 Thank you. Thank you, Mary-France. Thank you for inviting me to this conference. It is a great pleasure and an honor. I had the pleasure actually, some years ago, 1995, actually the spring of 1995, to share an office with Gaudemont at Jussieux. And I didn't really know him before that, and he was just an absolutely charming and extremely interesting person. And it was a great pleasure to know him, to meet him at that time. Zeta functions. I guess I chose the title because these days, perhaps the notion of a zeta function is one of the mathematical concepts that is most closely associated with Gaudemont, I guess because of the volume with Jacques. And so what I would like to do is to talk, this would be sort of an introduction to the program of Langlands, which goes by the name of Beyond Bendoscopy. And it is, I guess it's a strategy or maybe a dream of Langlands that is probably close to 20 years old now, to try to attack the general principle of functoriality by means of the trace formula. But to apply the trace formula in a very different way than it has ever been applied before, to combine the trace formula with automorphic L functions. Well, I won't talk about the general sort of premises of this program. It's quite a lot. I mean, to just describe that might be two or three lectures. So I'm going to talk about a very specific question, which is probably the first major step that would have to be accomplished. This is, I mean, there are other possible ways of looking at it, the Beyond Bendoscopy, but the proposal, the actual proposal of Langlands, what I'm going to talk about would be, I think, the first major step that would have to be accomplished before you could go on. And it has to do, well, it has to do with a part of the trace formula. And specifically, it's, there are problems for the most basic terms in the trace formula on the geometric side of the trace formula. And these somehow have to be solved first before one can look at the more exotic terms. So let me just recall some of these basic terms, the primary terms in the trace formula. So this is just going to be an approximation of the trace formula. And so in particular, I will confine it to GLN. Actually, I think maybe I'll take it for GLN plus one. The parameterization is a little simpler. The things I'm going to talk about are a little bit simpler to describe in the case of GLN plus one. It's over Q. So again, it's a very straightforward case. And I would take a test function F. Trace formula, of course, depends on a function. So a test function F on GA. And let's divide out by a central subgroup to make it have an actual discrete spectrum. Z plus it would be the, let's say, the real scalar matrices in which R is a positive real number. This is a subgroup of GR, the real points of the group, which is a subgroup of the idyllic points. So it's a central subgroup of GA. And so we'd be interested in functions that are Z plus in variant. So the general trace formula is a general kind of expansion between terms that have some sort of geometric origin, distributions that have some sort of geometric origin. Actually, it's probably better if I put it on a separate board. In fact, I will. Yes, I'm trying to budget my space here. So let's put it here. I've got the stick. Yes, no, I've seen other people use the stick very well. So I've got that in mind. But the trace formula, it's a formula, I'm going to, this one's a little more brighter, easier to see. So it's a general expansion between a general identity between a sum of geometric terms, a sum of spectral terms. Now, what the, in some sense, the primary geometric terms. So this is equal to, so the whole thing is equal to an elliptic expansion, elliptic regular terms, plus a whole lot of more complicated supplementary geometric terms. So that's what this is equal to. And this is equal to the discrete part of the, basically the thing that you really are interested in, the discrete part of the spectral decomposition. Let's write that as i2 of f plus supplementary spectral terms. So that's equal to this. And the terms that have to really be understood, the basic problems that have to be understood before you can do anything else, and in fact some of the most important problems have to do with these basic elliptic regular terms, which I recall are of the form, it's basically what matches what you'd get for the trace formula of compact quotient. So a sum of elements gamma in elliptic regular conjugacy classes of g, of a volume term, and an orbital integral. So that is just, that is just this thing here. And these things, this stands for elliptic regular conjugacy classes in gq. This thing I recall, so that's what this is. This is the volume of centralizers, so that's the volume of the quotient of g gamma, for a given gamma. That's the quotient of z plus times g gamma sitting in g gamma q, sitting in g gamma a. That's what this thing is. And this thing is the orbital integral. So this is equal to the integral taken too much space for the volume quotient, and I need a little more space just to recall the orbital integral. So that's the integral of g gamma a divided into g a of f of x to the minus 1 gamma x dx. I recall that's what this term is. And let me just, I won't say too much about these either. This is what you really are interested in. So that this is the sum over pi in, let me write it as pi two of g multiplicities of pi in the l two, the discrete spectrum sitting in the l two of g a modulo gq times the character, the trace of pi of f. Now that's just, that's, I'm just talking about what this is here. And this, by this I just mean the set of irreducible representations which are contained in the discrete spectrum of l two z plus gq g a. So these are, these are what you really want. This is the, this is the essence of functoriality to understand relations among these representations in the discrete spectrum as the group g varies in a, in a very natural way. That's what functoriality applies to. And these are some, in some sense the parallel terms to the discrete spectrum on the spectral, on the spectral side. So this is, that's what this is equal to. And this is, this is, I'm not writing this very clearly. That's what this is equal to. Plus, plus supplement, let's call them suplets. Let's, so, so this is just equal to this. And I'm not saying anything about these supplementary terms. All right. What I'm going to do is I'm going to write, I'm going to write the elliptic regular part. I'm going to, I'm going to write this approximation little symbol. This is equal approximately, well, to i two of f. And so this is the famous, the notorious pretend that they're equal symbol. But I'm not going to talk about the, even the spectral side. I'm going to just describe the initial steps that you would like to apply to the primary geometric terms, this thing here. So what I'll do is I will put this at the top, and we'll just have it there. I'll come back to it at some point, but try to discuss what we, the kind of things we would like to do with those terms. Those terms have somehow always been treated as a kind of black box. They have been endoscopy. The theory of endoscopy, for example, compares these terms for a given group G with another group, maybe an endoscopic group, G prime. But one doesn't sort of look at the internal structure so much of these terms. I mean, there have been questions. But what is needed here is to really understand, to really understand that there seems to be an internal structure to these elliptic regular terms, that really has to be understood before one can proceed further. So perhaps the first thing to say about them is a proposal or suggestion in a paper of Frankel Langlands and Go. And it seems a harmless enough suggestion, but it's kind of a notational slash kind of philosophical proposal. It's to replace an element gamma, a regular elliptic semi-simple element in GQ to replace it by its characteristic polynomial. This is just for GLN. So you'd have to take the construction of Steinberg and Hitchin in general. But I'm just talking about GLN. So it really is just a characteristic polynomial of the matrix. Yep. P gamma of lambda, the determinant lambda i minus gamma. So let me write it lambda n plus 1 minus a 1 lambda to the n and so on plus minus 1 to the n a n lambda plus minus 1 to the n plus 1 a n plus 1. So we can also parametrize it by a where a is this vector of rational numbers. This would be the determinant of the matrix. So let's write it, let's keep that separate. So it's numbers, let's say b1 up to bn a n plus 1 or just a vector b and a number a n plus 1 where b, this b is going to be an arbitrary rational number and bn plus 1 has to be a non-zero rational number. Well I say arbitrary but that's not quite right because I'm dealing with elliptic conjugacy classes and that's to say precisely that this characteristic polynomial is irreducible over the rational numbers. So we are then the proposal in this paper of these three authors is to identify an element gamma with an element a in qn cross q star. It was the property that this polynomial p a of lambda is irreducible over q. And gamma up to conjugacy of course. Gamma, yes so did I, yeah these are supposed to be conjugacy class, semisimple regular conjugacy classes elements gamma whose centralizer is a torus and that translates elliptic elements translates to the requirement that the characteristic polynomial be irreducible. Well let me make an even further simplification, the essential problems can be seen even in this further simplification. Let me write, let me suppose that f is a product of an Archimedean function and a product of piatic functions where f infinity, so f infinity then would be a smooth function of compact support. So I'm dividing up by the center and where f infinity, upper infinity is the unit function in at all of the piatic places. So the characteristic function so let me write it like this the i upper infinity or in other words the product over p of ip where this is the characteristic function so ip is equal to so this would be the characteristic function of the product over all p of gl n plus one of zp. So that's a perfectly good function to put into the trace formula. Okay I'll cover up the trace formula just momentarily and we'll pull it down when we need it. Then if I do this gamma and and furthermore I would like to restrict myself simply to so I'm going to take the test function to be this restricted form and let me consider only those gamma such that o gamma of f so that's going to be the product of o gamma orbital integral of gamma of f infinity times the product over all p of orbital integrals um of these piatic unit functions and let me just simply restrict myself to elements gamma for which those are non-zero and then by so doing I would then be considering pardon me oh yes I'm sorry yes I think I called it this is orb and so by making that restriction then I'm asking I'm looking at elements gamma that are bijective with vectors a so let me write it like this vectors a that consist of pairs where but b is now an element not just in q to the n but rather z to the n and epsilon is then the determinant of gamma and so that would be equal to plus or minus one integral matrices yes so I guess what I want to say is that this is is a really fundamental I mean it seems to me this is a fundamental change in outlook in the past we're thinking of the elliptic terms in the trace formula as parameterized by extensions of degree n plus one and then elements in those extensions that well that whose centralizer is is just a torus here we're looking we're thinking of them differently we're thinking of them in an entirely elementary way as polynomials parameterized by these integers so integral polynomials with constant term plus or minus one which are irreducible over q so these are kind of this kind of like the olden days maybe in galois theory where you don't know what the fields are you just are thinking of polynomials fields are somehow hidden there there are things that you don't know about and you're just going to try to work with your bare hands with polynomials whether I mean it's intriguing to try to imagine when you try to analyze the structure I mean I don't have any evidence for what I'm about to say but it's intriguing to think that if you're going to try to analyze the elliptic part of the trace formula in these terms it could be that the actual galois group of the polynomial that you're considering is well it's of course a subgroup of the symmetric group but the actual what actual galois group you get it's intriguing to wonder what that might have to do with the with the trace formula so maybe it's I mean it might have kind of has a flavor of 19th century galois theory theory of equations maybe galois resolvance but I don't have any evidence for that but it's it's sort of intriguing and so one consequence of this change of outlook is uh so I mentioned that Langland's proposal for beyond endoscopy concerns combining the trace formula with automorphic L functions now if I were talking about sort of the sort of general ideas that you hope might follow from this I would be talking about the spectral side and automorphic L functions of the the idea of Langland's is to combine the trace formula with automorphic L functions of the representations that are supposed to occur on the spectral side so I'm not going to be talking about that what I'm talking what I want to talk about here are L functions but not on the spectral side more elementary things simply the zeta functions of the terms that occur on the geometric side so those are the zeta functions that I'm we're in my title for start for start for start so zeta functions okay so for that suppose you're given an element gamma as above a pair b epsilon we of course have the field that comes from gamma where the coordinates of gamma lie this is what we're not going to be looking at explicitly so much as I've said from now on but we do have this field so of course it's q of lambda modulo the ideal generated by p gamma lambda so e over q is an extension of degree n plus 1 and it has the property that the centralizer group that I mentioned at the beginning is the multiplicative group of e but we have something else now if we're going to be changing this little this kind of philosophical change we have something else besides e we have r so this is I'm going to write it as r of gamma and so this is not q of lambda modulo this but z of lambda modulo p gamma of lambda the ideal generated by this so this is contained in the ring of integers of e and so this is I didn't know this before I started thinking about these things I didn't know this term but very basic term this is called an order in oe oh is that dedicant I see I see so he was onto this so so so what's relevant to both of these situations and is particularly important I guess is the discriminant so the discriminant of the order there's a discriminant of the order and there's also a discriminant of the field I want to take their absolute values in both cases and then this is the discriminant it's the same thing as the discriminant actually of the polynomial characteristic polynomial p gamma of lambda this is the discriminant of the field and the two are not equal there's an index of the order that is positive at the square of a positive integer so this is as sigma r is a positive integer and I guess this is often called the index and those objects seem to be absolutely fundamental to understanding what we would like to we would like to understand of these of these terms all right so zeta functions we've got the dedicant zeta function of the field e so I recall that that's equal to it has an Euler of course has an Euler product the product over all primes of local dedicant zeta functions local components and so that would be the product over p of the product over all prime ideals in e above p so p in oe prime so script p divides little p of one minus the norm e over q p to the minus one that's the Euler product of the static and zeta function s s yes minus s so that's the Euler product of this zeta function it also has a Diraclet series zeta e of s so it's a sum over ideals integral ideals you know f excuse me oe of norm e over q of l minus s which you then can write out as a sum by taking these norms as a sum as an actual Diraclet series as a sum of over positive integers n and it has a functional equation let's write it like this lambda so I'm calling this zeta of e let me write lambda of e of s that's equal to lambda e of one minus s where lambda e of s is equal to so it's just to you the way the notation that I've used here does not include the Euler fact the Archimedean Euler factor that would need be needed for the functional equation so it'd be zeta you could write it as zeta e of infinity times zeta e over s and so this is an Euler factor given by various gamma factors but the zeta function I really wanted to talk about I'm not going as fast as I should be but the zeta function I really wanted to talk about is not the Dedekin zeta function but it's a it seems completely new it's by this is by it's by Zhiyun so it's Yun's zeta function of the order r so this certainly was completely new to me it's about three or four maybe five years old he credits actually Collins paper of Colin Bushnell for motivation but his zeta function is a generalization of the zeta function which reduces to the zeta function when the order is equal simply to the ring of integers o e in the in the in the field so it's a it's let's write it as zeta r of s r being this order and it it has many of the same properties it has an Euler product zeta r p of s so he actually so I'm not going to give you the I won't give the definition of it but these Euler factors are given by a Diraclet series as these are and this is this is this local Euler factor for the Dedekin zeta function is a Diraclet series it's a generating function whereby you count the number of um o e modules of a given length of a given size and you you count the number of such things and you raise that you that would be the coefficient and you multiply that by p to the minus n s where n is the length of those modules this is defined in the same way where you take you don't take o e modules namely ideals but you take our modules and I think you count them you ask for our modules which are contained actually in the dual our check which is the dual of of this as an our module over itself and that's something that would contain o e I'm not comfortable with these things I didn't grow up learning the details of commutative algebra but there's a generating function he defines it in a simple way as a generating function and by which you count certain our modules according to their length a big day no pardon me it might have been known to us is that so is that so okay you mean you mean this this zeta function okay well let me let me wait a while I got there's some punch lines that he introduced a very uh so um I did not check okay so um in any case zeta r of p of s um it should be very closely related to the corresponding local factor for the datacan zeta function it's actually differs from it by a polynomial of uh in p to the minus s so it's it's of the form um p r I'm going to write it as p r of p of q to the of p to the minus s so it's a it's a it's a model it's a modification a multiplicative modification of the local factor for the datacan zeta function it's a local factor where uh for a polynomial p r of t um so this would be this is a polynomial with constant term excuse me constant term one and with integral coefficients sorry uh of degree basically given in terms of the discriminant or the local component of the discriminant so there's a polynomial of degree so I'm going to write it like this it's a polynomial of degree two delta of p where delta is the valuation of the p part of this integer um uh this integer sigma this integer there so delta p um so delta or delta p or we if we want we can index it by r delta r p this is equal to the valuation um at p of sigma of r all right so that's that's what the he is introduced and the product so this is of course that's the local factor that is introduced initially that's this thing here the product um which is still concerning the local factor but the product and let's write it as zeta tilde r of p of s so you take basically you take the local factor but you actually rather than having a polynomial you have it with some negative power positive powers as well as negative powers so p to the delta of s times this polynomial that's the new factor in the in the function p r p at p to the minus s so this is a this is a sentence this is a long sentence the product of this thing we know what that is this is the thing that comes in here and you've put its value at p to the minus s that product um satisfies two things it has a functional equation it has its own it's just a local factor but it has its own functional equation zeta r p tilde of s is equal to zeta r p tilde of one minus s so it's got its own the Euler factors of an L function or a Dedek and zeta function don't satisfy that functional equation but this obstruction or this difference between the Dedek and zeta functions local factor and the Yun zeta functions local factor does satisfy this functional equation oh do I have I forgotten a tilde up here I have yes I think it's I think it's okay I think it's all right in front of the maximum yeah I think I think that's all this is the definition or this is the property of this this this is the property of this slightly different oh yeah I forgot something yes yes you're now I've got this with this thing here no no I think I think this is what I want I think this is fine yes yes all right okay all right that's that's impressive all right and now here to me anyway this was that this was this was what was really surprising so this local factor satisfies a functional equation but it also has a very interesting property for its value at one so the local this local correction factor has the property that's value at one is equal to something that is very central to the trace formula namely it's the orbital integral of gamma at one p it's a chaotic orbital integral which seems like a real surprise what happens when you take full part the full ring of intentions you have the orbital integral of an element gamma whose first assertion one assertion one it's just one it's just one identically equal to one so the at that case the dedicated the yun zeta function reduces to the dedicated zeta function it does and I think he uses maybe one the normalization that you may not want to use eventually but I think it's the normalization which assigns volume one to the centralizer of gamma to assigns volume one for example yes yes are you assuming this real part of this is greater than one in the first sorry assertion one are you assuming real part of this is greater than one here but this is this is for all s for all s I mean it has fun it has it's an analytic function it's a meromorphic function yes yes it's a product overall prime p that's that's right it's just a modification of the it is a modification of the data can zeta function all right so what is this from one one immediately leads to an identity well and it leads to a functional equation of the global of yun's global l function where lambda r of s is basically it's it's again you tack on essentially the same Euler factor actually you need to modify it very slightly by this number attached to the discriminant of the order zeta r to the s zeta infinity so you need to modify the or the Euler factor of the dedicated zeta function slightly by just the power of this thing but then just from this property you see that you're going to get a same fun the gamma factor remains the same it's just infinity you see nothing yes yes all right and let me just also add that let me add that from two I'll put it here maybe so from two I'm just rewriting basically that relation to in terms of the original Euler factor so the orbital integral of gamma at p is equal to p to the delta times zeta E p of one to the minus one zeta r of p one so this is the Euler factor at one for the dedicated zeta function this is the Euler factor at one for the dedicated zeta function this is the Euler factor at one for this new zeta function it's mod of multiplied by this normalizer and that is the orbital integral so the orbital integral is basically at least at value one it measures the difference multiplicative difference between this new kind of zeta function and the dedicated zeta function all right just a couple of remarks I'm I'm this is taking me longer than I expected but I just a couple of notes this function this zeta function in a special case has been known to analytic number theorists for some time in the case g equals gl2 it was introduced by zagie well essentially it's the same thing um zeta zeta r of s it's really really the quotient of what I have up there divided by the Riemann zeta function and so what one gets in the for in the case of Dedek and zeta function is just a quadratic Diraclet Diraclet L function but modulo that difference this was um for gl2 this was introduced by zagie and other people have also studied it in 1977 for quadratic extensions of q pardon me quadratic extension you know we might do that both both both um so this was yeah this was a starting point um um oh I guess what I would also say um is um so this is a a compound remark the local factors um let's say zeta r p of s and the orbital integrals zeta tilde r p of one these were they have these have simple formulas and these simple formulas were a starting point um for the recent work the the part of the thesis of ali altu so he did some extremely important work for the case of gl2 and I'm just going to say they were a starting point for his work in establishing what I'll say in establishing Poisson summation for gl2 I may not get I'm not going to get as much said as I had hoped so I'll say that Poisson summation you've got the global orbital integrals are parameterized by elements in z m z m they're distributions that are parameterized by z m and the question uh the question that was later then posed in Langland's go and frankle can you apply Poisson summation to that sum over z n in the case of gl2 it's just a sum over z and uh altu with some extremely clever um you can't apply Poisson summation at all as it as it stands just yet but he did several very interesting things and was able to apply Poisson summation in the case of gl2 in the case g equals gln or gln plus one um the orbital integrals so this would be zeta tilde of r of p of one um perhaps also the local factors so this the orbital integrals there's less information here than are in the local factors because this would just be the local essentially the local factors at s equals one but the local factors so zeta tilde of r of p of s these I think it's fair to say uh certainly in the case that the order we're talking now about a local order so we can ask whether it's uh whether it is elliptic or whether it's hyperbolic uh elliptic case would be the main one um and then we can ask whether it's unramified tamely ramified or wildly ramified and certainly up to tamely ramified um these seem to have explicit formulas and you're going to need such things very much so explicit but very rich I'm going to say um richly rather richly complex functions I haven't checked everything but um I think it's pretty fair to say that these uh that can be derived I've certainly done enough special cases to make me believe that uh up to the tamely ramified case these I expect to be able to derive such things from well from uh some two very striking papers of valse berger germs so this is not the title of the paper but it's germs of peatic orbital integrals for gln I have three minutes left so what I would like to do what I would have um I've been my three minutes um I want to return to the trace formula okay so that's well it's up there um all right um so it's up there I elliptic regular f is then going to be equal to a sum the way we've originally written it as a sum over gamma volume of gamma the orbital integral of gamma f infinity and the product overall p of orbital integrals gamma of a unit element in that algebra all right so the idea would be to rewrite each of these three terms you see this is very uh this is very nicely uh matches what we would get from yun's zeta function so this the way we have set it up we have taken a rigid function at the peatic place no room for varying it just the characteristic function of the unit this is the only thing that would be varied and so we would regard this as a linear combination of distributions um in f infinity with coefficients built out of this and this um so we've got this one would like to uh I'm going to just have to say a few words to finish off this is an Archimedean orbital integral um these of course were studied by Harris Chandra and um it's best to normalize this by multiplying it by the square root of the absolute value of the vial discriminant now that fits very well with what we're dealing with here the vial discriminant has the same absolute value as the as the discriminant of the characteristic polynomial of gamma and so um uh you multiply it by that and then you get something that is close to what Harris Chandra studied um it's a function that has very mild singularities this um uh well you you do two things with that you apply um I'm following the strategy of l2 that he used for gl2 you use the class in Dirichlet's class number formula to rewrite this so this is basically um the regulator of the uh uh extension e of degree n plus one and you rewrite that as a product of the square root of the absolute value of the discriminant of the field extension uh times the value or the residue of the Dirichlet l function at one this you substitute in for uh Yoon's uh orbital integrals there and you see that that that matches very well with the values uh with with the discrim with with the value of the Dirichlet l function or rather its residue at s equals one and if you put those two together you have the residue of the Yoon's a function at one and so those two go together as a as a one sort of common coefficient times the orbital integral of this so in that context the question raised by Langland's and go and Frankel uh was to so so you're going to have a sum over elements not gamma but you're going to have a sum over elements b epsilon in z n cross plus or minus one of a of a distribution in a variable function f infinity with very simple but uh difficult coefficients namely uh basically the residue of this uh or if you prefer the the value at one of the quotient of the Yoon's a function by the Riemann's a function um but the question is can you apply that these guys raised can you apply the Poisson summation formula to that sum over b well not certainly not as it stands you've got your Poisson summation formula you're not allowed to have coefficients you could just have to have a sum of a Schwartz function over points in a lattice you can't have coefficients and you've got coefficients here and they're very bad coefficients they're not defined for for all real numbers they're just defined arithmetically for integers um what else um uh there was some particularly trenchant point I meant to mention which has escaped my mind uh but uh uh in any case that's the question that was raised and out to dealt with there's about three or four problems that are raised um uh and he dealt with them for the case of gl2 one after another very nicely um this the value at one of the zeta function um it's given only by a conditionally convergent series uh that's very bad for doing analysis with but he uh replaced that value of one by uh using the approximate what's called the approximate functional equation which uh replaces it at the price you pay is to have some extra truncation kind of functions or modification functions which you tack on to this and get a more complicated expression involving the orbital integral of f infinity but then you get an absolutely convergent series of uh where in which you take a sum over l over the positive integers you then uh and then there's further questions um uh but in any case I hope that one can um solve these questions I mean there's there's that's considerably more to be solved here than in the case of gl2 uh one would have to be able to successfully use laltz-berger's formulas for the orbital integrals one would want explicit formulas and um um uh but I I hope it should be possible to um oh yes I'm sorry the point I wanted to raise was they said they said look why don't we apply it would be great if we could apply Poisson summation formula to the sum over z n people have thought about this in the past but not for they didn't use the base of the Hitchin Steinberg Hitchin vibration that is to say they didn't replace gamma by its characteristic polynomial what they did was they simply took gamma to be an element in the multiplicative group of a field extension but then they tried to apply Poisson summation formula to the to the for each of those extensions of degree m Langland's actually used that for gl2 in his book on base change but it turned out not to I mean it looked very I mean it somehow seemed quite promising but it didn't it it it somehow has not it it's not quite the right thing um so this does seem to be the right thing I mean I mean by applying Poisson summation you are basically replacing this variable the coefficients of the characteristic polynomial your rate replacing it by a by a dual or additively dual variable by applying Poisson summation and you hope that that would be more easy to compare with the spectral information which of course are dual variables too and the specific problem you would like to settle by doing this is to be able to subtract away from the geometric side the contribution on the spectral side of the representations which are non-tempered the representations that occur in the discrete spectrum which are non-tempered the kind of manipulations that Langland's proposes in beyond endoscopy um are are contingent upon the geometric side being a tempered distribution so these guys actually made a big deal about trying to subtract away the contribution on the spectral side for the residual discrete spectrum the discrete spectrum automorphic discrete spectrum which is not uh tempered which is not cuspital in the case of gln and uh it seems likely um Al2 did this for gl2 it was just the one dimensional representations and he really showed that they corresponded to the spectral variable and after Poisson summation formula uh the there they would just be dealing with z1 and so the spectral variables are parametrized by z and it's the value of that spectral variable at zero that corresponds to the trivial representation on the on the uh the trivial one dimensional representation the one non-tempered representation in the discrete spectrum of gl2 which he showed corresponded to this spectral variable and uh he subtracted it away and thereby obtained an action a distribution on the elliptic uh regular uh part which I should say is tempered you don't know it's tempered until you prove Ramanujan's conjecture but which which ought to be tempered anyway I have gone a few minutes over I am terribly sorry thank you I have a conjecture it should correspond to divisors of n plus one and so it should correspond to terms uh uh you're going to have n plus one spectral variables parametrized by z and it should correspond to terms that are zero have zeros in those and then the remaining terms should be a diagonally embedded uh tempered spectrum so it corresponds to mogul and valse berger they they're they characterized the discrete non hospital spectrum for glm and that would seem see you can certainly you hope to see it and that's there's a very natural as I say conjecture as to what it ought to be namely n tuples n plus one tuples of uh integral uh integers um such which have a zero at uh regular places corresponding to a divisor of n plus one of all divisors we arrange over all divisors of n plus one on the spectral side you expect to see an analog of this you know no no this is purely for the zeta functions the l functions or zeta functions I mean these are more simple than the the automorphic l functions that would come on the spectral side it's it's purely uh for the zeta functions on on the geometric side yes one more question um that plays a certainly plays a role I mean even I mean even in the case of gl2 when you have applied plus on summation um uh it's no easy matter to isolate to prove that the spectral contribution from from the spectral variable at zero to correspond that it gives the characters of the trivial representation and the gamma functions play and their residues I mean it's a bunch various residues of these things that come in and it's the they do play a critical role in in improving that that that you get the trivial one-dimensional representation that out to get the trivial one-dimensional representation yes yes one more question so we'll use some of the speakers thank you thank you