 Good, so We'll start so This is one of the actually the academic highlights of the whole year for ICTP and So let me just give you an introduction first of all is this this is the What's called the salam distinguished lectures that we have started in 2012 and so maybe just to remind you the first one was Professor Neymar Kani Hamid who gave a talk called past present and future of fundamental physics The second set of lecture was given by Professor B Bialik from Princeton entitled the physics of life searching for theories and the first set of lectures was given by super sax death from Harvard also and Entitled is theory of quantum matter from quantum fields to strings and And usually we organize them to coincide with the birthday of Abdul Salam, which was 29th of January, so this is a way to honor the founder of the institution The set of lectures this year has been organized It's candy Founded by the long-term Collaborator of ICTP, which is the Kuwait Foundation for the advancement of science and we have a great pleasure to have one of the Total mathematicians in the world. I will say to agree to give those lectures So it's a don's a gear and let me say some words about done As one of the directors of the Max Plan Institute of mathematics in Bonn in Germany And it's a now we are very Honor and pleased to know that I mean he's distinguished as a staff associate of at ICTP So he likes very much ICTP and he's spending a good percentage of his time with us, which is a wonderful news for all of us Don is His main research interests are number theory modular forms arithmetic geometry quantum invariance in three-dimension topology One of his major results Is a joint work with Benedict Gross the gross like year formula as the result is known It relates the the derivative of the L series of an elliptic curve at s equals to 1 to the height of certain points on the curve constructed by a figure This theorem implies cases of the famous bridge and swinard on dire conjecture and it is an ingredient to Doreen Goldfield's solution Of the equally famous class number problem of Gauss Don's again. He's won the cold price in number theory in 1987 and the von Stout price in 2001 Over the last couple of years we have been seeing don here often and he has been lecturing at ICTP and Collaborating with our staff. He has been a great friend of ICTP in the last Years and I also have learned to know many of those particular properties of qualities of don who are just outstanding I have to say is Don is one of these persons who knows about everything who knows Very cultured person than that. So it's beyond all what I said about mathematics Constructions, he knows a lot of mathematics, but a lot of many other things He speaks so many languages. I don't know seven or so and And then he's a extremely good pianist I managed to see him once we were going for dinner to take the hour piano in Adriatico building. I was playing a beautiful Beethoven sonata. That's a perfect thing So so you can see that he's is much more of a young what being a famous mathematician. So so we have today Don he's going to give this Five sets lectures from today to Friday And in general as I say the topic is some modular forms. I forgot exactly the titles The magic of modular forms the magic of modular forms. Okay, so let's welcome John So, thank you very much Fernando anyone who has heard me play piano will know how to discount also the rest of what you were just told So it was it sounded good, but the reality is a little less so Who other remarks about the introduction he said some of the things I wanted to say about coming here but I'll say them anyway and also You gave me the opportunity to give the first lectures I'll ever have given which will be less technical than the introduction Because I won't have any derivatives of L series related to the birds to an entire contract So this will be meant for people who aren't yet in the field and I hope will make you interested So I wanted to say that I'm really delighted and honored to have been asked by the ICTP to give This series of lectures named after the wonderful person who founded it as you just heard from Professor Kevin I've been coming here for several years. I think the first occasion I was a co-organizer of conference with Lotta Götze and Catherine Bregman. Maybe it was not the first and During those years. I've really come to love this place For its extraordinary mixture of a very very high scientific level But never forgetting its mission, which is to make sure that science is for everybody and not for some kind of a lucky Elite and all of this without any kind of self importance It's really a very wonderful place been coming here often It's you just heard since a few months ago I've been affiliated with the Institute as a staff associate and I hope to be connected with it for many years to come So that's a general comment another preliminary comment before I start when I was a young man, which is in the previous century I Was I have been told that I had it sometimes spoke too fast or wrote badly on the board Of course now I'm old and experienced and that never happens But if I should slip by accident, please You know scream and say it's too fast or we can't read it So these lectures are meant to be informal you can interrupt you can certainly ask questions I certainly hope you will at any point and After the lecture I've been told it's a tradition to have a separate after the normal Question session I think there's a break and then we come back and if students who maybe don't dare speak up When they're older people there have questions They're free then and I hope many will will come I really look forward to that if it happens Then I want to start it's a five Course lecture series and today's the first it's called classical modular forms and Some of their applications in arithmetic. So I'll try to explain that first maybe two crucial preliminary points I'll be using the board only so as I've said before if you can't hear or see come closer So modular forms are a function of the variable and some people most people call it actually everybody calls it tau Except that a lot of people don't they call it Z or Zed and I'm unfortunately in both camps and half of my papers Do one and half the other so I'll use tau But if I slip and say Zed you get it another chance to yell and ask me to change it back And sometimes I might do it on purpose to see if anybody's paying any kind of attention So if that doesn't happen, I'll be very sad and start crying Okay, the second crucial point as I said this is meant I hope it'll be fun for people who do know about modular forms But it's mostly meant as a first introduction for those who haven't or haven't seen the many ways You can use them and I hope that getting to know modular forms will change your life So to assist in that process Long ago not just for this lecture series I wrote a book or a third of a book called the one two three of modular forms So the first third is by me and that's the one it's multiple forms of one variable It's a classical modular forms as today part two is by young gruny. It's about Hilbert modular forms of two variables much more advanced and part three by van der Herr is about Siegel modular forms which have three complex variables but The whole book is very nice, but the first chapter is meant to be very elementary And I hope you'll everybody will buy at least one copy Keep it under your pillow and give it to all of your friends when you visit them for dinner. It's much better than wine Okay, so that's just preliminary comments So just a few words first what modular forms are they'll be hearing a lot about them in the abstract I wrote and I have to read it Multidore forms are functions having an infinite group of symmetries and many beautiful properties So I hope to convince you that the second assertion is an understatement. It's really extraordinarily beautiful properties and many many The famous number theorist and algebraist Martin Eichler who my god to know very well because we together developed a theory of so-called Jacobi forms and wrote a book about it and it's there's an apocryphal quote I don't know if he really said it by him that there are five basic operations in mathematics addition subtraction multiplication division and modular forms, so I don't know if he really said it, but they're certainly very beautiful and they have many many Appearances in the rest of mathematics and mathematical physics Not only in the past but in recent years actually more and more it's increasing and I've chosen five of those Themes for the five lectures of this series and I chose five that are close to my heart because I've worked on all of them But of course there would would have been many more So Before I write down any technical definition I should say that the definition of a modular form there are really two ways of looking at them and It's exactly because the two ways are not quite obviously the same that in fact It's not obvious at all that you're talking about the same thing until you start doing it And it's exactly because of that that the theory is so fertile So a modular form is already said will be a function of one complex variable called tau unless I forget that I called it z and It will have as I said in the abstract an infinite group of symmetries That means that there's an infinite group which sends tau to other values of tau and the function f is either Invariant under those transformations or transforms in a very very simple way. So one point of view is f of tau with an infinite group of symmetries In other words as I said there infinitely many transformations, but it's fine at degenerate It's the only need to find that number to understand it Where you send tau to some tau prime at f of tau is either equal to f of tau prime or differs from it by a simple Scaling factor So that's from one point of view and that's why they're very deep objects because there's this hidden infinite group of actually Not a billion infinite group of symmetries But from the other point of view f will be a power series just a power series But not in the same variable in a new variable And that's the one thing you have to remember q will always be I'll mention it many times But it will always be eat the 2 pi I tau I'll come back to it when I need it And so you can also think of this function as a power series in a completely different variable and As a power series it's the reason it's interesting is because am is often and you'll see many many examples already in today's lecture An interesting or very interesting Arithmetical function so these so typically functions that number theorists want to study and you'll see many examples appear sometimes fairly obviously and sometimes very very mysteriously as the Coefficients of the expansion of this other variable q of a multireform So here you see the infinite symmetries here You don't see any symmetries, but you see interesting coefficients and now the word magic in my title This is multireforms Which I'll usually abbreviate for laziness and f's but the magic part is That from the second point of view It's you get easy proofs automatic proofs a computer programmable proofs of identities and Again, you'll see many many examples among these coefficients So again these coefficients a n will typically be something very interesting But I might friends have a second multireform g equals some b n q to b n and have guessed That a n and b n are the same because the first 20 values agree that a n comes from you know Not theory or from somewhere and b n comes from some other part of mathematics or physics and I see no way to prove it But if I know that both of them are multireforms There's a principle that tells you that it's automatic to check that if a few terms are equal They're all equal so it's easy. I could even say trivial or free proofs There's no work involved at all once you know that things are multireforms and that you only get from the second point of view So that's somehow the mechanism why multireforms are Set up to both be deep because they have this infinite non-abillion symmetry group and to be useful because they have Expansions whose coefficients are numbers that we care about and at the same time that it's very very easy to study from the point of view of identities So I'm going to give you many many examples and then you'll see how these principles work It'll all come this is a non-technical introduction, but it's either one So sometimes n is from zero to infinity Sun as it starts at a negative integer. It's both I didn't want to be that precise. This is just saying the two points of view No later. This is with two introductory sentences. This is I'm going to be lecturing for 10 hours There'll be plenty. This is just to say that there are two points of view And it has not to do with proof there. I'll come to it later, please I'll come to it later when I get to that part This was just a brief survey of like an introductory to remarks what I'm going to do So lectures now start and so there are these two points of view one and two and so I start with point of view one By the way, can everybody see the board? I hope that no part is cut off But if for instance there, I'll try to use the central part, but when I start having to erase I may have to move a little So I start with the variable tau well I've been telling you that it's a complex variable and indeed it is But it's only in the that's meant to be a German Like that H is the upper half plane so in the complex plane if you have zero here and I somewhere here Zero wouldn't be in it. Then it's all points which are which have strictly imaginary part So these are the tau that have strictly imaginary part And it's very well known that the the group of I don't have to give it a name SL to are of two by two matrices So these are two by two matrices with the term that one ad minus bc equals one with real coefficients This thing acts on the upper half plane So I'll have g tau is a tau plus b over c tau plus d And then that's a group action if you do it twice with g1 and g2 It's the same as you've did it once with g1 times g2 So you get a group action and the reason this is interesting Well, what we're interested in is a subgroup and I'm going to fix for simplicity. I'll fix the notation That in SL to are I take SL to z which is the same thing But now it's matrices which I'll also often call gamma to emphasize that there in the discrete group ABC They are now integers so this group of course also acts and I'm interested in functions essentially in functions which are invariant under this action So that means that I'm interested in dividing by this group actions I'll write h multiple gamut should maybe be on the left for the quotient So these are the equivalence classes of gamma and I'm sure many of you have seen but I won't use it a picture of this famous fundamental domain if you take here the point i and Here is 6th root of unity The other corner is a cube root of unity minus 1 plus i squared of 3 over 2 and these vertical lines Then every point in the upper half plane is equivalent under gamma to a unique point of this demand Except the points on the boundary these two points get identified with each other and here opposite points get identified So the picture of H mod of gamma would look like this It's got one cusp because often infinity gets thinner and thinner in the so-called hyperbolic metric And it is two little small singularities with the total angle. It's not 360 degrees But 180 or 120 which are those two bad points, but you don't need that picture for anything But what is important is that the quotient H mod of gamma is what some has called m1 the Multi-lyse space and one One would be more correct the multi-lyse space of elliptic curves And so that's going to lead to the definition of modular form since I want to explain it So we have two levels of functions a modular function this already a definition now a motor function is Simply a function a holomorphic function for us holomorphic function f Which goes from H to C, but it's invariant under gamma so it actually goes from H motor gamma to C So f of atel plus B over C tel plus D equals f of tau now Because of this identification you can think of this as f such an f is the same as a function on The space of all elliptic curves So I want to remind you very briefly what an elliptic curve is I'm working over the complex numbers e is an elliptic curve That means it's a it's a curve, but overseas, so it's actually a surface to Riemann surface Plus it has a group action And Then it's very easy to know that you know all surfaces have a genus the number of handles And if the genus is one it's a torus and it's very easy to check that anything that has a group action Has to be a torus at all. It looks like a torus And there's a marked point because it's a group So there's an origin and then you can show very easily from that that the e is actually the quotient of C motor L so L is Is some lattice for which I can choose a basis a omega 1 and omega 2 So here's the complex plane and here's on the go on and here's on the go to Go on go to go on plus on the go to and then you have the whole lattice of you know all linear combinations And when I divide by that well I just identify this side by this side and this side by this side like in kind of pack manner of the modern Equivalence where the little man when he waters off the right edge of the screen comes into the left edge of the screen So you get C motor L Okay, so therefore I can also say If L is the set of all lattices So a lattice is just such a discrete subgroup in C Then my F I can identify with a function from L to C But there's a slight subtlety if you multiply the whole lattice by a constant a complex constant So you scale it and maybe rotate it then If you take C motor L P times L or just L Then those are isomorphic curves. That's very easy to see you simply send Z to C times E and that gives an isomorphism So if I want isomorphism classes, which is what I do This is the multi space isomorphism classes Then I should have the property that F of C times the lattice is equal to F of L So now it's actually a function on L multiple C star So that's what a multiter function is But that turns out to be too restrictive and the title of my lecture series is not the magic of both of the functions But a multiter forms. That's a slightly bigger class, which I want to explain here So from this point if you have lattices a multiter form multiter forms Is more general a multiter form of weight so multiter form will ever wait Which typically will be an integer And sometimes a half integer you'll see examples of both already today So there's a certain number called the weight and if the weight is zero Then that's exactly the case. I just told you that's what's called a multiter function Okay, so if I think that from this point if you have a function on lattices Then I again look at function lattices, but now they don't satisfy that they're invariant When you rescale the lattice, but they're homogenous of some degree which I call minus K The minus sign is more convenient because the examples then all get positive weight. Okay So this is my basic definition, but that's a little awkward to work with and so the first trivial remark to make is the following if I have any lattice at all and I call it z omega 1 plus z omega 2 remember. I'm allowed to rescale it. So if I just multiply by Omega 2 inverse Then it will have a special form the new basis the second element will be 1 and the first one I'll call tau so tau is equal to omega 1 divided by omega 2 and If I always choose omega 1 to be to the left of omega 2 in the order I mean one of them is to the left of the other then tau will be in the upper half plane So in this way, that's why I only have to look at tau, but now watch what happens when you change omega 1 Omega 2 to omega 1 prime Omega 2 prime where I make a linear change of variables a omega 1 plus b omega 2 c omega 1 plus d omega 2 well Here a b cd are again integers With the term and one and so that means I've just Changed my basis over z. So it's the same lattice. So f has to be the same function F of it's the same lattice. I mean, I don't even have to write it There's it's an empty statement f of L this f of L, but now if I divide through Then I see that the tau will now go to tau prime, which is omega 2 prime over Omega 1 prime over omega 2 prime and so it's a omega 1 plus b omega 2 Divided by c omega 1 plus d omega 2 and I take out a tau everywhere I see that I get exactly the form that I had before That's why I told you that h modulo gamma with this action of gamma classifies elliptic curves But now if I take this equation then I see that I can translate between f and f which is Capital F was my function of lattices little f is a function on the upper half plane and the translation is f of once I have f Then I define little f just by considering Capital F on the special lattice generated by tau and 1 But conversely if I have little f then I can get f on any lattice at all with any basis By using the scaling property to say it's omega 2 to the minus k times little f of omega 1 over omega 2 So in this way I have a passage and I can translate this very confusing thing of functional lattices To just a function of a complex variable, but when I do that then because of this Scaling factor omega 2 to the minus k. It's a one-line exercise Which you should do maybe not while I'm talking but tonight when you're falling asleep if you've never seen it But then this definition transforms into the actual definition That I want to use a function a holomorphic function f From h to c is a modular form of wave k Say an integer for the moment It'll have to be an integer and even an even integer if my group is all of sl2c if it transforms Not the way it did for a modular function, which was simply invariant But before we adjust f of tau, but now there will be this famous Transformation factor called the automorphi factor c tau plus d to the k So this is just a translation of this So now if at least to find it it's still a bit abstract, but starting from now there will be Examples and you'll see how it works and actually you can forget all of this because as I've already said the second point of view With explicit expansions is the one we'll use and that's completely elementary But if I didn't tell this point of view you wouldn't have any idea why the other thing works so I didn't want to skip it but It'll be mostly the other point of view that I use But now I can already give examples still following this point of view So examples But these are called the Eisenstein series. So it's defined on lattices Let's see gk of a lattice is very simple. I simply take all elements of the lattice Which are not the zero element and I so I sum up 1 over omega to the k So here k should be bigger than 2 In order for absolute convergence, you could do something for k equals 2, but I'll skip it It's more technically clip So this is a good function if k is bigger than 2 you check easily that it's absolutely convergent and therefore the sum makes sense It doesn't matter in which order I add up these numbers on to the minus k I'll get the same thing and of course this key property is obvious if I scale the whole lattice by 3 I multiply every omega by 3 and so I multiply this thing by 3 to the minus k So that's completely okay, and then this corresponds to a function that I'll just write with gk of tau which is there for the sum Essentially 1 over m tau plus n to the k There may be a half depending how you know if I do it this way inside that again m and n different from 0 Just put a prime meaning you don't you don't take 0 to the minus k you take everything else So this is a function and then because of what I told you since this scales the right way This automatically scales the right way and so do I have to write it? Maybe I do multiple forms of weight k notation Mk is the vector space of multiple forms of weight k It's finite dimensional and and gk is an element of it However, it's clear that if k is odd then omega and minus omega cancel. So we actually only get this for For the 4 6 8 and so on and I want to show you that g4 and g6 are somehow More important than the others. In fact, there's a theorem Very easy theorem, but very important that the ring Which is usually called m star which is just the sum of all of these mk's you allow all multiple forms of all weights I didn't say something. I lied a little bit, but you'll I'll correct it in a minute this ring is Simply freely generated by g4 and g6 So every multiform is it happens is a polynomial in these two Special ones and so although I've only given you two examples You actually at the moment have all examples at least for gamma is SL2z. I Want to mention now already that you also look at multiple forms and other groups also higher dimensional But in particular in one variable tau The gamma might be replaced by subgroup of finite index and many of my exams look like that and then there are many more Multiforms, I don't believe that if you've seen g4 and g6, you know everything, but that's the sort of crucial thing No, yes. Oh, no, I don't need it because here in particular m0 is Certainly what I want to ring with identity So I don't want to start it m0 1 is an excellent multiple form But m2 is simply the zero space and so are all the old ones. There is no g2 But there's still a space. It's just the zero vector space So g2 is not a multiform. It's what's called quasi multir and I'll talk about it in a later lecture But it's not actually in there So m2 is zero and the first m4 is g4 I'll give you more examples of this in one minute But I want to explain still the last thing the connection with elliptic curves Why g4 and g6 play a special role and this is something that I Think even among the diploma students most people will have seen and certainly the more advanced So which is the virus trust equation? Yeah, the modular form is defined by a bcd, right? Yeah, I didn't understand a modular form is defined by a bcd No Not at all the multiform is written a multiform is a function. It's there's no a bcd and the definition It satisfies this for every for all a bc and d which are integers With a d minus bc is one, but but they vary that's not part of the definition There's no a bcd in the definition. It's a single function. Well k is a property of the multiform So you have multiple forms of weight four of weight six, but it can also not vary There are many multiple forms of weight a hundred. It's not a unique property It's like the height of a person you could have two people are six feet high. This is just called You know, it happens to have the weight which is an invariant is this number k So there are many many multiple forms of a given weight, but a multiform has a weight just like a human being has a height Well, they're different. It's like two people with the same height. I mean, they're just different They're different functions. They simply you know, I mean one of them satisfies f1 Let me take an example. Here's an element of SL2z because those are integers that determine this one So I have one function for instance of weight 12 and I'll give examples in a moment Which satisfies this and I have a second function with the same equation. Well, they're completely different functions Just a symmetry. It's like you have many even Let's take the word even in in high school mathematics. You have f of x is f of minus x So it's always called even that there are many such functions like x squared and cosine x I mean, it's just a property. It doesn't it doesn't fix the function I mean, I thought in fact I've given this formula here So you see that there are many many multiple forms of weight k, but anyway, you'll be seeing examples I think everything will be clear I hope very soon as I go on and you see more examples and then you see how it works Because of course I'm going through the definitions quickly. I don't know. There is no form of weight 2 it's 0 No, I'm coming to the virus trust function out. It's not it has something to do I'm going to explain now not all the virus trust function to the function of a different variable It's not even a function of the same variable. It's completely different. I'll come to it now I'm about to explain it. So that's a very good question. The question was the connection with the virus trust function So I'll say it right away. There's a theorem Due I mean, there's many parts so to speak. It was many years of research the virus trust theorem and it says let e be an elliptic curve Over the complex numbers so remember I already told you that means that it's a curve Which means it's actually a Riemann surface and it has a group structure Which means automatically that top largely it has to be a tourist nor their genus except genus one is permitted for a group And it has an origin and it in fact looks like see we already said I and that's easy that it is this form But virus trust proved Two wonderful things so you proved first of all E has a model the model, you know, this is only up to isomorphism It's just an abstract Riemann surface of a complex variety So you might choose different equations for it and each such equation is called the model and this is the famous virus trust form Here a and b are constants in C and there's a restriction which is you can forget which says that the cubic on the right does not have repeated roots But if you take generic a and b take two coms like three and eleven so that would be typically An elliptic curve and the claim is that you can always so here x and y vary And if you think of this here if I draw the real picture, but of course, it's really complex This cubic might have three roots and then the the curve will look like this So I just take so this would be e in this case it would have two real components, but of the complex numbers So it's a theorem of virus trust that you can always find a model Which is that a cubic in x and you can normalize the constant the quadratic term is zero equals the square of another variable So in other words, this is the graph if I do it. This is simply the graph In in the usual sense of x y coordinates of the function y is the square root of x cubed plus a x plus b So that's a theorem, but the second part of the theorem is if E is what I already said I've already given you an example which the lattice which I could normalize One of the generators to be one so it's z tau plus z then Up to stupid constants I'll just put a star as people do in mathematics for boring constants This constant is the function chief it now it becomes a function of tell because Calc can vary these two constants Which are the crucial constants to define the elliptic curve are Exactly the two Eisenstein series. I've introduced. There's some stupid factor in front like you know hundred and forty pi to the six There's something I don't care So in other words these g4 and g6 which are molded forms are very special because they define this and now to ask the Question about the virus trust function. I don't want to go into this at all in detail But if you have a function then virus trust to find a function if you have an elliptic curve See over L He defined a function which depends of course on that lattice P of z and it's Approximately the sum over omega in the lattice of One over z plus omega squared. It doesn't quite convert. There's a little trick to make it converge I don't want to talk about it because I'll never come back to this and now what he shows is that the actual parameterization this but this function is Invariant under the lattice Obviously from this because I'm just taking all shifts and therefore it's actually a function from C multiple L And so now I can go into C times C. So X and Y Capital X and capital Y in the following way I said Z equal to the virus trust P function Sorry X and I said why equal to half the derivative of the virus trust P function And then if you do that virus trust show that those two funds identically satisfied this equation And that's how he got part two from part one again If you haven't have seen elliptic curves This may be useful if you haven't it's for the next time that you see them it will ring a bell I'll never talk about it again And I would not like to answer questions about this because it's it's not my subject It's just a side remark for those who know the virus trust function. This is the connection that in the famous a Equation P prime squared is PQ plus you know for PQ plus G2 P plus G3 This is the G2 and G3. So if you know elliptic curves, this is meant to be enlightening But I don't want to talk about elliptic curves and that will never come back So now I want to get to the main point which is point of view to a Point of view to as I told you is that We have a Fourier expansion or a Q expansion So I now come to that and now everything should become much more explicit and much more elementary the hard part is gone And I'll never use that again from now on it will be just numbers So the second point of view I had a one in in a circle Which was the first point of view with the infinite group of symmetries Now the second point of view as already mentioned or maybe I didn't mention the group SL2 Z certainly contains the matrix 1 1 0 1 and so if you look at this famous transformation a Tell plus B over C. Tell plus D in this case It's just that and the autobiography factor the the C. Tell plus D is just one so I don't need it So what I get is that this is true not only for modular functions, but for all Modular forms capital M F is always modular form for me not not your function So every modular form is translation invariant that means that if I look at it In the upper half plan forget the fundamental demand I drew before but just make strips from 0 1 2 then whatever the function does here It simply repeats With period 1 just like cosine or sine it's a periodic function But then just as we know very well from Euler that the fact that cosine of 2 pi x Has period 1 Means that this can be written as a combination of e to the 2 pi i x and e to the minus 2 pi i x So in the same way any function any holomorphic function of tell which is translation invariant Can be expanded and of course the the converse is treatedly true by Euler's theorem e to the 2 pi i equals 1 Oilers formula this means that f is a function of q Where I already wrote q, but I'll write it again q is e to the 2 pi I tell But because of Euler's formula e to the 2 pi equals 1 if I change tell by 1 q doesn't change So if I have a function of q then of course it's invariant, but conversely it's very well known that if I have a function Of tell which is invariant It's a function of q and our priority n is an integer, but I only want that answers the earlier question about Laurent series I didn't say it before because it was more awkward in that language But I only want to allow functions with positive coefficients or a positive Exponents or zero but not negative sometimes what looks at the others, but I won't need them Okay, so that's well for multiple functions you do need them, but for multiple forms you don't so holomorphic multiple form will look like this Maybe that looks abstract for the moment. I just used you know a big theorem before yay But actually now everything is extremely easy So now I start with examples of multiple forms, but now in this language, which is much much easier So the first example is the same example. We already had Eisenstein series, but now in the new world So I'm going to define an infinite series of explicit functions and to avoid confusion with the GL called them e So e4 of tell I still call it a function of tell, but I write it in terms of q and Q remembers e to the 2 pi I tell So well first, I'll write it the in the shortest formula that I can No, I'll write it out. So the well To make it exactly like I did before I want for a expansion the a n here will be a 0 will be 1 and the other Coefits will always be divisible by 240 in this strange example and The coefficient of q to the n for n positive is 240 times sigma 3 of n so sigma 3 of n is the sum Over all positive divisive n that's a m divides m of m cubed and you can imagine that sigma 5 of n is the sum of m to The fifth so this is my first example of what I said in the introduction. This is an arithmetically interesting function Rinse it's it's got multiplicative properties as an example if I take a prime Then I get this but if I don't have a prime it'll be bigger So this is true if and only if n is prime So you can use it as a test for primality if you want I mean it's already in a function that number here It's care about it's not one of the most important, but it's already a nice example now So explicitly if I write out the first few of these I'll just to take two because I don't remember more But of course your computer will give you a thousand in a split second. It's that and similarly I take e 6 of tau Which is 1 minus 504 don't worry where these constants now put the sum in the usual place Don't worry where the constant 240 minus 504 come from it's some easy form for each case is a constant and for each k there's always such a constant and So I'll do one more. I have a reason So the next one is plus 480 and then it'll be the sum of course n positive sigma 7 of n q to the n So each e k will be a multiple and then sigma k minus one. So this will start q and the next one I actually don't remember, but I have it written down 6192 oh Yeah, it's coming in four seconds. I just wanted to write down what they are So there exist multiple forms and what I claim here I've just given any complete an explicit formula you cannot see anything from this formula You can see that it would be nice to understand this function because this is an interesting function But you don't understand. Why did I make this definition? Why plus 240? Why anything? But the fact is that in fact each e k is some constant Which is completely explicitly known times g k in other words if you start with the g k Which I did define in a sensible way by taking the sum 1 over m tau plus n to the k That is obviously moderate had the moderate property But you can't see that it has interesting coefficients But it's an elementary calculation given in every book of course also this one you compute the for a expansion And you find that up to some boring constant. This is a multiple of this e k So therefore this is a moderate form and this is a moderate form and this is a moderate form But now for the first time you can see the magic and so you know I hope that you you haven't seen this before you'll you'll like it It's the very very most trivial example, but it already shows you how powerful the theory is Namely if you are very good eyes, I mean mathematical eyes not to ocular eyes Then you might have looked at this power series with coffers 14861920 and said wait. That's funny that power series begins Exactly like the square of this power series. Can anyone notice that? So if you square this you get 1480 2 times 240 and then 240 squared plus twice 2160 believe me is that So you might guess that e8 is e4 squared Right that would be a reasonable guess because the first three coefficients work Then you ask your computer and the first 10,000 coefficients work that wouldn't normally be a proof in mathematics But here it is approved the proof is now automatic simply because Well, I actually already told you that every moderate forms of polynomial in e4d6 and this is weight 8 the only possibility is e4 squared But then we give you the full Property so that dk be the dimension of mk Which is a known number? Roughly k over 12 So the dimension grows roughly linearly like k over 12, but it's an exact formula. So we know the dimension Okay, and then we have the magic principle. I can give it even in two forms one If f and g are two multiple forms of the same weight k We're with the question is again You've guessed an identity that two multiple forms are equal like e4 squared and e8 But they better have the same weight because if they have different weights They will never be equal and you were just wrong with your guess So let's assume right from the start that they're of the same weight and if the first Dk Fourier coefficients Do you write f as some a and q to the n g as some b and q to the n and then you look at a n I Want the first so many free coefficients should agree So in other words, I've a n equals bn for n equals zero up to dk minus one Then f equals g So in our case the dimension is one you just have to look at one single coefficient Once this one agrees then all the rest are automatic and indeed the first three you can see pretty much in your head And the other version I won't insist, but it's the same if f and g are in mk and if f of tau i equals g of tau i for again dk distinct points of H Well in the fundamental domain that says that this is modulo gamma then Again, they're equal so to identify a form you don't need to know an infinite amount of data You need only a usually very small number of initial Fourier coefficients or a small number of values And that's what I met before about a computer proof But you don't it's not a computer proof in the sense I don't mean that the computer finds the proof you simply need a computer because most people can't multiply numbers to a hundred digits in Their head, but it's simply when you it's a trivial numerical verification So it's like checking when two polynomials are the same if you give me two polynomials of degree 20 And you think they're the same, but they're written differently each one is a polynomial and other polynomials But I know I can see that their degrees at most 20 then I can simply check by hand or by computer But I calculate that 20 values of these polynomials are 21 agree And then they're equal and the point is that modulo forms are just as precise just as rigid as Polynomials and so you have this magic principle that if you just look at the beginning of an identity then you get the whole identity for free so now I've gone various I You know I hope in quite a lot of detail on this introductory part and now I want to give you a series of Applications to number theory. I have eight here. I hope I'll get through them all and each one I'll say just fairly briefly how it follows So I'll stop talking about multiple forms I'll now tell you eight beautiful theorems of number theory and Each one comes from multiple forms. I'll try to say very briefly how it does in each case, but I may or may not succeed I'll make a slight pause to erase the board and That gives you time to breathe in case I have started talking fast after all Okay, so now comes the promised applications to number theory. Okay, so the first one is a rather boring one It's not a very interesting theorem, but I'm putting it down because it shows exactly how the thing works So the theorem is for all n your natural numbers positive integers Sigma 7 of m remembers the sum of divisors of n to the power 7 and the claim is that's the same as sigma 3 of n the sum of the divisors to the weight 3 plus 120 times the sum L from 1 to n minus 1 of sigma 3 of L the sum of the cubes of the divisors Time sigma 3 and mine cell is kind of a stupid formula But still it means you can get sigma 7 out of sigma 3 so an example of the example if I take n equals 2 You'd have the device of 2 or 2 and 1 So 2 to the 7 plus 1 to the 7 should be 3 to the 7 plus 1 to the 7 plus 120 Times 1 to the 7 times 1 to the 7th, and you can see this is 128 plus 1 is sorry I must have fallen asleep. The 3 was here And this is also 3 not that it matters in the case of 1 and so this is 8 plus 1 plus 120 And I think everybody can check that that's true but it's true for every n and the proof is So I can give you the entire proof and it takes you know Less than a line it takes a 1 1 centimeter and the proof is also 1 centimeter I'll just add the word trivially This is true simply because they're both multiple forms and one coefficient degrees and you're finished So you don't have to do any work There are elementary proofs of this one of my friends actually a doctoral thesis My wrote his master's thesis on that but it's a 20 page proof. So to prove this directly can be done, but it's very hard but now You know you need nothing it just falls into your lap That's kind of a stupid example because nobody really cares about writing sums of seven powers of advice in terms of sums of cubes of Devisors, but now comes a theorem which is historically Actually, it's it's several theorems I'll write a corollary which is already a famous theorem. So again for n a natural number Define R2 of n To be the number of representations This is a famous question of number theory and going back to Diafantes in the second century and to Fermat in this 17th century. It's the number of representations of n as a sum of two squares so a and b are in Z You have to be a little careful when you count like for five you might think naively There's only one way to write five is the sum of two squares It's four plus one but actually there are eight ways because first of all, it's also one plus four I mean a and b are labeled and secondly, I'm not telling you a squared I'm telling you a and b so a could be plus or minus two squared and b could be plus or minus one squared in the Same here. So this is a four. That's four. They're actually eight. So for instance R2 of five Should be eight. So we'll see in a minute that comes out and similar I define R4 of n equally a famous question of number theory as The number of representations of n as a sum of four squares then I've explicit formulas in these are absolutely wonderful number theoretical formulas completely elementary You can show to a good high school student just this formula in particular for R4 R4 is always divisible by four and it's the sum over all the devices of n which are odd So for instance one is always an odd divisor and then you take minus one to the power d minus one over two And that gives you the example. So let's do it for five five is two devices one and five One minus one over two is zero five minus one over two is two They're both even so I get plus one plus one is two and the four and I get the eight that we already had and the other Form is no more complicated. In fact, it's slightly easier R4 of n is the sum d divides and this time instead of not being divisible by two. It's not visible by four and Now it's simply the sum of d That's an absolutely wonderful theorem and the crime that gives closed formulas for these rather deep numbers the number of ways of writing Say a large number. So as an example if I have a prime So if n is a prime then R4 of p Well, the divisors are one and prime of course there one and p and so you'll get sorry There's an eight that I forgot you'll also always get eight p plus eight So that's completely amazing that the number of ways to write an odd prime say hundred and thirty one as a sum of four Square is exactly eight p plus eight for every prime p So that follows and and a corollary of this of course You'll notice that among the odd among the device of n is always one which is not just by four So this at least eight. So it's positive. So our four of n is positive. So every n is A sum of four squares. That's one of the most famous theorems of elementary number theory extremely Tricky to prove if you've never seen it that was Lagrange's theorem in the 18th century But the proof is very hard Where is here this proof of a much stronger theorem not just that it's positive But then it's exactly this is actually very easy. And so let me show you in two lines how you prove it, of course One has to prove things to get the whole proof. But let me just tell you the mechanism We take fate of tau, which is the sum of all integers of q to the n squared So that means one plus q plus q to the fourth plus q to the ninth and so on all squares except that it isn't Because as I just said when you have right four is the square It's both two squared and minus two squared. So you have to count everything twice except for the one Then it's a fact that this but that's very classical It was proved by Jacobian 1840 or something that theta is a multi-form in the sense that I haven't quite defined a way to have On some subgroup of index something in SL 2z. So therefore theta squared and theta fourth are some kind of multi-forms not for our gamma But for some other gamma of index in this case 6 in this case 12 in the original game in SL 2z Okay, but if you think about it a second that you'll see immediately that the Fourier expense of these just from the definition or These numbers that's what I meant when I said that the coefficients of multiple forms could be interested Arithmetic the interest in multiple arithmetic functions here The nth coefficient is the number of ways of writing n is a sum of four squares one of the very very classical questions of elementary number theory and so in fact They're both Eisenstein series So the Eisenstein series I defined above is just for the full group SL 2z But you can also take subgroups and then they change the form to slightly But remember that for the Eisenstein series ek with the sum of the divisors of m To the power k minus 1 so if the weight is either 1 or 2 You'll either have no power or the first power and you see here This is d to the 0 and d to the 1 but there's some congruence conditions about d mode for so Roughly, this is how it goes So but believe me this is very easy to prove and then you get a proof in three lines After a little preliminary some multiple forms of this rather deep theorem of number theory Again, I want to go on because I have many more examples. Yeah No It's a very good question. It's very hard. The problem is if I take theta to the sixth You can even do odd powers. It's yet harder if I take theta to the sixth It's certainly a multi-reform and theta to the tenth every even power But it's not always an Eisenstein series remember the principle only said if F and G are two multi-reforms at the same weight and if numerically you see that they begin the same way You can prove it but if they're different, you know that happens And so if I want to prove that theta squared is some it's some Eisenstein series of weight one But with a prime some other group and fade to the fourth is some other Eisenstein series I have to guess the Eisenstein series look at the first few coffees and then I can prove it Instantly by the magical principle, but if I take theta to the hundredth This will be the corresponding Eisenstein series weight 50 plus lots of other terms called cost forms and they're simply present So they mess up the formula So it's simply the method works, but the result is simply not true But again, you don't have to do any thinking the the calculation just tells you when you have identities like that And when you don't there's a very natural question There are hundreds of papers studying the higher cases by studying those cost forms and trying to make sense of the formless But these are the only really easy ones Okay, so I want to continue with my examples. So example three is absolutely crucial It's not as beautiful in the sense that I'm going to show you a function that if you haven't seen modern reforms You haven't seen before and so you'll say why should I care, but I'll show you the property that should make your care So let's define Delta of tau to be again in terms of q. It's given by an infinite product 1 minus q to the n to the power 24 So it looks very strange, but it's that's what it is Okay, so this is the so-called discriminant function of Ramanujan function. It has a q expansion Which starts with coefficients 1 minus 24 to 52 and then if you go on a little you find that the coefficient of q to the 6th is minus 6,048 and Ramanujan So we call these coefficients tau of n not the same tau q to the n so Ramanujan. Well, I should write him out He's a pretty important guy In 1916 observed You know, he just looked at these numbers and he said wait a second minus 6,048 He's not a random number. It's the product of minus 24 and of q and of 250 to in other words the product tau of 2 times tau of 3 is tau of 6 and So then he guessed and he checked by hand. Well, obviously there were no computers. He checked that more generally You have what's called multiplicativity Which means whenever m and n are co-prime integers then these coefficients multiply Now that was one of the shortest lived conjectures later I'll talk about a conjecture that took a thousand years to prove But this one was proved by so this was conjecture and theorem same result proved by more Del one year later So this is really true not just up to 30 Ramanujan checked it up to 30, but it's really true Now that's a wonderful wonderful property Even if you haven't seen this you must see that something amazing is going on Why should this be the product of these it looks completely wild? And so I want to say a few words about that because although it's a little less I mean I can't go into it because it would take us way too far a field But it's the most important part of the whole theory So I'll just say that hecka in the period between the world wars developed a theory and In particular the main theorem is all multidiforms not just for gamma But for any gamma star has a basis. It's uniquely spanned of I'll call them hecka forms Which means exactly this property that if you relate to the Fourier expansion a n q to the n then AM a n is equal to AM n whenever m and n are co-prime It will call such a thing a hacky form and the point is they're not rare This behavior of Delta is not some kind of a fluke to the contrary Everything is so let me give you an easy example. So examples of course Delta But also if I take 1 over 240 times e4 Well the constant term is 1 over 240 but that doesn't play any role here And the other terms remember we're sigma 3 of n the sum of the cubes of the divisors And it's very easy to see using the Chinese remainder theorem or something that sigma 3 is a multiplicative function So the e delta and e4 over 240 are typical The hecka forms and you see in weight 12 Sorry in weight 4 this is all you have anyway believe me that there's always a basis So if I took weight 12 it would be Delta in some constant times e12 and they would both be hecka forms And they span the space so you always have such a basis And I just want to say because it's so important that this is true And this is the basis of Two things that I'm sure you've heard about One is Jackie Langman's theory, which gives an interpretation of the whole theory of multiple forms in terms of the Representation theory of the Adele group gl2 and then generalizations to other groups not just gl2 over the Adele's So Jackie Langman's theory and the so-called langans program, which I can't go into at all that would I could give a nice lecture at the same level explaining many examples of it, but I don't Want to because I wanted to do five different themes So I'm just saying that this multiplicative property is absolutely crucial. So now let me take a few other examples of different types So just again very briefly So now I'll write a general theorem So we're now to example 4 This is another class not Eisenstein series, but they're called Theta series So if lambda is a lattice But not before we had lattices in C So they were two-dimensional, but this will be D dimensional capital D for some D Which could even be odd but think of even to simplify things like two-dimensional before and it has it's a lattice So the vectors in it each lambda lambda has a length But the length will usually be the square of an integer So I prefer to use the quadratic form which is the length squared which I want one where these lengths are integers There are plenty of such lattices like the famous E8 lattice from the root theory of the algebras Okay, then it's a theorem And not a difficult theorem that the associated series if you just take all L in the lattice and take Q to this quadratic form Then this is a modular form Of weight exactly D over 2 on maybe on some group Some group of SL2Z and if I write this out Then you'll see well, there's only one term Q to the zero because there's only one lattice of Vector of length one, but otherwise This has a Fourier expansion with arithmetic the interesting cofists. I promised our lambda then I won't write it It is just the number of vectors whose length whose square length is n It tells you how many vectors there are on let's say big ellipse of size n So that's a general theory and this is application I'll just say to for instance well the theory of quadratic forms of course, but also the coding theory and therefore this is actually quite important in Some parts of applied mathematics, but there are many other places where this plays a role and a really easy example Where D is odd is if D is one and the lattice is simply Z and Q of L is simply L squared Then this theta lambda is just like called theta before it's the same theta that I used Here somewhere wherever it's gone the sum well Maybe it's gone, but it was the sum Q to the n squared So that would be a typical example, but these are higher dimensional and higher weight examples So I just mentioned that because there are whole books by various good mathematicians on Coding theory and the theory of lattices and quadratic forms and they crucially use This theta series property that you can use the theory of multiple forms and it gives you many results that would be very hard to get without that about You know hamming links and things like that Hamming distances so let me skip on How am I doing with? Time oh, it's wonderful. I still can give my remaining examples so Here's an example also in arithmetic and I like it very much and I even have one contributes But I'll mention with dick Rose not the the same gross, but not the same theorem that for none to mentioned Here I want J of tau is a modular function Remember a modular function is weight zero so I ever since I introduced both reforms all of my exams at higher weights But the easiest way to get something weight zero is to divide two things of the same weight So remember we already a delta. I just introduced it weight 12 e4 at weight 4, so it's cube has weight 12 So there's a modular function in other words. It satisfies J of a tau plus b over C Tell plus D is simply equal to J of tau no factor or if you wish Equivalent since the group we know is generated by these two things. It's invariant under these two transformations Minus 1 over tau and tau plus 1 so this is some well-known modular function You can compute it well you can compute it like that or it has an Expansion which plays a role in the famous moonshine story that I might say something about and I think it's Wednesday so it has some Interesting coefficients which start one seven forty four one nine six eight eight four Okay, and now the wonderful classical theory, so 19th and early 20th century theory is The special values of a modular function or modular form I'll mean the values J of tau where tau Satisfies a quadratic equation over the rationales and so these special values are always Algebraic numbers in fact algebraic integers That's completely amazing. So this function is this very very surprising property. Let me give you a couple of examples and and Sorry, how is any type? It's a theorem. So tau is anything. There's no special equation if Tell us our special value. I thought I wrote it. Yeah, so here it's written Tells satisfies any quadratic equation the over Q with rational coefficients So you take any quadratic equation you want like for instance tau square plus one equals zero then tau would be I But if I put tau square plus two is zero then tau would be I squared of truth There's no equation for tau. There are many examples of this theorem. It's a theorem So any theorem there should be in many special cases to take any quadratic equation with rational integer coefficients Which has a solution the upper half plane and compute J of tau Then the claim is and let me do these two in fact. I'll do two more. So J of I is simply 17 28 a number We already saw But J of I times the square root of 2 which is this number turns out to be 8,000 So what the theorem says is without knowing anything just because I squared of 2 Satisfies a quadratic equation tau square plus 2 equals 0 automatically J of I squared of 2 is an integer and Similarly J of 2 I would satisfies the tau square plus 4 equals 0 is that when I've forgotten 287,496 on the other hand if I continue J of 3 I I didn't say it's always an irrational Integer, it's an algebraic integer and so in this series the first example Can't read it. So I hope this is 0 and not a 6, but it's one of the other It's an integer plus an integer times the square root of 3. So these are quite complicated numbers Nobody said this was going to be easy, but they are integers or algebraic integers always Okay, that's an absolutely wonderful theorem, but then I want to mention this context because There was an observation that was made and then proved by Gross and myself in the mid 80s and For some reason these numbers have been very famous and tabulated I think the first big table was from 1928, but nobody'd ever noticed the following observation that all Values and all differences of special J values are highly factored in a completely explicit way It's actually the only theorem I know in all the mathematics Where a number that you construct in some other way you can give its complete prime decomposition up or you can say exactly Which primes occur and to what power so I'll just give a numerical example with the numbers We already have if I take J of 2 I minus J by that's a difference of two special values So it's 2 8 7 4 9 6 minus 17 28 and if you factor that you find 2 to the 3 times 3 to the 6 times 7 squared so even though the number itself is A quarter of a million it is no prime factor bigger than 7 and as I say the complete theorem tells you exactly That the primes occur will be 2 3 and 7 and will tell you exactly the x-1. It's a complete theorem So that's kind of amazing. That's not a theorem about non-modern things. It's not an application It's a theorem about modern things named with these special values But it's a kind of an amazing fact and it actually is connected with it goes much much further and It's in fact a special case of the general theorem that's an under mentioned that grows and I proved which had many consequences In dive fountain analysis, but this one as it stands doesn't Okay, so I'm nearly done. That was number five So after five should come Six so now I want to give applications to die fountain equations And all of my last examples would be that so this is the oldest part of number theory It goes all the way back to die fountain in the second or possibly third century AD. Nobody knows exactly when I found his lived so I'll give to straight Equations are actually doing injustice myself. I need more examples six a Is a problem that was posed in an Arab manuscript at the name somewhere of year 1972 972 so well over a thousand years old and also in a famous book by Fibonacci the Lieber quadratorum in 1225 so this is a pretty old problem, it's more than a thousand years old and The problem is a number n is Called congruent. It's not the usual mob congruent if it's the area of a rational Right triangle so Pythagorean trying with rational sides But let me give you an example instead of talking a lot if you take n equals five That was the problem that was posed in the Arab manuscript and solved 250 years later explicitly by Fibonacci if you take five Then you take the triangle the sides the short side is three over two the long side is 40 over three My button uses 41 over six and you check that 41 over six squared is three half square plus 40 over three squared So it is a right triangle as I claimed and the error of course for bright triangles half the product 40 over three times three over two Oh course 20 over three because it's 40 over six so 20 over three so half of the product is half of ten It's five so the area is indeed five So the question is if you give me a number How can you determine if it's common? Of course if you just think of this triangle, but it took 250 years to find this and if you take n equals 157 I found it But then the the numerator denominator it exists, but they have a hundred digits almost a hundred digits So even with a computer you can't just find it by searching you need a theory So the question is how can you find those triangles and I'll just say briefly that there's a formula I won't write it. I was going to it takes one minute if anyone wants to ask I can put it This was there's an explicit formula Given by tunnel that associates to n a certain integer and if that integer vanishes Then it should be solvable and if it doesn't vanish, then it's not solvable So there's a complete answer and it comes completely from the theory of modular forms Example 6a it's a completely different equation. I'm just saying they're both classical. I found the equations This is Sylvester's problem He asked which numbers and then later more which prime numbers and later which numbers Are sums of two cubes so we already had remember the The equation that I counted the solutions with my R2 it was write the numbers as sum of two squares But you can also ask is n the sum a cube plus b cubed Well, this is no longer the nth coefficient of a modular form But rather amazingly the theory of modular forms still lets you solve it and the reason is because this equation Even though it doesn't look like it is an elliptic curve and so it fits right in and so I'll mention two theorems So theorem by Saatchi 1986 if p is a prime p means prime 9k plus 2 so 2 mod 9 then 2p is a sum of a cube plus b cubed and The theorem which I do want to mention because it's joined with my boss here my immediate boss Fernando Diego's not give it. Oh, so we proved this in 1995 Is if p itself is prime? This was 2p Sylvester asked about primes and turns out the only difficult case is 9k plus 1 the others at least model of the Famous conjecture, you know exactly which primes work and only depends on p mod 9 But here some as it works and some as it doesn't and our theorem here was that p is a sum of two cubes Even only if p divides a certain number Bk where it's the same k and I'll just give a little table These are just some numbers one two minus one hundred and fifty two six eight four eight They grow very quickly, but there's an elementary recipe the proof is not elementary uses multiple forms But there's an elementary recursion that gives these Bk So it's a simple sequence of numbers that you can again show to a high school student They're completely computable and then if you have your 9k plus one You just look if p divides the kth element of the sequence and if it does it should be a sum of two cubes And if it doesn't it isn't Okay, so those are examples of dive fountain equations and now I want to come to the Final one, which is of course the most famous dive fountain equation of all time, which is Fermat's last theorem so The final application But I'm going to split it into two parts Because both are absolutely important theorems of number theory. So my example 7 now if I haven't discounted This is the so-called TW conjecture Which also is called the TW theorem? That's not as most of you know because T and W you couldn't decide if they proved it or not But because it's a different T and W. This is Tanayama in a famous remark in some conference proceedings in Japan in 1955 this is Vey in a famous paper in German in 1967 and also Shibori is involved, but in my view not as part of the state in this conjecture And this is of course Taylor and Wiles So this is 1994 so much later that it became a theorem and the theorem is this let If I take two integers or rational numbers, it doesn't matter again for a cube plus 27 b squared not zero That's just a detail and I look at the same equation. We had before Remember that's how elliptic curves look so I look at the elliptic curve with rational coefficients or integer coefficients And the theorem is the theorem of that I've already said Tanayama a previously conjecture in one version There are two versions. I'll state the other says that E this elliptic curve Always I mean many cases were known before like exactly the ones I needed for those previous theorems were already known But it always has a multidare parametrization So that's very simple. It's just like parametrizing the circle by cosine theta and sine theta It means you take two transcendental functions cosine sine which satisfy the equation of the circle x squared plus y squared equals one So what it means is you have x of tau is a multidare function and some group Y of tau is another multidare function on some other group and they identically satisfy for every a and b You can find two such functions which identically as functions of tau Satisfy this equation and the equation we're talking about So you can trace out the elliptic curve just by letting x and y be two multidare functions of tau So this is a fantastic theorem. It was a conjecture for many years and another equivalent version, but it's not at all Obvious, but it had been proved before the theorem was proved. It's equivalent to say that there exists a Multidare form somewhere of wage to which the hacky form remember that meant it had multiplicative coefficients and AP is Simply P minus the number of solutions of The equation y squared is x 2 plus x plus b, but now it's in integers more than a P So if that doesn't make sense to you, but believe me that this is the kind of thing in arithmetic algebraic geometry Otherly want to study the equivalence of the two versions is not trivial, but it's true So that's what they proved and now I come to the last and most spectacular application Probably that not the forms will ever have because it was the most famous open question of number theory for 100 years or something example 8 is Fermat's last theorem So we have the theorem Which is usually ascribed to Taylor and Biles, but really not because they did the last step and I'll show you the previous work was to think of a special curve that The Frenchman eloquax in the German Freifam and then serr Formulated a very precise statement about it and rip had proved it There was about ten years before Tanya man V and they proved They proved that if you had the Tanya man V conjecture This is the Tanya man V theorem. So this is Taylor and Biles, but they showed That the Fermat's last thing would follow from the Tanya man V conjecture But they hadn't yet proved Fermat because Tanya man V wasn't established But when they when Biles did his work he built on that so this part was already done And I'll just take one minute to show you the mechanism and then I'm finished So proof well except the proof is 300 pages directly and another 2000 if you put in all the background so the proof If I think everyone knows what Fermat's last theorem says But if what it says is that the following doesn't happen that you cannot afford integers A, B and C non-zero and n bigger than 2 such that a to the n plus b to the n equal c to the n That's what Fermat claimed was true and even claimed it a proof although I think he dropped the claim since 30 years after he claimed that he never claimed it again But anyways became very famous because of that and so the idea is this then you look at and this is the curve That was invented Independently I think I'm not sure if Fry knew about eliquash so eliquash and fry Invented this curve the elliptic curve and remember an elliptic curve. It's just an equation y squared is a cubic in x So here's a very simple cubic. I take x times x minus a to the n times x minus c to the n and That was the curve discovered in a slightly different context by eliquash And then fry realized that it should have the problem about to say Sarah made it very precise and ribbed proved it about 10 years later or a few years later This elliptic curve has properties that mean that it cannot Have a model of parameterization It's simply and that's very very deep But there it uses this multiplicative properties a hacky form and all kinds of things Deformations of Galois representations, but if this solution existed you'd have an elliptic curve Which does not have a multiple parameterization, but ten years after ribbit had proved this theorem So this is this is the theorem of ribbit based on this other and that's before He proved that if you had that that's his theorem, but based on a conjecture of Sarah which implied it and he proved that conjecture Once you have this theorem then you're done because if you had such a solution He would be an explicit elliptic curve with no multiple parameterization But now Taylor and Wiles proved that every elliptic curve does have a multiple parameterization So this elliptic curve doesn't exist, but I wrote it down just using this solution So the solution doesn't exist and that's Fairman's last theorem So that I hope gives you an impression of the breadth and scope of the kind of problems of number theory that you can Attack with multiple forms and each of the other lectures. I say I'll have a different area of applications like differential equations Not theory. It won't be there won't be any more number theory. It's good or bad But that's today's lecture. So thank you very much Thank you very much done. It was a wonderful talk questions Yeah, yes number to the eight power plus whatever I mean morally the format here and Which is false? So can I prove according to the same? If it is a general way if you're lucky. Yeah, well Wiles is theorem. This is a general theorem every elliptic curve has a multiple parameterization But then you need some very clever friends like a little gosh and ribbons and fry and and sir Who prove that your particular equation if it had a solution would give an elliptic curve that contradicts that and that's very hard You have to think of you have to think of how it works So the method is general if you can use your special and there are many examples down the literature There's a whole industry so I could give you examples of similar Fermat like equations like sometimes it's a simple equation But you can sometimes do things like put to a and plus b to the nc to the n and show that that is no solutions So there are variants, but if you put You know if I put here 2,193 times 80 then nobody knows how to do it because from that one the corresponding elliptic curve You cannot prove has contradictory so it's an extremely technical thing to show the proof of this I Did this I've court. This is a joke called the proof This is the theorem that read the proof and it's a very very deep proof and of course the time the M of a theorem is even Deeper but to make the connection you have to construct an elliptic curve study deeply the Galois properties of its so-called Associated Galois representation and show that they're not compatible with deep theorems that are now known about those elliptic curves They do have a model of parameterization, then you're finished, but I can't say at all a priori which equations So in principle, yes the method has a wider scope, but it's still very limited and in most case Of course it won't work. It's a very very special tool I always thought it was a joke to me Fermat's last there was famous because you know you can tell people on the street But this was the big thing but I once asked wiles after you prove this. He's a very close friend I've known him since we were both very young and I asked him I said surely wiles, you know the press they keep talking fair My fair my fair my surely you didn't care about fair ma. You wanted to prove the big prize Which is the Tony I'm away and thanks to you know Ken ribbitt who's a friend of both of ours that would imply he said no Don You're wrong When I was seven I heard about Fermat's last theorem. I decided to be a mathematician I decided to spend my life learning with every mathematics. I needed to prove Fermat's last theorem. That was my goal I wasn't even interested in multiple forms. I couldn't care less about the time the arm of a conjecture But when ribbitt proved that by proving the time of a contract you could get Fermat Then I became a specialist and I worked for eight years and I proved it So in fact to me it's kind of a random application because it's a very special Accidental equation only his door would be important This is a very deep and important thing and it doesn't usually apply to other equations But luckily it did apply to that and luckily wiles cared enough about the special case to do the work to prove this 10 years of his life More questions the question myself the proof that the g4 and g6 Essentially span the whole set of model forms. Is it very difficult or something? It's not difficult at all Let me answer them in three lines It's actually very very easy So I told you that if we take dk the dimension of mk that there's an explicit form Then I told you it's approximately k over 12 But actually I can write it down exactly if t is if k is the form 12 r plus 12 r or 12 r plus 4 or 12 r plus 6 or 12 r plus 8 or to remember it's always even Then the dimension is r plus 1 and if it's 12 r plus 2 the dimension is art So we have an explicit formula now you just check that if you take polynomials in g4 and g6 Then the number of monomials in g4 and g6 of that weight is exactly this number So therefore it's the right number and it's very easy to show that they're algebraically independent And so this ring is contained in that and there's the same dimension to your fingers So it's it's a three line proof But of course you have to do this there's a little bit of work But let's say the whole thing at a very elementary level for students in my book It's it's one page so it's not a deep theorem, but it's not an obvious it is a theorem But not some deep thing and it's very special if I take a subgroup like gamma 0 of 113 Then the ring is still finally generated, but it might have 25 generators and lots of relations It's just an accident that here it just happens that it's freely generated That's not a deep property of multiple forms. It's it's this particular case But it's always an explicitly computable finally generated ring with generated some relations here It happens to have no relations Thank you another point is You had expansions the Fourier expansion of the modular forms Yes, and you said that the coefficients have this arithmetic arithmetic properties But can you I didn't quite say that I mean of course they well there could the coefficients are numbers So they have properties I said often the coefficients are numbers that are interesting to you That's a psychological question. I can't guarantee that you will be interested in every multiple form I will be because it's my job, but you won't and I just said very often you get wonderful functions And I gave any examples of representation. There's a sum of four squares But is it true and it's easy to see they are always integers Well, of course, they aren't always integers because if I multiply g4 by pi then they're always integers times pi Right, so they aren't always I mean it's an extra space over C But what is true and in this case? It's trivial to see is that there's a basis Which does have integer coefficients and it follows for instance from that theorem because g4 and g6 except for the E4 and E6 up to a denominator 240 or something had integer coefficients, but it's a general theorem of Shimura for any group of this sort So there is indeed a basis, but then of course if you take a stupid linear combination with transcendental coefficients Then you get those coefficients, but yes It is a general fact that mobile reforms can be generated and I also told you heck is theorem Which says and that's not some random base. It's unique basis Heck is said that the heck you forms remember the ones where a m n is a m times a n They form a basis an actual basis. So in mk. They're exactly DK of them not more not fewer It's not that there is a base at that form and they always have integer coefficients But in general algebraic integers, but still integers and so there is a very very strong arithmetic theorem that's I hope answers but of course not every form because of the Skating by Constance very good. Thank you very much done. So if there are no more questions, we'll ask Everybody to go outside for the reception and then the students to come down but before doing that. Let's all thank Don again