 Yeah, so thanks a lot and thanks for the invitation and give me the occasion to talk in this seminar So as the title indicates This is going to be about an I think number theory and a distribution and with a lot of harmonic analysis and this is About joint work Currently still ongoing with Arthur Fourier and Raviya Frazane and we hope to have a manuscript. That's more or less Complete and readable in the in the next few weeks about what I'm going to talk about So let me begin with something that I think everybody loves In mathematics not only number theory, which is the Euclidean Fourier transform and then we'll discuss about geometric versions of that and then generalizations to different situations so I I start with the simplest case so the free transform on the Real numbers so formally speaking we have a function from the real numbers to a complex numbers and then we define another function by integrating f of x with The kernel exponential minus 2i pi xy With respect to x of course there needs to be conditions to make this work But in principle, this is the classical Fourier transform and this is one of the most important objects In mathematics not only pure but also applied mathematics Let me summarize a few of its formal properties and then a few of its more refined and attic properties So just of course free inversion which is that you can recover the function by More or less doing the same thing up to a sign in the kernel Of course again, this may or may not make sense point-wise, but in principle, this is only something Which we feel is true Then very important second property is the planche-relle formula which tells us that in the L2 space The Fourier transform is an isometry And sometimes depending on the normalization it might be isometric up to a scalar but with this normalization that I've written down here It is exactly an isometry Another very important Property which looks a bit more complicated at first, but it's really an essential one is the compatibility with convolution Which says that you can identify The product of two Fourier transform as a free transform namely of the convolution product So the convolution products and x again formally because it might not exist To the product to the integral of the product of f of y g of x minus y with respect to y So these are more or less formal Almost algebraic properties and it has lots and lots of other More analytic properties. So for instance Things like stationary phase which I'm not going to write down in Detail, but many of you know what stationary phase does it allows you to give asymptotic formulas Functions defined as Fourier transform or for other oscillatory integrals really Let me mention another one Which is Absolutely essential in many many analytic applications The uncertainty principle which again takes many many forms and I'm not going to write any in a specific way But roughly speaking. It's a more analytic way of saying that the Fourier transform of a delta function Is the constant function one? Okay, except that this does not always make sense point wise It might make sense distribution wise and in a smoothened form this becomes This can be transformed into various forms of uncertainty principles Many many other things which I don't write down and it has applications, of course This is definitely not an object of pure beauty that people look at from far away and don't try to do anything with it and Here I'm not going to even try To write the applications You can fill in the blanks with whatever is your favorite application So there's a famous saying of Martin Eichler that the fifth operation is modular forms I think the more proper way would be to say that the fifth operation of mathematics is really a Fourier transform In one form or another Okay, so this is the classical Euclidean Fourier transform and It has lots of generalizations So we're going to talk about finite fields soon, but the first thing I want to mention is the really general form of Euclidean Ability and commutative free analysis is Pontiwag in duality So you start with a locally compact group G Then maybe a function on this locally compact group Then you have the group of characters so continue some morphisms from G to the unit circle and You define the Fourier transform Which is now a function on this group, which a priori is different from G by doing something similar But integrating with k of x instead of exponential 2i pi x1 Okay, and these are lots of similar properties and For instance, this is what gives rise to the million transform so maybe the case apart from the Real line that's most commonly used although the Positive real numbers is isomorphic to the real line the million transform is a version of that okay so that's the classical Fourier analysis and Now what I want to discuss the really the proper beginning of the talk is Starts with a letter of the lean to cash done in 1976 Which you can find online if you Google for letter of the lean to cash down you will find a link to copy of it It's quite an interesting reason you observe that there is a So let me write it as a slogan first. There is a geometric incarnation of the Fourier transform Over finite fields and I should say over additive groups of finite fields, but let me say over finite fields And I will discuss a bit later. What was the lean's motivation? It's for magnetic number theory and it goes at the more precise form of this Slogan is that so we are given finite field K and then I will always write K bar for an algebraic closure and KN for the intermediate extensions of degree n inside K bar That's just notation. We're going to use we fix an additive character of K and It's going to be useful to think that we've also fixed an isomorphism of C with some QL bar where L is does not divide the characteristic of the field but you can just think of the complex numbers for To understand a bit what's going on at least Then we need an extra object, which is I'm going to call a coefficient object and It can be different things. So there are more or less equivalent ways of viewing it. So you can think of it as a Galois representation of the field of rational functions In one variable of a K you can think of it as a an elliptic shift or something even more complicated but concretely If you Want to think really in terms of concrete and attic number theory properties this as a trace function and This is really what is of interest to an attic number theory at least and in fact it has trace functions for every extension field KN So complex valued or QL bar valued functions Defined on KN and here and we may be given example So for every polynomial F in K of X There is such an M with TM X Again, yes, I'm going to lose Hopefully have enough space equals Psi of the trace From KN to K of f of X Okay, so these are somehow some of the simplest Trace function that correspond to some of the simplest of these coefficient objects Okay, this is all data which Is a priori given and what they'll improve or observed So I say observe because really given the formalism that E and got and the the algebraic geometry school and Developed in the previous few years. This is really just an observation that things fit well together So there exists another coefficient object. So this is the geometric Fourier transform Okay, so it's again of the same kind You can think of it as a Galois representation essentially or as an analytic sheaf or something like that and the point is that its trace functions Are the discrete Fourier transforms of The trace functions of M so with some normalization which I'm not going to put in right now But we sometimes is convenient. So it's the classical discrete Fourier transform with Y Psi of the trace of XY So let's look again at an example to see immediately what this tells us So if we know that we can construct the objects that I described in the previous example So very simple coefficients where the trace function are exponentials of polynomials Then this gives you new objects new coefficient objects color representations or something with the property that the Corresponding traces. Let me just do it over the base field and let me even take K equals FP and Psi of X is the standard additive character, which I write as E of X over P then this Discrete Fourier transform becomes Over FP just a classical additive exponential sum. So E of f of X plus X times Y over P so the kinds of sums which come up all the time in an antique number theorem and I Think it should be said at least for me that the fact that such a thing exists. So as I said Given the formalism of algebraic geometry that existed at the time. This is really not so hard It's just putting the formalism correctly together But the fact that there is such a geometric for a transform is something quite extraordinary if you think in terms of modular forms because the These coefficients you could also think that their trace functions are eigenvalues of a cooperators on On on the on the GL2 or on some GLR of the field of fractions of the finite field K Then you're constructing a new automorphic form with completely different Behavior where the eigenvalues are the Fourier transform of the given eigenvalues that that's quite a remarkable Remarkable thing Okay, now what the linear already observed in his very first letter is that this is indeed an analog of the Fourier transform, so we show the version of inversion. He showed the versions of the Planche-Rell formula Which takes the form that? Essentially takes the form that if M is irreducible Then the same for the Fourier transform and Conversely if you think of Galois representations for instance There's something very important is the convolution also works and this is also mentioned in the first letter of the link So I'm not going to write it here because I'm going to discuss it later and now you might ask about the further properties that I mentioned quickly above and about applications, so it's Something quite important in application. There's a version of stationary phase because it takes a very different form than what you might be What one is used to in the niting number theory, but no more showed that properly interpreted there is a version of stationary phase for this geometric Fourier transform On the other hand as far as I know, I mean, I don't know of any good analog of The uncertainty principle, I know some statements which look a little bit like uncertainty principles But I would guess that I don't There I would guess there must be some because this is such an important fact of the classical Fourier transform But I don't really know one for this geometry Fourier transform and for applications Well, I would say that There's many of them In a niting number theory So the lean mentions in his letter so apparently it was an answer to a previous letter from cash done but he had been seeing about it because of applications to estimates for some trigonometric sums and you can imagine that indeed as we've already seen with this geometric Fourier transform you get an algebraic formulation algebraic understanding of Classical families of exponential sums For which you may be able to use some tools from a algebraic geometry to say a lot of things and indeed This is what happened so you have Estimates by combining with the Riemann hypothesis Over finite fields You have a quick distribution statements which again I will discuss later in more generality, so I won't state And so on I won't state the result right now But it's not that so there's applications So Lomo also use that to Give new proofs of parts of the veil conjectures and a product formula For the signs of function equation for epsilon factors, okay, and something which For me and Fouvri and Philly Michel has been very important so our all series of works That we did in the last six or seven years Depends essentially on this geometric Fourier transform Okay, and there's an IT application so there's the This will function in arithmetic progression There's some other Level of distribution for Primes in arithmetic progression and so on and so on all kinds of analytic facts now depend essentially on the on the Fourier transform and and of course there are others and again I'm not trying to be exorbitant They are certainly less at the moment and the classical Fourier transform, but it's only is a very important tool In number theory Okay now What Really is the starting point of the work that Arthur and Ravier and I did is The question about what about other groups? So like there's a contraging duality for groups other than the real numbers Try to think about and we can see there are other groups with their own characters that we might want To to consider and for which are exponential sums which arise parameterized by such characters Which we might also want to study so Here the oops in question would be Technically commutative algebraic groups of a finite field, but let's take the simplest example would be we look at The invertible elements of care of one of its extensions Then we have multiplicative characters And if we have functions defined on the invertible elements of k or of its extensions we can construct so for coefficient m Not only can we construct the discrete Fourier transform? additively, but we and look at multiplicative character sums So some of chi of X Then the trace function of X Okay, now X is in K and star and so this come up in application Now we view these as functions of chi so This is really what should be called a discrete mailing transform and these come up naturally also in applications At least in I think applications now for an individual individual chi We still have the Riemann hypothesis of the linear which can give in principle Can give good estimates sometimes it's hard to prove but In principle it gives you an understanding that's often sufficient for applications But now what happens when chi is varying if you need to understand What happens on average over over chi like we know that We have a good distribution properties from delins Fourier transform. Do we have something like this for the discrete mailing transform and And examples of questions like this arose in particular So Evans has this question to Katz concerning sums like this so over over Fp so where X bar is the inverse of X modulo P and Various sums of that type arose in questions about quantum chaos by Rosensweig and Rudnick I won't write the sums that arose but they are somewhat similar maybe to the Evans sums and In a series of work of Keating Rudnick and a number of other You also have sums of that kind that arise naturally for occasions and also Sometimes with more than one character. So you have okay d times where these fixed so in that case the group the underlying group is not the Multiplicative group of a finite field. It's a product of actually many of them and of course you could consider also the product of additive group times a multiplicative group and in some cases you could also consider an elliptic curve the K point of an elliptic curve of an ability of variety and then the characters of that and and write at least write down some exponential sums And ask if you can say something about their properties Okay, and so the first thing I want to to discuss although it's not literally necessary for for the later discussion, but it's something which maybe is we're seeing is that Let's say for for this multiplicative character case There is no analog of the lens Results, so you might think to be more precise. So you might think because we know for the Duality three four five a billion groups that the group of characters of k and star is isomorphic to KN So functions defined or characters of KN are star are just the same Via such an isomorphism as functions defined on can star. So you could think maybe so given M Maybe there's an Maybe I should write M twiddle now. It's not an M ad like the lens one. There's another M twiddle again or On the multiplicative group. So the beyond k bar star with trace function After some isomorphisms Equal to the discrete man in transform. This would be very optimistic But that's a way that you would be able to generalize the lens work very easily Okay, and the fact is this is not true except in trivial cases Let me explain one reason why it's not true because it actually is quite nice And it fits with some of the things we we did with Arthur and and Ravier in more Complicated and fancy circumstances. So I don't really know who first made this observation I mean, I would guess that Deline must have known that there's definitely a version of it in a paper of boy Ashanko and Greenfield, but this something that was known much earlier than that I think the argument we we used to show that it doesn't exist is maybe new So the thing is that if if there was one so if m tilde existed It would have an L function like Galois representations or sheaves have L functions so this would be one way of defining it would be as a formal power series So exponential of the sum of t to the n over n and then you would have the sum over the characters And this would be the points of the algebraic variety on which this m tilde is supposed to exist and then You would have to have the sum of the corresponding traces which would be these these k of x times the trace function of of m and this would be a Rational function because that's true for any reasonable coefficient Galois representation sheaf whatever and This would have some kind of function equation now you can compute this just using Standard discrete for analysis over k and star. I mean you can see here Exchanging the sum you're going to get the size of k and star times the value at one of the trace function At the origin of trace function using this you will easily see that it is indeed a rational function But this is not it does not have a function equation except in digital cases okay, so To summarize the situation at that point so we have naturally arising at least families of exponential sum Let's say over multiplicative characters, but also over other Which are not additive groups We suspect or we might hope that we can understand them statistically We can do experiments. I mean at least let's say the epsons are very easy to compute and events did that and checked that it seemed to be distributed to some measure and Now the question becomes Where does it come from? So this counter example shows it cannot come from Simply applying the Linsik distribution serum somewhere. So it has to be something different that's happening so What can we say? So what can be done? I should say so as I mentioned these questions were They are natural and so both Evans and Zef Rudnick asked Nick Katz about their respective examples if algebraic geometry could say something about it and Katz was able to to find a solution For so for K and star in about I guess close to 10 years ago now and What we have done So with Arthur and and Ravier is so we generalize the approach of Katz Which is based on on convolution plus the ten I can formalism to any Commutative so here I'm going to give the technical connected Connected commutative. I should have said algebraic group For any finite fields, okay So in particular You have the multiplicative group which corresponds to this Multiplicate invertible elements in K bar or K n bar you have You could have the additive group to some power times a multiplicative group to Another power this would corresponds to having Some sphermatized by D1 multiplicative characters and D2 Elements of the field defining additive characters But you can also have elliptic curves or you could have a billion varieties more generally all products of these so the Kissing is we have something that's that's completely general So I won't try to explain how the the theory really works. So I mentioned the key properties the convolution So the point is if you look at the definition of convolution on an arbitrary finite group There is no character that shows up in the definition of the convolution of two functions Which means that the formalism of algebraic geometry will be able given two objects two coefficient objects to construct essentially a third one whose trace function is the convolution of the trace functions and Once you do that, I mean that that's something that again is that At the level of what the linear observed in 1976, but what cuts realize is that combined with Terrican formalism, which I won't say anything what it is Is a way to then associate a group To any coefficient on the multiplicative group any coefficient object on the multiplicative group And then he was able to exploit that and again the Riemann hypothesis to prove an equidistribution theorem for some over over the For some sparmatized by multiplicative character And what we've done is generalized this to in such extent that we can now do this for any of these Commutative connected algebraic groups and not just the multiplicative group or the additive groups So we are the precise statement Of the general equidistribution theorem that results Okay, so it's really the Very general equidistribution theorem for any kind of Expansion sums that you can write if they are parametrized by the characters of such a group. So Let's do this So we start with the data. So k k bar k n As above so finite field algebraic closure L is different from the characteristic of k is a prime number M is a coefficient object Which concretely you can think of as given by its trace function, which have to be some nice functions And again an example would be for any if you have a function On the sorry, I should define the group first. Otherwise does not really make sense So G is connected commutative algebraic group over over k Example to keep in mind for the case of multiplicative character is the multiplicative group Then you have a coefficient object on M and again A simple example that you can always think of is If you are if you have a function on the group With values in the defined line. So just a polynomial function of the coordinates of your group essentially And then you can find an M with trace function given by psi of f of x So psi n of f of x we are saying is composition with the trace Okay, again psi is a fixed additive character of k But this this side doesn't mean that we're just working with With the additive group. It's just a way of constructing coefficient objects Then what we show so there exists the following So there exists an integer r non-negative So maybe I should say to be precise for Not to claim that I understand what I'm writing here But so the the correct technical words For these coefficients objects for which the theory really works Of which these are special cases You have to have a perverse leaf And the way I'm writing it it should be of weight zero And uh semi simple Yeah, semi simple But that's just so that I write the words once Okay, then and so that the theorem is correct It's precise and correct. So there exists a non-negative integer There exists a group Which I'm going to call h sub m Side glr And you can think of it as glr of c So it's some group reductive group With the following property so Let first k m Be a maximal complex subgroup Of this h m So for instance h m could be the old glr of c Then k m would be The unitary group of size r over c Then let mu m be the probability r measure So the invariant measure on this compact group k m With total mass one Then the discrete mailing transform or Fourier mailing transform associated to this coefficient object on this group uh as sums As complex numbers become equally distributed Uh essentially like the trace of a random element corresponding to mu m Of k m so Write it first in words and then the precise statement. So the discrete Fourier transform of the trace function of m on g of k n Are distributed When n gets large Like the trace of g Where g belongs to k m is distributed Like mu m Okay, and the precise statement is the following So for every function f On the complex plane Continuous and bounded If you look at the following average So it turns out if you just look at a single finite field uh You might not get convergence. That's something that's well understood. That's already arose in the theory of the lean And we go around that by doing an extra Cisaro average That avoids uh having to make unnecessary assumptions so Okay, so we average First over the given finite field Uh the values of the function f add these discrete mailing transforms. So these are just the sum a chi of x t m x k n So just the discrete Fourier transform that you might Expect for the given group g Okay, so and here is a chi Then we do these extra Cisaro average to avoid complications And these converges to the integral of f of the trace of g with respect to this measure mu Over k m So this is the statement. So again Summarize a little bit So we have a finite field. We have these extensions We have some group on which the characters define interesting exponential sums Like the multiplicative group Some kind of coefficient object whatever that is but for our instance exponentials of polynomials And then these become always equidistributed at least in this Average sense with respect to a very special kind of measure Because it's a measure on the compact the trace uh on on the compact group Okay, and so Should there be a normalization here That uh t the Fourier transform of t. Are you not should you divide by the square root of the number of terms? Right. That's actually that's hidden in this Uh asking that the coefficient is a perverse shift of weight zero It turns out that this automatically gives you the right normalization It divides by the square root of the dimension of the support of the trace function That's actually quite quite efficient and quite nice Because it could be that the trace function of m is supported on a smaller dimensional sub variety Like for instance the diagonal in in a product And then you don't want to divine by the square root of the size of the group But by the square root of the size of the diagonal and the Assumption that you have a perverse shift of weight zero whatever that means automatically incorporates this the right normalization Even even in the sum of a chi Yeah, it's exactly so there's a normalization here in the average of a chi i divide by the number of characters So that's that's you have to do But in the sum of a chi to to define the discrete Fourier transform And you have this m which has this property of being a perverse shift of weight zero You don't need to do the normalization by dividing explicitly by the square root of the number of terms Okay, so as I said, so delin proved this in the case of the additive group to any power really or even any unipotent group commutative unipotent group Which occur in some analytic applications through vid vectors and things like that uh, and cats Did that for gm? and Plus some abelian varieties and some and some coefficients With some extra conditions So here we have a really general statement Which is really quite nice. I think So they're still work to be done to when you want to apply it because you need to compute Uh, especially this group hm and this And this integer are but at least in principle This is saying that in the realm of this kind of standard type of Expansion of some parameterized by characters of algebraic groups Uh, there is always some a priori equal distribution One might need to make it explicit, but there is always an a priori one And that's that's a I think it's a first step towards kind of Uh, a more general form of delin's geometric Geometric free transform We cannot quite have exactly the same thing, but we're having some of its aspects That that still makes sense now In that is work you don't need to average over the The degrees which you have in this chaser, right? Yeah, but in cat's book, he always assumed that the geometric and the arithmetic monodromic groups are the same And if we had this assumption, we could also remove the cesaro average But we don't the point is that it's quite hard to compute these groups and Uh, if you want to make this extra assumption, that's something that's could be very difficult to check So here we have the advantage that we have a statement with no assumptions at all concerning the monodromic groups, whether the arithmetic or the geometric ones are the same or not But if we knew this a priori, so sometimes if we compute the group We can see that they are the same and then we can also remove the cesaro average Okay, so I don't want to to go too much over time. So now maybe I should say a few words about applications and then I want to discuss questions so applications So There are some to well Equal distribution, of course, if you can compute the group you get equal distribution And and we have various examples where we we can do that. That's Okay, so I should say maybe explicit equal distribution Uh, so we can improve quite significantly some of the results of Of the the Keating-Rudnik type. So maybe in particular results of all Keating and and Rodity Gershon So the point being that their results is about a question where the natural parameter space is multiple multiplicative characters And they were able to deal with it by using Katz's theory By specializing all but one of the multiplicative characters and letting just one vary But if you do that then you get worse results than just looking at the very the family of all the parameter parameterizing characters and applying our results And uh Again, we hope to have more and maybe even better example or better applications Later, but at least the the series there now and the applications can follow Now, uh, as I said, I want to discuss a bit about some questions Um, so maybe the most important one is okay. So what about a more geometric? Maybe I'll just discuss that one to avoiding too much time more geometric version in the general case Okay, so we saw that literally speaking if you want the uh, the discrete main transform for instance would be corresponding to some Gallery representation. It doesn't work because there's this L function that does not have the right properties um But still we have we have an L function and this L function as Is a rational function. So some properties are there. We have a distribution. We have convolution. So It's still tempting to think that maybe there is more. Maybe it's just optimistic But maybe there's still something on the other side And here there's a little hint that I find very very interesting. It's raviere who pointed out to us Uh, if you look at the delin theory, so let's say Viewed as being exponential sums Over ga so Fourier transform of the additive group And here we would have the the ones over Over gm now cats And others had noticed kind of results Which is really not any formal As far as I know any really formal statement, but there's an analogy with Uh differential equations demodules so linear polynomial linear differential equations the the kinds of things um Which Such that basal functions for instance are solutions of now I'll do it this way. So if you have Now exponential sums over gm They are still an analogy with linear differential equations over c star and and this as I said has at least been used By cats, uh many times now So what we want to do is we want to take, uh, the mailing transform now For differential equation, there's a very very well defined being transformed And what's the image of the main transform for differential equations? You get different equations difference Okay, it's very easy to give a concrete example. You take exponential minus x that solution of linear differential equation Then here you get the gamma function And it has satisfies gamma s plus one equals s gamma s Okay, now we are wanting to do the discrete mailing transform of exponential sums over gm or things like that and what we're asking here is Is there something here and therefore we should also ask is there an analog Can we go from difference equation? What's the kind of finite field analog of difference equations? It's clearly not an obvious thing because the the link between linear differential equations and exponential sums is of course not Obvious either But it's very tantalizing to think that something is going on here I mean at the very least we can ask the following question now Suppose we start with an object over gm where we see clearly what is the corresponding od Then we're going to have a different difference equation from mailing transform on the top It will have some kind of tanic and there is a group associated to that Is it going to be the same as the group that cuts associates to objects over gm? And then we can try to do the same with other Other groups instead of gm and again This is something which is very tantalizing and for the moment. We don't really know what what's going on And I think I will stop here, but I might be to take questions, of course