Kenneth A. Ribet, "A 2020 View of Fermat's Last Theorem"

good afternoon I'm Jill Piper and it's my great pleasure and honor and my role as president of AMS to introduce the distinguished speaker of this lecture Ken ribbit professor of mathematics at UC Berkeley served as president of AMS from 2017 to 2019 and is giving this invited address from a past president on a subject veremos Last Theorem that has inspired mathematicians for well centuries research unfair mahse Last Theorem resulted in mathematical ideas that have transformed the field a body of work to which Ken ribbit has made fundamental and exciting contributions over the decades of his career his research in number theory has earned him extraordinary honors some of these include the fair MA Prize in 1989 elected membership in the American Academy of Arts and Sciences and in the US National Academy of Sciences an honorary Doctorate from Brown University in 1998 his undergraduate alma mater I also want to share with you that Ken has been widely recognized in the mathematical community for his talent for and love of teaching and for his mentorship of students more than two dozen PhD students by now ken has been involved in AMS service and governance for many years and he's contributed quite a bit but let me just point out one of his signature contributions as president namely his leadership for the campaign for the next generation co-chaired with Gen Tabak and Rob Lazar spelled Ken reached out to members to the leadership of the Society identified donors and helped to create the campaign committee he was truly the outward face of this tremendously successful fundraising campaign for the for interests the next generation which are close to his heart as a direct result of Ken's and others efforts AMS now has a three million dollar and grow endowment to permanently give small but impactful grants to early career mathematicians so please join me in welcoming Ken ribbit [Applause] and I want to thank you so many of you for coming out to what will prove to be a very technical discussion of advance number theory I'm glad that management directed you to the emergency exits in case they're useful I I'm going to talk about pheromones Last Theorem and it's really impossible to talk about that subject without starting with pheromones equation so we know what the theorem says it says that when you consider nonzero integers or if you prefer positive integers and you take perfect nth powers you can't get to perfect nth powers to add up to a third nth power for n bigger than two for N greater than or equal to three and when I give general talks on pheromones last theorem as I did quite recently at Brown University where I was introduced by Jill Peiffer I point out that it's possible to have two perfect squares sum up to be a third perfect square 9 plus 16 is 25 and the theorem says that you really stop at perfect squares so this theorem was apparently stated probably stated in a marginal note that peered the fair ma road and his copy of Diophantus sometime in the 17th century and it's probably a fair statement to say that he wrote this marginal note about three hundred and seventy-five years ago we don't have the actual marginal note because his copy which passed into the hands of his son Samuel de fair MA cannot be found we generally believe that fair ma as amateur mathematician no longer believed that he had proved the statement which has become known as his theorem but it's not his theorem he spent a considerable amount of time using his method of infinite descent which basically means mathematical deduction induction to talk about a special case of the theorem he proved for example that the sum of two perfect fourth powers cannot be a perfect square and therefore in particular cannot be a perfect fourth power and because of that and the fact that the theorem for given an implies the same statement for any multiples of n as exponent if you think about the factorization of positive integers n you realize that powers of two are ruled out and numbers bigger than two will then be divisible if they're not powers of two by a prime number bigger than or equal to three so fair Amma's work reduced the theorem to the case X to the n plus y to the N equals Z to the N where n is a prime number that we might as well call P and P can be three or more but then in the next century Euler wrote down an argument for n equal three and he did that by factoring equations in what we now say is the ring of integers of the field of third roots of unity what you get by adjoining minus one plus the square root of minus three over two and the fact that he did that certainly reduced the theorem to the case of a prime number bigger than or equal to five and not only that it kind of set the tone for further work on fair mahse last theorem for the next 300 and well say 250 years where people basically developed as needed the modern theory of factorization in algebraic number rings so fair Moz less theorem not only was a very old problem what I was finally settled in the 1990s but it had been a tremendously fruitful problem where people had developed new techniques in mathematics as they attempted to solve it and I hope you can see to the bottom of the slides it looks like you can this problem was open until the 1990s but I did just allude to the fact that it was settled in the 1990s it was settled in 1994 and the name that's associated with the resolution of Fermat's Last Theorem the single name is that of Andrew Wiles who's now a professor at Oxford he was then a professor at Princeton and he announced his proof at a workshop in June 1993 that was at the Isaac Newton Institute in Cambridge England I was present at that workshop and you probably can't see the small writing on the slide but this is a scan of the original program that I found in my office last week and what's remarkable is that on the first three days of this workshop Andrew gave the morning lecture the first lecture so he gave three lectures in a workshop where typically people would just give one lecture and then cede the stage to somebody else and he he announced the resolution of fair Maas Last Theorem on Wednesday June 23 I'm looking down at my slides he's lecture was between 10:00 and 11:00 in the morning and then unfortunately after his lecture I had to give the next lecture in the workshop which was not kind of it was a hard act to follow and one thing that happened during the period between the two lectures is that I spoke on the phone - Gina Cullotta who wrote up the story for the New York Times I was hiding under a desk because it was so noisy out in the corridor and the next morning in the New York Times there was a front-page article on fair mahse Last Theorem that mentioned the names of the people who had been associated with the proof and basically said everything is over we've now proved faramarz Last Theorem there was so by the way this these slides are punctuated by photos that I took of various people who figure in the story this is a photo that I took in Norway when we were at Annabelle symposium in June 2002 and that's Andrew Wiles in the Setting Sun now after the announcement that Last Theorem had been solved which was in June 1993 there were several of us who were asked to be referees of the manuscript that was not public but it was given to five or six different people and one of the readers was Nick Katz our friend from Princeton who sent lots of email messages and faxes to Andrew because Nick was in the US and Andrew was in Princeton it was in Europe and each time that Nick had a question he got an instant answer and then there was one question where there was no answer and Nick it actually found a gap in the manuscript and the proof that it took people a really long time to realize was a significant problem so for the immediate time after the announcement in July and August and September many of us were doing interviews and telling people what a great result this was but in fact there's been a problem that was kind of festering and it took a while before the mathematical community realized that the proof was not complete and the situation remained apparently static for pretty much a year and then in the early fall of 1994 richard taylor who had been Andrew Wiles s thesis student before this had started worked with Andrew and together they put together a new manuscript with some new techniques that did exactly what the original manuscript failed to do at one particular juncture and so by October 1994 it was clear that the gap had been filled and the proof had been completed and then there was a new wave of jubilation and this time it really stuck what I'm going to do now is in a couple of bullet points or Roman numerals just kind of say what the outline of the proof was with the results of Wiles and Taylor Wiles together and then I will give you lots more information about the elements of the proof so I'm trying to sketch out some skeletal proof and then fill in some of the details so the first thing is that you do it by contradiction so you suppose the fair mas Last Theorem is false that means there are integers a B and C none of them 0 where a to the B plus B to the P is equal to C to the P and you can and do assume that P's at least 5 and then what you'd like to do is derive a contradiction from that and there's a little technical point which is that you can shuffle around the order of a B and C and change some of the signs so that the statements that I'm about to make are actually literally true and I'm not stretching the truth all that much but the major construction which is the one displayed equation on the slide is that you start with a B and you don't even need C because C is defined implicitly in terms of a and B and you construct some exhilarate which is a new cubic equation and this equation is an elliptic curve and elliptic curve is just a plane curve defined by a single cubic equation if you like and this is called a FRA I looked a curve so in a minute I'll tell you about Gerhard Frey Frey was the person who suggested making this construction and said that he thought that it could lead to an ultimate proof of Fermat's Last Theorem and it turns out that you don't really need to kind of think about a B and C as integers exactly from that point of view what you do is you think about the solute to curve and you derive its properties and you discover the two of them are opposite from each other you get a contradiction so that's the way the proof went and the main point about the contradictions so what's special about this particular cubic equation well if you look at the right hand side you have a cubic polynomial and if you remember how to compute the discriminant of a cubic polynomial you take the product of the three differences among the roots and then you square the result and if you look at the roots the roots are 0 a to the P and minus B to the P and the difference between minus a to the P and B to the P up to sign a see to the P and so that up to sign the polynomial has a discriminant which is ABC all to the power to P and it's a perfect piece our so this is very unusual peas in your mind you know I said it's at least five but think that it's you know at least a hundred million or something because by the time this was all announced using various criteria starting with Karma's criterion from the 19th century people doing computer calculations had verified that fair Moz last theorem was true up to exponent you know into the minions and so the numbers involved are presumably very very big although in the end you prove they don't exist at all it's kind of a funny thing to talk about but nonetheless you have this amazing cubic polynomial whose discriminant is a very high power and a piece power in particular and then there was a kind of a race to prove that this elliptic curve is not what's called modular and what does it mean to be modular well I'm going to tell you that in a few minutes and the idea of how to prove it was not modular was set out and some manuscript actually a letter that was written down by jean-pierre Serra and this was a very public letter people made Xerox copies there was no internet yet really and everyone understood that you had to prove a certain property of the silifke curve in order to check that it wasn't modular and the work that I did on fair Moz Last Theorem was to show that this property was correct so I proved basically in 1986 although my manuscript didn't get published for several years that E is not modular so modular is a single buzzword meaning associated with modular forms and the contradiction is that in 1993 in the 1994 Wyles together with this taylor wiles new manuscript proved that the elliptic curve is modular so it's both non modular and modular and that's the contradiction and in fact when Wiles and Taylor Wiles work they didn't work with that particular elliptic curve they worked with a very wide class of elliptic curves called semi-stable elliptic curves and they proved that all semi-stable elliptic curves over the field of rational numbers are modular and what I said a few minutes ago about shuffling around a b and c and changing signs was exactly to make it true that the resulting elliptic curve was semi stable and that's the category that while started working with in the late 1980s and if you have kept track of all these Roman numerals you'll see that we have a contradiction well I really just told you that the thing is both modular and non modular and therefore the game is over there was no elliptic curve so before it's defining terms I'll show you Frey here's Gerhard Frey and someone you may recognize as the person standing in front of you this is from June 2016 three and a half years ago at a conference at the Fields Institute in Toronto and now we go back and talk about the basic objects in my skeletal outline first of all there's this idea of an elliptic curve so elliptic curves are kind of fundamental in lots of areas of mathematics and outside of mathematics including physics and I'm not sure what else they're used very widely in cryptography in different ways and they are given simply by a single equation in most cases Y squared equals a cubic in X and you saw an example of that where the cubic was this funny cubic with a very smooth as one says discriminant but it could be anything you know it could be y squared equals x cubed plus X plus 1 or something like that and the main thing that you want that the cubic should have a nonzero discriminant that's what makes the elliptic curve a non singular curve in the sense that you don't have simultaneous vanishing of partial derivatives at any point on the curve and an elliptic curve is given by such an equation and technically it's the curve defined by that equation together with a single point on the curve which can be taken to be the unique point at infinity so when you write y squared equals f of X you're defining a curve in the affine plane but in algebraic geometry it's much better in most cases to work with projective space instead of affine space and if you look at this type of equation in the projective plane there's a single point on it that is not on the affine plane that's called the point on infinity and that's the distinguished point that completes the definition of the elliptic curve the elliptic curves the curve defined by this equation plus this point and infinity and we already saw one example but here's a picture of another y squared equals x cubed minus x plus 1 and it's probably big enough that most of you can see that there are six points that are darkened in on the screen one of them is gold and the other five are blue that's because at Berkeley the colors of the golden bears are blue and gold and the gold point is one of two points with a special property that I'll describe in a minute but there are other points that aren't darkened for example zero one and zero minus one I'm not sure why didn't Arkin them and it's actually quite remarkable I wrote down this curve pretty much at random although I knew that the discriminant of the right hand side had a very small absolute value namely the discriminant is 23 and it turns out that this particular lip to curve has a remarkable number of integer points there are twelve of them and you can see on the screen for example three plus or minus five and then off the screen four points two of them with x-coordinate five and two of them with x-coordinate fifty-six mostly if you look at examples of elliptic curves you'll see a fair number of points with rational coordinates but the denominators are typically quite big and this is a nice example because the denominators are one in some points that you can draw and one of the things that you can do with an elliptic curve is you can take the equation for the elliptic curve which is a literal equation you know something equals something and you can regard it as a congruence modulo different prime numbers 2 3 5 7 5 whatever 11 and every time you do that you get a curve and for the most part the curve will have a non zero discriminant although in the example that I just gave if you look at the thing as a congruence mod 23 the discriminant all of a sudden will become 0 and then you don't get an elliptic curve mod 23 but you do get an elliptic curve mod 3 5 7 11 and all but finitely many Prime's and when you do get an elliptic curve you have what's called good reduction otherwise you have bad reduction and this particular example that I gave you with Y squared equals x cubed minus X plus 1 has good reduction except at two and twenty three twos problematic for it turns out and 23 is problematic for the reason that I just said namely the discriminant is minus 23 and then when you reduce this curve you can start doing finite things because you get a curve over a ring with just a finite number of elements the integers mod 3 or 5 or 7 but before I tell you about that which is the key to explaining what modular means I'll make an attempt of telling you in some sense what semi-stable means when you have bad reduction there are two kinds there's additive and multiplicative and a curve is semi stable if none of the reductions happens to be additive so every reduction is either good reduction or multiplicative reduction and in the example that I just gave the reduction at twenty three is multiplicative and the reduction at two is additive so the particular elliptic curve that I gave you as my running example is actually not a semi stable elliptic curve and there's a numerical there's a numerical quantity that goes along with this it's called the conductor of an elliptic curve I should have said in numerical invariant and the conductor is divisible by exactly the primes at which the reduction is bad and it's divisible to the first power when the reduction is multiplicative and to a higher power typically only the second power if the reduction is additive and so for example this elliptic curve has conductor 96 4 times 23 and another way to say that an elliptic curve is semi-stable is to say that it's conductor is square-free okay so that's some of the vocabulary and to describe modularity now I just have to tell you about this finite situation that you get when you take the original Congress the original equation and you regard it as a congress if you regard the thing as a congruence then you're in a situation where the ring is the ring of integers mod L which has only L elements 0 1 2 3 up to L minus 1 and if you look at the pairs x and y there are only L possibilities for X and L possibilities for y so there are l squared possibilities for the pair and then there's the point on infinity so this elliptic curve could have at most L squared plus 1 points and if you think about another curve over what we call the field with L elements that just means the integers mod L you could just have the projective line and the points of the projective line are 0 1 2 3 up to L minus 1 that's L points together with a point infinity so you get 2l plus 1 and when you take the difference between the L plus 1 and the number of points that this elliptic curve happens to have you get a number that we think of as an error term and that's because of a theorem of hasa that tells you that it has small absolute value basically the square root of L whereas we're comparing two numbers of higher-order numbers that are roughly L or L plus 1 and this error term is negative when the elliptic curve has lots of points and it's positive when the elliptic curve has relatively few points and then what you do you kind of keep book on these error terms you get an error term for every prime of good reduction so in the case of the example for 3 5 7 11 13 17 19 you only don't want to take 23 because that's bad but then you go to 29 and so on you have this infinite list of numbers and the modularity means that this infinite list appears somewhere in the appropriate book in the sense of you know proofs from the book it's the book of modular forms and modular form is is you can think of as a power series I say it's a Fourier series it's given in powers of a variable that's typically written q and q in the subject is an abbreviation for e to the 2 pi I tau or Z where the tower Z is a variable and the complex upper half-plane complex numbers with positive imaginary part and the modular form that is associated with this elliptic curve starts as indicated on the screen and for example if you look at the coefficient of Q to the 7 that's a negative coefficient that means the elliptic curve has lots of points it has more than 8 points and how many more than 8 well for more than 8 so it has 12 points modulo 7 and this is not too surprising in a certain sense because the elliptic curve had already quite a few points with integer coordinates of course these points with integer coordinates could collapse mod seven two of them could be the same as two others for example there was a 56 on some previous slide and modulo 756 is the same as zero but then you can get other points that weren't the reduction of integer points and the twelve is a kind of reasonable number for curve like that to have mod seven I'm just trying to tell you that the numerix are not super surprising and what happened in 1994 as I already said when I had those Roman numerals up on the screen is that Wiles Plus Taylor Wiles the package of the two articles that they wrote prove that all elliptic curves with square free conductor over the rational fields over the rational field were modular they were associated with a modular form and then there was a little kind of revisionary period between 1994 and the summer of 1995 where a number of different authors in different articles expanded the range of modularity and by July I think it was of 1999 this group had proved that all of the curves semi-stable or not over the rational field are associated to modular forms and the names associated with this work between 1994 in addition to richard taylor is Christoph bribe Brian Conrad and Fred diamonds and the work modulo three turned out to be the stumbling block in the last article in the series so they really proved modularity and now I want to go back to a lift of curves and tell you a little bit more about the structure or properties of elliptic curves so then I can make sense of parts of the method that intervene when you prove modularity so the biggest thing is that an elliptic curve is a group if you have two points on the elliptic curve you can make a third and the basic idea which is seen only in small types so to speak on the slide but a lot bigger on the next slide is that if you have a cubic curve and you have a line and you intersect the line on the curve that line will intersect in general three points of the curve and if two of the points have rational coordinates then the third will also have rational coordinates because you you know it's some business where the sum of three roots of a cubic is one of the coefficients of the cubic and we're dealing only with equations with rational coordinates it's not true in general that the some that if you start with two integer points you get a third integer point but it is true with rational points and the formulas for how to add two points involve denominators and you can find these formulas for example in Joe Silverman's book on the arithmetic of elliptic curves and so here what happens well let me spill the beans here the set of rational points of this elliptic curve happens to form an infinite cyclic group and this infinite cyclic group has two generators like all infinite cyclic groups and one of the generators is the point P with gold and the other generator will then be minus P which is below it and one thing to understand is that negation in this situation means reflection through the x-axis and the other thing is that when you take two points and you find the third point using this business with a straight line the third point is not the sum of the two points but it works out to be the negative of the sum of the two points and so for example if we were to take this point P and try to add it to itself that would mean that we kind of take the tangent line at P and then the third counting multiplicity point that the tangent line would strike on the elliptic curve would be the point minus 1 comma minus 1 so then 2 times P is not minus 1 - one but it's the reflection of that through the x-axis which is minus one comma one and so on if you go to to the left of the point P that's actually twice P and if you fool around with this construction which is called the chord and tangent construction we just had a tangent and when you when you connect two distinct points that's the chord you will see that the lowest point that smart for P really is the sum of two P in itself or you can add P plus P and then continue that have three additions and you get to the four P so the main point which is now written in words on the previous slide but I just said it is that summing to zero and zero is the point at infinity you if you want to have a group you have to say what the zero element of the group is and the best way to start is to take the one point that you know namely the point at infinity and then with that convention three points sum to the point and infinity if and only if they're collinear so this elliptic curve is a group and you can have a group of rational points you can have the group of real points you could have the group of complex points and you could have the group of points modulo seven eleven seventeen or twenty nine whatever you like so this is a completely algebraic construction that enables you to add on the curve and now here is something that is kind of technical which is that you take a positive integer N and our discussion n might be three or it might be the prime in the exponent of fair mahse Last Theorem and you look at the set of points on the curve with complex or algebra coordinates it doesn't matter such that n times the point is the origin and you say what is that well that's an abelian group you're solving some equations you can see fairly easily there are only a finite number of solutions because you're just solving some polynomials have known degree and it turns out that you can visualize this group very well if you remember virus theory from the back of Al Force's book where an elliptic curve can be written as a complex as the complex plane modulo a lattice and then this group that I'm describing is just 1 over N times the lattice a slightly dilated lattice modulo the lattice and the lattice is just a free abelian group of Rank 2 and what you get when you make this construction is of something of Rank 2 over the set of integers modulo n and so the thing is kind of very very concrete and if you want to understand what these points are and a specific example you can do that you can use software like sage which was written founded by William Stein and I wanted to mention at some point in this talk because a lot of a lot of the concrete work that people do in the subject really comes from the tools that he introduced so the first bullet point is just that it doesn't matter whether you use cube R or C and that's because you're just solving algebraic equations with rational coefficients and the second thing is that you get something that's free of rank two over the cyclic group with n elements the group of integers mod n so that you can see by virus stress theory and the thing that's kind of really fundamental in the subject is that because you're solving equations with rational coefficients what happens is whatever you get as the solutions are per muted amongst themselves by the Galois group of the rational field and so the Galois group of the rational field kind of commutes the set of n squared elements but it permutes it in such a way that you respect the addition on this group G of n because the addition involves writing down equations whose coefficients are rational numbers and they won't be disturbed by the Galois group and so instead of getting you know some huge Extension whose Galois group might be the you know the symmetric group on N squared elements instead you get a gala extension whose Galois group is just a group of matrices because what you're doing is you're taking automorphisms of something free over the cyclic group of order n and so you get two-by-two invertible matrices modulo n so what you get is called a representation because the representation of a group is a home morphism from the group into a group of matrices this is called a ghoul representation because you have a Galois group and in particular you can take the example of a Frei curve that was introduced by imagining that there is some solution to pheromones equation and you can take a of n where n is equal to P and then you get a Galois representation mod P which is kind of mapping into two by two matrices invertible matrices modulo p and you can ask whether or not in some basis this representation is upper triangular then the representation would be called reducible and it turns out there's a really hard theorem of Barry Mazur from 1977 that in fact this representation is irreducible it cannot be made upper triangular and this has nothing to do by the way with the particular example it's true anytime you have an elliptic curve over the rational numbers that is given by y squared equals a polynomial with rational roots like this one has factored cubic Mazur proved that you cannot get some subgroup stable by Galois of order P and this is true for any P bigger than or equal to five and it's certainly you know kind of even more true if P is gigantic like three million as you imagine the situation to be when you're talking about fair MA and I don't want to kind of get stuck on technicalities but the point is that this elliptic curve if you have the theorem of Wiles and Mazur while at Taylor wiles is actually modular it's associated with a modular form and you can transpose that information to the situation where you just think about the Galois representation the Galois representation is also going to be associated to a modular form and the association is very simple it's that when you look at distinguished elements in the Galois group associated to random prime numbers Frobenius elements and you look at the matrices that you get the traces of those matrices should be given by the coefficients of the modular form of course looked at only modulo P because this is a representation modulo P and the whole point is that this elliptic curve has a conductor which is probably immense because the numbers a B and C are immense and therefore you might imagine that the representation well it has some conductor that you can define you might imagine that this conductor is also gigantic but what happens is that because the polynomial defining the FRA curve has this discriminant that's a piece power the conductor of this particular representation is minuscule it's just the number two and there is a conjecture that Sarah wrote down in 1987 to clarify things although it really grew out of correspondence that he had with John Tate long before where he said you know if you have a representation that has some minimal looking properties as this one does it should be associated with a modular form and the modular form that is associated with a representation with conductor two should be a modular form of level two and the whole point to Ser is that there are no modular forms of level two and the whole contradiction can somehow be channeled and to looking at this representation and so Sarah's challenge which he wrote down before he really published this article but kind of roughly at the same time was somehow trying to take and lift a curve with huge conductor and therefore associating to it a modular form with very large level I'm not sure that I said what the level was I probably removed the relevant slide but it tells you what subgroup of SL to Z you have to look at large level means a very small subgroup what Sarah said is you should take this thing and there should be some method of somehow parlaying this modular form a very high conductor into a modular form of conductor 2 and that's what I did using my interests in Galois theory and modular forms I thought that I would somehow take a tiny break from the technicalities by telling you about two of my interests that I found Facebook knew about me it says that modular form is a local business for some reason and what I proved my contribution to the whole thing in the mid-1980s was to show that if you knew that row came from a modular form of very high level and it had this very small conductor then you could somehow whittle away at the conductor incrementally and get down to a conductor to and get the contradiction that way and what I did this idea of moving from high level to low level of course it's called level lowering and sometimes it's called level adjustment and now just to kind of summarize the situation 25 years ago roughly is that we knew that a counter example of more than 25 years ago in 1986 we knew that a counter example to fair ma would give you a Frey curve that could not be associated to a modular form and if we somehow knew that this modulus Frey curve were modular then we'd get a contradiction so that was what we had many years ago in 1986 and then Taylor Wiles and Wiles came along and they proved that semi-stable elliptic curves are modular and that completed the contradiction we showed that the solut occur could not have been written down with the properties that we ascribe to it and now another tiny break from the technicalities is that when the proof was written up there was a conference at Boston University organized by Gary Cornell Joe Silverman and Glenn Stevens two of the organizers are here at this conference and all of the people associated with the proof explain their parts of their contributions at a conference that was quite long in the summer of 1995 at Boston University and now what I could try to do is to tell you from the point of view of 1994 how Taylor Wiles and Wiles proved that the elliptic curve was modular but what I'd like to do is kind of go through these slides very very quickly and and just give you some of the highlights so one of the highlights is that you have to have a modular form coming from somewhere in their way of doing it and what they did was they said even though we're interested in the mod P representation attached to this elliptic curve we're going to start with a mod 3 representation which we know a lot about and we know a lot about the mod 3 representation because the image of it is inside of a group that's so small that the image has to be solvable and using the work that had been published not too long before by Langlands in his very long book base change for GL to using that plus an extra contribution of Gerry tunnel you could actually prove that the mod 3 representation was modular and here's a photo of Gerry tunnel that I took at the Abell symposium in Minnesota in honor of John Tate our common advisor and what Taylor and Wiles do is they say knowing initially that the mod 3 representation is modular we're going to go up powers of three and that the mod 9 and mod 27 in mod 89-81 representations are all modular and they do this by a new technique that they introduced using the deformation theory of Barry Mazur and they actually get that the entire package of mod 3 to the N representations are all modular and technically you can say you can use the language of andrey vey and John Tate you can say that they prove that the 3 attic representation was modular because all of the 3 power representations are modular and here's another photo this is right after Wiles as announcement don't want to dwell too much on the photos because there isn't a lot of time left but the whole point is that what they proved is called a modularity lifting theorem and that starts with a modularity of a mod 3 representation or it could be a model representation but they take L equal 3 because that's what they had and they deduce the modularity of an entire package and technically what they do is they use some techniques of commutative algebra plus you know many many more techniques that they introduced for this purpose to prove the isomorphism of two rings so one of them is a universal deformation ring that's called script R and one of them is just the heck of ring associated in the situation which is a ring associated to those representations that are associated with modular forms and they proved an R equal T theorem and so the proof that they gave as presented in 1995 involved a number of components so one of the components was my level lowering from 1986 another was the theorem of Langlands and Gerry tunnel that started things with the modular forum and the new contribution was this R equal T theorem the modularity lifting to get that the elliptic curve was modular and there's another little image at the bottom of the slide it the handwriting of Andrew Wiles who wrote to me in the 1980s I trust that you were busy preparing or demonstration of pheromones Last Theorem that's on a postcard that he sent me and I did not make this up and I don't have time to talk to you about the deformations but I do want to say that they proved in the end that some natural ring homomorphism which a priori is surjective is fact as an isomorphism and they proved our equal T in this way and what they did is they launched an entire subject so mark Eason who many of you know professor Harvard was one of the first to prove many more are equal T theorems with varying hypotheses in different contexts and in preparation for this lecture I consulted a number of people who I thought were more expert than I am one is Frank Caligari from Chicago who said that in looking only at the very top tier journals and he has high standards he could find at least 25 articles where the main result is an R equal T theorem and he has a blog called persiflage and so one thing that happened beyond 1995 was this kind of whole fluidity of proving R equal T and another which is maybe not as widely appreciated is an article that Richard Taylor wrote around 2000 he spoke at the ICM in 2002 I believe and the article is remarks on a conjecture of Fulton and Mazur and what he did was he introduced an entirely new - called potential modularity where you start with something that you want to prove to be associated with modular forums and you maybe can't do it but if you make a small extension from the ground field of rational numbers to a finite extension that you can make totally real then you can get the thing to be modular and it will be associated with Hilbert modular forms and this tool plus all these are equal T theorems enabled check our car a and jean pierre van Thanh beer j2 proved in 2008 the whole conjecture that sara had proved in that sarah had published as a conjecture in 1987 and as you must have inferred from what I said if you knew Sara's conjecture then this contradiction would just come immediately you wouldn't need my level lower and so in doing so you know even just formally they simplified the proof although they used a lot of the tools that were then raining including I believe my techniques certainly the techniques of Taylor and certainly all these are equal T theorems so I'm out of time I just want to give you kind of a executive summary of the rest which is that now all of the people who really work in the subject are so familiar with the tools that if you try to ask them how to prove modularity how to prove pheromones Last Theorem as Michael Harris said in an article in quantum magazine 10 different young people will tell you 10 different proofs they're already kind of an enough pieces on the chessboard to checkmate the King in in lots of different ways so the proof of Fermat's Last Theorem has changed in the sense that it's gotten much much richer and there are lots of ways to do it and whether or not you want to say this is a new proof or a different proof really depends on your perspective but I think one of the objectives in the subject was to reduce the reliance on my techniques which a lot of people didn't really study when they were studying the r equal T theorems and also the reliance on the Langlands results that really were analysis more than algebra and so it's kind of up in the air whether you consider that there's a new proof or a different proof or a modified proof the final thing that I want to say with just two photographs is that in case you were wondering I really did enjoy being president of the American math Society for two years just as many of my colleagues that Jill seems to be enjoying her presidents and her successor Ruth Charney I'm sure will enjoy being president and I just have to photos that I want to conclude with one was a photo that was taken at the 2012 JMM at the Springer booth where the members of the nominating committee asked me whether I would run for president or I asked me whether I would run for counsel and I agreed to run for counsel so I've been to many council meetings my last council meeting was Tuesday of this week it has been an honor and a pleasure to be the president of the American math Society thank you thank you can that was a beautiful lecture I'm afraid there's no time for questions as we get ready for the prize session but please join me in thanking Kent again [Applause]

Channel: Joint Mathematics Meetings

Views: 11,231

Rating: 4.9126639 out of 5

Keywords: mathematics, Kenneth A. Ribet, Fermat's Last Theorem, Joint Mathematics Meetings, American Mathematical Society, JMM, JMM2020, JMM20

Id: mq9BS6S2E2k

Length: 52min 21sec (3141 seconds)

Published: Mon Feb 03 2020