Yuri Tschinkel, Height zeta functions

00:05 - We’re still in this series of talks on rational points and Manin conjectures.

00:12 - And today, we’re very happy to have Yuri Tschinkel speaking to us about height zeta functions.

00:19 - And is it all right if we record your talk today? YURI TSCHINKEL: Yes.

00:24 - RACHEL PRIES: Oh, thank you. All right.

00:26 - Well, go right ahead. YURI TSCHINKEL: Thank you for the invitation and the introduction.

00:33 - So the plan for the talk– I’ll discuss some background and then review some analytic tools.

00:42 - And I wanted to show you one example– the simplest example– showing how these tools are applied.

00:51 - The basic problem that you’ve already heard of in this seminar is counting rational points on smooth projective varieties over number fields with respect to some height functions that are attached to very ample line bundles, together with a choice of metrics, heights.

01:12 - The counting problem is, how many points of bundle type do you have? And on suitable Zariski open subsets.

01:21 - And this, of course, is the subject of a series of conjectures by Manin, Batyrev, Peyre, and others, and different generalizations in different contexts.

01:31 - The main idea, of course, at the time this subject was introduced, was to connect new developments at that time– new developments in the Minimal Model Program to classical questions in analytic number theory– in particular, the circle method, the outcome of the circle method.

01:50 - So Marta and Will gave very nice overviews in previous talks touching upon torsors, special subvarieties, thin sets, freeness, and some geometric analogs of these conjectures, in the number-field case, replaced by a function field of a curve over over a finite field.

02:11 - So there are many, many interesting ideas and directions in this area– things inspired by certain topics in this overall framework.

02:25 - I just wanted to give you a very incomplete list of things that have not been mentioned as much in the previous talks, perhaps.

02:33 - So the theories of extensions to counting, let’s say, integral points– so you count integral points on quasi-projective varieties, these give a framework explaining those asymptotics– Campana points, more recently.

02:51 - There is a whole chapter– I don’t know how to call it– many, many papers applying ideas from ergodic theory, mixing, dynamical systems to these kind of counting problems, in particular, in the context of integral points on homogeneous spaces, so starting with a paper by Duke-Rudnick-Sarnak, and Eskin-McMullen, and then Oh, Mozes, Shah, Gorodnik, Maucorant, and others.

03:27 - There is a series of works exploring geometric consistency due to Lehmann-Tanimoto, and also Sengupta, more recently.

03:34 - There is a motivic framework where people talk about motivic height zeta functions– Chambert-Loir and Loeser, in particular.

03:44 - There are some very, very new ideas– homotopy theory ideas– due to Manin and Marcolli.

03:52 - They just posted the paper very recently. And finally, something that hadn’t been applied yet, or sufficiently, let’s say– namely harmonic analysis of spectral expansions on spherical varieties, where the analytic theory has been worked out by Sakellaridis and Venkatesh in their book.

04:15 - But I think applications to counting problems are still forthcoming.

04:22 - So most of my work in this area over the years was joint with Franke, Manin, Batyrev, Strauch, Chambert-Loir, Shalika, Takloo-Bighash, Tanimoto, and Gorodnik.

04:33 - So I will focus today on some very basic things that are common to most of our papers.

04:39 - So I will review the analytic tools algebraic tools, and then just discuss the very, very simplest examples to show you, at least in the simplest case, how these things are applied and used.

04:54 - All right. So what kind of analytic tools do I have in mind? Tauberian theorems, convexity into shapes and forms, harmonic analysis, p-adic integration.

05:07 - Then, of course, when you do the work, at some point, you have to prove convergence and integration by parts.

05:14 - Geometrically iterated residues in brackets because I won’t have an example of that.

05:22 - But there is actually pretty big machinery helping and complex analysis in higher dimensions can be very relevant.

05:35 - All right. So I’m starting from the sort of from the end.

05:38 - Tauberian theorems– refer to the following situation.

05:42 - You have some kind of generating series, a_n’s that are of interest to you, some kind of number theoretic function– let’s say divisor function or the Moebius function– well, it’s a divisor function.

05:53 - So then you look at an over n to the s. So I will assume an answer bigger than 0.

05:59 - Assume that the function is holomorphic for real part as big as something, has an isolated pole at s equals a, has some order b with leading coefficient c at that pole.

06:11 - And then you can conclude that the sum, over all the an’s, for n less/equal than B, is asymptotically– the constant that you see– it’s a leading pole over some kind of a gamma b, B to the a, and then log B to the b minus 1.

06:35 - All right. So this is a bridge going back and forth between what we’re interested in, primarily– namely, asymptotics of the number of points of bounded height and analytic properties of some generating series.

06:48 - This is a common thing. And there are all kinds of versions.

06:52 - You get better expression– you can write the polynomial of degree b minus 1 a polynomial in the logarithms if you assume additional information about the behavior of your function f of s in vertical strips a little bit to the left of the pole at s equals a.

07:12 - So OK. Now, how do we apply this? Of course, you’re going to apply this to the height zeta function.

07:19 - So the sum over all rational points– there are finitely many points of bounded height.

07:26 - OK. So HL is the height. This is sort of our…

07:31 - and then the analytic properties of this height zeta function will translate into asymptotic properties of, well, this N of B, which, in fact, depends on U, the appropriate subset that you pick in the set of rational points, on the metrized line bundle L that gives us a height function, and on B.

07:54 - So that’s a context. And I will come back to this in the application later.

08:00 - Another thing– when you hear convexity complex analysis, there are several shapes of convexity.

08:06 - Suppose you have a function that is– here, I write Z– that is a function that’s holomorphic in the tube domain over some connected open subset of R to the n.

08:20 - A tube domain means you take this V plus i times Rn.

08:25 - So you’re going up in imaginary in the full thing.

08:28 - But in the reals, you prescribe what it is.

08:32 - So now there is a beautiful theorem in higher-dimensional complex analysis which says that, then, you’re automatically holomorphic in the tube domain, which you claim by closing up– looking at the convex envelope, convex hull, of your domain V.

08:49 - So this is the kind of convexity that is used in the machinery of the height zeta function.

08:56 - How are we using this? Well, suppose you have a function maybe in several complex variables, and you think that it’s going to be holomorphic in the interior of some cone, and it’s going to have poles along the faces of the cone, and so on and so on, and you want to get meromorphic continuation, you want to get through the walls that describe the cone that you’re looking at.

09:21 - An application is going to be the cone of effective divisors.

09:25 - So the way you approach this, you look at– you analyze this function just in the small neighborhood of the vertex of the cone.

09:35 - And that’s enough. If you can prove meromorphic continuation just out of the cone at the vertex, it will automatically propagate towards the whole space.

09:46 - This is essential because of the second thing that I’m going to show you.

09:50 - Namely, in applications, when you do estimates or you have error terms, you have zeta functions where the main term– you want to control the error terms– appropriate bounds, frequently, you can obtain only very, very locally.

10:10 - And it turns out that this kind of convexity allows you to prove what you want just from this analysis of very, very localized.

10:20 - POONEN: So in that statement, you’re starting with a function that’s already– it’s already meromorphic or holomorphic inside– in the interior of the cone.

10:30 - Is that right? YURI TSCHINKEL: Yeah.

10:31 - So you start with something in the interior of the cone.

10:33 - You analyze the poles very, very close to the vertex– like, in the epsilon neighborhood of the vertex.

10:41 - And you can get epsilon as small as you like.

10:44 - And then, automatically, it will propagate along the whole cone.

10:50 - AUDIENCE: OK. You’re saying, because you have the– because the convex hull of the union of that neighborhood with the interior of the cone contains the neighborhood, it’s all closed YURI TSCHINKEL: Exactly.

11:00 - That’s exactly– POONEN: OK. YURI TSCHINKEL: –the point.

11:02 - And it’s important that you can make the– all the higher derivatives and so on and so on in analysis of toric varieties– all of this only works in a small neighborhood of the vertex.

11:17 - Now, the second convexity chapter refers to the following.

11:23 - This is what’s known as the standard in the Phragmen-Lindelof theorem.

11:26 - So suppose you have a function where you have kind of a priori bounds in vertical strips– some a priori bound that it’s not growing too fast.

11:41 - And suppose you can get a polynomial estimate, like t to the k1, on one vertical line.

11:52 - So for fixed sigma 1 and t varying, you have an estimate t to the k1.

11:59 - And then at the other end, you have an estimate t to the k2, all right? So then it turns out that, in between, for all sigma in this interval between sigma 1 and sigma 2, you get an estimate t to the k, where that k is a linear sink between k1 and k2.

12:23 - So you should think about it like this. So there is a linear function.

12:26 - So you fix its value in one corner of the interval, the other corner of the interval.

12:31 - You draw a line. And that’ll tell you how the function grows on a vertical line with a fixed sigma here.

12:41 - So this is, again, essential for estimates, computing integrals, and so on and so on.

12:47 - It’s a general theorem. How do we apply this? Well, we apply it to the Riemann zeta function to get– AUDIENCE: Can I ask you, is that s there on the previous slide– is that uniform in the– YURI TSCHINKEL: In– AUDIENCE: –sigma? Sorry.

13:03 - YURI TSCHINKEL: In the sigma– whether that O is uniform in sigma? I don’t remember the– let’s see.

13:11 - It might be. Well, it might be.

13:18 - I don’t know for sure. I have to look it up.

13:23 - AUDIENCE: OK. OK. Don’t worry.

13:25 - YURI TSCHINKEL: I taught it in my class– AUDIENCE: –I would have guessed, too, it should be uniform YURI TSCHINKEL: Yes.

13:30 - Yes. It’s– YURI TSCHINKEL: Yes.

13:33 - Yeah, I think it is. Yeah. You’re right, of course.

13:40 - But it is uniform, yeah. So this is applied to the zeta function in the neighborhood of 1.

13:50 - So the point is that you can make it as small as you like if you make sigma close enough to 1 where you get your first pole.

14:04 - So that’s essential, OK? Now, how do we prove this? You look at functional equation.

14:13 - You have, of course, the Riemann zeta function is absolutely convergent for real part s bigger than 1.

14:20 - For any sigma at, like, 1 plus something, you know that you’re bounded by a constant in a vertical strip.

14:30 - And then using bounds for the gamma function, you get the growth rate– let’s say, the line sigma equals 0, real part s equal 0.

14:40 - And then you just draw the line. And that shows you that– well, I think, at ¹⁄₂, you get, like, ¹⁄₄ plus epsilon, convexity bounded.

14:49 - At close to 1, you can get any epsilon to here.

14:56 - So this is used as well. Now, once you know it for the Riemann zeta function, it bootstraps.

15:03 - You can look at Eisenstein series where, in the functional equation, you find products of Riemann zeta functions and gamma functions.

15:12 - But the point is that, once you know it for the previous ones for the Riemann zeta function, then you get similar estimates for Eisenstein series– again, in the neighborhood of the poles, right? But the first time you hit the pole, for an Eisenstein series, you get similar estimates.

15:31 - And then you can iterate and iterate. Next chapter.

15:36 - Fourier analysis. So we have a locally compact abelian group.

15:40 - We have some character. We have a closed subgroup.

15:44 - And then Poisson summation formula reads, integrating your function over a subgroup is the same thing as integrating the Fourier transform over the dual thing The dual thing are all the characters of G which are trivial on H. And the Fourier transform is, well, with the right chi or chi bar.

16:04 - It doesn’t matter. And now, of course, measures.

16:08 - You have to measure the h here. You have to measure the g here.

16:11 - So you have to suitably normalize the measures everywhere.

16:16 - So the issue is when you do these measures, but, of course, integrability.

16:19 - You can’t just put in anything you like. You have to have some conditions on the function that make sense.

16:27 - So how is this applied in situations of interest to us? Well, so G is either the additive group or the multiplicative group, and X an equivariant compactification of G.

16:41 - So then it turns out that you can extend the height function from rational points, from G of k, to adelic points, G of Ak.

16:51 - So then you write down a height zeta function.

16:54 - You sum over all x in G of k. But you fix the G. And so, in this way, you get a function, well, for real part as big enough, which is actually in L2, GA mod Gk.

17:10 - So here, it’s completely abelian for now. You can write down the Poisson summation formula.

17:16 - And then you can rewrite this in the form that I described– some of the Fourier transforms of these things.

17:26 - But before I go there, it turns out that you can do it in much bigger generality.

17:34 - You don’t need to assume commutativity of your linear algebraic group.

17:37 - So X is a smooth projective compactification of any linear algebraic group.

17:43 - So we can assume that X is a smooth projective.

17:45 - D is a boundary. We can assume it’s normal crossings after various blow ups and so on.

17:53 - So it turns out that, in this generality, you can define something like a height pairing, comparing between adelic points and complexified– write in terms of sum over a certain product over all the alpha– C of the alpha.

18:12 - And what it is, this thing that I defined there– locally, it’s just a distance to the boundary component D alpha.

18:20 - So let me put it like this. So when you write, locally, your D alpha is a varnishing of some coordinate– let’s say X alpha equals 0– then p-adically, you can simply write the distance to the boundary as well– its p-adic value of X alpha.

18:44 - And you call it H of D alpha– local height with respect to the alpha, or something like this.

18:51 - And then you take a product over all the components.

18:53 - So this measures the distance to the boundary locally, p-adically and over the reals, complex numbers, but also adelically.

19:02 - Now, once you have that– so now we have a function– several complex variables labeled by the boundary components, and again, the sum over all rational points.

19:12 - And again, we can expand spectrally. Now, if the group is GLn so OK.

19:23 - When I write sum, even for the multiplicative group, it’s not actually a sum.

19:28 - There are integrals of this. But let’s say it’s formally a sum over irreducible unitary representations.

19:36 - And then you project your thing onto representation spaces.

19:39 - And you can now analyze these Z pi’s. And that should be enough for now to say what it is.

19:49 - So if we expand it in Fourier coefficients and then in spectral expansion.

19:56 - But there is one representation that sticks out.

20:00 - And that’s the trivial representation. And so what I would do for the trivial representation– we’re just computing the integral– that heights that we have defined times dg.

20:11 - And the hope, of course, is that this contribution from the trivial representation gives the main pole.

20:24 - And then by Tauberian theorems, the main asymptotic term for the counting problem– so that’s the hope and expectation, that this is the most important thing in the spectral expansion.

20:43 - Now, when I put it like this, you see it’s analogous to the idea that you count lattice points in some domain.

20:52 - And if the domain is nice enough– let’s say it’s a circle, a ball– then the number of lattice points should be equal to the volume.

21:00 - So this is what this stands for. And of course then you do classical Fourier analysis like R mod Z, and you pull out the first term in the Fourier expansion.

21:12 - This is what it is. The first term is the volume of the domain.

21:17 - And all the things that you’ve heard about the applications of the torsor method, the Manin’s conjectures– so all of this is showing directly, using different techniques, that the number of lattice points on the torsor is equal to the volume of the domain that you have on the torsor.

21:38 - But why is this progress– we had some Dirichlet series, you know, height zeta function.

21:52 - And then we pull out some kind of p-adic integral.

21:54 - Well, what do we do with it? Well, it turns out that, well, it’s an adelic integral.

21:59 - It’s a product of p-adic integrals. But it turns out that integrals of this type can be computed in closed form using just the geometry of the boundary.

22:09 - So you have the boundary components, all right? And you can write down an answer simply in terms of what you see in the boundary.

22:18 - Now, again, this is a great thing. You can have a closed uniform expression for almost all places– finitely many places where the issues of bad reduction and so on and so on can be ignored most of the time.

22:37 - And so this is actually a great step and stepping stone or block of the papers that you write, you extract– you regularize what you see for a trivial representation.

22:58 - So this integration– adelic integration– splits naturally into p-adic integration.

23:06 - And there is a paper– a long and fundamental paper– that Antoine and I wrote, “Igusa Integrals and Volume Asymptotics. ” So let me just briefly talk through the main things in there.

23:21 - So as I said, what we’re integrating are functions that are sort of [INAUDIBLE] to the boundary functions.

23:30 - Now, of course, you have to talk about measures.

23:35 - Measures, measures. These functions are related to their local heights.

23:42 - They come from the metrization of your line bundles and so on and so on.

23:46 - But what’s relevant for the evaluation of the integral is the following.

23:52 - You see, if we are working a non-closed field.

23:55 - So the boundary strata– they could be irreducible over the non-closed field, but geometrically reducible.

24:02 - So in fact, they’re seeing Galois orbits of the boundary.

24:06 - So this is the algebraic stratification that they’re seeing.

24:09 - Of course, the Galois action is going to preserve the boundaries.

24:14 - So you see the boundaries permutation modules, so to speak, you know orbits of the Galois group on these components.

24:25 - But now what happens is that, you see, when you take a component defined over your field– local field– let’s say it’s the reals or the p-adics– you see, the component could be defined over the field.

24:36 - But it may not have points over that field.

24:39 - And that’s also relevant for the computation.

24:41 - So you could have a conic over the reals. There’s no real points.

24:46 - And that could be part of your boundary. And so there’s a whole language that takes into account– we call it Clemens complexes.

24:54 - You have the algebraic Clemens complex that keeps track of the boundary and all intersections and mutual intersections of intersections– so these strata DA and then the open pieces of the strata, OK? And then you have to keep track of the analytic structure, whether or not these pieces have points locally.

25:16 - Anyway, there is a way to package everything and understand everything.

25:20 - And this is what we call the geometric Igusa integrals.

25:24 - And we have a whole framework dealing with this.

25:29 - All right. So now [INAUDIBLE] loaded into the machine, what’s left? So we are analyzing these height zeta functions, right? So we have to match analysis with geometry.

25:43 - So the geometry, as was explained, I guess, in previous talks, says that the height zeta function should be holomorphic, and the cone of effective divisors shifted by the anticanonical class.

26:00 - Now you have to extract it from analysis. You have to match what you’re seeing– the complex analysis– with your conjecture.

26:09 - Do the terms that you obtain– do they actually– and this is a very, very nice and strong interface in that, of course, when you do examples, you make mistakes.

26:21 - But, well, if it doesn’t match what you’re seeing here, then you match it its there.

26:26 - So anyway, it’s self-balancing. It’s self-checking because these things come from very, very different corners, so to speak.

26:36 - And the conjectures say that they have to match, so they have to match.

26:39 - So you run your computations until it matches.

26:42 - Anyway, you have to match the zeta function and complex analysis with the prediction of the Manin Batyrev conjecture.

26:51 - But then you have these formal expressions– spectral expansion, Poisson summation formula.

26:58 - And you have to show that it converges. And that’s actually very non-trivial.

27:02 - So even if you’ve correctly identified the leading term, and you know all the poles, and you say, well, this is it.

27:09 - I’m done– but wait a minute. What happens to the rest when you sum over all the other Fourier coefficients? Well, each individual, of course, could have a smaller pole or a pole more to the left.

27:24 - But all of them together– who knows? And that’s why uniformity that Bjorn mentioned, is important and crucial.

27:33 - So that’s a description of the general machine and the general framework.

27:40 - Now I want to give you one example– the simplest case– namely, a compactification of the affine line by p1.

27:49 - So I’m going to show you how to count points on p1 using these kind of tools.

27:55 - And I’m going to do it step by step. So of course, we’re going to look at characters of the adele, so you have to start with local characters– the character at infinity you know what it is, its, you know, e 2 pi i ax.

28:09 - But what about p-adics? OK. So we’re looking at characters of the p-adics.

28:13 - We have some kind of p-adic a, p-adic x. And then we can take out its– whatever it is in Zp.

28:21 - And then once you take out the p part, you’re left with a rational number.

28:26 - And you know what it is. 2 pi i into a rational number is clear what it is.

28:31 - So this is what your character is, p-adically.

28:36 - So you have some kind of characters. The Qp, Zp are self-dual.

28:40 - You have Haar measures. You normalize Haar measures so that over the p-adic integers, you get 1.

28:45 - Multiply this– YURI TSCHINKEL: –p to the n, you get 1 over p to the n.

28:48 - And you look at the Zp star. You take 1 minus 1 over p.

28:53 - AUDIENCE: Can I ask what is the meaning of when you say Zp is self-dual? What does that mean? YURI TSCHINKEL: Well, so you’ll see, it’s like– all right.

29:03 - So Z and R are self-dual. When you look at the characters that act trivially on Zp, they happen to be labeled by Zp.

29:12 - That’s what I mean by self-dual. AUDIENCE: I thought the characters are indexed by Qp mod Zp.

29:17 - YURI TSCHINKEL: Yeah. So– AUDIENCE: The dual of Zp is Qp mod Zp– YURI TSCHINKEL: Yeah.

29:20 - Yeah, you’re right. Yeah, you’re right.

29:21 - Yeah. So Qp mod Zp. That’s right.

29:24 - But yeah. AUDIENCE: OK. OK.

29:26 - YURI TSCHINKEL: So now, they also put this measure at infinity.

29:31 - We have some measure on the adeles. We have a character from here to there.

29:36 - We have a product of these characters– local characters.

29:39 - So we get our adelic character. So now I have Q and AQ.

29:46 - And now I have self-dual, what I wanted in the first place.

29:51 - So the characters of A trivial Q are Q. All right.

29:55 - The Poisson summation formula, therefore, takes the form, we sum over rational points– sum over Q, f of x.

30:04 - And that’s a sum, again, over rational points– f hat of a.

30:08 - So f hat of a is an integral, over the adeles, f of x character.

30:18 - Now, of course, it’s just a form of expression.

30:21 - So you need convergence. You have to have convergence here, convergence there, and so on.

30:26 - Now let’s come to our counting problem– geometric counting problem.

30:29 - We have P1 of Q with representative x0, x1, primitive nonzereo 0 up to plus/minus 1.

30:40 - The height function applied to this representative of rational point is simply square root of x0 squared plus x1 squared.

30:48 - So the counting function is x0, x1, height less/equal than B.

30:53 - And so we want the asymptotic for that. And that’s, of course, nothing but the Gauss circle problem– counting lattice points on a circle with a slight twist that we are now looking at lattice points with coprime coordinates.

31:06 - So we’re assuming that we’re looking at primitive vectors in there.

31:10 - And you’re not 0 or excluding 0. All right.

31:15 - You’re translating this into the adelic language.

31:19 - We have a product formula. And the coprimality tells us that if you write it for the representatives that we looked at– x0 and x1 coprime– it just means that the maximum of the p-adic relations is 1 for all p.

31:37 - So once we know that– so in other words, now we can rewrite the problem as follows.

31:47 - So let’s introduce these height functions. We have the local height function Hp is simply maximum 1 and xp.

31:54 - And this x is going to be a rational number.

31:57 - It’s going to be x0 over x1 or so. And H infinity the thing at infinity is 1 plus x squared.

32:05 - And so this is exactly the same problem that we had before.

32:11 - But the advantage is, before, we had to fix an integral representative for the rational point.

32:16 - But now we don’t need to do that. X is simply a rational point.

32:21 - P1 of Q is a complement to the point at infinity simply A1 of Q.

32:27 - So now when you look at it like this, you realize that the local height Hp is actually invariant under translation by Zp, right? So you can add anything, like a p-adic integer, and nothing is going to happen to this local height.

32:44 - So that’s important. So now we want the asymptotic of points of bounded height.

32:49 - And now we’re looking at all x in Q rather than x0, x1 [INAUDIBLE] square.

32:56 - So x in Q so that H of x is less/equals than B. And H is this.

33:01 - All right. So we look at the zeta function.

33:05 - It converges absolutely to a holomorphic function.

33:08 - Real part is bigger than 2. That’s easy to see.

33:12 - And then we write a Poisson summation formula.

33:15 - And so we had a sum over rationals. And again, we get a sum over rationals.

33:20 - And then you ask yourself, well, what did we gain? Why is this sum better than this sum? And the point is that– and of course, H hat is simply the Fourier transform of the height function with respect to the character.

33:36 - So why is it better? We started with a sum over Q. We have a sum over Q. The point is that Hp is invariant under Zp.

33:48 - And that happens because p1 is actually an equivariant compactification of the additive group.

33:55 - And this always happens when you have an equivariant compactification of a group or a homogeneous space that the height functions that you get– local height functions– they’re actually invariant under the action of the complex subgroup.

34:11 - And so therefore, the only characters which survive in this expression here– the only characters psi A that matter are those where this A is a p-adic integer for all A.

34:29 - Well, if you’re a p-adic integer for all A, you’re an integer.

34:33 - So A must be in Z. So rather than summing over Q, we are actually summing over Z.

34:42 - And why is this great? It’s because, now, if isolated the trivial representation corresponding to A equals 0 from all the others.

34:52 - So it’s the lattice over which we are summing, not Q. You don’t have representations accumulating to the one that you’re looking at.

35:03 - So let’s analyze it one more time. So we have the trivial character, which is this integral.

35:10 - And then we have the nontrivial characters.

35:12 - So let’s look at what happens for each of the terms.

35:16 - So we want to integrate our function. And for that, we introduce some little tubes into our domain.

35:27 - So U0 corresponds to xp less [INAUDIBLE] x.

35:35 - And then we fix the p-adic valuation of x. So this is the volume that corresponds to this domain– this little tube there.

35:45 - And you integrate the local height against Qp.

35:49 - Well, it’s a constant equal to 1 if xp is less or equal than 1.

35:58 - So this is just integrating 1 over U0. And now we integrate to cover all j bigger/equal than 1.

36:05 - And so what is it? It’s just 1 plus this series, p to the js– p to the j is our height– and the volume of this domain, Uj.

36:17 - And then, well, you know how to sum these things.

36:19 - I give you the volume. This is what comes out.

36:24 - And now, of course, at infinity, we’re just integrating this.

36:27 - And it’s some kind of gamma functions. It’s something.

36:31 - So the Euler product gives you zeta s minus 1 over zeta s, and then times gamma sinc.

36:38 - And where is the pole? It’s at s equals 2.

36:40 - And what’s a leading term at this pole? 1 over zeta 2.

36:44 - And of course, now that you think about it, they’re interested in primitive points.

36:49 - Primitive points are those where not both coordinates x0, x1 are divisible by p.

36:55 - So the probability is, like, 1 over p times 1 over p.

36:58 - It’s 1 over p squared. The complementary probability– 1 minus 1 over p squared.

37:03 - You take a product over all p’s. You get exactly 1 over zeta 2.

37:07 - So this is how you get that part. But of course, that integral here– that will simply give you the volume– the area of the circle.

37:17 - So now, here, again, the characters unramified– Qp mod Zp, as Bjorn pointed out.

37:26 - So that’s your characters. Unramified– we are trivial on Zp.

37:31 - Those are the only characters that matter. And then we have these kind of dualities, characters trivial on Q [INAUDIBLE] so A is in Z.

37:42 - So when A is in Z, there are some primes that divide A. Those are bad primes.

37:49 - Let’s not think about them for now. But let’s suppose that p does not divide A, and let’s try to compute our local integral.

37:57 - Of course, the character’s unramified. The first integral– it stays.

38:01 - You’re integrating the function 1 over Zp. You get 1.

38:06 - Then this is a height– p to the minus sj. This is the sum over all those little tubes.

38:12 - And then we have an integral of this character, and we want to understand what it is.

38:18 - And it turns out that, here, only the first term survives and contributes minus 1 over p to the minus s.

38:26 - So let’s do this. Let’s look at this proof very quickly.

38:31 - So rather than looking at Hp of x equal to p to the j, let’s look at the set where we are less/equal than p to the j.

38:40 - And then let’s try to integrate our character phi over that set instead, all right? Well, of course, you do the usual thing.

38:50 - You change coordinates. And then what happens is that if p doesn’t divide a, you’re integrating a a non-trivial character with complex roots.

38:58 - Therefore, you get 0. But for the first term, you get 1.

39:02 - And so therefore, when you look at V1 minus V0, well, you get 0 minus 1, depending on i bigger/equal to or equal to 1.

39:14 - And that gives you that integral. So it’s really basic exponential sums.

39:21 - They’re basic character sums. Now, when p does divide a, we don’t want to compute, because, I don’t know, maybe p squared, p cubed divides A. Who knows? So we just estimate.

39:32 - So at finitely many places, you write some kind of estimate.

39:35 - It doesn’t matter. You simply replace the character by 1.

39:39 - And then we get something like this– p divides a, 1 over p to the epsilon.

39:44 - And here, you can get a bound, like logarithm 1 plus a– something like this.

39:49 - I should have written absolute value of a because– well, anyway, it’s a positive integer.

39:53 - But in our applications, a, of course, is all integers.

39:57 - All right. So here, I write absolute value.

40:00 - So OK. Now what do we have? The main term, we understand.

40:05 - And this is the rest of the Fourier expansion.

40:09 - It’s a sum over all a. It’s a product over p not dividing a– something completely uniform, like 1 over zeta p– and then a product over p dividing a of something that is– who knows? But we can estimate this.

40:27 - And then we see this. And so now the issue is, OK, we can estimate this product over the bad primes and you know, say, 1 plus a to the delta it doesn’t matter.

40:41 - Anything will work. But we need to show that the sum over a converges.

40:47 - Of course, this 1 over zeta s, where we look at s equals 2– of course, this is holomorphic.

40:52 - There is no issue. So we have a holomorphic piece.

40:56 - And then how do we know convergence? And how do we know convergence of Fourier series, in general? You want to do integration by parts.

41:06 - Just two seconds. Low battery. So you do integration by parts.

41:12 - And you do it as many times as you like. So you put the e to pi i under the dx.

41:18 - And then you take the derivative of this. And the beauty is that when you take the derivative of this, what comes out is something that’s, again, like this.

41:28 - And this is a completely general fact. So for arbitrary compactifications of additive or multiplicative group and linear algebraic groups, when you have a vector field going towards the divisor dj, and you integrate with respect to– integration by parts with respect to this local derivative– then what comes out is, again, going to be integrable, if the previous thing was integrable.

41:53 - And the local height is integrable. As you can see, we’re looking at s equals 2 around here.

41:59 - And so, all right. When you do it as many times as you like, you pull out that a.

42:07 - When you put it under the dx, of course, you get 1 over a.

42:11 - So you get as many 1 over a’s here as necessary.

42:15 - So here, you can get an exponent that’s any N.

42:21 - And here, you have some fixed delta, which is small.

42:23 - And so sooner or later, this thing will, of course, take over this.

42:29 - And it’s going to show you that the rest of the Fourier expansion is simply holomorphic near your first pole.

42:40 - And so that’s the main term. And then your Tauberian theorems kick in.

42:45 - And all is good. So this is the simplest case, OK? And it’s elementary, in a sense.

42:55 - But once you’ve done it, and you’re used to geometry and working with local charts and so on and so on, it can be done much more generally.

43:14 - And so here is a list of results that happened over the years.

43:18 - Of course, the first paper was some flag varieties height zeta function [INAUDIBLE] analytic properties are known.

43:25 - This is the work of Langlands. Strauch looked at twisted products of different flag varieties.

43:32 - So then Victor Batyrev and I– we looked at compactifications of tori equivariant horospherical varieties with Strauch.

43:40 - Additive groups, bi-equivariant compactifications of unipotent groups quite generally, Heisenberg group, compactifications of semi-simple groups of adjoint type, De Concini-Procesi varieties, and so on and so on and so on.

43:55 - So the method of harmonic analysis applies in many situations and establishes all of the conjectures, not just the one about anticanonical heights.

44:11 - You can get it for the other heights. And the advantage, let’s say, over the torsor method is that once you know how to prove it, it works over all number fields and very very generally.

44:23 - Now you can ask, well, what’s left? Well, any open problems? So here is an open problem.

44:28 - Look at the ax plus b group, an extension of Gm by Ga.

44:33 - And we don’t know the full conjecture for arbitrary compactifications of this group.

44:39 - It’s a surface. We have some partial results with Tanimoto, but the general case is still open.

44:48 - All right. I think just in time– 1:45.

44:51 - Thank you. [APPLAUSE].