 This video will be about the Selberg trace formula. So let's start by looking at an example of it. So the Selberg trace formula is given by this expression here. If you look at it more closely, you see that it really looks like a rather complicated and gruesome mess. And so what I want to do is explain the underlying idea behind this, which is really quite simple and how you how you would get such a complicated expression out of this. So it's used for studying quotient spaces such as you might take SL2 of R, which is 2 by 2 matrices over the reals and quotient out by SL2 of Z, and you want to understand this space here. Well, let's start by looking at a simpler example. So I'm just going to take G to be a finite group, and acting on finite set S. Let's suppose the action is transitive, so S is equal to G modulo H, where H is a subgroup of G. And the problem is the following. What we want to do is understand the action of G on the space of all complex functions on S. So this is a finite dimensional vector space with a basis of points of S and G acts on S, so it acts on this, and we want to know what is this representation of G. In particular, we want to know its character, where the character of G is just the trace of G on the space C of S. And this is actually rather easy to work out. Suppose G is an element of the group G will act on S, and it acts with various orbits. For instance, it might have an orbit of size 1, and it might have an orbit of size 3 here. And if we write out the matrices for these orbits, you see the matrix here is 1, and the matrix here looks like 0, 1, 1, 0, 0, 0, 0, 1, 0, and what you notice is that this is trace equal to 1, and this is trace equal to 0. And what you see is that whenever there's an orbit of size 1, we get a contribution of 1 to the trace. Whenever there's an orbit of size greater than 1, we get a contribution of 0. So this is just equal to the number of fixed points. Of G on S, which is just the cardinality of the fixed points are sometimes indicated by S with the G op there. And you can rewrite this in various ways. For instance, you can write the number of fixed points as 1 over H times the number of elements A in G with A, G, A to the minus 1 in H. Or you can rewrite it as a sum of over conjugacy classes of H, because for each element of H, you can look at the number of points A such that this is that particular element of H. So anyway, this formula here for chi of S is essentially the Selbow trace formula for a finite group G. And there are various generalizations of this. So the action of G on C of S, well, this is an example of an induced representation of G. So it's induced from the trivial representation 1 of H, where H just acts on a one-dimensional vector space. And in general, if H acts on V, then we can obtain a representation of G in several ways. For instance, one is to notice that G acts on, we can take the group ring of G and tense it over the group ring of H with V. And this will be an induced representation. We can interpret this as sections of a vector bundle over the space G over H. So the special case where V is just a one-dimensional vector space C, this vector bundle would just be the trivial vector bundle and sections are just functions on G over H. And there's a formula for the character of this. It's given by Frobenius. And it's very easy. What you do is you just take the character of the space V. So this is a function on H. And you extend it to be zero for elements that aren't in H. And you just take the sum over all elements G and G modulo H of all conjugates of this. So roughly speaking, this function here is not invariant under conjugation by G. And if it's a character of something you want it to be invariant under G, so just make it invariant in the most obvious possible way just by taking the sum of all conjugates under G. And this is Frobenius's formula for the character of an induced representation. It's a generalization of the formula I had on the previous page where you just take the number of fixed points. And now the Selberg Trace formula, all we do is we take G to be SL2 of the reals. And we take H to be our favorite subgroup. For instance, H might be SL2 of Z. It would be a typical example. Or we might take H might be the fundamental group of a Riemann surface, a compact Riemann surface. And if you've got a compact Riemann surface, a genus greater than one, then its fundamental group is naturally a subgroup of SL2 of R. And you can look at this case. So what we're really doing is we're looking at all functions on the modulo H and trying to understand this as a representation of the group G. And all the Selberg Trace formula consists of is working out the character of this in much the same way as we did before. However, there are certain complications that there are three cases. If G over H is finite, this is essentially trivial. It's the case we've done before where you just count number of fixed points. If G over H is compact, then this is sort of easy. Well, it's not easy, but it's easy by comparison to the non-compact case, which is sort of the hard case of the Selberg Trace formula. So what I'm going to do in the rest of the lecture is explain what the various complications you get are when G is infinite and how you deal with them. So the first problem is G is not a discrete group in general. Well, this is a problem because for the Frobenius formula, we quite often have to sum over all elements of G. And if G is infinite, then the sum is generally infinite, especially if G happens to be a topological space. Well, you fix this in the obvious way. Instead of taking a sum over G, you have to integrate over G. And if G is a locally compact group, there's a sort of reasonably well-defined left invariant integral, so that's okay. However, we get several problems. So first of all, the integral might be left invariant. So the integral over F of G is the integral over F of A of G, but not right invariant. So if you're summing over, sorry, I can't spell invariant, if you're summing over a group, then summing is both left invariant and right invariant. One of the extra complications you get in the non-discrete cases that the integral might be left invariant, but not right invariant. This doesn't actually happen for SL2 of over the reels, but it does happen for groups like the subgroup of all matrices like this inside SL2 of R here. Left integration over this group is not the same as right integration, which is an extra complication you have to deal with. The second thing is that the integral has to be normalized in that there are various ways of choosing the integral. So if you're summing, it's obvious how to normalize. You just say each element of G is weight 1. But if you're integrating, you know, you can multiply the integral by constant, and it's not quite clear what to do. For SL2R, this isn't too difficult. In more general cases, like higher-ranked groups over the Adels, normalizing the integral is actually quite complicated. There's a standard way of doing this due to Tamagawa, which is called Tamagawa measure, whose definition isn't actually too difficult. The trouble is it's quite hard even working out basic properties of the Tamagawa measure. Like you might want to know what is the measure of G over H, and actually working out the Tamagawa measure of G over H is a rather hard problem known as the vague conjecture, which took several decades to solve. Also, if G isn't discrete, you can't really use matrices. So for discrete groups, you can represent a line of transformations, a matrix consisting of elements Mij, and then you can take things like Bi is sum over Mij of Aj, where you're summing over all elements J inside a group or inside some finite set. If you are working with non-discrete groups, you need to use a kernel, which is really just a matrix, except X and Y are not elements of a discrete set, they're elements of a topological space. And then you replace this expression by integrating kxy times f of y dy, and this gives you a function g of x. So kernels behave just like matrices. In fact, kernels and matrices are really the same thing, that they have different names and different notation for historical reasons, because they were invented before Lebesgue integration. When you do Lebesgue integration, you realize that these are both just integrating over a set, which is discrete in this case and continuous in this case. So the next problem is we need to know what are the conjugacy classes of g and h, and if g and h are finite, they're a finite number of conjugacy classes. The problem for things like SL2 of r is there are infinitely many conjugacy classes, so you need to divide these conjugacy classes into families. For example, there's one obvious one, which is this matrix here, then you can have the matrix like that, but then you could have a conjugacy class consisting of matrices that look like this for this element non-zero. Will these form a single conjugacy class, but then you can have conjugacy classes represented by matrices like this, and here we have a full family of conjugacy classes, because this element is not conjugate to another element like that, unless a equals b or b to the minus one. So we have a family of an infinite number of conjugacy class, and then there are still more conjugacy classes, for instance, we could have a conjugacy class that looks like this, where c is cosine of some angle theta and s equals sine of theta, and we also want to know what the conjugacy classes of the subgroup h are, and when h is equal to SL2 of z, it gets even more complicated, although that's not too bad, we can handle SL2 of z, but the problem is h might be some sort of congruent subgroup, like it might be the matrices a, b, c, d, the c congruent to 0 mod n, so this subgroup turns up quite a lot, and the conjugacy classes of this group here are really rather painful to describe or think about. In fact, even describing a set of generators for this group is not very easy, so in order to deal with that, well the problem is if we're looking at SL2 over a field, we can sort of handle the conjugacy classes, if we're looking at SL2 over a ring that isn't a field, it's rather difficult to see what the conjugacy classes are, so one way of dealing with that is moving to adels, so instead of looking at SL2 r modulo SL2 z, what you do is you look at SL2 of the adels, modulo SL2 of the rationals, and it turns out these two quotient spaces are sort of fairly closely related, if you understand this one you understand this one, and now the great thing about this is this number here is a field, so the conjugacy classes can be described, and they're a bit of a mess, but it's easier than doing it for z, and the problem is you've replaced r by the adels, so the conjugacy classes of this group here are now more complicated, and this is somehow the price you have to pay for changing z to the rational numbers, what are the adels, well the adels are more or less the product of r times a sort of restricted product of all periodic numbers, so now instead of working out the conjugacy classes of SL2 of r, we also have to work out the conjugacy classes of SL2 of the periodic numbers for all primes p, so we have rather large numbers of different sorts of conjugacy classes, and this is one of the main reasons why the cell both trace formulas rather complicated, that we have a terminate for each sort of conjugacy class, and if you look at the complicated mess I had here, then roughly speaking each line corresponds to a different sort of conjugacy class in SL2 of the reals, or SL2 of your discrete subgroup, so the next complication is the character of a representation of g is a distribution, not a function on g, and so what's going on here, well what we want to know is what is the trace of g on b for g and element of our group g, and the answer is very easy, this will usually just be infinite or undefined or something, it's very difficult to make sense of it if g is an infinite group and v is an infinite dimensional vector space, so what we do instead is we look at action of some sort of group algebra on v, so we might take f to be a smooth function of compact support on g, and if we're lucky then f will act on v, what you do is you sort of integrate the action of g over g according to f, so what we can do if v is a well-behaved representation is find an action of smooth compact support functions on g, and this might have a trace, so what we do is we get a map from smooth compactly supported functions on g to real numbers which are just the trace of f on this representation, and a linear map from smooth compactly supported functions to real numbers is a distribution, and sometimes if you're lucky this distribution is actually really a function, there's a famous rather deep theorem of Harris Chandra which says that irreducible representations of semi-simple d groups are very often represented by distributions that are really just locally integrable functions, but sometimes it's not a locally integrable function and it really is a distribution, so as long as these functions are of trace class you're okay, and if g over h is compact these functions often are compact operators and are of trace class, so we're sort of okay, we get well-defined distributions. The real problem in the cell bow trace formula comes when g over h is not compact, and then the problem is that the action of f on space v is also not a compact operator, so it usually doesn't have a trace, or a trace is infinite or something, it's very difficult to make sense of it, so how do we deal with this? Well let's first of all take a look at a couple of examples, suppose you take the group ring of r over z, now this can be, if we take all nice functions on r over z, say we take you know smooth functions or l2 functions or something I don't really care, under reasonable conditions we can write this as a direct sum of periodic so it should be integer n for n in z, so we can write a reasonable function as a linear combination or an infinite linear combination of periodic functions, and this is just a Fourier series expansion, if we try and do this for functions over the reals we can sort of try writing them as a Fourier integral, so we get a sort of integral over e to the 2 pi i x y times something, and this is the Fourier, this is essentially the Fourier transform, and the trouble is this space is not a direct sum of the subspaces generated by these, it's really a sort of integral of them, and when you decompose sl2r modulo sl2z, suppose we take l2 of this, then this is the sum of a discrete part plus a sort of integral of a continuous part, so the continuous part is sort of like a Fourier integral, and the discrete part is sort of like a Fourier series, and both of them actually occur here, now the discrete part, the functions on g tend to have trace class, so we can define the character of an action on the discrete part, so the problem is that we need to figure out what the continuous part is and kind of remove it, now the continuous part is given by things called Eisenstein series, well if you've done modular forms you've come across an Eisenstein series that looks something like this, you've sum over all c d that are non-zero over c tau plus d to the k, now the Eisenstein series that occur here are a slight variation of these, so these are holomorphic functions, what we really want is sum over c d not equal to zero, one over c tau plus d to the 2s, and you notice these are now real analytic but not holomorphic, and these converge or the real part of s sufficiently large and there are two problems here, first of all we need the Eisenstein series in a region where they where they don't actually converge, so we need to analytically continue them, now this is the Eisenstein series for s l 2z and it's not very difficult to analytically continue that one, analytically continuing them for more general groups is much harder was done by Selberg for s l 2, for higher rank groups it gets to be a bit of a nightmare and the analytic continuation was done by Langlands in this notoriously difficult manuscript, so the first step is to analytically continue the Eisenstein series, you can then use them to find the continuous part of the decomposition and you sort of use them to subtract off the continuous part and you're left with a discrete part where operators have a trace and the Selberg formula now comes by finding the trace of continuous of smooth compact support operators on this discrete part, so to summarize what the Selberg trace formula consists of, you take the Frobenius formula for an induced representation and you make the following modifications for it, first of all you change all sums to integrals because your group is not discrete, secondly we have to work with distributions rather than functions because the character of a representation is usually a distribution, third we use, we tend to use Adele's to handle the conjugacy classes, actually if you're just doing s l 2 you can usually get away with that using Adele's but for higher rank groups using Adele's is essential otherwise trying to deal with the conjugacy classes is just hopelessly complicated and finally we have to use Eisenstein's series to get rid of the discrete part, to get rid of the continuous part, so that's all you have to do and the good news is that this has all been written out in detail by people, by several people for instance Hedgehall has a book giving the detailed description of almost all possible cases, the bad news is that Selberg, that Hedgehall's book looks like this, it's two volumes and together they cover about 1300 pages giving you the Selberg formula in its full generality where you allow non-trivial representations of H and allow H to be non-co-compact and so on