 Donc, j'espère que j'ai suffisamment motivé les utilisateurs d'Adel, donc je ne sais pas. Donc, juste ce que je veux dire est qu'à ce point, en regardant les Adels et ce que nous avons fait sur eux, nous avons fait des théories de set, donc pas beaucoup de structure. Donc, maintenant, peut-être que je veux discuter... Donc, la structure additionnelle qui vient de l'Adel, qui peut-être peut-être faire de l'analysie. Donc, je vais commencer maintenant par regarder juste les Adels. Et plus tard, je vais revenir pour faire plus précis ce que j'ai discuté à la fin de la parole. Donc, maintenant, je vais juste commencer par A, donc le ring des Adels. Donc, c'est le suivant ring. Donc, c'est le set des classes de complétions de Q. Donc, c'est V, c'est juste le set de primes et c'est une complétion qui est rétente d'une infinité, qui correspond à la complétion réelle de Q. Et donc, le ring des Adels est le suivant du produit restricat sur tous les places des complétions variées. Donc, il peut être rétendu à l'air par le produit restricat sur toutes les pièces de la QP et donc, j'ai écrit l'air par AF, qui sont les Adels finissimes. Et AF, donc, c'est le set de séquences des éléments de paix-à-dique pour paix-varying et ce qu'on requiert, c'est que l'axe P belongs to the ring of paix-à-dique integer ZP for almost every P. So, you see, this is completely analogous to what we introduced, but we defined this for the general linear group. So, the condition that we add is that instead of having just paix-à-dique numbers, then here we had the condition that we had a sequence of matrices and we were requesting that almost all of these matrices had paix-à-dique integral coordinates. So, A and AF, these are rings. So, what other interesting set can we find into... Okay. So, and we have embeddings of... So, we have Q, which embeds into this ring of Adels. Simply if you take a rational element, then you send it to the constant sequence and because Q has a finite denominator for almost every prime, XQ will be a paix-à-dique integer. So, we have this embedding and before we have discussed also this embedding, which is the same, so the diagonal embedding and... So, the constant embedding and... Okay. So, we have... So, we can also see if we want the ring of Adels inside... So, at least the additive group of Adels inside the finite Adels, inside the additive group of Adels by sending a sequence XP to the sequence of the XP at all every prime and then we put zero concerning concerning the R factor. Okay. We may also embed... So, for... We may want to embed also the paix-à-dique field so we can write delta P to be the embedding of QP into either A or AF which is you have a paix-à-dique element and you send it to the sequence with XP prime equals zero if P prime is not P and... Okay. So, this ring... So, one important feature is that this ring... So, it's a product or it's a so-called restricted product of topological rings. These are even topological fields and even with metric. And so, okay, one may want to give to this ring also a topology which is somewhat related to the topologies of its factors and so, you cannot... So, it's not a good idea to give this ring just the product topology because it would not have very good properties but instead one puts on it the Adélique topology topology and a basis. So, you define the topology by defining the open set or a basis of open sets of the form. So, the shape. So, an open set or a basis so, you take an open set of the shape product of... over all V of omega V in QV is open and what we ask is that omega P is ZP for almost every prime. So, I remind you that in the periodic topology the ring of periodic integer it's both open and a closed subset of QP so, it's even an open and compact subset of QP and so, this and likewise, you give the finite Adélique the same similar topology and it makes AF a closed additive subgroup of A and each QP becomes closed also closed inside the full ring of Adélique. Ok, so, a few things that one can say now that we have this topology so, then we have this subring of AF which is just the full product of the ZP and then this... when you restrict the Adélique topology to this full product then you just obtain the tick-off topology because all these factors are compact and so, this is an open compact subring of AF and it is maximal it's the maximal and maximal it's the only one, in fact so, it's the maximal open compact subring of AF and maybe some things which can be interesting do I say this now to say it's maximal yeah, and let's say maybe just to be more concrete about these bases so, bases of neighbourhoods of the identity element for the additive group structure so, of 0 in AF is given by the open and in fact, open and compact is given by... ok, it's a sort of abuse of notation so, yeah, ok, no, it's correct it's just this product NZ for N an integer so, when you vary N you take these things and the more and more N is divisible the smaller and smaller your neighbourhood is ok, so, various properties which are important for A is that, so, if we look at Q so, and it's this property that Q embedded into A is a discrete subgroup and, so, the one reason is the following so, if you is discrete so, if you intersect Q so, here, my embedding is delta that I mentioned here so, if you intersect Q with so, this open set so, what are you looking at? you are looking at rational number which are periodic integer at every prime which means that rational number which is everywhere a periodic integer is an integer and then this integer has to be between minus 1F and 1A of 0, so, it's a you see 0 is isolated by this neighbourhood so, when you get a discreteness and, so, it's discrete but it's not too small nevertheless, because so, it's a sort of exercise so, if you look at this quotient of additive groups so, what you obtain it's that this quotient morphique to R modulo Z so, it's an exercise so, in particular, this is okay, this is infinite but this is compact okay, so, and because this thing is compact it means that M of Q is also compact yeah, so okay on the other hand, yeah just for the record if you look at the embedding of Q into AF then, you can show that this Q embedded this way is dense and, again, this comes this is a consequence from the Chinese reminder CRM and a description of this of this basis of open set of neighborhoods of zero okay, so so, depending on whether you have the adults including the real number or not including them you see that Q has very different behavior okay, so these are some of the basics and now, what I want to do construct a bit more complicated objects and I want to come to GLN of A and to G of A for G group of an orthogonal group so, okay so, if you have A then, you can consider the product AN with the Adelic product topology and, okay, so the topology and you can prove a lot of so these properties, if you adapt them they remain true so, in particular so, QN into AN is discrete because it's just because of the product situation and so on so, I want to play with this and so, okay so, now, what I consider is V so, I consider V naphine sub-variété naphine-variété naphine-variété so, here this is n-dimensional naphine-space over Q okay and so so, remind you V it's defined by an ideal of polynomial so, and it's a set of n-upples such that which are cancelled by by all the polynomial which are contained in some ideal defining the variété in an ideal of of polynomials et so, what you may want to consider is you have this ring you have this this ring which is a Q-algebra and then you have this variété so, you may want to define V of A to be the set of X in now AN such that P of X equals 0 for all P in IV okay, and so here, this is a ring and this is a polynomial whose coefficients are in Q so they are in particular in S so it's legitimate to compute this polynomial and polynomial functions are continuous on the adults because the adults is a topological ring so, this which is a subset of A to the n is closed okay it's a closed subset and so then, this variété has a closed subset of Rn inherit the adelic topology so, and you may have a concrete description although the set may be empty but nevertheless you can write it as as a set of Xp with Xp in V of Qp and Xp in V of Zp for almost every P it may be the case that this set is empty nevertheless and you may look at V of Q and because of this discreteness V of Q will be if even could be empty but V of Q will be discreet as well will be discreet as well okay so the most important example for us will be the case where V is the general linear group so I remind you that the general linear group is an affine variété so it it's a closed sub-variété of the product of n times sign matrices times the affine line and the ideal defined by of gln it's the ideal generated by the polynomial determinant of gij times t minus 1 so t is the coordinate of a1 and gij is the coordinate of the matrix element okay so if you take a matrix and you ask that this matrix times t minus 1 equals 0 it means that t times the determinant won't be 0 and in fact t will be an inverse of the determinant so and then so the the gln that you get in that way which you see again as a subset of this product so which is a isomorphic to aq n squared plus 1 acquire the adelic topology and is locally compact yeah maybe I did not say compact topological group okay so yeah I don't remember you have to say this so when you equip the adels with the adelic topology the great feature that you get is that your topological ring is locally compact and separated so it's very good properties so locally compact separated oh maybe it's okay it's being separated put it into topological groups and so you can do now the same with any algebraic subgroup so linear algebraic group because you may see this group as a closed subvariative of this affine variety and you you give it with so you consider g of a gln a which is closed so locally compact and just by okay so just an example just to if g is the spatial orthogonal group it's ideal so it's so g of g it's come from the equations so transpose of g time q time g equals equals q and determinant of g is one so here q is the matrix of the quadratic form into the canonical basis and then you have this equation which is a a set of polynomial equation in the coordinate of g and this is again a polynomial equation so you have something which is closed okay any questions so for people who are not used to this okay so everybody used to this yeah it admits a basis of pre-compact neighborhood of identity say because it's a group so you it contains very small compact subset so it's a sort of minimal thing that you want if you want to do reasonable topology so topology I can understand at least okay and so when you have the embedding so g of q embeds into g of a and it's discrete on the other hand if you look at g of q embedded into g of a f it's not so it's not necessarily dense it could be quite big but not maybe not dense what are the interesting object that you might want to see so g of a so or let's say g of a f it's this restricted product so I just repeat once again the definition for any linear algebraic group so I wrote this sometimes so g of zp so which are the matrix which coordinate in zp and maybe a more canonical way to write this g of zp is to write it as g of qp as the stabilizer of the square lattice in g of qp so you have this and then you have inside this you have g of z hat which is just a product of the g of zp which is an open compact subgroup not necessarily a maximal one so if you are ready to change coordinates you may adjust the square lattice to have a maximal compact subgroup but anyway so just at least what you can say is that this gzp is a maximal open compact for ah no no I need the condition I will erase this because of the unipotence so if the group is semi-simple ok so and again so what is interesting is that g of af admits a basis of a neighborhood of the identity formed of open compact subgroups so that's very useful to have a very very small open subset which are in fact groups and which are more over compact these are the principal congruences subgroups which are so let me call it kf of n so for n greater than 1 kf of n is a product over kp of n with kp of n is the set of gp in let's say g of zp so integral matrices with gp congruence to the identity mod n ok so if n is invertible the condition is void so it would be g of zp but if n is divisible by p it says that gp is periodically close to the identity element and so when n becomes more and more divisible you get a set shrinking set of of open compact subgroups ok and then you can just translate this neighborhood to translate them to any point so to have a nice neighborhood ok so now I so this is merely local information so now I want to come into some important finiteness so due to Borel and we will make the link very quickly so for all kf into g of af an open compact so when we look at the quotient g of af divided by g of q and divided by this kf and so this quotient is finite and it's it's cardinality so which I will write h g of kf is the class number of this open compact subgroup so I will come to what we have done earlier but just a small remark if you want to prove such a statement it is sufficient to prove this statement for any open compact subgroup so if known for one this holds for all because if you have so two open compact subgroup kf and kf so the intersection it will be an open compact subgroup and because it is open and compact it's so it's contained to the intersection and of finite index in both in both of the big groups so so in a sense to prove this you have a choice of the open compact so it's just formal ok so now of course you will have recognized what I something I have said earlier so if if g is the orthogonal group and and kf is say g of l at so it's a product of the stabilizer of the LP for P lattice so this in that case the fact that this double quotient is finite so this is this is Hermit Minkowski's theorem why because you have g so the genus I recall you the genus of the lattice l so this is the orbit of l under this action and so it's centrally identified with the quotient by the stabilizer the quotient of g of af by the stabilizer of the lattice which is g of l at and then if you make a second quotient so this here you have the genus here you have the set of genus classes and you know that this is finite so the theorem of Borel it's a wide generalization of the Hermit Minkowski theorem but ok we will I will give a proof of this theorem now in a special case so proof if so for g is soq and so I'm asking q is q anisotropic so meaning that q of x is not 0 for all x in qn minus 0 so the quadratic form never vanish on a non-trivial rational vector ok so for this I will just recall something so I introduce so we so the adels it's tightly linked with the notion of lattice so I will translate this statement into some concrete statement about lattice for which we prove the theorem and even stronger so let me introduce l of rn to be the set of lattices in lattices in in rn so not necessarily rational basis so the Z-module generated by real basis of rn and ok so ok ok so as I said to prove the Borel theorem it is sufficient to do it for any open compact subgroup so I will take an open compact subgroup of that shape so I take l rational lattice l rational lattice ok and I consider the following map so from the adels into the real lattices ok which let's say I have a needle element gr times gf and I map it to the following so I take gfl so then I get a rational lattice and now I act by my real element let's say by minus one something like this ok and what do I get under this map so this map ok so what do I get here so the image is the union of lattices of real lattices which are in the gr orbit for the natural action so gln r act on the real lattices and so in the gr orbit which are in the gr orbit of some lattice in of some lattice in in the genus and and so so then really you have identified ok so you have this map and this map of course it's not injective because you can see that ok you can see that the group of rational elements if you define this map that way the gq act so maps does not do anything so what this map gives you it's an identification between the quotient ga modulo gq modulo g of lat and it gets identified with this set of lattices and or you could do it directly also it will be identified with the disjoint union of gr modulo gamma i ok ok let's say that so let me select a set of representatives of the genus classes ok so I take a set of representative of these genus classes and yeah even I don't even know that the genus classes is finite it's just a set of representatives and so what is gamma i so I don't want to make a mistake so it's gq intersected with i kf gfi i-1 yeah where gfi is such that it sends l to the lattice li ok and so you can check it's a set theory you can check that you have this identification and this identification it's a nomeomorphism of so this is a discrete subgroup of gr so this has a topology and you take this disjoint union inside so to speak inside the set of all lattices so and you get this so ok so now I'm going to use my hypothesis so up to scaling q we may assume that q of l the lattice so the quadratic form is integral on l and so now so for any x in l not 0 this implies that q of x in absolute value is greater than 1 because it's non 0 integer ok but so you can see that if this property is true it remains true for any so when it implies that this is true so q of l prime is integral for all l prime in the genus of l because if you want to check that something is integral you need only to check it for every periodic element so for any p for any prime number then the lattice l prime you cannot distinguish much l because they differ from each other by an orthogonal transformation so you have this property and so for any x in l prime and not 0 you have that q of x is greater than 1 but then if you just apply also a real orthogonal transformation you have that for all x in l prime prime in ge of l where ge of l is just this map that I have written there you have x not 0 so you have that q of x is greater than 1 these are all isometric to each other ok and this is very good because what it says because q is a quadratic form so this implies that for all x in l prime prime different from 0 then the Euclidean norm of x if this is big the Euclidean norm of x has to be big by some constant but which depend only on the lattice l and not on l prime so the second thing that that you should know is that you may look at the volume so the Euclidean volume of the lattice l prime so which is the which is rather the covial volume so the volume of Rn modulo l prime prime and what you can prove or check is that this volume is in fact equal to the volume of the original lattice so the volume is fixed so now what you have is that you have a set of lattices all of the same volume which have no small vectors so now you have malheur compactness criterion implies so you have to show that this set is closed but malheur compactness criterion implies that that this so which is a standard result in the geometry of number implies that the set is finite j a of l is compact so not only it shows that this quotient is compact but also that this union is finite so you get both in one strike any some questions so this is very basic the simplest case of proof of Borel's theorem but it's so our general is the use of malheur compactness so for which groups we need to find the isotropic group you do exactly the same thing so the torus so you get the finiteness of the class number of a number field no I'm not sure no no no but ok it's ok I don't know the proof of Borel so you can if you want to have the proof of Borel finiteness theorem it suffice to ask because it's published at AHS so appropriated finitude so you can even bring it back with you in the plane I don't remember the volume but I ordered it so for myself so then you see that so at least you see that if q is q anisotropic this adelic quotient so a modulo gq modulo glat it's a very nice space because it's a compact compact space even you can remove this quotient because then this thing will be compact because you quotient by a compact and but this set it has a very concrete interpretation it's a finite union of quotient of a real group by discrete subgroup and I remind you this is basically the space we wanted to study or we found when we were looking for the distribution property of representation of an integer by a quadratic form so I remind you the drawing ok but need some yeah ok it's not exactly so what did we add on what side if we think classically we had here a collection so we had made some gymnastics and what we have reached it's a collection of H X naught orbit so ok and what is ok we had this orbit since it was GX so for some representation GXI is a real matrix here defined by GXI of X naught equals XI over square root of D so our problem about equidistributing the representation was about looking at a set of HX naught orbit into this this union and on the other side we had to look so this thing was essentially so it's not exactly true because maybe it's ok you have this map basically an identification so it's more an inclusion and here on that side you have you take the and I want to be correct ok GR of GF prime and we have this and what are these here and GR prime ok so GR is a real matrix defined by GR of X naught equals XQ over D to the one half and GF prime it was a finite adelique adel such that G prime F of L equals L prime ok it's not always the case that this quotient which you send which you project into this quotient correspond to doing this but it's a subset of this quotient in general and under so some condition it can even be the fuller set of representation that you want to have so and just I remind you the notation so you have this G it was SOQ then you had produced out of the other principle representation of D by Q in some lattice L prime and this and we have this subgroup of G which was a stabilizer of XQ into G and so you can do exactly what we did here we have this quotient and we look at this an orbit of this subgroup into this big quotient ok so what do we want to do ok X naught in the very beginning of this ellipsoid I remind you we have RQD of R plus or minus 1 of Rn so it was a unit sphere or unit hyperboloid and this and if you pick X naught into this hyperboloid then you get an identification so this is a space under GR so you may identify it with GR with H X naught with H X naught of R which is ok I wrote X naught with this yeah maybe I should write X naught just to be consistent so maybe I switch a bit the notation in the previous talk ok so you make a choice just to identify this hyperboloid with a quotient so it's just arbitrary choice and it's a reference point so to speak ok so ok what are we going to prove in the end so to prove the equidistribution statement ok so GA and HE are locally compact topological groups so they admit left invariant R mesure G G GQ GQ GQ GQ GQ GQ GQ GQ GQ GQ et en fait dans notre case et puis vous avez new H on GA mode HE mode HEQ ok donc let's again assume that Q is anisotropic so what I will say is basically true but it becomes a bit simpler in that case so there is ok because these space are compact these measures so mu G and new H are finite again because you have finite in fact the measure mu Gs are also right invariant from general things so it's not true for a general linear algebraic group but left invariant R mesure is also right invariant in that case the group it's true so R R so right GA invariant and finite mesure ok and so you can normalize to be of volume 1 so we normalize to have mass 1 so so the measure of the total space is 1 which is always possible and so what do we want to do we want to prove that for any which is a continuous function on say GA mode GQ and what do we want we want this continuous function in fact to be a continuous function on the quotient by this so we will ask it to be G of L at invariant ok because this is a compact group this subspace is a closed subspace of the full space ok just almost I just write the statement I want to have so I have this and what I want to prove is that I consider the integral of phi of h d mu of h and so what I need to prove at the end is that this integral converges to the integral so as d goes to infinity and I remind you there are some additional possible additional conditions ok and so what happens is that it's possible here to take our functions phi to be a of a very special shape so this function here that I am considering so you can see these are continuous function on this union of quotient so these these are perfectly non-exotic objects I would say and these are automorphic forms ok and so what I will explain or describe next time is the necessary background to to understand this statement thank you again