 This video is the fourth and final part of a series of videos on my Wienberg lecture on Wienberg's algorithm and Katz-Mudi algebras. If you want to see the first three parts, there should be a link to them in the description of the video down below. So this part will be mostly about automorphic forms. So in the previous lecture, we described there was a sort of relation between hyperbolic reflection groups and Li algebras. This relation is a little bit mysterious. So there are plenty of hihobolic reflection groups that don't correspond to Li algebras or even Katz-Mudi algebras and plenty of Katz-Mudi algebras that don't correspond to hihobolic reflection groups. But there seemed to be a few cases when there was a sort of strong relation between them. The one we looked at in particular was where you take the even 26-dimensional Laurentian lattice. And this corresponds to a sort of generalized Katz-Mudi algebra with Dinkin diagram, the Leach lattice, plus some norm zero vectors. So this was the subject of the third lecture, so I won't say too much more about it just now. What we're going to discuss is that both of these objects seem to be related to automorphic forms. Again, this correspondence isn't exact. There are plenty of automorphic forms that don't correspond to Li algebras or hihobolic reflection groups. Not all Katz-Mudi algebras or hihobolic reflection groups seem to be connected with automorphic forms. But there are several examples suggesting there is some sort of interesting connection between these. So I'll start by just recalling what an automorphic form is. So the simplest examples of automorphic forms are just modular forms. And a modular form is a function f such that f of a tau plus b over c tau plus d is equal to c tau plus d k times f of tau here. And the imaginary part of tau is greater than zero, and there are some other minor conditions that I won't worry about too much. And what's going on is, well, we have this mysterious fudge factor here. If we ignore this fudge factor for a bit, you can see this is almost saying that f is invariant under sl2z. So here these matrices A, B, C, D are in sl2 of z, which is just 2 by 2 matrices with integer coefficients and determinant 1. And this acts on the upper half, plain by A, B, C, D, acting on tau is A tau plus b over c tau plus d. So without this fudge factor, we're just saying the function f is invariant under the action of sl2z. This fudge factor means that f isn't quite invariant, but transforms up to an elementary factor. What it's really saying is that f is a section of a line bundle under sl2z that is invariant, and line bundle is a sort of high level way of saying that f transforms up to this fudge factor. And an example of an automorph form we had in the previous lecture was delta of tau, which is q times product over n greater than 0 of 1 minus q to the n to the 24, where q is of course e to the 2 pi i tau. Now this function delta has some obvious symmetries. So tau of delta plus the delta of tau plus 1 is equal to delta of tau. This is just obvious because q is invariant under tau goes to tau plus 1. And there's a sort of hidden magical symmetry. It says that delta of minus 1 over tau is equal to tau to the 12 times delta of tau. And now you see this says that delta of tau transforms like this for the following two matrices. We have the matrix 1 1 0 1 corresponding to tau goes to tau plus 1, and we have matrix 0 minus 1 1 0 corresponding to tau goes to minus 1 over tau. Now these two matrices actually generate sl2z. So from this we can easily see that delta actually satisfies this relation for all matrices in sl2z. So delta is an example of a modular form. So what's an automorphic form? Well an automorphic form we just replace sl2z by a larger group. Now sl2z is contained in sl2 of the reels and what we're going to do is we're going to replace sl2 of the reels by some larger lead group and have a larger discrete subgroup in it. And then there's going to be some relation saying a function is invariant onto some fudge factor and the fudge factor will be more complicated and we'll see some examples of it fairly soon. So the first example of an automorphic form is going to be the denominator function of the Lee algebra I mentioned earlier related to the 26 dimensional even Lorentzian lattice. And we remember from last lecture that this had a denominator formula which looked like this. If we take sum over all elements of the vial group of some sine times omega of e to the somewhere tau n times e to the n row where tau n is the coefficients of the delta function and this is the product over alpha greater than 0 of 1 minus e to the alpha power of the multiplicity of alpha. So let's just recall what the various bits in it are here. Sum of tau n e to the n is just delta of tau which is q minus 24 q squared and so on. And the multiplicity of alpha was given by p24 of 1 minus alpha squared over 2 where p24 are the coefficients of 1 over delta which is q to the minus 1 plus 24 plus 324 q and so on. So this is quite complicated. And what this is doing is it's saying that a certain sum is equal to a certain product. And what I'm going to do is to show that if we think of this as being a function, we can think of this as being a function on the following space. We take ii 25 comma 1 and then we tensor it with the reals and then we have i times c where c is the positive cone. So you remember this is a Lorentzian space which has a double cone of norm zero vectors. And inside this we can see a sort of positive cone here which I'm going to call c. And we're going to say that this is going to be a function on the set of vectors whose real part is anything in this space and whose imaginary part is in this positive cone. You can think of this as being an analog of the upper half plane. So the upper half plane says the real part is in a one-dimensional vector space and the imaginary part is in a cone in the reals which is just the positive reals. So let's see why this thing is an automorphic form. Well you remember delta is an automorphic form because first of all it had an obvious transformation under tau goes to tau plus one and it had a mysterious one and the tau goes to minus one over tau. Well so our function here has some obvious transformations. First of all we can translate by elements of the lattice ii 25 one which sort of correspond to tau goes to tau plus one. We also have automorphisms of the lattice ii 25 one. Well we don't quite get all automorphism but up to a factor of two. Automorphisms of this lattice gives us elementary transformations. There's also a non-obvious transformation and we can get this as follows. So here I had this sum that I'm not going to write out again and it was equal to some product that I'm not going to write out again either. And what we notice is the sum is a solution of the wave equation. This is because if you look at the terms of the sum carefully you see that all these vectors appearing at her honor of norm zero and x of a norm zero vector basically gives you a solution of the wave equation. On the other hand this product, if you take its logarithm, the logarithm is singular when v is imaginary and v squared is equal to two. So here what's happening is the imaginary part lies inside this cone c here and inside this cone there's the hypersurface of vectors with v squared equals two. It's two rather than minus two because we're looking to imaginary vectors. And this product actually vanishes here and we can ask why does this product vanish? Well, what we do is we recall that if we've got a power series sum of a, n, z to the n and if the radius of convergences are and all the a, n's are greater than zero, then this has a singularity at r. So, you know, power series with positive coefficients has to have a singularity at the real point of its radius of convergence. Now, if you look at this product we can work out where it converges by using the Hardy-Romanagin-Radamaker formula for the asymptotic behavior of the partition function. Or rather, partitions into parts of 24 colors. Furthermore, if we take the logarithm of this, all its coefficients are, well, they're not all positive, they're all negative, but that's good enough. So, using this, we can see this product must actually be singular on this surface here. On the other hand, this sum here is non-singular, so this sum must actually vanish when v is imaginary and v squared equals 2. Well, now what we notice is that, let's call this function phi. So, we've got these two functions, we've got this function phi of v, so it's going, so this is a solution of the wave equation that vanishes on this hyper surface. On the other hand, if we look at the function vv over 2 to the 12 times phi of minus 2v over vv, we can check this is also a solution of the wave equation. I mean, this is just the transformation of the wave equation under a certain conformal map. And the fact that this function vanishes on this hyper plane means that these two functions actually have the same zero and first derivatives on this hyper surface. So, we can now apply the Koshy-Kolevsky theorem, which says that two functions that satisfy the wave equation and have the same zero and first derivatives on a Koshy hyper surface must be equal. So, these two functions are actually the same, but I guess I should have put a minus sign in there. So, here we're applying the Koshy-Kolevsky theorem. So, what we have now is a magical extra transformation of this function. So, this kind of corresponds to this transformation of the function delta. And what we saw is that for delta, these two transformations mean that we're actually transforming under the group sl2z. Well, if we put together all these transformations of this function phi, what we see is that phi is an automorphic form or the group. Well, what we do is we take an orthogonal group of the lattice 25,1 over the integers. Well, again, it's up to a factor of 2. I should really take a subgroup of index 2, but I won't worry about this. And this is containing a lead group where you just take, it's a 26-dimensional Lorentz group. So, it should be 26,2 here. Things mysteriously go up by one. So, what we have is an automorphic form for a group which you should think of. You should think of this as being an analog of sl2z contained in sl2 therials. And we can do this with several other hyperbolic reflection groups. So, we get the same for hyperbolic reflection groups corresponding to... Well, if you take the leach lattice and take the fixed point under some automorphism, that is your lattice rather like the leach lattice. And you can sort of go through and get a similar reflection group and a similar automorphic form. So, this has worked out in a few cases by Nieman and generally by Scheithauer, who showed that if you take any automorphism of a leach lattice with a non-trivial fixed point sublattice, then you can get a similar automorphic form for it. So, that shows there are some hyperbolic reflection groups corresponding to automorphic forms. Well, now we have the problem. What about Wienberg's groups? So, taking a fixed sublattice of the leach lattice only gives you some rather special hyperbolic reflection groups and these automorphic forms are rather special. They turn out to be automorphic forms of singular weight. And it turns out you can for all the groups Wienberg studied. And the answer is we take the form phi on ii26,2 which is actually an automorphic form on ii25,1 tends to see with imaginary part in the cone. And we can just pick a Dinkin diagram in the Dinkin diagram of ii25,1 which is just the leach lattice. So, if we take some Dinkin diagram let's call it D and we look at the orthogonal complement of D which is contained in ii25,1 so this will be some lattice. And then this automorphic form restricts to an automorphic form for the perp plus 110. It's only one slight problem this restriction is identically zero so although it transforms like an automorphic form this is completely uninteresting because it's a zero automorphic form. And the problem is that phi vanishes on the orthogonal complement of any root r. In particular D contains roots r so the orthogonal complement of any Dinkin diagram the automorphic form you get will be identically zero. So that seems to be a bit of a problem. Well there's a solution to it. We can differentiate by before restricting. And what we can do is we can differentiate once for each hyperplane that it vanishes on in D. So this will be half the number of roots of the root system of D. So D has N roots there will be N over 2 hyperplanes on all of which phi vanishes. So what you do is you sort of make phi non vanishing on all these hyperplanes by first differentiating it. And the effect of differentiating it increases the weight of the form phi. Well what's the weight of phi? You remember phi has this property that phi of 2v over vb is equal to minus vv over 212 times phi of v. Well the weight is just this bit here. And so what we get is automorphic forms for various lattices whose weight is a little bit bigger than you might guess. So let's have an example. Suppose we take Wienberg's reflection group for I-19-1. So he showed that the reflection group of this has a finite Dinken diagram which was a little bit too complicated to do by hand. So Wienberg and Kaplan-Skye got a computer to work it out. And the even sub lattice is given by the orthogonal complement of D6 in the leach lattice. So we take a D6 Dinken diagram in the leach lattice and take its orthogonal complement. By the way, in case you're thinking that leach lattice has no roots at all, we're thinking of the leach lattice as being the Dinken diagram of the 26 dimensional even Lorentzian lattice. So we get an automorphic form for I corresponding to I-19-1 of weight. Well it would be 12 plus 60 over 2. Well what's 60? Well this is the number of roots of D6. And similarly for all the other reflection groups of even unimodulatus that Wienberg studied, we can get an automorphic form of some weight. For example for I-21-1 which was the largest one, we get a form of weight 12 plus. Well this time we notice this is the orthogonal complement of D4. So D4 is 24 roots so we take 24 over 2. And these automorphic forms vanish on the reflection hyperplanes. So each of these reflection groups has reflection hyperplanes and the automorphic form very neatly vanishes on these hyperplanes. And if you wanted to describe all the places where the automorphic form vanishes, it actually vanishes on reflection hyperplanes. Not just of this lattice but you know we have to make this lattice bigger by adding on a little two-dimensional Lorentzian lattice. And in fact the automorphic form vanishes on hyperplanes of roots of this bigger lattice. So the automorphic form corresponds very nicely to the hyperbolic reflection group and tells you what its reflection hyperplanes are. Well if we look at I-21 something a little bit strange happens. Here this is the one where Wienberg showed the Dinkin diagram is infinite. And we get an automorphic form corresponding to it and this automorphic form vanishes on the hyperplanes corresponding to orthogonal complements of vectors r where r squared is equal to 1, 2 or 3. And these ones are not roots. So something more complicated seems to be going on. The automorphic form sort of notices the hyperplanes where the reflection group vanishes but it also notices some other hyperplanes that don't correspond to reflections of the reflection group. And this sort of seems to be related to the fact that the Dinkin diagram is infinite and somehow the automorphic form is kind of causes the automorphic form to pick up some extra zeros not corresponding to reflections. So you can also look at the example of even union modular lattices. So Wienberg looked at these two cases. So I-19-1 this corresponds to the E-10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. So this corresponds to the E-10 Dinkin diagram and recall that this corresponds to bigger Dinkin diagrams. I'm just going to sketch that looks like this. We take two copies of the E-9 diagram and join them like that. And the question is, can we find automorphic forms corresponding to these? And the answer is yes. And this is going to give us a bonus because it is also going to tell us what is the nice Lie algebra corresponding to this. So the question is, can you extend the E-10 cat's moody algebra to a bigger Lie algebra with a nice denominator formula? And the automorphic form tells you how to do this. So what goes on here is, again, we just restrict the automorphic form phi after differentiating, of course. And for this we get a form of weight 12 plus, well, we need to figure out what E-10 is. Well, E-10 is given by taking E-8 plus E-8 and then taking its orthogonal complement in the 26-dimensional ignorance in lattice. So we need to know how many roots does this have. Well, it is 248 plus 240 plus 240 roots. So just 480. So it would have weight, we have to add 480 over 2. And this gives us an automorphic form. And this automorphic form turns out to be the denominator function of some sort of generalized cat's moody algebra. So this cat's moody algebra, the real simple roots are just E-10, but it also has quite a lot of imaginary simple roots. Furthermore, this automorphic form again vanishes exactly on hyperplanes corresponding to norm 2 roots of this bigger lattice ii 10 comma 2. And this seems to be arguably the correct Lie algebra corresponding to the E-10 root system. It's a Lie algebra corresponding to a nice automorphic form actually done by the orthogonal group of this lattice. And of course you can do the same thing for this reflection group just by taking, just by regarding this as the orthogonal complement of E-8. This sort of shows that you can't really understand the E-10 Lie algebra without going up to 26 dimensions and then restricting the corresponding automorphic form. There are several other examples of finite reflection groups corresponding to automorphic forms. These were studied by Gretzenko and Nicolín several years ago who showed there were some other examples of hyperbolic reflection groups with finite Dinker diagrams that could actually be extended to correspond to certain automorphic forms. So I'll just finish by mentioning a few open questions. So the most obvious question is can we classify various hyperbolic reflection groups? For instance you can try calculating the ones corresponding to lattices over the integers whose reflection group has finite volume and there's been quite a lot of work on this by Nicolín and others. In fact I sort of heard a rumor that this has recently been done but I haven't yet managed to figure out what the details of this are. We shouldn't actually restrict the ones of finite volume because in some sense the most interesting cases don't have finite volume. For instance we have Conway's reflection group which doesn't correspond to finite volume. As I mentioned the analogs of these have been partly classified by Scheithauer and recently Brandon Williams, Yang, Wang and Sun have some recent preprints where they get quite close to classifying all the cases that look like this by classifying the corresponding automorphic forms. I might put a link to their paper in the description of this video. One of the interesting things they showed us is that the lattices you get are closely related to Scheliken's list of conformal field theories. So we can ask which of these hyperbolic reflection groups correspond to automorphic forms. So these classify rather special ones that correspond to automorphic forms of singular weight. But as we've seen in the examples there are quite a lot of automorphic forms that aren't of singular weight that also appear to be quite interesting corresponding to finite Dinkian diagrams or reflection groups whose fundamental domain is infinite volume. Another problem is find natural constructions of the Lie algebras. The problem is that although we can construct Lie algebras from this they're just given by generations and relations which is a rather untidy way of describing a Lie algebra. For some of the Lie algebras this is known. For example in the case of the Lie algebra of the 26th dimensional wrench in lattice there is a construction using the no-ghost theorem from string theory. And for a few of the other similar Lie algebras there are also similar constructions known. But for most of the Lie algebras we get corresponding to automorphic forms I don't think anyone has found a really natural construction that doesn't rely on just writing down generations and relations. Next we can ask what about analogs of II251 over other number fields. And sometimes you can find analogs over other number fields. For instance if you take a fixed point free automorphism of this of order three you can use that to make this into a lattice over the Eisenstein integers and that seems in some way to be a sort of Eisenstein integer analog of this. And then you can do things like look at reflection groups over the Eisenstein integers. And Daniel Alcock showed you could get several rather striking complex reflection groups related to this. And then I got a really crazy idea. So recently Biazowska managed to prove that the Lie lattice was the densest lattice packing, sorry the densest packing in 24 dimensions using various magic functions. So she had some magic functions associated with the Lie lattice and the E8 lattice. And we can ask, do analogs of these magic functions exist for fixed point sub lattices of the Lie lattice and if so what can you do with these functions? So yeah, so I think I'll leave it at that.