 All right, so thanks to everybody who came to this last lecture. So let me remind you where we stopped last time, and I'll try to also get to what we'll try to do today. So we defined Ziegeltransform, so we had the function f from rd to c, and the Ziegeltransform took a function like this, and it moved it to a function on the modular space of lattices, so g mod gamma, the modular space of the unimodular lattices in rd, unimodular meaning that they have covolume 1, so lattices, I'll call the lattice delta. And so what were the Ziegeltransforms, so there are several different flavors, they're essentially equivalent, and if you can prove something about one, you can prove something about the other ones, but the relationship between them is going to be useful, and so it's good to have names for all of them, so the main one was the full transform, if you give it a lattice, it just sums the function at all the vectors in the lattice, including the zero vector, then we had the reduced one where we summed over the same collection of vectors except for the zero vector, so all lattices have the zero vector, so we can just forget about it, and finally we had the primitive version where we summed only over the primitive vectors, so remember primitive meant that no other multiple of the vector except maybe minus the vector, it was in the lattice, no other smaller, shorter vector proportional, but smaller vector was in the lattice. Okay, so there are two properties, so let's write like this, so the Ziegeltransform was any of these three, essentially it took functions, let's say compactly supported functions on RD, and it mapped them to continuous functions on g mod gamma, and if we took the reduced Ziegeltransform, it preserved up to some scalar multiple, so here we have Lebesgue measure to a one of g mod gamma with harm measure, so it was a bounded function in this sense, and the other important fact was that we had an action, so both on RD and on g mod gamma, we had an action on both of these spaces by g, which I remind you is SLDR, and this transform was equivariant, so we could act on a function here and then move it by the Ziegeltransform, and it would be the same as just moving it by the Ziegeltransform and then acting on this side, so this we had this equivariance property which was important, and we said that we can also consider the action on measures, so a measure is something dual to a continuous function, if you have a continuous function, and if you have a measure, it gives a number to every continuous function by integration, so since there are duals, the measures are going to go the other way, so if you have a measure here, you can produce a measure on RD, so action on measures, so if mu was, let's say, finite or, well, let me say just reasonable because finite is not enough, g mod gamma, then we define the new measure as a measure on RD by the following formula, we said that, so a measure has to take a function from RD and give you a number, so it takes a function and it first moves the function by the Ziegeltransform and then applies the measure on the other side, so again, this is, so we got this transpose maps from measures on, so reasonable measures on g mod gamma to reasonable measures on RD, and so the reason why I'm being vague about what's reasonable is because we're only going to be interested in a particular measure on the left side, so on the left side, you have a very natural measure given by Har measure, so what we have is, so we're interested in the Har measure on the side, where does it move over here, and so we said that, so we know it is SLDR invariant because it's a SLDR invariant on this side, so it will be, when you move it and this preserves the SLDR action, it will be SLDR invariant on the other side, so we would like to understand what is, what is it, and on the other side there are only two SLDR invariant measures, so it is going to be either, it should be some linear combination of the Dirac delta mass at zero and Lebesgue measure, so what we would like to understand is, first of all, what are these numbers, what are these coefficients, and yeah, so okay, so the first fact is that, so and you can ask the same question for any of these three transforms, you can ask where does each of these transforms take Har measure, and there should be some relatively simple relationships, so the reason why the full Ziegel transform is useful, so I mean, just notice what I said, so same question for the primitive, the reduced Ziegel transforms, where does, where do they take Lebesgue, Har measure on the left side, and so the first claim is that if you take the full Ziegel transform of the Har measure, then it's going to be the same linear combination of delta mass and Lebesgue measure, so in other words, if you use the full Ziegel transform, then you get everything, then you get the same number on both sides, so you have the symmetry, so why, why is this so, so the proof, so remember we had the Poisson summation formula, which went like this, let me write this here, so it said that if you take the sum, so this is true for any, again, let's say compactly supported function, if you sample it over all vectors of a lattice, it's the same as sampling the Fourier transform over the vectors of the dual lattice, and the Har measure on G mod gamma is invariant by this evolution, delta goes to the dual lattice, delta dual, and so this means that Har measure, for example, is preserved by this evolution, but on the Rd side, so if you look on Rd, Fourier transform, so Fourier transform of the Dirac delta mass is Lebesgue measure, and the other way around, if you take your Fourier transform of Lebesgue, you get the delta mass, so this means, so you get that whatever this, the Fourier transform of Har measure is, it is invariant by Fourier transform, and since you know it's some, some combination of these two guys, these two guys have to have the same coefficient, okay, so at least we know that there's just really one number that we care about, so the other claim, the other two claims which are easy to check, but will require a little bit more time, so I'm not going to do it, but at least morally it should be quite obvious, is that if you take the, let's say, the reduced Eagle transform, so really the reason you get this delta mass at zero is because when you define this Eagle transform, you're sampling over the zero vector, so if you don't sum over the zero vector, so if you take, if you transform the Har measure with respect to reduced, so without summing over the zero vectors, then you are just going to get, this is just going to be proportional to Lebesgue, and the same thing for the primitive vectors, in other words, there's no Dirac delta mass at zero, and the reason, the way you can see this is as follows, so what you can do is you can test on some functions, right? So the reason why we, so far we're just doing formal manipulations, but the reason why these concepts are useful is because you can actually test them on functions, and the function you can test it on is a bump function at the origin, very small, a small function continuous of integral one that's supported in the neighborhood of the origin, then if you look at this, for most lattices, this is going to be zero, because most lattices don't have short vectors, and so you get that this, whatever this transformed measure is, it does not have much, too much mass near the origin, and as you shrink the neighborhood of your support, of your bump function, you go, you see that it goes to zero, so you see that the transform measure gives no atom to the origin, so that's why you don't get any atom, you don't get any delta mass. Okay, so now we really have just one number that we would like to understand. We would like to understand this, well, we would like to understand, yeah, we'd like to understand this constant factor, and I'm going to first give you the claim, and then, let's see, yeah. So let me say the following. So, we're going to have a claim, and then from this claim, we'll see the, I don't know if I want to number these, maybe I'll say claim one, is that if you take the, which way do I want to, sorry, let me try to organize them the right way so that we don't have to prove the same thing many times. No, okay, so let me say the following, that forget about the claims. So, first let's look a little bit about the relationship between the primitive and the reduced Ziegels transforms. So if V is in the lattice, let's say non-zero, then there exists a unique, let's call it L and V prime, where L is in, is a natural number, and V prime is in the primitive vectors, such that L times V prime is equal to V. So this is just saying that if you have a vector, there's a unique rescaling of this vector that makes it primitive. So if, let's say, V is a one, a d, then L is the GCD of these numbers, and V prime is just a one over L, a d over L. Okay, so this is simple, this is just the definitions essentially, but we have the following exercise that you have the vector, let's call it V zero, one, zero, dot, dot, dot, zero, and then you have the group gamma SLDZ, and the claim is that for any primitive V prime, and V inside Z to the D, so for any vector, any primitive vector like this, there exists gamma and gamma such that gamma times V zero is equal to V. So it says that if you have a collection of numbers like this, which are relatively prime, there's a matrix in SLDZ with this as the first column. This is what it's saying, that you can always complete a vector like this with relatively prime entries to a D by D matrix such that the determinant is one. So this is your exercise. So this tells you that basically, so the last two things I wrote here tell you that the lattice is essentially composed of two things, of the primitive vectors and then successive rescaling of the primitive vectors. So this means that the lattice delta is the union, sorry let me call it delta minus zero, is the union over L natural number, L times the primitive vectors minus the zero vector. So it's the union of such rescaling. And so this tells you that the reduced and the primitive Z equal transforms, there's a direct relationship between them. So if you denote by M lambda, the operator of just multiplication by lambda, X goes to lambda X, then there's a way to connect these two guys. So the Z equal transform is just a summation where you apply the primitive Z equal transform, but then before applying the primitive Z equal transform, you rescaled the function so that you sample it on the larger vectors, a rescale of multiple of the vectors. Okay, so this is good. So now we want to understand how these things, so let's see how they act on measures. So if you take the transpose, so the this is going to be the sum of the T, right? If you take transposes, you should reverse the order. And so, so what do we get? So since if you apply the primitive transform, the harm measure, you get Lebesgue times some factor A times Lebesgue. And what does multiplication by lambda do to Lebesgue measure? What happens if you multiply by lambda and you apply that to Lebesgue measure? How does it affect it? So it multiplies it, but how much? If you multiply, if you have a transformation in the plane, you multiply everything by lambda. What does it do to Lebesgue? Sorry? So it's going to be lambda to the power minus d, right? So the question is, is it plus d or minus d? It's going to be minus d because take a circle of radius, a sphere of radius one, you multiply to make it larger and that has volume one. So this is smaller. So this is lambda to the minus d times Lebesgue. So this series now tells you that if you apply the reduced transform to a harm measure, then it's the same as applying the primitive transform and then scaling it. So this is just going to be one plus d to the minus, sorry, plus 2 to the minus d, plus 3 to the minus d, oops, plus and so on, plus the primitive applied to harm, which is Lebesgue measure. So we know that this is Lebesgue measure. So you get this, so you get that, you get exactly this factor which looks exactly like the zeta function. So you get that exactly as zeta of d times the primitive part, if you apply it to harm measure. So now we just have to compute the primitive transform of this harm measure and then we'll see what happens. So there are two claims. Let me make the following two claims. So if you take the primitive transform of harm measure, so this is claim one, then the claim is that you get the volume of a lower dimensional quotient times Lebesgue on RD. So the claim, yes, so the first claim is that if you just transform the primitive harm measure, you get a lower dimensional volume, which you might hope to know, and then times Lebesgue measure. So this factor is the slower dimensional volume and the second claim is that if you, so this is what I should have done in the very beginning. So if you take the, just the full, you take the full transform, so this is including the zero vector, you're going to get, so the factor that we wanted is the volume of G mod gamma delta zero plus Lebesgue on RD. And it's also the same as volume of G mod gamma times delta zero plus the transform, the reduced transform of harm. Okay, so let me try to discuss first claim two and then, before actually discussing them, let me show you that claim one plus two give the following recursive formula, SLDR mod SLDZ. It's data of D times the volume of the lower dimensional one. And so you get the promised formula that this is just going to be the product of the Zeta functions, the product of the Zeta values. Okay, so why is the claim one and claim two give you exactly this? It's because, well, we said that the primitive one gives you the previous guy and then the reduced one is the primitive times the Zeta value and the full transform is just the volume of G mod gamma times this. So you get that the volume of G mod gamma is equal to Zeta of D times this. So if you put these three, these formulas together, you get this recursion. Okay, so claim two is a little bit easier. So I'll just maybe say it in words. So there are two parts. This was from, so there are two equalities. So which one am I proving? So this, can you say it louder? I can't say it here. Yes, so one and two are, so these are independent of the normalization of hard measure. Yes, this one. No, but this is claim two, but there's claim one, which is dependent on the normalization of hard. So claim one, so I'm saying that claim two I should have done in the very beginning and I apologize. As you say, claim two does not depend on the normalization of this. If I scale this side by lambda, this scale also scales by the same thing. So claim two is true in general. Claim one is what gives you a normalization. So the content of claim two is just that first these two coefficients are the same. And the first coefficient is the volume of g mod gamma. So let me clarify claim two again is that the coefficient delta not in, if you just take the hard measure is the volume of g mod gamma, which is independent of any normalization. And the other part is that the coefficient of delta not is equal to the coefficient of Lebesgue or ST of hard. So it's just these two claims, which are scale invariant that they don't depend on the normalization. So this one, this equality was done using Poisson summation, which I did earlier. This part, so why is this, so well, yeah, so this is, let me call it A. Yeah, so A followed essentially because, so again, you take a small bump function zero. And if you compute it, you see that this function is essentially, because you're using the full summation over the full thing, you're going to get exactly the, that the function is going to be, so you test the, you'll see that it should converge to the volume of, you call this f epsilon, right? Then, then basically this is going, you should, sorry, and by bump f epsilon is one at zero and supported in epsilon neighborhood of zero. So then, if you take this f epsilon and you test it again, so you compute the Ziegler transform, so Ziegler f epsilon is equal to one at most lattices. So then the integral of s f epsilon with respect to R on g mod gamma is approximately the volume of g mod gamma. But at the same time, this should equal the volume of the transform, which is the delta mass, but this is the same as, you know, the f of, f of zero approximately should be approximately f of zero times the coefficient which we're interested in. Okay? And so this coefficient has to be this volume. Okay? Now, I've been completely honest with everybody, up to these approximate sides, which you can estimate here. Okay, so the upshot is that we just have to do this computation. So we have to transform the primitive hard measure. So what is it that we have to do this? So we need to compute, so for claim one, integral over g mod gamma of the integral of, sorry, of the summation of vectors and primitive vectors f of v. This is what we have to compute, right? We have to take a test function and integrate it on g mod gamma. So over all lattices, you, and then you start summing over these vectors. Okay? So remember, we had this structure. So we had something called gamma p, the stabilizer one zero zero in gamma, which is SLDZ and GP to be, again, the stabilizer of the same vector one zero zero in G, which is SLDR. So you see that these guys are just, so GP is just SLD minus one R, semi-direct R to the D minus one. So these are matrices of this type. So it's one stuff here and then D minus one by D minus one matrix. Can people see, should I write this, that left corner, that corner again or can, this I said earlier, this is by this Poisson summation. So the equality of coefficients is by this symmetry. Okay. So we have to, so let me know the thought. So in particular, this implies that the primitive vectors, so by the exercise I gave earlier, this is just, let me see on which side I want to do the quotient. Does it matter? Gamma mod gamma p, right? So they're identified with this kind of cosets because gamma acts on the lattice and the stabilizer of a primitive vector. So we're really summing over this coset. So now let's define, so let's observe that G mod of gamma p is equal to the moduli space of, so if I took the full G mod gamma, then I'm taking a lattice and I'm forgetting the basis. Here we're taking a lattice and we're not forgetting the basis, we're just forgetting the last d minus one vectors and the basis. So this is the moduli space of lattices, delta tilde, with a distinguished vector, primitive vector. So if you like it's delta tilde and v, so it's a lattice, delta tilde and then you have a distinguished primitive vector in that lattice. Okay. So you can see now that this moduli space is kind of, it's a nice moduli space, mod gamma p, it maps to rd minus zero by just a straightforward evaluation, which is if you have the sliders delta tilde v, you just map it to v and it also maps to the full moduli space G mod gamma. And so now you notice that the integral that we have over there, so we have this integral over G mod gamma of the summation of f of v where v belongs to the primitive vectors. This is nothing but the summation, it's just an integral over a bigger space. It's, you see the fibers over this point are just the choices of these possible primitive vectors. So instead of taking an integral and then a sum, we can just write it as an integral over this entire space. So it's an integral over G mod gamma p of just f of v d of delta tilde v. So this is, so we really want not just an integral and then a sum, we just have one single integral. So now, but notice that f came from here, f is coming from here, so this f is constant on the fibers of this map. So what are the fibers of the map? So the fibers of mod gamma p to r d minus 0 are just essentially cosets. So what's r d minus 0? This is g mod gp, right? So if gp is that s of d minus 1 r times r d minus 1 then this is, gp is the stabilizer of a point in r d minus 0. So it's again a homogeneous space under g. So you see we have g mod gamma p mapping to g mod gp. So the fibers are just together in the right side. So hopefully it's something like this, gp mod gamma p, which is the same as s l d minus 1 r semi-direct r to the d minus 1 divided by s l d minus 1 z semi-direct z to the d minus 1. So you see now that this integral, so, and the other fact is that these measures that we're looking, higher measure here and higher measure here, and here you have Lebesgue measure, then they disintegrate in the right way. If you know, if with the correct normalizations they disintegrate in the right way. So you get that the integral over g mod gamma p of f of v is just the integral over r d minus 0 of f of v times the volume of this quotient gp mod gamma p. But here so on r d minus 0 we have the usual Lebesgue measure invariant, Lebesgue measure. So now about normalizations, the claim is that you can, if you normalize the measure in the right way on s l d mod r mod s l d z, you get normalize Lebesgue measure on r d minus 0 and then normalize Lebesgue measure on the next guy. So it's a computation, but they're induced natural measures on the next guy on s l d minus 1 r. If you start with one on the top, there's an, this vibration gives you induced measures. And so, and this, and finally, to just end, volume of gp mod gamma p is just the volume of s l d minus 1 r mod s l d minus 1 z because the volume, the normalization is such that the volume of r d minus 1 mod z d minus 1 is just 1. So this group is s l d r, semi-direct r d, s l d minus 1 r, semi-direct r d minus 1. And then these fibers, these are just story of constant volume. So they don't affect the integration. Okay, so we've proved the claim, we've proved the recursion formula with express the volume, the integral on this high-dimensional group by kind of lifting it to this guy and then pushing it down and showing that it's related to the previous volume, volume form. Okay, so we're done with Ziegels formula. So the conclusion is the following, that with the right normalizations, the volume of s l d r mod s l d z is just this product of zeta values. So now, since I don't have a lot of time, I'll just give two short applications which are quite nice and even if you don't care about these numbers, the applications are kind of interesting. So the first application is this. So the first application is suppose that you have k, a set, reasonable set, such that for any unimodular lattice, delta, k intersect delta is not empty. So suppose you have a set which intersects any lattice of co volume 1, then the claim is that the volume of k is also at least one. So this is kind of the converse of Minkowski's theorem which says that you have a convex set that intersects, you have a convex symmetric set and has sufficient large volume that it will intersect every lattice. This is kind of the converse. It says if you intersect every lattice, then you have to have some volume. And the proof is very simple. You just note the following. So take the Ziegler transform of the indicator function of k and this is at any lattice by assumption this is bigger than or equal to 1. And so the integral over g mod gamma of the indicator function of this transform is bigger than or equal to the volume of this g mod gamma. So here the normalization of the volume on this g mod gamma doesn't matter. But at the same time, since we know that if you take r, you get volume of g mod gamma times the bag, this was claim 2 which I wrote a little bit confusingly. But this claim was independent of the normalization of the volume on g mod gamma. It just says that take r by the full Ziegler transform, sorry, plus delta 0 I guess, but this is not so important. Then you find that the integral of 1k over rd minus 0 is bigger times the volume of g mod gamma bigger than or equal to the volume of g mod gamma. If you cancel it, you get the inequality on the volume of k. Okay. Are there questions? No. So let me, for the next application, I'll, we'll need just a very small strengthening of this. So again, k is, so now it doesn't have to be, it's not any set, it's a rate, like, so it's a symmetric respect to 0, respect to 0 set. And it is radio where, I guess, sorry, star shaped, meaning that if you have a point, it's, it contains all the points in between 0 and that point. So it's, star shape just means something. So that if you contain a lot, if you contain this point, you contain the entire line. So then, suppose that k intersects any primitive, any lattice except, it intersects it in primitive vectors. Then the claim is that the volume of k is bigger than or equal to zeta of n. So, sorry, it's even better, it's 2 zeta of n. So, I'll leave the proof as an exercise, but the point of the proof is exactly the same thing here, but now you apply the primitive Zegel transform. And you know that if a vector, if a symmetric, if I set like this, intersects the primitive vectors. It also, so if it intersects, if there's a vector v, then there's also a vector minus v, which gives you the factor of 2. And then the zeta of n factor comes from the Zegel transform factor that you got there. Okay, so you have this nice lower bound. So what can you do with such a lower bound? Well, one question which people about lattices have, people who study lattices have looked at for a long time is sphere packings. So how can you find lattices with, so we want to find a lattice with long shortest vector. What does this mean? It means that you, for any lattice, you try to, you look at what's the shortest vector in that lattice and you want that shortest vector to be as long as possible. So you want fat lattices. So in R2, you probably know that the hexagonal lattice like this is the best one. This is in R2. And then higher dimensions, not much is known. Okay, so the estimate which you can get, which is nice, is that, so using this method is the following. So let me just take the numbers. So there exists a lattice for this vector, bigger than or equal to what? 2 zeta of, so I apologize, I wrote D there, and there instead of D, or 1 over D. And here sigma D is the volume of the sphere of radius 1 centered at 0 in Rd. So you get this bound and this is, if you actually work it out, it's roughly a constant times square root of D. So it's not a, it's not a bad bound. It's definitely bigger than 1, or D large enough. And okay, so I don't want to go over time, so I'll just tell you what you do is you take, so set R as the minimum of all radii, or infimum if you like, of all radii such that the ball of radius R is centered at 0, intersects every lattice a non-zero vector. This is some number, it could be very small, right? This is the number we want, but it could be very small. But if you apply the bound that I just gave here, that the volume of this ball of radius R, since it intersects every lattice, it will particularly intersect in primitive vectors, the volume of that ball is going to be 2 times the zeta value, and the volume of the Rd, so the ball of radius R around 0 is just sigma D times R to the D. So you get exactly this bound on R. You get that R is at least this much. So you get, yes, I get this. I won't try it explicitly what these zeta values are, and what the volume is, but the explicit formula is, and this turns out to be roughly square root of D, which is not a bad bound. All right, so I'll stop here. Thanks for your attention of these four classes.