So see these lectures more as a teaser for people working in analytic number theory to invest the idyllic setting, because I believe there are a good deal of ideas from analytic number theory which can be transposed with great profit into idyllic setting.

So recall what we want to do. So driven by Duke's theorem, we have an orthogonal group.

And what I would say, it can be more general. So we have then an idyllic space. And so what I explained last time was that this idyllic space is end of with...

So let me call it G between brackets. And it admits a measure which is given as a quotient of the R measure on that group

divided by the discrete measure on this discrete subgroup. So call it R. And in that case, this R measure is... So this quotient measure is a right invariant.

And so I did not say it. And finite. And so then we can normalize it to be a probability. And so what I explained last time was that when the quadratic form is anisotropic,

this quotient is in fact compact. So there is no surprise that the measure is finite. But in fact, this is true in greater generality. And this is the so-called Borel-Darrich-Chandler finiteness theorem.

OK, no, it may not be always finite. So it's finite unless N is 2 and Q is isotropic.

Q is, I would say, isotropic over the rationales. But in that case, the orthogonal group is isomorphic to the multiplicative group.

And then, so the adelic point of the multiplicative group is just... So which is just gl1 of A.

So this is just the unit of the adeles, of the ring of adeles, so called idels. And then you see that this quotient, it's isomorphic to R star modulo gl1 of Z,

which is Z star, which is the multiplicative group of positive real numbers.

So here the R measure is, of course, not finite. But it is only the case. So here we will work mainly when Q is of long greater than 2.

I will explain if there is some small problem. And so we have this thing. And then we have a subgroup, so the stabilizer of a rational point.

And we want to, so then we have also the quotient. So then this H is an orthogonal group in n minus 1 variable.

So it may be that the measure is finite. And we will assume that Q restricted to the ortho complement of X is anisotropic.

Or that n is greater than 3 when we have no problem. So then we have again a finite R measure.

And so we want to take a function, say a continuous compactly supported function on G.

For our problem, we ask that this function is invariant by the stabilizer of the lattice we are interested in. And we want to show that the integral of phi along H, along this H orbit.

So actually we have to add a small adelic element which depends on our choice of this real point X0 and of the lattice L prime in which XQ sits.

And so integrate this, and we want to show that this is converging to the integral of the function but along the full group.

And even, so here in this, so even we may assume that the integral of phi equals 0.

Because when phi is a constant function, this is obvious because both measures are normalized to be probability measures. So we may even restrict to that case.

So under this condition, we want to show that this integral converges to 0 as d grows plus additional conditions that were mentioned in the first lecture.

OK, so I want now to describe this space. And so maybe, so just recollection on quadratic space of law.

So just will be useful because in a sense the most interesting case for this problem are the law. So the hardest cases are the law rank cases.

So when n is 2, and if you have a quadratic space of rank 2, then a quadratic space of rank 2 is always similar.

So which means isometric up to a scalar, it's always similar to quadratic space of the shape k times the norm of k for k some quadratic eta algebra.

So which means simply that either k is q times q or a quadratic field.

And nr is just the norm of this eta algebra, which so is the norm on the quadratic field or the product of the coordinates if you are in this situation.

So and then you can also compute what is the orthogonal group. Then SOQ, let's say, will be isomorphic to the algebraic group.

So it is algebraic of element of k of norm 1.

So and simply the action is you have lambda in k, so nonzero element, and you take x in k.

And then if you consider the map x gives lambda x, then this is orthogonal similitude of similitude factor, the norm of lambda. So this is the rank 2 case.

So if you are in rank 3, ternary quadratic space is always so similar to ternary space of the shape b naught of nr b where b is a degree for central simple algebra over q.

Meaning that b is either the algebra of 2 by 2 matrices or a quaternion algebra over q.

So nr is the norm, so in that case the norm is just determinant.

So b naught, this is the subspace of trace 0 element. And the quadratic form q is just either the determinant if you are in the matrix case or the norm of the quaternion algebra.

Let's say z times the involution of the quaternion algebra. So you have this description, and you have also a nice description of the orthogonal group.

SOQ is isomorphic to the algebraic group associated to the invertible element, the unit of b.

And here this is the center, so you take the quotient of this group by the scalar quaternion. So I will write it p b star. So it's an algebraic group whose points over q are just the element of the algebra,

the invertible element of the algebra itself divided by the scalar element. And the action is given by this, so if you have w in b star,

and if you take an element of trace 0, then you associate to this element its conjugate,

w z w minus 1, and then you get a new trace 0 quaternion whose norm is the norm of z.

So maybe I should say, for n is 4, there is an analogous description.

So either you have the quaternary space given by a quaternion algebra equipped with its norm, or, so this is if the discriminant of q is a square, and if it's not a square,

then you have a description which I will write b k plus times the norm,

where k is the quadratic, okay, it's slightly false, but let me skew off, in fact, of the square root of a multiple of the discriminant.

And so b k plus is a certain four-dimensional subspace inside b k,

and we equip this space with the norm, and then any quaternary quadratic space is similar to one of these two spaces. In fact, if you want to take k to be the split quadratic algebra,

which is what you would get, you have this description is valid uniformly, whether the discriminant is a square or not. So the orthogonal group SOQ can be described in terms of b k star,

of the action of this algebra of quaternions over k.

So this is standard. I think you can find this in Kitaoka's book on quadratic form. And the proof goes by considering the Clifford algebra associated to any of these quadratic spaces, so it's classical.

So it's just to mention that for a low-rank quadratic space, so what will come up, a central object, will be quaternion algebra and groups associated to quaternion algebra.

So now, so, automorphic forms. So I go back now that one has an idea of what the structure of G is.

Automorphic forms on G are nice functions on the quotient G A modulo G Q.

And I want to give examples maybe of automorphic forms for various kinds of groups. So just to have an idea, examples. So let's take the case of A, so the additive group, modulo Q.

So it can be realized as a, okay, can identify it with NA over NQ,

where N is the group of unipotent matrices.

So you have, so it's, okay, and I recall you, you have A mod Q mod Z at, which is identified with A mod Z as an additive group.

And so then if you take a character of this circle group, then because you have this identification, this psi of N corresponds to a character,

additive character on A mod Q, which is a trivial on Z at.

And maybe we just work out what happens exactly.

So if you let this computer consider XR in R inside A, so then you have, by definition, psi N of XR is E of N XR.

But what does it give if you compute, again, several elements? So now if you take XP in QP, so then your XP, you can decompose it.

So you take the periodic expansion.

So here this is an element of ZP. And this, this is an element of Q, because it's a finite, so, okay, AK, this is an element of Q.

And in that case, let's take phi N of XP. Then it's defined by exactly as E of minus N of the so-called fractional part of this periodic element.

And of course, ZP being contained into ZP at. If you evaluate psi N again, this element, you just get one.

Okay, so the reason for this computation is because you know that your character is trivial on Z at, and also it has to be trivial on Q, on the Q-embedded diagonally.

Maybe I want also to do another example. So which is now, let me do G is GL1. So G of A is A*. And so we have that, this identification,

G of A modulo G of Q modulo G of Z at. It's identified with, so in other terms,

if you give yourself a character of the multiplicative group of the positive real number, automatically you will get a character

on this quotient, so on A*, modulo Q*, which will be trivial over Z at*. And so the characters of A*, they are all of the shape, they are just the absolute value,

the real absolute value to some power.

And so once you give one of these, you obtain a character on this quotient. And so what is this character precisely? This character on A*,

the following of the shape is given by the following formula.

So you take now an idyllic element, invertible, and you map it to the following function, to the power of S, which is by definition the product over all,

times the product over all prime of the periodic absolute value of the P component to the power of S.

And so you verify that in that way you obtain a character, which will be trivial on Q*, and trivial on Z at*, and of the right shape. So this character here,

so the product of all the absolute values over all the components is called the modulus, or the idyllic modulus.

So this is another example of an automorphic form. And here you are on this quotient.

So other examples are given by Dirichlet characters. And OK, so suppose you have a Dirichlet character, so then you have the following isomorphism.

So you can show that it's just the same kind of reasoning,

that it's isomorphic to A*, modulo Q*, modulo KFQ, with KFQ is what we called previously the principal congruent subgroups.

So it's 1 plus Q Z at*, or if you prefer, it's the product over all primes not dividing Q of the ZP*,

times the product over all primes dividing Q of 1 plus Q ZP. So this is a multiplicative subgroup of Z*, and its index is precisely the cardinality of that.

And so in that way, from any Dirichlet character, you obtain, so T, multiplicative Dirichlet character, gives an idyllic character of A*, mod Q*,

which will be trivial on this open compact subgroup. And in fact, you have a bijective map between Dirichlet characters

in bijection with finite order characters of A*, modulo Q*.

Where are the reals?

And now if you want to obtain all the characters of this quotient of A*, modulo Q*, what you have to do is to multiply a finite order idyllic character with a character of that shape,

and then you get a description. I will finish the example on that board.

Now if you go further, if you take a finite extension of Q, a field, then this field defines an algebraic group,

an algebraic torus, which I know Tk, which is a subtorus of the linear transformation of k.

So views as a Q vector field, and simply the action is given by x in k and lambda in k*.

You have this linear map, give lambda x. So k star acts linearly on k by multiplication, and the k star is a group of rational points of some torus inside this GLk, which you may see

as belonging to GLn Q by choosing a Q basis of k. So in fancy terms, Tk is the restriction of scalar from k to Q of the multiplicative group. And what happens is that a Q character,

so classical a Q characters of k, correspond objectively to continuous characters of Tk a modulo Tk Q.

And so when you look at, say, Ziegel's lecture

or Hecker's lecture on the definition of a Q character, usually it's a big mess because your character has to satisfy some condition. You have a lot of condition of your character,

and these conditions are made to ensure that your corresponding idyllic character is trivial on the rational element. OK. And finally, last example, which will be OK.

f in Sk of SL to Z Hecker eigenform. OK. So then for you f of gamma z equals g of gamma z

to the k f of z. And so how does one associate to this Hecker eigenform

an automorphic form? So we observe that the quotient H mod SL to Z is identified with, so this is what Valentine explained

in a much more general context. And simply this map is given by g goes to, let's say, gi gamma.

So you have this identification. OK. And let me write this here as PGL2R modulo PGL2Z modulo PSO2R.

And here I can identify this with the Natalie quotient, PGL2A modulo PGL2Q, and here modulo PSO2R times PGL2Z.

OK. So it's getting very, very small.

So OK. You have this identification of the modular surface, with an Natalie quotient. So of PGL2 by the rational points. And on the right, you have the product of the compact group, compact real group, by this compact finite Natalie subgroup.

And now, and so how do I associate a function

on that quotient? Then it is because of all these identification, it suffice to define a function on, yeah, not exactly on that quotient.

So to my function modular from f of z, I associate f tilde of g defined by g of.

So it's a function on the group. And so what you can check is that you can check that for all gamma in, say, gl2z, f tilde of gamma g

is f tilde of g. OK. It's a positive data just to, for this to be well defined. So then it means that f tilde is a function on.

OK. And it's not a function on exactly this double quotient. But if you look at what happens,

so let's take kappa matrix. So where the angle theta is only defined modulo p. And then if you look at f tilde of g kappa, what you obtain is that it will be e minus k of kappa

of f tilde of g, where e minus k of kappa is e of minus k of theta.

So k is even and theta is defined modulo p. So this is well defined. And so this e minus k of kappa is a character of ps2r.

So what you get from holomorphic weight k modular form is a function on this quotient, which transform in a special way under the right action of this compact real subgroup.

And so in that way, you see that you obtain from f tilde gives you, so f tilde defines a function

on modulo pgl2z at transforming, which

is an eigenfunction for the right action of ps2r.

And the eigenvalue for this action in this is this character, which is determined by the weight. And if your modular form is a eigenform,

then the f tilde will be a genuine automorphic form. So I return now to g is so2q. And so we want to understand the space of continuous function on this quotient.

And we would like to understand the function, which are moreover invariant by some open compact subgroup. And for this, in fact, the starting point is to understand first the L2 space.

And this L2 space is a unitary representation

of g of a for the right action multiplication. So if you have a function and you have g,

g phi is the new function, which to g prime associate phi of g prime g. So you see, if your function is a function on ga, which is gq invariant, this new function will still

remain gq invariant. And the representation is unitary, simply because the measure that is taken here

is R measure, which is right invariant.

And so the main thing to do.

And so one thing which is good is that, which is why.

In fact, you can upgrade this from a representation of ga, and this is a standard thing, to a representation of an algebra.

For instance, the algebra of continuous compactly supported function of ga. So with the product replaced by the convolution.

So simply, if you take a function, continuous compactly supported. OK, the f is not the same f as a modular form. But if you take such a function, you can define the operator.

And so just by this convolution trick.

So suppose that your continuous function, which is a function on this product, is written as a product of a function on gr times a function on g on the finite adults.

And suppose that this function is smooth, and this function here. For instance, suppose you take that it

is a characteristic function of principal congruent subgroups. So this small neighborhood of the identity, which is an open compact subgroup. Then r f phi will be smooth in the so-called gr variable,

and k f right invariant in the j f variable.

So by this convolution trick, your L2 space of function, which are just measurable, is generated by very nice, smooth

functions. So then if you understand the L2 space, you will also understand very well the smooth functions, which is really what we need in the continuous functions. And also, it should say that the eche operators have

an idyllic description, which is the following.

So for p sufficiently large, then you can define the g p in g of q p.

Then you can define the eche operator t p, t of g p, to be the operator, the convolution operator, associated to the characteristic function of the double class.

And so, again, if p is sufficiently large,

this algebra of operator will be commutative and made of normal, even in that case, if g is an orthogonal group, self-adjoint operators. And so eche, because these are eche operators, eche eigenform,

so for, say, PG, for classical eche eigenform, the corresponding function will be eigenform for the family of all such eche operators.

OK, so I don't have much time, but I will try to explain what is the.

And so now I can write, OK, suppose that q is anisotropic and n is greater than 3, which is our case of interest. So this representation decomposes

into an orthogonal sum of irreducible representations. And so this orthogonal sum is following. So this part is very simple to describe.

It's a direct sum of one-dimensional vector spaces.

It's chi composed with the norm. So let's say q is anisotropic, which means that, I recall you, the ternary space corresponds to a quaternion algebra,

plus condition.

So this space is this, and chi, this function is, so you have a character, so chi, it's a character on the e-dels, finite order.

And so this function is a function with two g in, let's say, b star of a goes to chi of the,

so g corresponds to a quaternion, so you take the norm of this quaternion, ideally quaternion, you compose with the character, and you obtain a function.

OK, so here this is an infinite sum over all the eq characters, which can live there, or which are defined a non-trivial thing here.

And OK, L2 cuspidal is a orthogonal sum where p are so infinite dimensional representations,

infinite dimensional, so unitary representation of gl of a. But these representations are of quite a special type because they admit the realization made of vectors,

which are functions which are invariant of g of a. And so these representations are so infinite dimensional, cuspidal representations of ga.

So these are all, not all the unitary representation of that group would be too big. So if q is isotropic, you have a further component coming from Eisenstein series.

And so then we are interested in the v pi,

which admits in the pi, or the v pi, which admits g of L hat invariant vectors.

OK, not all of them. And there, there are only finitely many.

No, not finitely many. And so for such, OK, either the finite dimensional ones

or the infinite ones. So now to study this integral that we want to study,

this convergence, we may assume that phi belongs to one of the, that phi is either

of the shape chi composed with the norm, or phi belongs to one of these v pi. And so then you have a different situation. So this case is relatively easy.

So I will do the first case. So this case, so we are looking at the integral along h of chi of, say, norm of h.

And then there is a g x0 L prime dh. So this, you can factor out this term.

And so we are left just with this integral. And OK, and so what we, but then this is, and in a sense we would like to show that this thing goes to zero when d.

is sufficiently large. So what happens, and so for key, of course, for key, not trivial, not the trivial character. So then what can happen is that, so this key,

so just to exist, this key has to be a further 2. So then is associated to some quadratic field,

because it's a character of further 2, so it corresponds to a quadratic Dirichlet character, and so corresponds to a quadratic field. So what happens is that if H, which

I have recalled in the beginning, is associated to a quadratic field, to the same quadratic field, so then it

will imply that chi of the norm on H will be a constant equal to 1. And so therefore, this integral won't be 0. So then this explains the exclusion

from the theorem of exceptional square classes for D, because the D determines the quadratic field, which is bad,

because we would like the integral to converge to 0. So and just in a milli, so I'm very late, so I finish with just the remaining case.

So now if phi is in V pi is contained in L2 cusp,

so then there is a formula of values per j. And phi plus phi is nice.

But we may assume that then there is a formula of values per j, which tell you the following.

So the integral we are interested in, square, so can be written up to a constant

as a null function of a modular form

plus a sort of finite product.

OK, I will not write what this is. For f, eche-Eigenform, not necessarily eigenomorphic, eche-Eigenform with the same eigenvalues,

eche-p for almost every p.

And this is just the eche-L function of f, and the eche-L function of f times k. And this is the value of a quadratic character attached

to d times d to the 1-half. And so the fact that this integral over h goes to 0

follows among other things to the following bound, which

is called a subconvex bound plus bounds for these things that I have not written and no time to describe here, but which are much easier to get. So of course, I could have formulated everything

in adelic terms, but no time anymore. So this is a so-called subconvex bound. And so such a bound was obtained for the first time

by Ivaniak for holomorphic forms, from which Duke could write a great deal of ternary quadratic forms. And then Duke extended Ivaniak's bound to other kind of standard form, mass forms, and so on.

And so then, together, we shoot the pillow. Then they could get more general results, in particular, as a principle. So when the d was square-free. So to remove the square-free assumption, then you have to deal with these local things that I have not described.

And here, in that case, maybe this, OK. I should say that now there is a purely adelic treatment of these L functions, which is due to Venkatesh, which works over general number fields.

So in that way, you get the theorem that I have stated about ternary forms for over general number fields. OK, it's over time. So thank you.

Any questions or comments? Yes? I have a question about what's on the board, behind that board, about the appearance of the continuous spectrum or not. So I'm probably just confused about translating

anything in the classical picture. OK, so if Q is anisotropic, I thought whether or not the continuous spectrum shows up or not depends on whether G is compact or not. Yes, exactly. But if Q is anisotropic, it could fail to be isotropic, just at a finite place.

But the group could still be non-compact, and then? No, so if Q is anisotropic, Q over the rational numbers. So you could be isotropic at a real place or at some other finite place. But the global quotient, GA modulo GQ, will be compact.

And then its spectrum will be discrete, because all these operators, these RF will be compact operators. So the condition is that to have Eisenstein series is that you are isotropic.

And if you consider more general group, it means that your group contains a rational unipotent element. And then the Eisenstein series comes from these unipotent elements. Other questions?

And then your comment, can you give a few references for this one? OK, yeah. OK, so maybe references. OK, so for analytic number theorists, maybe if you want to get used to Adele's,

maybe just to start, I would suggest there is a recent paper of Browning and Pankaj-Vichet, who was there last week, who they do an idyllic circle method.

So there is no automorphic form, or the only automorphic form that could appear are the additive characters over the idylls that I have described. So that's an intro. So because they prove an analog of Skinner

on finding points on cubic varieties in maybe nine or 10 variables over a number of fields. So then, to have a presentation of automorphic forms over the idylls,

I would start with the book of Gelbart at Princeton University Press. So which is really a gentle book with, at the end, a very nice application. And then maybe I would follow this up with the book of Bump at Cambridge.

And maybe there are also the book of Goldfeld, which could be, OK.

Now, if you want to enter into the analytic number theory that enters into this proof, now there is no book yet. But I would suggest Akshay's paper.

I think it's something like 205 in Annals of Math. Oh, I forgot. There are nice books by Knightley-Elley.

For instance, they have a manuscript to give an idyllic treatment of the Kuznetsov trace formula. So if you do a classical modular form, this book is very interesting, very gentle. There are several books. And OK, so then you have Venkatesh.

And if you want to look deeper into the subconvexity problem, then this is my paper and with Akshay at. So probably you can get from the staff at IHES.

And OK, and then I should say, maybe some small advertisement, book in preparation. So the list of authors is not completely determined. But OK, so there is at least one author.

And I hope more book in preparation on Duke's theorem. So I would say in general. And so you can find the first free chapter of that book on my web page.

And this free chapter describe more or less all what I have explained in these lectures. So the first chapter about the adults will come maybe soon, maybe in one or two weeks. And then there will be further chapters.

OK.