Andrew Wiles: Fermat's Last theorem: abelian and non-abelian approaches

well thank you very much for that introduction and for the kind words it's such a great honor and a great pleasure to be here in Oslo and to receive this prize I've realized that there may be many people in the audience who are not fully trained number theorists let alone fully trained mathematicians so it may be that you don't understand more than the opening descriptions of the problem and perhaps the first few things I say about them but there is no better way to understand the mathematical experience than to understand just the question and the opening statements that's what we suffer every time we faced a new mathematical problem so I hope you'll join the Jane enjoy the genuine mathematical experience tour that will involve so this is the family occasion that X to the N plus y to the N equals said to the N should have no solutions as you all know FEM are annotated a copy of a Greek text of mathematics and in the margin he wrote it is impossible on the other hand to divide a cube into two cubes a fourth power and to fourth powers or likewise any power higher than the fourth into two like powers for this truly I have a wonderful proof but this margin is too small to contain it he wrote this we don't actually know when because it was only after he died that his son republished the original Greek text with these marginal notes in 1670 and it is only through his son's intervention that we can actually know what ephemeral is worsening so we can't precisely date it we don't really know whether he started to cross this out but for a long time people assumed he did have a solution and tried to find that solution myself included so the case N equals two of course there are many many solutions we learn in high school 3 squared plus 4 squared equals 5 squared 5 squared plus 12 squared equals 13 squared and so on these were known to the ancient Babylonians work on this problem of trying to extend this or rather refute this in the general case to show that there are no solutions for n greater than or equal to 3 generated an intern was inspired by many developments in number theory the field of nama theory in some sense started with Famer himself and there's a clear division in the attempts to solve this problem which was parallel to a division in the subject of algebra which was studied by árbol so as you well know I will prove the in solve ability of the general equation the general quintic and more generally higher degree equations and that marked an end of hundreds of years of study of algebraic equations of solving equations using radicals but this problem solving equations for radicals it's an algebraic answer to solving equations and what number theorists are looking for is a much more arithmetic issue but actually we get stuck at the same point that was resolved by árbol so what if Emma actually achieve so what we have written evidence for of the case is N equals 3 and N equals 4 so in the case any three although it's not at all obvious you can restate the problem as showing that there is no right angled triangle where the sides have rational lengths all three sides and where the area is one so it's not obvious you use the formula for the area of a triangle and you can deduce this using pythagoras theorem that it's equivalent to the case N equals three of famous loss theorem so firm our claim this in letters several times in his life particularly letters to Digby on khakha V the case N equals four he showed and actually described the proof the case actually something even stronger that the sum of two fourth powers is not a square Thermo describes the method and he called it the use of infinite descent so unfortunately we know rather little about exactly how he wrote out his proofs who thought about his proofs because he was still living in an era with rather limited publication so in the Middle Ages when equations were being studied prior to arbol in the 14th 15th centuries mathematicians in Italy would challenge each other by having mathematical joules where they would set problems to each other and their reputation their status and their careers really depended on the outcomes of these duels and I'm sure you know the famous history of the solution of the cubic well there was a certain leftover inheritance of this in the way fam'ly discussed mathematics he would issue challenges to other mathematicians particularly the English mathematicians which they stickily failed to understand or solve most cases so that's part of the reason that we don't have the detailed proofs because he said his results as challenges rather than explaining the proofs well this method of infinite descent I tried to trace where it came from and actually in 1636 family knew nothing of number theory he must have encountered some problems from the book of diophantus and his very first question actually to rob evolve another French mathematician was actually an extremely elementary question where he didn't understand the proof but it actually reduces to a very famous classical ancient classical proof of the irrationality of the square root of two so just to give an idea of how infinite descent works I will show you how he might have thought of that proof in terms of of infinite descent so here is an actual proof for you theorem the square root of 2 is not a rational number so what you do is you you start off by supposing the square root of 2 is a rational number so you can write it as a quotient a over B where a and B are integers like 7-elevens whatever you choose a solution where a and B are both positive well you can rewrite that as a squared equals to B squared and now a is even because it's twice B squared so a is twice another integer substitute that in a equals to a zero in the original equation and you get 4a 0 squared is 2 B squared cancel the 2 you get 2 ace 0 squared is B squared that now tells you that B is an even number so B is twice B 0 ok now you do the same thing again so you substitute in B B's to be zero and you get a zero squared is to be zero squared and now you've got the square root of two is actually a zero of a B zero so what you've done is you've taken a solution to the equation and shown that actually there's a smaller solution and this new one has the a zero strictly smaller than a well then he says well a smaller solution gives a contradiction because you can't keep getting smaller solutions forever there's no infinite descent so this is not how this is typically presented but I imagine this is how Pharma came to the idea of infinite descent and it's very striking that actually this idea of infinite descent was basically the foundation of all attempts for on fam'ly until the late 19th century so now I'm going to jump mathematically to the next successful partially successful attempt which was by coma in the 1840s so there was a lot of drama in this period where in the French Academy people announced proofs which were fallacious and there was a lot of excitement but actually it turned out to be they turned out to be wrong and they were wrong in a way that had already actually been explained by coma and published previously a few years previously so the idea is you again try and find assume you have a solution to the equation and then try and deduce from it a smaller solution so what you do is you take the equation the general equation so it said to the P is X to the P plus y to the P well that actually can be factored as a product of P factors like this where Zeta is e to the 2 pi over P is a piece root of 1 so this is a using complex numbers we can just factor this as a product of linear factors now the idea is if you take a number which is a p f-- power and you write it as a product of other numbers if those numbers are co-prime to each other then each one itself has to be a p f-- power for example if i take in terms of cubes if i take 8 times 27 that number there's no other way of breaking it up as a product of cubes what does that mean that means that essentially if these numbers are co-prime to each other then each one should be a piece power well what we mean by P powers well we have to work in arithmetic of Z Zeta so we have to allow not just ordinary integers but combinations of powers of Zeta and then miraculously what you can do is you have this completely formal identity X plus Zeta Y times 1 plus theta minus X plus Zeta squared Y is Zeta X plus y well what's the use of that well the point is that if each of these factors were peeth powers then this is a piece power times 1 plus theta minus a piece power is Theta times a piece power it's very similar to the original equation and the idea of Kama is well supposing we start with a solution here and now we probably going to start with a solution in Z Zeta instead of a rational solution we can deduce that each these factors of P powers and hence we get another equation which is very close to the original one maybe we can use infinite descent if we start with a solution we deduce in this way that there's actually another solution which is smaller than the original one and again try infinite descent well at first sight this looks very tricky because this 1 plus Zeta is in the way the satyrs in the way but it turns out that if you really study the arithmetic of Z Zeta you can make this work these are units 1 plus Zeta and Zeta are both actually units in this what we call ring of numbers and said Zeta provided something it's true about ideal class groups so the problem is this principle that if you take a number like 72 that it's 2 times 2 times 3 times 2 times 2 times 2 times 3 times 3 there's a unique way of factoring any number in our arithmetic unfortunately in general in this arithmetic there's no unique way of factoring so this brought come at a great development the invention and study of ideal class groups in general so the idea is you take ideals in this ring said Zeta that is abelian groups that sit between some multiple of Z Zeta and said say to itself such that they're preserved under the action of the ring so alpha times that a billion group is again inside the abelian group and then you have special ones which are called principle ideals which are just multiples of one particular element so in this way you have certain things attached to this new arithmetic this arithmetic of Zedd Saito well what he he showed was that if you study what's called the ideal class group that is the quotient of the group of ideals that were principal ideals it's not quite a group but the quotient is a group ideals mod principal ideals he proved first that famous last theorem is true if P doesn't divide the order of this group so he gave a very explicit criterion for proving Fermat's Last Theorem well because it could be quite hard to calculate this size of this class group so he gave a second criterion which is even more amazing in a way which is that P divides this border of this class group if and only if P divides the product of these Bernoulli numbers b2 up to be P minus 3 these Bernoulli numbers are the numbers you get by expanding T over e to the t minus 1 as a power series and they're the coefficients of that power series 1 plus the sum of B MT to the M over N factorial so he gave this amazing analysis not only of the equation but actually of the obstacle to to solving the equation all attempts at pheromones last serum up to the end of the near the end of the 20th century really hinged on this technique and on trying to refine these properties of these class groups was the something about class groups who are missing can you prove more about them because unfortunately this wasn't enough it's very very hard to determine when P divides these and it certainly happens often in fact one of the things I studied my last fling as it were with fellas Last Theorem by the old methods when I was working with Mazur at Harvard we analyzed the fine structure of this class group to actually determine a relation between each individual binary number and some piece of this class group but these methods were not enough to actually master the original problem it's still there were gap between what we could even hope to do and a resolution of the problem by this method seemed completely inaccessible so this is as I said how how their subject proceeded for a long time now I've called this the abelian approach to farmers Last Theorem so why do I call it a billion what's it to do with our ball well it's to do with abelian extensions so árbol the named abelian is attached to extensions of the rationals or of father fields where there is AG our group and the action is a billion so for example here and we were using arithmetic with this Zeta or Zeta P I'm writing here P through to one so that's a root of this equation X to the P minus one over X minus one this well it's just another way of writing it from root of that serum so it's a billion because the gawe group of cues a to P over Q it's the rationals and join the Zeta P is described as the abelian groups said mod P said across that's the multiplicative group of said mod p see how does that work well a typical autumn orphism in here will ten take Z P to some other P through divinity it has to be a power of side V write it as the alpha power and that goes to L mod p so this this describes the Galois group explicitly and the gower group is abelian this describes it as an abelian group and this Sigma L came to be known much later if L as a prime is the Frobenius at L usually write it fro bell well these are abelian extensions and this for a long time became perhaps the central study of number theory partly inspired by a theorem of Kronecker and Weber they didn't work together in fact they were separated by quite a long time this took a long time to actually get properly approved that in fact all abelian extensions of the rational numbers are generated by roots of one so that's two statements really one that feels generated by roots of one our abelian and that's the easy part and the hard part is that every abelian extension arises that way well this takes us the Kronecker weber theorem to nearly the end of the 19th century the first part of the 20th century was spent in generalizing this abelian Theory to other fields and this is known as class field theory and this is really the sort of centerpiece of of number theory for much that certainly the first half of the 20th century so to describe a billion extensions of any number fields let me just take an example whoops sorry an example of F is the field generated by the cube root of two so what you want are a billion extensions of that field so it's actually very hard to write down some simple description like using roots of unity so that we haven't really mastered in any generality although it's still a problem an open problem what we can say is what exists so we can say that there is an extension which is similar to Q's a tepee plays the same role and it has the property that the GAO a group of this field which I've called F P so the FP just means it's it's just a labeling for this field which I claim exists the GAO a group of FP over F is isomorphic to the integers of F modulo P times the integers of F the multiplicative group of that modulo the image of the units so Oh F of the integers that said cube root 2 and this group Oh F cross plus for the units or the positive units of Oh F so this took the first 25 30 years of the 20th century but to really formulate it well took at least until the 50s even 60s it's much more complicated when the ideal class group of Oh F is not one so in the cases that fam I was worrying about I mean not fair mile but common was worrying about it would be much more complicated so of course when you've spent so much time developing the theory of abelian extensions one wants to move on to the non abelian extensions and this is the great split and in the second half the 20th century a huge program was developed to try and understand these extensions so they're basically equations over the rationals are over another number field where the gower group is not a billion where it's perhaps even a non abelian simple group and the fundamental description of this what it should be it's still conjectural it comes from what's called the Langlands program so there's a decisive shift to this in the 1960s and there are many many problems in this area which many people work on and we've made a little progress in this context but not very much moment it's actually easier to describe the parallel progress that this generated on the Fermo problem itself so I'm going to switch back now so this was how number theory is developed but now the non-abelian approach to to FEM I itself so this big breakthrough occurred in 1985 and was suggested first by Gerhard Frey and then it was the actual connection was completed by ribbit a year later so the idea is this and it's a completely different idea supposing you have a solution to a firm our problem okay new idea suppose eight the people speed to the PSC to the P P I'm just going to assume P is prime and famine is enough to do it for prime exponents so we usually just restrict to those then you manufacture artificially this new equation which is equation elliptic curve y squared is X X minus a to the P X plus B to the P okay it seems harmless enough it's odd because you take the discriminant of this cubic it's the probe that's of course the product of the difference of the roots squared so it's a to the P B to the P and then a to the P plus P to P which is C to the P by hypothesis so the discriminant turns out to be a perfect piece power okay so elliptic curves have been studied a lot and this seemed very odd to find a discriminant which is a perfect piece of power and in fact what Frye realized was that such a curve should not exist discriminant should not be peeth powers so he proposed a way that you might try and prove this it was a bit tenuous at first but ribbits as I said confirmed that in fact if you assumed what's called the modularity conjecture the taniyama-shimura of a conjecture has various names then in fact you would have a concrete way of proving Fermat's Last Theorem if you could show there's no elliptic curve this form you prove Fermat's Last Theorem and to show there's no elliptic curve of this form you can use the modularity conjecture so this gave a completely different approach so just to give a little hint about this I need to just say a tiny bit about elliptic curves again the origins of this subject are in arbol arbol was the first to really understand and talk about these doubly periodic functions he died at 26 maybe he could have done all this Pete lived longer it was so fast so supposing you take an equation this form y squared is X cubed plus ax plus B what elliptic functions enable you to do is to describe the complex solutions to this as a group it's C modulo a lattice the lattice being just an abelian group of the form set plus Z times another generator this tower is not itself rational and then this quotient actually forms the complex points of an of this curve well one thing that's very odd about this is that this is actually a group that's not at all obvious when you look at this equation so the solutions to this equation actually form an abelian group and if you draw seem odd the lattice in the plane and it's basically the interior points with some of the boundary described like this okay in particular you can ask about points on this group of finite order and the points of order P to the N had the structure Zed mod P to the N Z plus Z mod P to the N said funnily enough this was something that baffled árbol for a while he didn't understand why when he looked at this he got N squared roots instead of n roots I mean now it's clear but he wrote in his correspondence him he was mystified by this he understood it after a little while but so if this equation is actually defined over the rationals you can actually get the Galois action on these points so the gamma group the colonel of P to the N that's the C mod p to the anvils see my PT on the gawe group gives an action so it injects and she L to of said mod p TN said so what does this okay this is background on elliptic curves we have quite good enough to understand how to apply this to the FEM I equation yet so I want to just specialize the case P equals three so this was some sense the first big step well the first step of I described yesterday in tackling fair MA was to understand was to choose an arithmetic approach which as I said there's different ways of approaching this modularity conjecture through geometry through arithmetic through analysis so I chose arithmetic nur's that's what I knew about but then the big choice was actually to go to P equals three because if you look at the three division points so I've put them on these are the nine three division points in here algebraically you can actually describe these if you take the equation y squared is X cubed plus ax plus B these are actually the points of inflection just in classical calculus terms if you calculate the points of inflection of that equation remembering you're not just talking about real points of inflection there's actually eight of them and then you get a ninth one its infinity they actually themselves form a group there's nine of them and they generate field Q III and the gawe group of Q III over Q then acts on GL two of said mod three set so this is just the same as in the case P canapé I'm just specializing to P equals three what is special about this is that okay it's not a billion anymore this is not an abelian extension of the rationals but happily for me it's also a solvable group this group she also said mod three said is actually a double cover of the symmetric group on four letters it's a solvable group so where as arbol thought in terms of there was a division between equations up to the Quint up to the quartic which where the equation was solvable and equations of high degree in language of Galois which might not be solvable I'm actually considering three distinctions as the abelian groups the solvable groups than the non solvable groups the non solvable groups are extremely difficult and as I said arithmetic hasn't got there yet but this realization was that actually this in-between stage a solvable stage where I could perhaps squeeze through okay so that's elliptic curves and then the modularity conjecture so I have to say what it is tries to link what are called modular forms or modular functions with elliptic curves so modular functions have been studied in the nineteenth century they're actually a generalization if you like of the exponential function the exponential function has the property that it's periodic if you take e to the 2 pi I said it takes the same value when you move from Z to Z plus 1 I said plus 2 to Z plus 3 and so on and this is the basis of Fourier series and Fourier analysis and so on but again that is the abelian situation and the non-abelian situation is when you move to functions which are invariant not under Zed under the abelian group but under a non abelian group in this case a subgroup of SL to set so these functions yes they have a Fourier series they are invariant on the sector's to set plus one but they also have this other property that if you take F of a a plus B over C Z plus D its well we allow something slightly more general in the case we want is C to the Z plus d squared times F of Z that's whenever the matrix ABCD is an element of SL to Z with the C divisible by some integer n so for all ones that ABCD with C divisible by n and just as long as you can choose an end where it works that's what matters so this is a generalization the non abelian generalization of exponential functions we get modular functions like this so everything is moving from the abelian situations for non abelian situation so the modularity conjecture of tonyamarie shamora was that there's some correspondence every time you have an elliptic curve over the rationals you can actually make a module form of wait - that's the two here so I have the final weight is but just means this factor here and some level that's the end so the way to means here on the level here it doesn't really matter in detail at the moment what they are and what is this correspondence supposed to do it's supposed to say that when you take the gawe group from the gawe action on the p to the n division points so view that in CL 2 of 0 or P T&Z because that's what the action on the p TN Division points looks like under that the Frobenius of L will I described to you earlier there's a notion of a Frobenius so I described it in the abelian setting on roots of unity it takes data P 2 Zeta P to the L it's actually a bit more tricky to describe it in this non abelian setting but there is a generalization due to frobenius has its name but for every prime L you have a Frobenius of L and under this correspondence it goes to an element with trace al al is the ail from here so the Fourier expansion of this function tells you about the arithmetic of the elliptic curve so this is the arithmetic of the elliptic curve the gawe action on these p to the n division points this is the modular form you look at its Fourier expansion and there's some magical relation between the Al here and the Frobenius in the arithmetic so this conjecture seemed when it was first proposed in a vague form by taniyama got made more precise with shamora but it was very paper of 1967 where he pinned down the level n and proved a very beautiful theorem about it that enabled people to test it so they could test this conjecture numerically and they got overwhelming evidence people were convinced this was true so now when ribbit had proved that the modularity conjecture this one implied fellas last theorem we now could be really confident the FAMAS last theorem was true that it would be proved and that it could be proved this way because this modularity conjecture it's not something mathematics can put aside Fama's last theorem could have been like perfect numbers it's put to one side and we can get on with mathematics without it but we can't get on without this this is at the core of the transition of number theory from non abelian mathematics from a beam in mathematics to non abelian mathematics so famous Last Theorem was back right on center stage okay so what is what is the outcome of this well as I said as soon as I heard that ribbit had proved this connection I was riveted and spent my time working on this so why does the case P equal three why is it so valuable well the point is what have we got so we have fry and then ribbet prove the modularity conjecture implies pheromones laughs theorem so now the problem just proved modularity conjecture okay and as I said when I started trying this the first after the first step and taking the right choice making the right choice to go into arithmetic I think the first real conceptual conceptually important step is that if you look at the three division points our website put gl3 through BG L too said mod three said p GL 2 said mod three said is isomorphic to s 4 this is a solvable group and when Langlands had proposed this general program for understanding non solvable extensions there was one case that he could really understand and that is precisely this when you have gower groups of the rationals in two or even any on the field into a solvable subgroup of GL 2 and s 4 is such a subgroup of p GL to s bar is such a subgroup so what Langlands and actually this particular case is an extension of Langdon's theorem by Gerry tunnel this is connected actually with the modular form it's not the one that we want associated to an elliptic curve it doesn't solve the problem right off but at least it gives you some tenuous link between the theory of elliptic curves and this theory of modular forms he used based this on his theory of base change and it uses the fact as I said that s4 is a solvable group so this this was a remains the first step there is no way around this step at the moment we have no way of jumping straight to non solvable extensions the only entry point we have is why these solvable extensions in the middle so a billion extensions we understood in the first half the 20th century we've pushed that a little further to these solvable extensions like s4 and we can use that to get a little bit further but we really have not mastered these completely in solvable extensions these simple groups of any kind well let me just say a few words about this generalization of Langlands which lots of people work on in number theory it's really as I said the the main thrust of modern number theory because even working on other problems like elliptic curves and any kind of generalization of elliptic curves you want to solve equations you're going to have to use this ultimately so it's a generalization of class field theory remember class field theory describes a billion extensions here the problem is to describe all Galois extensions of the rationals or any number field and roughly speaking and I'm being little dishonest but roughly speaking it generalizes modular forms to automatic representations roughly speaking we use functions invariant instead of under subgroups of sl2 but under subgroups of GL n to classify n dimensional representations of the Galois group so it's a very beautiful idea that you can do this now it's lightly dishonest because you really have to use a Del's and pathetic representation theory but this is back to classical number theory where you use analysis and now also naturally representation theory to study number fields we've been very lucky to have the assistance of algebraic geometry for 50 years and it's um sometimes has blinded us the fact is Langlands proof did not use algebraic geometry it's just analysis and representation Theory Lang was proof for the s4 case and this general case is still completely baffling to us so let me just mention there's a little progress so one of the simplest ones you might try and understand is the Gower group the representation the Gallagher of Q barbecue Tucci L too said model said so this group is not solvable but if you assume the determinant is odd and rows are reducible then row is associated to a modular form and the sort of way I've been describing the proof uses similar techniques again the entry point you have to enter by this these solvable extensions so you enter this theory using Langlands theorem using also the reducible case but then a very very ingenious induction on primes they managed to push these techniques to prove the modularity for Association to modular forms of these kinds of extensions so this corresponds to extensions of the rationals with certain kinds of Galois group sit inside gl2 of said model set but this is still a very very special case of the language program unfortunately the method doesn't seem to work in the general case the general case would be gln or even any other group and any number field in place of the rationals so we are at a stage we've feel we've mastered abelian setting we would really love to understand the non solvable even a simple group setting we're just piecing together some results about them the solvable case but we have a few isolated results like this in the genuinely non solvable case but we're still at the very beginning of this great enterprise the Langlands program thank you very much [Applause] [Applause] we have time for some questions thank you very much for exciting lecture you mentioned that one of the key ideas and the whole presentation was Fry's idea to construct an elliptic curve which is now called the Phi curve so I told me at the time that he thought this was some kind of joke to reduce one unsolvable problem to an even more difficult and more and solve of a problem obviously it was proved wrong but I would like to hear your idea and how much he perceives as a joke or whether you felt immediately this was realistic and there was a chance sorry who said it was a joke by himself he did well he felt it was a joke to reduce something unsolvable to something even harder yes I didn't know he said that but actually there was an even more painful story I think that he'll aguar some years earlier had studied similar equations and had wondered if they were related to famine and he'd shown them to a very eminent mathematician who said that's ridiculous he might be able to do far more that way and so he'd given up working on it according to him sir yeah yes I think that does happen I have to admit when fry made this proposition I actually was very skeptical not about modularity conjecture but about what ribbit proved so I was very surprised we're very happy when ribbit proved it I thought it was just wishful thinking but yeah I would like to return to the elementary and the beginning yeah you said that thermo himself solve the problems for N equals equal to three and four and I suppose that after that there was a third for solving it for individual numbers or classes of numbers bigger than four five for example yeah could you say a few words so how that yes the first one to really pick up on it was Euler because the proof for N equals three had not been written out so Euler did the case N equals three I think he made a mistake the first time but I think he corrected it so the next one was N equals five and that was very that was very tricky you can try and do it the same way but it's much more complicated and I believe the story is that I quite like this story with my age that dares lay with his early 20s and did one part of it the what's called the first case I think of the problem it's since I mean it's now called so he did one case but he didn't complete it but he somehow was talking about it and a meet I believe it was at 70 years old I've got the right way around came in and beat him to it so I think it's but those two so I think the case that equals five was done the case N equals seven I think this precipitated the problems of the French Academy because I think LeMay claimed a very complicated proof and was announcing it to the French Academy and some of the other mathematicians like Co she got very excited and said yes I can do that too and I can do this and unfortunately I think there was this was where the mistake was the unique factorization problem he hadn't sorted it out he'd made some assumptions so I think eventually the case seven was done in hands-on methods but I've forgotten the details but anyway there was an attempt at various no prime numbers but then when comers work became known comers methods did all Prime's less than 100 except for 37 59 and 67 unfortunately commas Methos couldn't even be used to prove it worked for infinitely many primes I don't think so it's it's stalled I mean it looked very promising for a while but uninstall [Applause]

Channel: The Abel Prize

Views: 182,084

Keywords: Abel, Abel Prize, The Abel Prize, Andrew Wiles, Sir Andrew Wiles, Fermat's Last theorem, Fermat's theorem, math, mathematics, mathematician, number theory, abelian, non-abelian, arithmetic

Id: 4t1mgEBx1nQ

Length: 53min 11sec (3191 seconds)

Published: Mon Jan 27 2020