorem would probably also imply that an elliptic curve existed that was not modular.
Frey showed that there were good reasons to believe that any set of numbers abc proof reading papers, n capable of disproving Fermat's Last Theorem could also probably be used to disprove the Taniyama—Shimura—Weil conjecture. Therefore, if the Taniyama—Shimura—Weil conjecture were true, no set of numbers capable of disproving Fermat could exist, proof reading papers, so Fermat's Last Theorem would have to be true as well. Mathematically, 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.
Thus, according to the conjecture, any elliptic curve over Q would have to be a modular elliptic curveyet if a solution to Fermat's equation with non-zero aproof reading papersc and n greater than 2 proof reading papers, the corresponding curve would not be modular, resulting in a proof reading papers. If the link identified by Frey could be proven, then in turn, it would mean that a proof or disproof of either Fermat's Last Theorem or the Taniyama—Shimura—Weil conjecture would simultaneously prove or disprove the other.
To complete this link, it was necessary to show that Frey's intuition was correct: that a Frey curve, if it existed, proof reading papers, could not be modular.
InJean-Pierre Serre provided a partial proof that a Frey curve could not be modular. Serre's main interest was in an even more ambitious conjecture, Serre's conjecture on modular Galois representationswhich would imply the Taniyama—Shimura—Weil conjecture. However his partial proof came close to confirming the link between Fermat and Taniyama.
In the summer ofKen Ribet succeeded in proving the epsilon conjecture, now known as Ribet's theorem. His article was published in In doing so, Ribet finally proved the link between the two theorems by confirming, as Frey had suggested, that a proof of the Taniyama—Shimura—Weil conjecture for the kinds of elliptic curves Frey had identified, together with Ribet's theorem, would also prove Fermat's Last Theorem. In mathematical terms, Ribet's theorem showed that if the Galois representation associated with an elliptic curve has certain properties which Frey's curve hasthen that curve cannot be modular, in the sense that there cannot exist a modular form which gives rise to the same Galois representation.
Following the developments related to the Frey curve, and its link to both Fermat and Taniyama, a proof of Fermat's Last Theorem would follow from a proof of the Taniyama—Shimura—Weil conjecture—or at least a proof of the conjecture for the kinds of elliptic curves that included Frey's equation known as semistable elliptic curves. 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.
Hearing of Ribet's 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.
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 [it]. Wiles initially presented his proof in It was finally accepted as correct, and published, infollowing the correction of a subtle error in one part of his original paper.
His work was extended to a full proof of the modularity theorem over the following six years by others, who built on Wiles's work. During 21—23 JuneWiles 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.
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. Proof reading papers 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, proof reading papers.
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 proof reading papers one part was affected, proof reading papers.
Wiles spent almost a year trying to repair his proof, initially by himself and then in collaboration with his former student Richard Taylorwithout 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. Instead of being fixed, the problem, which had originally seemed minor, proof reading papers, now seemed very significant, far proof reading papers serious, and less easy to proof reading papers. Wiles states that on the morning of 19 Septemberhe 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 attempt using Iwasawa theory could be made to work if he strengthened it using 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 proof reading papers refereed paper: [13] [17].
I was sitting at my desk examining the Kolyvagin—Flach method. It wasn't that I believed I could proof reading papers it work, but I thought that at least I could explain why it didn't work. Proof reading papers I had this incredible revelation. I realised that, the Kolyvagin—Flach method wasn't working, proof reading papers, but it was all I needed to make my original Iwasawa theory work from three years earlier.
So out of the ashes of Kolyvagin—Flach seemed to rise the true answer to the problem. It was proof reading papers indescribably beautiful; it was so simple and so elegant. I couldn't understand how I'd missed it and I just stared at it in disbelief for twenty minutes. Then during the day I walked around the department, and I'd keep coming back to my desk looking to see if it was still there.
It was still there. I couldn't contain myself, proof reading papers, I was so excited. It was the most important moment of my working life.
Nothing I ever do again will mean as much. On 6 October Wiles asked three colleagues including Faltings to review his new proof, [19] and on 24 October Wiles submitted two manuscripts, "Modular elliptic curves and Fermat's Last Theorem" [4] and "Ring theoretic properties of certain Hecke algebras", [5] 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 " have proof reading papers a truly marvelous proof of this, proof reading papers, which this margin is too narrow to contain". As noted above, Wiles proved the Taniyama—Shimura—Weil conjecture for the special case of semistable elliptic curves, rather than for all elliptic curves.
Over the following years, Christophe BreuilBrian ConradFred Diamondand Richard Taylor sometimes abbreviated as "BCDT" carried the work further, ultimately proving the Taniyama—Shimura—Weil conjecture for all elliptic curves in a paper. InDutch computer scientist Jan Bergstra posed the problem of formalizing Wiles's proof in such a way that it could be verified by computer.
Wiles used proof by contradictionin which one assumes the opposite of what is to be proved, and shows 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 semistable curves are modular, proof reading papers, by proving that the Galois representations of these curves are modular.
Proof reading papers aims first of all to prove a result about these representations, that he will use later: that if a semistable elliptic curve E has a Galois representation ρ Ep that is modular, the elliptic curve itself must be modular. Proving this is helpful in two ways: it makes counting and matching easier, and, significantly, to prove the representation is modular, we would only have to prove it for one single prime number pproof reading papers, and we can do this using any prime that makes our work easy — it does not matter which prime we use.
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 semistable 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's lifting theorem or modularity lifting theorema 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 semistable elliptic curves Ebut 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, proof reading papers. Wiles showed that in this case, one could always find another semistable elliptic curve F such that the representation ρ F ,3 is irreducible and also the representations ρ E ,5 and ρ F ,5 are isomorphic they have identical structures.
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