 Yeah, so as I said in the first lecture, I'm roughly dividing this course into three parts. The first two parts are rather short. They are on SON1 and SP2N. And then I'll move to the third part, which is PGLN. And we were in the middle of the discussion of Siegel modular forms. And yeah, so what I would like to discuss briefly at this point is HECA theory for Siegel modular forms, of course, only in the un-rammified case, the rammified case. I don't know if this has ever been worked out, but it's certainly very complicated. So gamma is the group SP2NZ, so the analog of the full modular group. And HECA operators are parametrized by double cosets of the form. So we fix a prime P. Fix a prime P. Gamma and then a diagonal matrix with entries P to power a1 up to P to power an. And then P to power r minus an, P to power r minus a1, 2n elements and gamma. And we can order them such that a1 is greater than 0 and so on. And it's in increasing order up to an. And then it continues to increase. So an is bounded by r over 2. And r is some integer. So r is basically the degree of the HECA operator. In particular, if r is 1, then there is only one such HECA operator. That's simply Tp. And by slight upuse of notation, I just write the double coset. That's just 1, 1, 1, 1, 1, and then P, P, P, P. There are several HECA operators of degree 2. Ti of P squared is of the following form. I have a couple of ones, i of them. And then I have a couple of P's, n minus i of them. And then again, a couple of P's, n minus i and P squared. In particular, so this is for 0 less than i, less than n, with possible equality. In particular, if i is 0, then this is just P, P, P, P, P, P. So this is the identity. So T0 of P squared is the identity. And it turns out that Tp and then TiP for 1 less than i, less than n minus 1, generate the P part of the HECA algebra. So every HECA operator in P is a polynomial in these operators. They have some nice properties. There are Hermitian with respect to the inner product that we defined there commutative. And of course, by the Chinese remainder theorem, everything is multiplicative. So if I take two different primes, then everything is nicely multiplicative. But if I stay with one prime, then the HECA relations become very complicated. HECA relations are very complicated. This is a very, very complicated combinatorial problem to write down a linear. So if you multiply two HECA operators, you can write them as a linear combination of other HECA operators. But the linear combination is very complicated. And I don't think, so to my knowledge in general, there is no closed formula or not even an easy algorithm to compute what the product of two HECA operators is. Here is a simple example, T p squared. We would like to write this as a linear combination of degree two HECA operators. And this is T2 p squared plus p plus 1 T1 p squared plus p to the power 3 p squared p1 times the identity. But this is the simplest of all non-trivial HECA relations. And they become completely crazy if you want to write down linear combinations for products of HECA operators. What am I missing? You don't write a double costed for T i of square. Oh, yeah, yeah, yeah, yeah, OK. OK, for holomorphic Ziegler modular forms, the Ramanujan conjecture is known, just as in the case of genus 1. Yes, yes, genus 2, Ramanujan conjecture, is known. This is work of Weissauer. But as I said, there is no really clear connection between the Satake parameters and the Fourier coefficients. And so what does this mean? It means that the Satake parameters have absolute value 1 unless the form f is a lift. And I will explain what a lift is in a moment. All right, L functions. I'm not going to go in any detail. I will just mention there are two types of L functions. There is the so-called standard L function, which is an L function of degree 2n plus 1. And here, the analytic continuation and functional equation are known. And there is the spinor L function of degree 2 to the power n. And here, the analytic and functional equation is known in the case n equals 2, but not in general. My impression is nevertheless that the spinor L function is used more often than the standard L function, in any case. I'm not going to go into more definitions. But certainly, in the case n equals 2, there is a degree 4 and a degree 5 L function. OK, finally, lifts, because they produce some arithmetically interesting objects. The most famous lift is in the case in n equals 2, the cytokoro cover lift. This is the case. n equals 2. And this has been generalized to arbitrary, I think, even n or so that I will discuss this later. So the idea is the following. We take a classical holomorphic cusp form of genus 1 and weight 2k minus 2. So this is the space of holomorphic cusp forms on the upper half plane, the usual upper half plane, of weight 2k minus 2. And by the theory of half integral of modular forms, as developed basically by Shimura, you can associate to this half integral weight modular form with the following Fourier expansion. It has some coefficients that I normalize to be of absolute value, roughly 1. So the exponent is k over 2 minus 3 over 4 e of mz. And this is a half integral weight form of weight k minus 1 half of level 4. And it is in Conan's plus space, which means that m is congruent to either 0 or minus 1 mod 4. So only half of the coefficients occur. And then, Wald's Po-Jay's theorem tells us that these coefficients are central twisted L values of the original function f. So this is up to normalization. I mean, g is only defined up to a scalar multiple. But I can choose a normalization such that the coefficient is the central L value, chi d, at least for d, a fundamental discriminant. Yes. Yeah, for non-fundamental discriminants, there is also a formula, but that's a bit more complicated. And then I can attach to g, or these coefficients, a Ziegler modular form as follows. So I proceed on this blackboard. To this, I can attach a Ziegler modular form capital F of z. It has a Fourier expansion sum over matrices t, a t, e of trace tz. And that's a Ziegler modular form of weight k for sp4z. And the at's are given explicitly. And in certain cases, they can be described quite easily. It's determinant of 2t to the k over 2 minus 3 over 4 times c where c is this Fourier coefficient determinant of 2t. And this holds at least if minus 1 to the power k minus 1 times dit 2t is a fundamental discriminant. And if not, then there is a slightly more complicated formula that describes these Fourier coefficients. Yeah. So the Fourier coefficients are up to normalization. Well, they're basically the Fourier coefficients of this half integral weight modular form. So capital F is the so-called cytokuro cover lift of either f or g. This is also sometimes called the mass special char, because Hans Maas investigated this a lot, and he called it special char. So special whatever, I guess it's untranslatable. But if you find this phrase special char, then it refers to. Or sometimes it's also called the mass space. So the subspace of capital F of this form inside the whole space of Ziegler model. I mean, it's of course only a very small subspace of forms that are lifts. But this is sometimes called the mass space inside the space of all Ziegler modular forms. OK, so these lifts are quite interesting. Let me mention two results. There is Ichino's period formula. So what you can always do, this has nothing to do with lifts, what you can always do if you have a Ziegler modular form, you can restrict it to the diagonal. And by this I mean, if you have a point Zx plus Iy in the Ziegler upper half space of genus 2, then you can write this as x1, x2, x2, x3 plus Iy1, y2, y2, y3. And you can project this onto the diagonal x1, x3 plus Iy1, y3. And then this lives on two copies of the usual upper half plane. There is a complex number x1 plus Iy1, and then there's a complex number x3 plus Iy3. So the restriction to the diagonal restricts to two copies of the upper half plane. So you get a function that lives on two congruence quotients, and what you can do now is you can take the L2 norm of the, so both of these are, of course, equipped with an inner product, and you can take the L2 norm of the restriction relative to the L2 norm of the original function. And if you equip everything with probability measures, then it turns out that this ratio is given as follows. It's pi squared over 15 L3 half f. So now f is a lift. So f is a lift of this f over there, and f is assumed to be hecka normalized. So little f is a hecka eigenform. There is a unique way of doing this, putting the first Fourier coefficient to be 1. 1 sims square f times 12 over k minus 1, and then a sum over an orthonormal basis. So this is an orthonormal basis of sk. So notice that f has weight 2k minus 2, but here we are summing over forms of weight k. 1 half sims square phi times f. So this makes good sense. I mean, this sum has size k, and on Lindelof, all of these are essentially bounded. So you divide by k, and all of this is also bounded. So roughly speaking, the mass is rather evenly concentrated on the diagonal. But it's a very beautiful formula. It compares the mass on the diagonal with the complete mass, and it's given as a mean value of certain degree 6 L functions. OK, so that's one thing. And the other thing is that the L function, so the spinor L function, L spin of this lift f agrees with the original L function of the cusp form little f up to some simple factors. zeta of s minus k plus 1 times zeta of s minus k plus 2 times L fs. OK, and this cytokoro cover lift has been generalized to arbitrary values of n. Well, it was conjectured, I think, by Duke and Imamoglu, and it was then proved by Ikeda. So this is called the Ikeda lift, generalization to arbitrary n. This is the Ikeda lift, which has similar features. But it's more complicated, for instance, in the sense that for cytokoro cover, one can give the Fourier coefficients exactly, even for non-fundamental discriminants. This is a complete disaster in general. There are formulas, but they are not manageable. OK, so finally, a bit of literature for further reading. A very good book, and I think one of the standard references, is Freitag's book, Siegelsche Modul Formen, which unfortunately is in German. Nevertheless, I think it's one of the standard references. There is a shorter book that basically contains a subset of Freitag's book by Klingen. This is in English, introductory lectures to Siegel Modul Forms, or something like this. There's a very old book by Siegel, which is called Symplectic Geometry. And it was, in fact, a research paper that appeared in the American Journal of Mathematics. But it's like 100 pages long, and then they decided to make a book out of it. So it's the exact copy of the original paper. This Siegel's book contains much of the underlying geometry. OK, any questions? Is the analytic continuation and functional equation known for the spin or L functions in general? No, no. I think for N equals 2. And I'm not sure what the state of the art is with N equals 3. Perhaps there is some partial, I don't know, but certainly not in general. OK, any other questions? It was a lot of works of Conan and Zagir about Schumann relief from the congruence subgroup. And we start not the grammar reforms or model groups, but congruence subgroups. And then leave them for half waveforms. And then I wrote us here. Yes, yes, sure. So I mean, I'm not going to go into the greatest, most greatest detail. So certainly you can do this for congruence subgroups as well. But I mean, this nice formula of Conan and Zagir, to my knowledge, exists for the full modular group. Although, of course, the general framework holds in generality. In any case, this certainly is possible for congruence subgroups. There is nothing special about the modular group, except that everything is simpler. For these two NL functions, I mean, how should one think about the fact that it's two different ones, that the answer reflects the underlying aspects of the form, or if you think about it? Well, combinatorially, it's a different combination of the Satake parameters. I mean, they are, of course, defined by Euler products. And the Euler factor is a certain combination of the Satake parameters. And you can do this in different ways. Well, at least for SP4, it's also there's two fundamental representation of SP4. And these are the Langlund's NL functions, but actually, it's two fundamental representations. So I'll prove it, they'd be quite independent. OK, well, then this ends my short discussion of Siegel modular forms. And so I'm now going to move on to PGLN. OK, so here's the setup. Our group G is PGLN of R. I won't mention Adelts. In this lecture, everything is either over R or over QP, but no Adelts. OK, it has an Iwazawa decomposition, NAK. N are the unipotent matrices. K is PON. Gamma, for us, is just SLNZ. Again, you can talk about congruent sub-cruise, but I'm not going to do this. H, this is now a third H. It's not the Siegel upper half plane. It's not a hyperbolic upper half space, but it's a model for G mod K. The dimension of H is N minus 1 times N plus 2 over 2. And we choose the following coordinates. We choose the Iwazawa coordinates coming from A and K. So a point z, little z, perhaps, is x times y. And x is 1, 1, 1, 1, 1, something. And y is given by 1, y1, and so on up to y, N minus 1, y1. OK, and double use the vial group. OK, so for N greater than 2, strictly bigger than 2, there are no holomorphic forms. So everything is a mass form. And so we really think of these forms as mass forms. And so we talk about differential operators. D is the algebra of G invariant differential operators on H, and that's the center of the universal enveloping algebra. And this turns out to be a polynomial ring in N minus 1 variables. And this can be viewed as polynomial functions on the dual of the Lie algebra of A. So A is the group. German A is the Lie algebra. This is the dual vector space. And I take all symmetric powers. So I can think of this as polynomial functions. And they should be invariant under the vial group. This is the so-called Haas-Chandra isomorphism going from here to here. So A is the Lie algebra. And it's isomorphic to R to the N minus 1. But we view it as a hyperplane in Rn. I mean, this is a reflection of the fact that we are working with the projective group. So we are working with a trace 0 hyperplane. Trace 0 hyperplane. So a typical element in A has n components, and they add up to 0. Trace 0 hyperplane in Rn. And this isomorphism here sends a differential operator to, well, a map lambda d that goes from A star to C. And for instance, the Laplacian goes to the map that sends mu, which is mu 1 up to mu n. So this is now an arbitrary element in A star to n cubed minus n over 24 plus sum of the squares. To each operator in script D, there is attached a polynomial. And this polynomial tells you that if you have an eigenform, an automorphic form with these spectral parameters, then the eigenvalue under the corresponding operator is this polynomial. OK, so for me, always mu is the notation for an element, well, and potentially the complexification of the dual of the Lie algebra, say, modulo of the vial group. These are the spectral parameters, or Langland's parameters, of an automorphic form. So each automorphic form comes with such an n tuple of spectral parameters. And if you want to know what is the eigenvalue of this automorphic form with these spectral parameters, then you just plug the values into the respective polynomial under this isomorphism. OK, so for me, everything is normalized such that the Ramanujan conjecture says that the muses are real. Other people normalize differently. For instance, in Goldfeld's book, the unitary axis is 1 over n plus i a star. And some other people, other authors, say that the unitary axis is i a star. This is a matter of taste. For me, the unitary axis is a star. OK, as you know, the Ramanujan conjecture is not known, not even for n equals 2. And the best general bounds are imaginary part of mu j is bounded by 1 over 2 plus 1 half minus 1 over n square plus 1. And for n equals 2, better bounds are known. So are for n equals 3 and 4. But in general, these are the best bounds. And this is due to Luol Rodnik and Sarnak. OK, so how many mass forms are there? How dense are they, in some sense? And this is measured by the Harris-Chandra C function. There is a general definition for the Harris-Chandra C function for any Lie group in terms of root systems. I'm not going to go into detail here. Let me just say that in our case of the group Pgln, the Harris-Chandra C function is given by a constant times the product 1 less or equal than j, less than k, less or equal than n, g of lambda j minus lambda k, where lambda is lambda 1 up to lambda n in A star, perhaps star c. Well, let's define it only on A star. Well, yes, that's right, except that lambda, for me, is an integration variable. Later, I integrate over lambda, and I keep mu fixed for my favorite automorphic form. But yes, they play the same role. And g is a quotient of gamma functions that you can simplify, and it just looks like x times tangent hyperbolic of pi x. So the tangent hyperbolic, of course, only plays a role if x is very small. Otherwise, this is essentially 1. So g of x, for practical purposes, g of x is x. Well, in fact, it's absolute value of x, because if x is negative, then also the tangent hyperbolic is negative. So this is bounded by product of 1 plus lambda j minus lambda k. OK, so unless there is some conspiracy between the components of lambda, if you choose a generic lambda such that all these differences are roughly of the same size, then this is the norm of lambda to the power n choose 2. If you're at the walls of the vial chamber, when some of these are perhaps identical or very close to each other, then, of course, the spectral measure is a bit smaller. So this drops at the walls of the vial chamber, of the vial chambers. So the density of mass forms is a bit less at the walls of the vial chamber. And there is a vial law, and this was proved by Müller and Lapid, 2009, that this spectral density really measures the density of mass forms. So the number of cusp forms with spectral parameter lambda plus o of 1, so you take a ball of size 1 about a given parameter lambda is of size c lambda. So what I wrote down here. So I mean, the precise statement is a little different. So the ball has to be sufficiently large in order to get a lower bound. And then if you expand the region, then they actually get an asymptotic formula with a power saving error term. So this counts the number of mass forms. For instance, in the case n equals 2, you get the usual vial law and this then holds in general. OK, so I mentioned earlier, so I'm using these blackboards rather randomly. So I mentioned earlier, for n greater than 2, there is no discrete series. We only have mass forms. However, we do have lifts from holomorphic forms for n equals 2. We do have lifts from holomorphic forms on GL2. So you can take a holomorphic form on GL2 and, for instance, take the symmetric square and then you get a mass form on GL3. OK, any questions? OK, then next thing I want to discuss are Whitaker functions. Whitaker functions. OK, Whitaker functions have two arguments. Well, they have an index and an argument. And the index lives on AC star modulo the vial group. And the argument lives on H, this H here. And the value is a complex number. OK, and they are defined as follows. We take an x in A and I write down the first off diagonal as x1 up to xn minus 1, and I don't care about the rest. And then I can define a character. Theta of x is just E of sum xj. This is a character on n. Character on n. And then a Whitaker function satisfies the following properties. w mu of xz is theta of x times w mu of z. So it transforms by multiplication from the left with respect to this character for all z in H, x in n. And it satisfies differential equation. If I take a differential operator d in script d, then this is lambda d of mu. So lambda d is this map that I defined up there, times w mu of z for all d in d. OK, so this is what I want to call a Whitaker functions, by no means clear that such objects exist and so on. But they do exist. And you can actually explicitly construct them as follows. So in particular, these are all eigenfunctions of this algebra script d. Construction is as follows. You integrate over n a function i that I'm going to define in a moment, i mu. You take the long vile element, w long, times u times z theta bar of u d u. u is the harm measure on n. And i mu of z depends, in fact, only on y. It's independent of x. It's a product. It's basically a power function. j from 1 to n minus 1, yj to some power lj of mu, where these are linear forms in the mu that I'm not going to write down. So lj is a suitable linear polynomial. Basically the idea is, so you choose lj in such a way that this power function, i mu, satisfies exactly this differential equation. And then the Whitaker function inherits the differential equation from the differential equation of i. You don't have to take the long vile element. You can also take other vile elements here. And then you get degenerate Whitaker functions that come up in the constant terms of the Fourier expansion of Eisenstein series. So yeah, if you take other vile elements and the long vile element, you get degenerate Whitaker functions. Degenerate Whitaker functions for other vile elements. For other vile elements, will the integral be convergent? Sorry, what's that? For other vile elements, other than the long element, the integral will be convergent? The what? Ah, for GL2, there are two. Well, yes. I mean, probably you can't take the trivial vile element. And for GL2, there is no other vile element. But I think if you take a non-trivial vile element, then it converges. OK, so you can write this down. In principle, it's completely explicit. It's an integral over the unipotent group. But these Whitaker functions, we have to admit, they are very poorly understood. OK, as an example, the case n equals 2, well, then we can write down explicitly the Whitaker function. So the Whitaker function has an argument here, which is a pair of two numbers that add up to 0. So it's mu minus mu. And then traditionally, it's just mu. There is no extra information. But nevertheless, z is up to scalar multiplication, square root of y, e of x, k i mu of 2 pi y. And then it's a matter of taste, whether you further normalize this or not. I like to normalize it by multiplying by cosine hyperbolic pi mu over 2. But again, that's a matter of taste if you want to do this or not. The Bessel-K function decays exponentially in mu, so you can compensate for this by multiplying with the cosine hyperbolic. Or you don't have to do this. And OK, so Bessel functions are fairly well understood. So this is a Whitaker function that we can handle. But in general, for n greater than 2, it's very poorly understood. So if you take the construction you gave for n equals 2, do you get this cosine hyperbolic? You just literally take the sitting line. I forget. So the cosine hyperbolic, if you include it or not, is a matter of whether you take the so-called completed Whitaker function or not. I have to look it up. I forget whether this integral produces the completed Whitaker function or not. But we do have some partial knowledge on general Whitaker functions. And what we know is the following. So you can recursively get a degree n Whitaker function by integrating a degree n minus 1 Whitaker function. Then if you repeat this, you can get down to degree 2 Whitaker functions, which are Bessel functions. So general Whitaker functions are integrals over Bessel functions. So it can be expressed as iterated integrals over Bessel functions over k Bessel functions. But of course, if n gets big, then you get a whole bunch of integrals, and things become very complicated. But in principle, there is such a formula, and that's eurostat, eric state. I would believe in the spirit of Emanuel's talk that this is a reflection of the fact that the GLN closed-term insum attached to the long vial element, at least in certain cases, can be written as a product of GL2 closed-term insums if the moduli are pairwise coprime. And I would assume that this is some weak form of reflection of this fact that in certain cases you can reduce the GLN case to products of GL2. For n equals 3, a bit more is known. So for n equals 3, we have a fairly explicit. The iteration of the communicative function? I'm not sure. I have to look it up. I think it's more complicated. You get arguments u. Well, perhaps you get u and 1 over u. Well, I don't know exactly. I think it's not just a multiplicative convolution. It's a bit more complicated than that. But you can look it up in state's paper. We know the double-mellon transform of, so if we keep x fixed, then in the variable y, I have two entries, y1 and y2. So the double-mellon transform in y, y1 to the power s1, y2 to the power s2, dy1 dy2 over y1 y2, is known. And it's, in fact, what you, well, perhaps I don't know if you would expect this, but certainly it's a ratio of gamma factors. So that's fairly simple. And then we have a very important formula. That's a very strong tool. And it's called state's formula. This is the Archimedean rank in Selberg theory. It's the product of two Whitaker functions. If you take w alpha of y and w beta of y bar, determinant of y to the power s, and then the correct measure on the group A. So in the classical case, this is a product of two Bessel functions, but now you take a product of two general Whitaker functions. And this is the sort of rank in Selberg ratio of gamma factors. So it is what you would expect. So gamma, OK, gamma r is the usual thing. I'll define it in a second. It's s plus i alpha, I hope I get the signs right, alpha j minus beta bar k. I hope that's correct, where j and k run from one to two. And so basically you take the Langland's parameters and combine them each with each. Divided by whatever you expect from rank in Selberg theory, gamma r of ns, so that's kind of zeta of 2 in the classical case. And then product, so here you get a kind of symmetric square l function at 1 gamma r 1 plus i alpha j minus alpha k gamma r 1 minus i beta bar j minus beta bar k. So I don't know if this is readable or not. Hopefully it's not readable because it's perhaps slightly wrong. And perhaps there is a constant involved. So if you want to get the constants right, that's also a complete nightmare because everybody normalizes slightly differently. And you can be sure in the end you will be off by a power 2 pi. OK, so what is gamma r? Gamma r is the usual thing. Gamma r of s is pi to the minus s over 2 times gamma of s over 2. OK, let me just write down one more formula to wrap this up. And then I'm done for today. In GL2 there is the so-called Contorovich-Lebedev transform. If you integrate the Bessel-K function against a test function and then integrate it back, but this time over the index, then you get the original function back. And this holds in general. And this was recently proved by Goldfeld and Contorovich. So if you start with a test function f and you integrate it against a Whittaker function and you call this f hat of mu, then you can recover f from f hat. So if you integrate f hat of mu against the Whittaker function and now you integrate with respect to mu d mu over the spectral measure, OK, I said that lambda would be my integration variable. Now it's mu, whatever. Over the A star, then you get f back. So this is the Whittaker functional theorem, right? Yes, yes, yes. I'm just, isn't that was proved to be by Wallach? OK, that's true. That's true, but yes. OK, yeah. Goldfeld and Contorovich give a very explicit formulation of this. But certainly you find an abstract version of this in Wallach's book and in Wallach's work. Is this really Wallach? Yes, OK. OK, fair enough. Didn't you say yesterday I'm going to conclude the session? No. OK, anyway. So that's it. Are there any questions? OK, well, that's it for today. Does the speaker really ask for questions? But if you just want to take questions only from the organizers, then ask the questions now. Yeah. So regarding the leaps, this might be a stupid question. How many percent, what is the percentage you can get in the leaps from which you can get from a lower order? In general, you mean? In a special case, for instance. Well, I mean, if you look at Weil's law, so for GL3, in if you take, so here you're sitting in A star. And in a ball of radius 1 of size t, you have roughly t to the three guys of mass forms. And on the diagonal, so the Gehlberg-Jacke lifts by Weil's law for GL2, they are of size t. So you have t lifts sitting exactly on the diagonal, but in the whole square of size 1. So this is t, t plus 1, t, t plus 1. So in this ball of size 1, you have all together t to the three mass forms. So the lifts are, in some sense, negligible.