 J'ai commencé, je vois que j'ai definitivement perdu Enrique, j'espère que je ne perds pas n'importe qui n'aie plus, donc pour éviter cela, peut-être que je répète ce que l'on a fait l'hôtel d'hôtel d'hôtel donc donc notre gros but c'était d'investir ce intégral comme principal qui est pour déterminer donc à quel point le set de représentation d'un intégral de forme quadratique inelatis n'est non-empté et donc nous avons un sort d'un candidat naturel pour une équivalence pour un critérien pour pour ce qui est ce que ce que j'ai appelé donc ici c'est c'est représentable par q inel et ici c'est ce statement, donc ce qui est un peu plus long mais ce qui est élémentaire pour vérifier donc donc donc l'équation donc nous voulons solver cette équivalence q du x equals d donc ici nous nous demandons pour avoir une solution pour cette équivalence dans l'élémentaire l et ici nous avons nous demandons pour avoir une solution pour cette équivalence dans l'espace réel et dans l'élémentaire de l'élémentaire l attaché à l et ici nous avons dit que c'est localement représentable par q inel et donc le but c'était de donner la condition suffisante pour ce qui s'est passé et donc j'ai mentionné le résultat général et donc pour étudier cette question et spécialement pour étudier cette condition nous avons introduit une famille de lattices qui sont close à l'élémentaire donc qui était appelé le le q génus de l et informément c'est le set de lattices donc l'élémentaire de lattices rationnelles qui sont localement isométriques à l et donc nous comme j'ai expliqué ce set peut être réalisé à l'orbit de l'élémentaire sous l'action d'un groupe où donc j est l'élémentaire de l'élémentaire de cette forme quadratique et donc gf a f est le groupe de points adhéliques de points adhéliques de j qui est un set de séquences gp gp in g of qp ok and gp in g of zp for almost every p so because we have say because any lattice can be completely recovered by the by the data of its collection of local lattices we can act on each local lattices and create a new lattice so in a sense why why the set of the genus is is natural to consider because so when you are aware of the existence of such an action of even the the general linear groups on the space on the set of rational lattices you see that the lattice which are in the in the q genus of L are exactly the the set of lattices which satisfies this condition if this condition satisfies for L so just because if this set is non-empty for some for LP then this set will be non-empty if you replace LP by any local lattice isometric to LP so in a sense it's the most the most the most natural set to consider when you look at this at this condition the most natural family of lattices et so yesterday we finished on a consequence of the so called regular as a principle which is when you so the as a principle is when you is the equivalence of two conditions like this where L replaced by the space QN and LP replaced by the local vector space QP to the N so basically as a consequence of this definition so it's just essentially mechanical the as a principle implies that if so I should say D is locally representable by Q in L is equivalent to D is representable by Q in sum L prime in the genus of L a priori we don't necessarily have representation in the lattice we want but we have one representation in one of the on these closely very closely related lattices ok so this is this is good but ok so there is one thing you see so this set the genus of L it's still an infinite set of lattices ok because because really you can make a lot of so genus of Q is infinite but as I as I explain this problem as I would say additional symmetries but if say RQ of DL prime is non empty then this is a case for any other lattices which is globally isometric to L prime so for any L prime prime in the now in the rational G orbit of the lattice L so really the question of representability it's not about L prime but rather global isometric class and now we have really a fundamental result so of Hermit and Minkowski which say the following so which say that if you look at the set of global isometric classes in the genus so that set is finite so so that's maybe I will continue with the definition so the cardinality of that set so the cardinality of that set it may be noted HQ of L and it's called the let's say the Q class number and so this theorem of Hermit Minkowski of course it generalize the Gauss theorem on the finiteness of the class number of binary quadratic form so and when later we will see a further generalization of it in the context of adels and algebraic groups but so so then if we say this and add to this remark about symmetry loosely you could say so the integral as a principle holds up to finitely many obstructions so being that ok you the as a principle gives you that you have a representation in some of these lattice classes but maybe not the one you might want but there are only many finitely many possibilities if the integral as a principle fails but at least in particular so if the class number of the lattice is one so if the genus of L is reduced to the global isometry class of the lattice which is obvious then the integral as a principle holds and so there are a number of quadratic form for which this class number is one so examples is when n is 2 or 4 then you have the square quadratic form so the Euclidean quadratic form and in that way you recover the CRMs of Fermat Le Gendre and Lagrange about sum of 2 squares 3 squares and 4 squares another example is another example is ok it's a discriminant quadratic form so this one has class number 1, 2 so ok so there is a bit of but maybe I should say but in general the class number is greater than 1 so then you have to improve so maybe I will just say a few words of what happens so maybe I should say there is one case which is maybe easier than the other so if q is indefinite so the class number may not be 1 but hq of L is a small small what do I mean by small is small in terms of so the natural numerical invariant which measures the complexity of the quadratic form in terms of the discriminant of the quadratic form so relative to the lattice L so you take a basis of L and then you form what's the name the gram matrix and you compute the determinant of it so I will define it as discriminant even if maybe it's not ok so of the size of the discriminant in fact if it is indefinite the class number which be basically 2 to the power maybe less than 2 to the number of maybe prime factors of the discriminant so it's very small but if q is indefinite and then hq of L grows polynomially with the discriminant and even so you have even a precise formula so for this you need to make some primitivity assumption on the lattice L say that it has to be maximal if q is integral on it ok so it's a vague statement so but yeah maybe it should be n minus 1 so something like this so it's pretty quick quick grow so ok so then you see in general you basically cannot escape the other principle for the most general quadratic forms so you cannot escape these obstructions excepted in a few cases always even for ok ok ok so it's good ok ok never the less adélic treatment enables you to completely ignore the issue of being indefinite or indefinite so ok it may simplify things create difficulty on the other side so what do I want ok yeah so now I want to explain a strategy to to cope to handle these obstructions and ok just for the second this we don't need it so ok so I remind you what we had that we had a representation in some lattice l prime which was given to us by the as a principle so we have at least one and we will start from this representation to build a new ones ok and the way to do this so we will consider h the subgroup so we take the stabilizer of this vector in the orthogonal group so this is an algebraic subgroup of the orthogonal group which is an orthogonal group in n minus 1 variable because it's orthogonal group of the quadratic form when you restrict it to the auto complement of the vector x ok xq is not 0 so you introduce this subgroup and now look at this orbit so you take now the adelic point of this subgroup and you consider the lattice l prime which is of course contained to the full l prime which is just the q genus of l so and so consider some lattice in that subset of the full genus so what do you have so so what does it mean so you have that let's say write l prime prime to be hf l prime so which mean hf is a collection of local of matrices contained in the local group attached to h and so you have prime prime p is hp times l prime p for every p but now you have the equation that xq is in l prime p because of this so and if you apply hp to each of these and because xq is contained in all the local lattices then that xq is also in l prime prime and of course xq this vector has not moved at all so it means that q of xq is d so it means that xq belongs to the set of representation of d by l prime but what is important here is that l prime prime this new lattice does not necessarily belong to to the global isometric class of l prime a priori because you have not acted by a rational matrix you have acted by a nadelic element ok so potentially one has created a new representation in the sense that l prime may be new may not be an obvious representation that you could have gotten from this representation just by acting trivially by global isometry ok so now suppose because you have a new lattice suppose that this lattice l prime prime belongs instead to g of q of l so then it means what it means that xq belongs to gq l for some for some global element then you have xprimeq which is gq-1xq which is a vector belong to l ok so what you have is that so let's write it that way so we started from a lattice l prime and out of this lattice we collected a family of lattices which is contained in the genus and the genus I recall you it's the union say of global isometry classes so if this orbit meets the global isometry class we are interested in we have obtained a representation in the lattice we want ok so it's a bit strange because we start from a representation so from a vector so we do not move the vector a priori we move the lattice but of course so you see that when you really get new representation because this q is twisted by this global element ok and so that's to prove this so the goal is so show in fact what we will show is somewhat stronger because there is no reason to that l has more special role that any other lattice in the genus will show that so if you look at this orbit of l prime so show so give sufficient condition so that when you look at the orbit at this orbit and you projected on the genus so what you want to show is that this projection map is subjective so then you see this is a statement about sort of it's a dynamical statement you have a space on which a big group act so in fact the space is a homogeneous space the genus is a homogeneous space of under the action of this group because it's orbit and you take a subgroup of of your of the bigger group and you look at an orbit in this subgroup and you want to know does the orbit the space ok is it so it may be a bit disturbing this way of seeing so is there a question so ok so this is a strategy that can be so the claim is that so what one needs so with respect to what I have explained in the beginning we need to show that there exists X so I remind you so basically the statement was that the integral as a principle holds whenever D is sufficiently large and satisfies a set of additionnal conditions anisotropic prime so so what we need to show is that for for D large enough so plus additionnal condition which were spaled in there exist D there exist a representation XQ such that H F so this orbit is growing so becomes bigger and bigger so here is a silly example where you may have an orbit which is not growing which a priori you cannot exclude because ok so as a principle does not tell you much about this except this it's existent so example where it will not work assume that is growing as D grows so example suppose that D is of the shape D note D1 square ok so then suppose you have a nicks note such that you start from a representation such that Q X note equals D note so then you see that X is D1 X note so which is a non primitive vector so is a representation of D but but the stabilizer which is H so let's say H of X is just a stabilizer of X note so because it's a scalar multiple of this does not as D1 goes to infinity so you have some small problem related to imprimitivity but so this can be solved or if you are happy with additional assumption say I don't know if you are happy with saying that D will be square free or we'll have a bounded evaluation of every prime this will not occur so I will ignore this is merely a technical point so so under the condition I have given you can always find a good representation out of the one you started from which will be an imprimitive vector so maybe not imprimitive but at least with when this group will be growing could you please trace how the right hand side is H of A raw depends on just general state so H will depend on D could you please just show the because XQ depends on D so if D moves and again and you avoid such situation your torus will move so is it just XQ if XQ is imprimitive basically you set the D XQ and that will the dependency of H A raw L prime yeah but you have to choose XQ and if you don't want to make a choice assume say that D is square free then as D varies on square free integers you will have a good variation of H too ok something clear so this covers the exceptional cases no ok so because so in the case of exceptional square classes so ok so what I will explain is that then this such a statement that you cover the full space will be so when we will prove something stronger that in fact the orbit when projected there will be equidistributed so we want to prove that it is equidistributed meaning that we will have to integrate the orbit along a test function on that space and so for most test functions so the one associated to cuspidal automorphique form there will be no problem but you have also a test function which are characters associated to characters and in this case it may be that if the D is in a bad square class your character will be trivial so your character will be globally non-trivial but will be trivial along the orbit so you cannot have a sort of equidistribution so in this representation the exceptional square classes will come from the so called the residual spectrum of the adelic group so but I so in order to know these extra obstructions you have to know the shape of the lattice L because then you need to know which are the characters which can occur and so you know how to so you can you tell me that you have conditions to the statement you made that if this big enough and you only look at primitive elements primitive elements is it true even ? so here I have a traditional condition oh yeah so no no but the group still the group H will grow and it will feel a good portion of of that space but it will avoid some finite finite places ok so so maybe so there is a so this strategy ok there so besides the Duke's there is a very nice place where this strategy is implemented and which goes at higher level but in which really it's so I say it's a dynamical statement but there is a place where it gets really dynamical is in the paper ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 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