3 Articles on the proof of Fermat's Last Theorem Newsweek, New York Times, and FXKMS@acad3.alaska.edu. ========= Internet Amateur Mathematics Society Newsletter 5 IAMS@quack.kfu.com Fermat's Last Theorem Proved If you haven't heard it yet, Dr. Andrew Wiles had claimed that he proved the famous Fermat's Last Theorem. Here I have three articles, one from Newsweek, one from New York Time, and the other one from FXKMS@acad3.alaska.edu. Brief Summary of Dr. Wiles' Proof, From FXKMS@acad3.alaska.edu If E is a semistable elliptic curve defined over Q, then E is modular. It has been known for some time, by work of Frey and Ribet, that Fermat's Theorem follows from this. If u^q + v^q + w^q = 0, then Frey had the idea of looking at the (semistable) elliptic curve y^2 = x(x-a^q)(x+b^q). If this elliptic curve comes from a modular form, then the work of Ribet on Serre's conjecture shows that there would have to exist a modular form of weight 2 on $\Gamma_0(2)$. But there are no such forms. To prove the Theorem, start with an elliptic curve $E$, a prime $p$ and let \rho_p : \Gal(\bar{Q}/Q) \to \GL_2(\Z/p\Z) be the representation giving the action of Galois on the p-torsion E[p]. We wish to show that a certain lift of this representation to GL_2(Z_p) (namely, the p-adic representation on the Tate module T_p(E)) is attached to a modular form. We will do this by using Mazur's theory of deformations, to show that every lifting which `looks modular' in a certain precise sense is attached to a modular form. Fix certain `lifting data', such as the allowed ramification, specified local behavior at $p$, etc. for the lift. This defines a lifting problem, and Mazur proves that there is a universal lift, i.e. a local ring R and a representation into GL_2(R) such that every lift of the appropriate type factors through this one. Now suppose that \rho_p is modular, ie there is some lift of \rho_p which is attached to a modular form. Then there is also a hecke ring T, which is the maximal quotient of R with the property that all modular lifts factor through T. It is a conjecture of Mazur that R = T, and it would follow from this that every lift of \rho_p which `looks modular' (in particular the one we are interested in) is attached to a modular form. Thus we need to know 2 things: 1. \rho_p is modular 2. R = T. It was proved by Tunnell that \rho_3 is modular for every elliptic curve. This is because \PGL_2(\Z/3\Z) = S_4. So (1) will be satisfied if we take p=3. This is crucial. Wiles uses (a) to prove (b) under some restrictions on \rho_p. Using (a) and some commutative algebra (using the fact that T is Gorenstein, `basically due to Mazur') Wiles reduces the statement T = R to checking an inequality between the sizes of 2 groups. One of these is related to the Selmer group of the symmetric sqaure of the given modular lifting of \rho_p, and the other is related (by work of Hida) to an L-value. The required inequality, which everyone presumes is an instance of the Bloch-Kato conjecture, is what Wiles needs to verify. He does this using a Kolyvagin-type Euler system argument. This is the most technically difficult part of the proof, and is responsible for most of the length of the manuscript. He uses modular units to construct what he calls a 'geometric Euler system' of cohomology classes. The inspiration for his construction comes from work of Flach, who came up with what is essentially the `bottom level' of this Euler system. But Wiles needed to go much farther than Flach did. In the end, under certain hypotheses on \rho_p he gets a workable Euler system and proves the desired inequality. Among other things, it is necessary that \rho_p is irreducible. Suppose now that E is semistable. There are 2 cases: 1. \rho_3 is irreducible. Take p=3. By Tunnell's theorem (a) above is true. Under these hypotheses the argument above works for \rho_3, so we conclude that E is modular. 2. \rho_3 is reducible. Take p=5. In this case \rho_5 must be irreducible, or else E would correspond to a rational point on X_0(15). But X_0(15) has only 4 noncuspidal rational points, and these correspond to non-semistable curves. If we knew that \rho_5 were modular, then the computation above would apply and E would be modular. We will find a new semistable elliptic curve E' such that \rho_{E,5} = \rho_{E',5} and \rho_{E',3} is irreducible. Then by Case 1, E' is modular. Therefore \rho_{E,5} = \rho_{E',5} does have a modular lifting and we will be done. We need to construct such an E'. Let X denote the modular curve whose points correspond to pairs (A, C) where A is an elliptic curve and C is a subgroup of A isomorphic to the group scheme E[5]. (All such curves will have \bmod 5 representation equal to \rho_E.) This X is genus 0, and has one rational point corresponding to E, so it has infinitely many. Now Wiles uses a Hilbert Irreducibility argument to show that not all rational points can be images of rational points on modular curves covering X, corresponding to degenerate level 3 structure (i.e. \Im(\rho_3) not \GL_2(Z/3)). In other words, an E' of the type we need exists. (To make sure E' is semistable, choose it 5-adically close to E. Then it is semistable at 5, and at other primes because \rho_{E',5} = \rho_{E,5}.) New Answer for an Old Question, The proof's in the putting By Sharon Begley with Joshua Copper Ramo Newsweek, July 5th 1993, Pg 52 Copyright 1993 NEWSWEEK, INC.: 444 MADISON AVENUE, N.Y, N.Y, 10022 All Rights Reserved. "I have found a truly wonderful proof which this margin[of my notebook] is too small to contain." So asserted French mathematician Pierre de Fermat in 1637, in what became the biggest historical dare in mathematics. He was referring to a beguilingly simple assertion about numbers that had intrigued mathematicians since Roman times. Fermat died 18 years later, never having gotten around to writing down his "admirable proof". That should have been a tip-off. The greatest minds of the next four centuries tried to find this proof and failed so abysmally that Fermat's last Theorem, as the assertion was known, became the top unsolved challenge in all mathematics. Now it may have fallen. Last week, in a lecture at Cambridge University in England, Andrew Wiles of Princeton University announced that he had proved Fermat's Last. Within an hour number theorists were spreading the word from London to Boulder to Berkeley through a joyous hail of E-mail. "There is a sort of euphoria", exalted Princeton math department chairman Simon Kochen. "Euphoria because we lived to see this." Pure mathematics is a sucker's game. It lures the curious and confident with its seeming simplicity only to make them look like fools. Consider the equation 9+16=25. It can be written 3^2+4^2=5^2. More generally, one can write that a number squared (multiplied by itself) plus a second number squared equals a third number squared. Now the inquisitive are hooked like rubes in three-card monte. Try to find three whole numbers that fit the equation x^3+y^3=z^3. That is, pick numbers for x and y, multiply each by itself twice (like 3\times 3\times 3), and find a third number z which, when multiplied by itself twice, equals to the sum of the x^3 and y^3. There is no such z. That's what Fermat's Last states. Nor are there any numbers that fit the equation where the exponents are anything greater than 2. That's what Fermat claimed to have proved. Fermat did, in fact, prove his assertion for exponents of 4. Leonard Euler, the great Swiss mathematician, proved it for exponents of 3 in the 1790s. France's Adrien Legendre proved it for exponents of 5 in 1823. A few years ago a computer proved it for everything less than 30,000. Things were looking pretty good for Fermat's last, but none of this constituted a persuasive proof. What if the number right after the last number checked turned out to falsify the theorem? Only a rigorous proof would do. Their inability to find one, especially in the face of Fermat's taunt, drove mathematicians crazy. Too shy: Wiles approached Fermat's last Theorem the way one would a skittish horse -- obliquely. He started with a 1984 finding that if there are any numbers for which Fermat's equations holds, then the solutions can be fashioned into something called an elliptic curve. Then wiles noted a 1987 proof by Ken Ribet of the University of California, Berkeley, that any such elliptic curves could not be of a certain type. (Those who can't balance their checkbooks can drop out here.) When Wiles proved the contrary -- that the relevant elliptic curves are of this type -- he had shown the "if" he started with to be wrong: there are no numbers that make Fermat's equation work. Just as the Frenchman said 356 years ago. This all took Wiles 200 pages. Wiles (who describes himself as too shy to talk to the press) combined ideas from number theory, topology and other disparate fields and basically "three the kitchen sink" at Fermat's Last, says Kochen. "Among theorists there is often a sense that something just looks right. It's the general feeling that [Wiles's proof] looks right," he says. Mathematicians will not know for sure until they check every line, a process that could take years. Thousands of other claims to have proved Fermat's Last Theorem have fizzled. But if Wiles has triumphed over the historical dare, his proof would promise a huge advance in number theory. It is a field of almost pristine irrelevance to everything except the wondrous demonstration that pure numbers, no more substantial than Plato's shadows, conceal magical laws and orders that the human mind can discover after all. At Last, Shout of `Eureka!' In Age-Old Math Mystery By Gina Kolata The New York Times, Thursday, June 24, 1993 Copyright 1993 The New York Times More than 350 years ago, a French mathematician wrote a deceptively simple theorem in the margins of a book, adding that he had discovered a marvelous proof of it but lacked space to include it in the margin. He died without ever offering his proof, and mathematicians have been trying eversince to supply it. Now, after thousands of claims of success that proved untrue, mathematicians say the daunting challenge, perhaps the most famous of unsolved mathematical problems, has at last been surmounted. The problem is known as Fermat's last theorem, and its apparent conqueror is Dr. Andrew Wiles, a 40-year-old English mathematician who works at Princeton University. Dr. Wiles announced the result yesterday at the last of three lectures given over three days at Cambridge University in England. Within a few minutes of the conclusion of his final lecture, computer mail messages were winging around the world as mathematicians alerted each other to the startling and almost wholly unexpected result. Dr. Leonard Adelman of the University of Southern California said he received a message about an hour after Dr. Wiles's announcement. The frenzy is justified, he said. "It's the most exciting thing that's happened in -- geez -- maybe every, in mathematics." Mathematicians present at the lecture said they felt "an elation", said Dr. Kenneth Ribet of the University of California at Berkeley, in a telephone interview from Cambridge. Impossible Is Possible The theorem, an overarching startment about what solutions are possible for certain simple equations, was stated in 1673 by Pierre de Fermat, a 17th century French mathematician and physicist. Many of the brightest minds in mathematics have struggled to find the proof ever since, and many have concluded that Fermat, contrary to his tantalizing claim, had probably failed to develop one despite his considerable mathematical ability. With Dr. Wiles' result, Dr. Ribet said, "the mathematical landscape has changed." He explained: "Your discover that things that seemed completely impossible are more of a reality. This changes the way you approach problems, what you think is possible." Dr. Barry Mazur, a Harvard University mathematician, also reached by telephone in Cambridge, said: "A lot more is proved than Fermat's last theorem. One could envision proof of a problem, no matter how celebrated, that had no implications. But this is just the reverse. This is the emergence of a technique that is visibly powerful. It's going to prove a lot more." Remember Pythagoras? Fermat's last theroem has to do with equations of the form x^n+y^n=z^n. The case where n is 2 is familiar as the Pythagorean theorem that the squares of the lengths of two sides of a right angled triangle equal the square of the length of the hypotenuse. One such equation is 3^2+4^2=5^2, since 9+16=25. Fermat's last theorem states that there are no solutions to such equations when n is a whole number greater than 2. This means, for instance, that it would be impossible to find any whole numbers x, y and z such that x^3+y^3=z^3. Thus 3^3+4^3, (27+64)=91, which is not the cube of any whole number. Mathematicians in the United States said that the stature of Dr. Wiles and the imprimatur of the experts who heard his lectures, especially Dr. Ribet and Dr. Mazur, convinced them that the new proof was very likely to be right. In addition, they said, the logic of the proof is persuasive because it is is built on a carefully developed edifice of mathematics that goes back more than 30 years and is well accepted by researchers. Experts cautioned that Dr. Wiles could of course have made some subtle misstep. Famous and not-so-famous mathematicians have claimed proofs in the past, only to be tripped up by errors. Dr. Harold M. Edwards, a mathematician at the Courant Institute of Mathematical Sciences in New York, said that until the proof was published in a mathematical journal, which could take a year, and until it is checked many times, there is always a chance it is wrong. The author of a book on Fermat's last theorem, Dr. Edwards noted that "even good mathematicians have had false proofs." Luring the World's `Cranks' But even he said that Dr. Wiles's proof sounds like the real thing and "has to be taken very seriously." Despite the apparent simplicity of the theorem, proving it was so hard that in 1815 and in again 1860, the French Academy of Science offered a gold medal and 300 francs to anyone who could solve it. In 1908, the German Academy of Science offered a prize of 100,000 marks for a proof that the theorem was correct. The prize, which still stands though has been reduced to 7,500 marks, about $4,385, has attracted the world's "cranks", Dr. Edwards said. When the Germans said the proof had to be published, "the cranks began publishing their solutions in the vanity press," he said, yielding thousands of booklets. The Germans told him they would even award the prie for a proof that the theorem was not true, Dr. Edwards added, saying that they "would be so overjoyed that they wouldn't have to read through these submissions." But it was not just amateurs whose iamgination was captured by the enigmatic problem. Famous mathematicians, too, spent years of their lives on it. Others chose never to get near it for fear of being sucked into a quagmire. One mathematical genius, David Hilbert, said in 1920 that he would not work on it because, "before beginning I should put in three years of intensive study, and I haven't that much time to spend on a probable failure." Mathematicians armed with computers have shown that Fermat's theorem holds true up to very high numbers. But that falls well short of a general proof. Tortuous Path to Proof Dr. Ribet said that 20th century work on the problem began to grow ever more divorced from Fermat's quations. "Over the last 60 years, people in number theory have forged an incredible number of tools to deal with simple problems like this," he said. "As the tools became more complicated, they took on a life of their own. People lost day-to-day contact with the old problems and were preoccupied with the objects they created." Dr. Wiles' proof draws on many of these mathematical tools but also "completes a chain of ideas," said Dr. Nicholas Katz of Princeton University. The work leading to the proof began in 1954, when the late Japanese mathematician Yutaka Taniyama made a conjecture about mathematical objects called elliptical curves. That conjecture was refined by Dr. Goro Shimura of Princeton University a few years later. But, Dr. Katz said, mathematicians had no perception through the 1950's to 70's that this had any relationship to Fermat's last theorem. "They seemed to be on different planets," he said. In the mid-80's, Dr. Gerhard Frey of the University of the Saarland in Germany "came up with a very strange, very simple connection between the Taniyama conjecture and Fermat' last theorem, " Dr. Katz said. "It gave a sort of rough idea that if you knew Taniyama conjecture you would in fact know Fermat's last theorem." He explained. In 1987, Dr. Ribet proved the connection. Now, Dr. Wiles has shown that a form of the Taniyama conjecture is ture and that this implies tht Fermat's last theorem must be true. "One of the things that's most remarkable about the fact that Fermat's last theorem is proven is the incredibly roundabout path that led to it," Dr. Katz said. Arcane to the Arcane Another remarkable aspect is that such a seemingly simple problem would require such sophisticated and highly specialized mathematics for its proof. Dr. Ribet estimated that a tenth of one percent of mathematicians could understand Dr. Wiles' work because the mathematics is so technical. "You have to know a lot about modular forms and algebraic geometry," he said. "You have to have ollowed the subject very closely." The general idea behind Dr. Wiles' rpoof was to associate an elliptic curve, which is a mathematical object that looks something like the surface of a doughnut, with an equation of Fermat's theorem. If the theroem were false and there were indeed solutions to the Fermat equations, a peculiar curve would result. The proof hinged on showing that such a curve could not exist. Dr. Wiles, who has told mathematicians he is reluctant to speak to the press, could not be reached yesterday. Dr. Ribet, who said Dr. Wiles was shy, said he was asked to speak for him. Dr. Ribet said it took Dr. Wiles seven years to solve the problem. He had a solution for a special case of the conjecture two years ago, Dr. Ribet said, but told no one. "It didn't give him enough and he felt very discouraged by it", he said. Dr. Wiles presented his results this week at a small conference in Cambridge, England, his birthplace, on "Padic Galois Representations, Iwasawa Theory and the Tamagawa Numbers of Motives." He gave a lecture a day on Monday, Tuesday and Wednesday with the title "Modular Forms, Elliptic Curves and Galois Representations." There was no hint in the title that Fermat's last theroem would be discussed, Dr. Ribet said. "As Wiles began his lectures, thee was more and more speculation about what it was going to be, " Dr. Ribet said. The audience of specialists in these arcane fields swelled from about 40 on the first day to 60 yesterday. Finally, at the end of his third lecture, Dr. Wiles concluded that he had proved a general case of the Tatiyama cnjecture. Then, seemingly as an after thought, he noted that that meant that Fermat;s last theorem was true. Q.E.D. People raised their cameras and snapped pictures of this historic moment, Dr. Ribet said. Then "there was a warm round of applause, followed by a couple of questions and another warm round of applause," he added. "I had to give the next lecture, " Dr. Ribet said. "It was something incredibly mundane." Since mathematicians are "a pretty well behaved bunch," they listened politely. But, he said, it was hard to concentrate. "Most people in the room, including me, were incredibly shell-shocked," he said.