Thank you. Thank you, Mary France. Thank you for inviting me to this conference.

It is a great pleasure and an honour. I had the pleasure, actually, some years ago, 1995, actually, the spring of 1995, to share an office with Gaudemont at Jussières. 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 Jacqué. 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 Endoscopy. 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. Even 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.

There are other possible ways of looking at it, Beyond Endoscopy, but 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, 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 the 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 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 invariant. 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... I've got the stick, yes. 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 and a sum of spectral terms.

Now, 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,

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, 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 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...

I've 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. This is what you really are interested in. So that this is the sum over pi in, let me write it as pi 2 of G

of multiplicities of pi in the discrete spectrum sitting in the L2 of G A modulo GQ times the character, the trace of pi of f.

Now that's just, I'm just talking about what this is here. And by this I just mean the set of irreducible representations.

Which are contained in the discrete spectrum of L2 Z plus GQ GA.

So these are what you really want. This is the essence of functoriality to understand relations among these representations in the discrete spectrum as the group G varies in a very natural way. That's what functoriality applies to.

And these are in some sense the parallel terms to the discrete spectrum on the spectral side. So that's what this is equal to.

I'm not writing this very clearly. That's what this is equal to. Plus, let's call them sublets. 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

the elliptic regular part. I'm going to write this approximation little symbol. This is equal approximately, well, to I2 of f. And so this is the famous, the notorious

pretend that they're equal symbol. But I'm not going to talk about 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 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 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 Frenkel, 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. Is there a joint representation?

No, this is just for GLN. So you'd have to take the construction of Steinberg and Hitchen in general. But I'm just talking about GLN. So it really is just a characteristic polynomial.

Of the matrix. Of the matrix, yep. p gamma of lambda, the determinant, lambda i minus gamma. So let me write it lambda n plus 1 minus a1 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 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 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's the property that this polynomial p a of lambda is irreducible over q. And gamma of the conjugation goes.

Gamma, yes, so did I? Yeah, these are supposed to be conjugacy classes, semi-simple, 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 p-adic, a product of p-adic 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 at all of the p-adic 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 i p, where this is the characteristic function.

So i p is equal to, so this would be the characteristic function of the product over all p of gl n plus one of z p.

So that's a perfectly good function to put into the trace formula. I'll cover up the trace formula just momentarily. We'll pull it down when we need it.

Then if I do this, gamma, 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 of these p-adic 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 looking at elements gamma that are bijective with vectors a. So let me write it like this. Vectors a that consist of pairs

where 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 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, whose centralizer is just a torus. Here 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 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 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 trace formula. So maybe it's, I mean it might have, it kind of has a flavor of 19th century Galois theory, theory of equations, maybe Galois resolvents.

But I don't have any evidence for that, but it's sort of intriguing. And so one consequence of this change of outlook is, 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 general ideas

that you hope might follow from this, I would be talking about the spectral side and automorphic L-functions. 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 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 were in my title. For a start, for a start, for a 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 Dedekind? I see, I see. So he was onto this.

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, the 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 the square of a positive integer. So this is, 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 understand of these terms.

All right, so zeta functions. We've got the Dedekind 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 Dedekind 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 1 minus the norm E over Q P to the minus 1. That's the Euler product of the Dedekind zeta function. S, yes, minus S.

So that's the Euler product of this zeta function. It also has a Dirichlet series, zeta E of S.

So it's a sum over ideals, integral ideals in 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 Dirichlet series,

as a sum 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 1 minus S, where lambda E of S is equal to, so it's just the way the notation that I've used here does not include the Euler fact, the Archimedean Euler factor that would be needed

for the functional equation. So the zeta, 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 Dedekind zeta function, but it seems completely new.

It's by Zhiwei Yun. 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 Collin's 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, OE, in the field.

So let's write it as zeta R of S, R being this order. And it has many of the same properties. It has an Euler product, zeta R P of S.

So I won't give the definition of it, but these Euler factors are given by a Dirichlet series as these are.

And this local Euler factor for the Dedekind zeta function is a Dirichlet series. It's a generating function whereby you count the number of OE modules of a given length, of a given size, and you count the number of such things

and you raise that. 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 don't take OE modules, namely ideals,

but you take R modules. And I think you ask for R modules which are contained actually in the dual R check, which is the dual of this as an R module over itself.

And that's something that would contain OE. 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

by which you count certain R modules according to their length, how big they are. I wonder whether it was known to Hasen. Pardon me? It might have been known to Hasen. Is that so? Is that so? Okay. You mean this zeta function? Okay, well let me wait a while.

There's some punch lines that he introduced. I did not check. I did not check. Okay. So in any case, zeta R of P of S, it should be very closely related to the corresponding local factor

for the Dedekind zeta function. It actually differs from it by a polynomial in P to the minus S. So it's of the form P R. I'm going to write it as P R of P of P to the minus S.

So it's a modification, a multiplicative modification of the local factor for the Dedekind zeta function. It's a local factor for a polynomial P R of T.

So this is a polynomial with constant term 1 and with integral coefficients 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 2 delta of P where delta is the valuation of the P part

of this integer sigma here, this integer there. So delta P or if we want we can index it by R delta R P.

This is equal to the valuation at P of sigma of R. All right, so that's what the P 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 which is still concerning the local factor

but the product, let's write it as zeta tilde R of P of s. So basically you take the local factor but you actually rather than having a polynomial

you have it with some positive powers as well as negative powers. So P to the delta of s times this polynomial that's the new factor in the function P R P at P to the minus s.

So 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 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 1 minus s.

So it's got its own. The Euler factors of an L function or a Dedekind zeta function don't satisfy that functional equation but this obstruction or this difference between the Dedekind zeta function's local factor and the Yun zeta function's local factor

does satisfy this functional equation. You don't have zeta multiplied by zeta R P of s until the tilde. Oh, have I forgotten a tilde up here? I have, yes.

I think it's okay. I think it's all right. Yeah, I think that's all right. This is the definition or this is the property of this slightly different...

Oh, yeah, I forgot something. Yes, I've got this thing here.

No, no, I think this is what I want. I think this is... I think you're fine. Yes, yes. Your memory is 30 years old. All right, okay, all right. That's impressive. All right.

And now here, to me anyway, 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 1. So this local correction factor has the property

that its value at 1 is equal to something that is very central to the trace formula. Namely, it's the orbital integral of gamma at 1P. It's a p-adic orbital integral, which seems like a real surprise.

What happens when you take for half the full ring of integers? You have the orbital integral of an element gamma.

For the first... Assertion 1. Assertion 1? When r is the ring of integers? It's just 1. Just 1. Identically equal to 1. So in that case, the Eun's zeta function reduces to the Dedekind's zeta function.

This requires a normalization on the right-hand side. Can you say what it was? The orbital integral? It does. I think he uses maybe the normalization that you may not want to use eventually,

but I think it's the normalization which assigns volume 1 to the centralizer of gamma. Assigns volume 1, for example. Yes, yes. Are you assuming this real part of S is greater than 1 in the first part?

Sorry? Are you assuming the real part of S is greater than 1? Here? This is for all S. For all S. I mean, it's an analytic function. It's a meromorphic function.

And the product, depending on zeta, is over what? Over all prime p of OE? Yes, it's a product over all prime p. That's right. It is a modification of the Dedekind's zeta function. All right. So what is this from 1?

1 immediately leads to an identity. Well, it leads to a functional equation of the global, of Jung's global L function, where lambda r of S is basically,

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 Euler factor of the Dedekind's zeta function slightly by just the power of this thing. But then, just from this property, you see that you're going to get the same... The gamma factor remains the same.

The gamma factor remains the same. At infinity, you see nothing. You see nothing. Yes, yes. All right. And let me just also add that.

Let me add that from 2. I'll put it here maybe. So from 2, I'm just rewriting basically

that relation 2 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 1 to the minus 1 zeta r of P 1. So this is the Euler factor at 1 for the Dedekind's zeta function.

This is the Euler factor at 1 for the Dedekind's zeta function. This is the Euler factor at 1 for this new zeta function. It's multiplied by this normalizer. And that is the orbital integral. So the orbital integral is basically, at least at value 1, it measures the difference,

multiplicative difference, between this new kind of zeta function and the Dedekind's zeta function. All right. Just a couple of remarks. This is taking me longer than I expected.

Just a couple of notes. 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 Zagier. Well, essentially it's the same thing. Zeta r of s, it's really the quotient of what I have up there divided by the Riemann's zeta function. And so what one gets in the case of Dedekind's zeta function

is just quadratic Diraclet L function, but modulo that difference. This was for GL2. This was introduced by Zagier.

And other people have also studied it in 1977. For quadratic extension. For quadratic extensions of Q. Pardon me? Quadratic extension. Both, both, both.

So this was a starting point. Oh, I guess what I would also say is, so this is a compound remark.

The local factors, let's say zeta r p of s, and the orbital integrals,

zeta tilde r p of 1, these have simple formulas. And these simple formulas were a starting point

for the recent work, 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 Poisson's summation for GL2.

I'm not going to get as much said as I had hoped, so I'll say that Poisson's summation,

you've got the global orbital integrals are parametrized by elements in Zn. Zn. They're distributions that are parametrized by Zn. And the question that was later then posed in Langlands, Goh and Frankel,

can you apply Poisson's summation to that sum over Zn? In the case of GL2, it's just a sum over Z. And Altu, with some extremely clever, you can't apply Poisson's summation at all as it stands just yet,

but he did several very interesting things and was able to apply Poisson's summation in the case of GL2. In the case G equals GLn, or GLn plus one, the orbital integrals,

so this would be zeta tilde of r of p of one, perhaps also the local factors. So 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,

certainly in the case that the order, we're talking now about a local order, so we can ask whether it is elliptic or whether it's hyperbolic. The elliptic case would be the main one. And then we can ask whether it's unramified,

tamely ramified, or wildly ramified. And certainly up to tamely ramified, 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 richly complex functions. I haven't checked everything,

but I think it's pretty fair to say that these can be derived. I've certainly done enough special cases to make me believe that up to the tamely ramified case, I expect to be able to derive such things from,

well, from two very striking papers of Walzberg, germs. So this is not the title of the paper,

but it's germs of p-adic orbital integrals for GLN.

I have three minutes left, so what I would like to do, in my three minutes, I want to return to the trace formula.

Okay, so that's, well, it's up there.

All right, 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 nicely, matches what we would get from Jung's zeta function. So this, the way we have set it up, we have taken a rigid function at the p-adic 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 in f infinity with coefficients built out of this and this.

So we've got this. One would like to, I'm going to just have to say a few words to finish off. This is an Archimedean orbital integral. These, of course, were studied by Harris-Chandra, and it's best to normalize this

by multiplying it by the square root of the absolute value of the Weyl discriminant. Now, that fits very well with what we're dealing with here. The Weyl discriminant has the same absolute value as the discriminant of the characteristic polynomial of gamma. And so you multiply it by that,

and then you get something that is close to what Harris-Chandra studied. It's a function that has very mild singularities. This, well, you do two things with that.

You apply, and I'm following the strategy of L2 that he used for GL2, you use the Dirichlet class number formula to rewrite this. So this is basically the regulator of the extension E of degree n plus 1,

and you rewrite that as a product of the square root of the absolute value of the discriminant of the field extension times the value, or the residue, of the Dirichlet L function at 1.

This, you substitute in for Eun's orbital integrals there, and you see that that matches very well with the values with the

value of the Dirichlet L function, or rather its residue, at s equals 1. And if you put those two together, you have the residue of the Yun-Zeta function at 1. And so those two go together as one sort of common coefficient times the orbital integral of this. So in that context,

the question raised by Langlands, Goh, and Frenkel was to... 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 Zn cross

plus or minus 1 of a distribution in a variable function f infinity with very simple but difficult coefficients, namely basically the residue of this, or if you prefer, the value at 1 of the quotient of the Yun-Zeta function

by the Riemann-Zeta function. But the question is, can you apply, that these guys raised, can you apply the Poisson summation formula to that sum over b? Well, certainly not as it stands. Poisson summation formula, you're not allowed to have coefficients.

You just have to have a sum of a Schwarz 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 all real numbers. They're just defined arithmetically for integers.

What else? There was some particularly trenchant point I meant to mention, which has escaped my mind, but in any case, that's the question that was raised,

and Al II dealt with, there's about three or four problems that are raised. And he dealt with them for the case of GL2, one after another, very nicely. The value at 1 of the Zeta function, it's given only by a conditionally convergent series.

That's very bad for doing analysis with, but he replaced that value of 1 by using what's called the approximate functional equation, which replaces, the price you pay is to have some extra truncation kind of functions

or modification functions which you tack onto this and get a more complicated expression involving the orbital integral of f infinity. But then you get an absolutely convergent series in which you take a sum over L,

over the positive integers. You then, and then there's further questions. But in any case, I hope that one can solve these questions. I mean, there's considerably more to be solved here than in the case of GL2.

One would have to be able to successfully use L'Alpeurge's formulas for the orbital integrals. One would want explicit formulas. And, but I hope it should be possible to...

Oh yes, I'm sorry. The point I wanted to raise was they said, look, why don't we apply, it would be great if we could apply Poisson summation formula to the sum over Zn. People have thought about this in the past, but not for, they didn't use the base

of the 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 for each of those extensions of degree n.

Langlands 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 somehow has not, it's not quite the right thing. So this does seem to be the right thing.

I mean, by applying Poisson summation, you are basically replacing this variable, the coefficients of the characteristic polynomial. You're replacing it 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, representations that occur in the discrete spectrum which are non-tempered. The kind of manipulations that Langlands proposes in Beyond Endoscopy 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 tempered, which is not cuspital in the case of GLN.

And it seems likely Al II did this for GL2. It was just the one-dimensional representations and he really showed that they corresponded to the spectral variable after Poisson summation formula. 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 trivial one-dimensional representation, the one non-tempered representation in the discrete spectrum of GL2, which he showed corresponded to this spectral variable

and he subtracted it away and thereby obtained a distribution on the elliptic regular part which I should say is tempered. You don't know it's tempered until you prove Ramanujan's conjecture, but which ought to be tempered. Anyway, I have gone a few minutes over.

I am terribly sorry. Thank you. This was a very excellent talk. Do you have questions? Is there speculation of higher dimensions

from what the non-tempered? I have a conjecture. It should correspond to divisors of N plus one. And so it should correspond to terms. You're gonna have N plus one spectral variables parametrized by Z and it should correspond to terms that have zeros in those

and then the remaining terms should be a diagonally embedded tempered spectrum. So that corresponds to Maglin and Vals-Bergier? It corresponds to Maglin and Vals-Bergier. They characterize the discrete non-cuspital spectrum for GLN.

You can certainly, you hope to see it and there's a very natural, as I say, conjecture as to what it ought to be. Namely, N plus one tuples of integral integers which have a zero at 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, do you expect to see an analog of this unit there?

No, no, this is purely for the zeta functions, the L functions or zeta functions. I mean, these are more simple than the automorphic L functions that would come on the spectral side. It's purely for the zeta functions on the geometric side.

One more question. Is the geometric contribution attached to gamma the residue of the global zeta? That certainly plays a role. I mean, even in the case of GL2,

when you have applied Poisson summation, it's no easy matter to prove that the spectral contribution 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 of various residues of these things that come in and they do play a critical role in proving that you get the trivial one-dimensional representation,

that Altium gets the trivial one-dimensional representation, yes, yes. More questions? So you signed the speaker. Thank you, thank you. Thank you.