 Thank you Philippe for the introduction and so I have watched a lot of number theory web seminars but it is my first one as a speaker so I am very happy but I am a bit frightened by the perspective. Okay so I will discuss so joint work mainly with Manny Hakka, Manuel Lutti and Andreas Wieser and hopefully I will also have enough time to speak of extra development which is also joined with Ricardo Menares which I will briefly touch near the end. So I start very basically with an elliptic curve over a field is so an algebraic curve of genus 1 and it has a plane embedding and it admit cubic equation which is given here at least so it's in a simple form because if one assume that the characteristic is different from 2 and 3 and otherwise it has more terms and okay so the elliptic curve has a basic invariant like the discriminant which is nonzero because the curve is a smooth and attached also to the discriminant comes the J invariant which is given explicitly by this formula and the J invariant classifies the elliptic curve up to isomorphism at least isomorphism over the algebraic closer of the field not necessarily over the field over which it's defined but it will be sufficient for us. Also I recall that the elliptic curve has a structure of an algebraic group in the plane embedding the group law is given by the usual chord and tangent construction and the zero element is the point at infinity so here I spend a lot of time of designing this picture so I want to show them to you and so this is the sum of two points and this is the double of a point which is constructed via a tangent okay so this is the basic object we want to consider and so when you have an elliptic curve so an invariant is the ring of endomorphism of the curve so of really group morphisms and so for me the ring of endomorphism is really the ring over the over the algebraic closure so not over the base field so that this ring is really as big as as it could be so and then you have basically three cases so first the ring of endomorphism is reduced to z and then it means that the characteristic of the field is zero then the ring of endomorphism could be bigger if it is bigger than z it can be an order in an imaginary quadratic field and so this order which I will write by od it's z plus multiple of the ring of integer of the quadratic field q of square root of d and the multiple is the discriminant divided by the discriminant of the field called the fundamental discriminant and in that case one says that the elliptic curve has complex multiplication by that quadratic order and then the characteristic can be either zero or positive and finally the last case is when the ring of endomorphism is is a rank for the the module it is a maximal order in in a quaternion algebra the quaternion algebra is so what happens is that necessarily the characteristic of the field is positive so say it's p so then the quaternion algebra is the quaternion algebra which is ramified at p and infinity so which is definite and ramified that just at the prime p and the ring of endomorphism is a maximal order in this quaternion algebra in this case the curve is called super singular and in fact it is it can be it admits an equation isomorphic to a curve defined over f p square so I will pass now to the case of positive characteristic and then so I assume that my field is containing the complex number I choose an embedding and then the complex point of the curve can be identified with Riemann's surface of genus one that is the quotient of c by lattice lambda and this this identification is via the map which to map which to an element in the torus in the complex torus associate the value of the of the value of p function and its derivative so it gives the two coordinates of naffine point and so if one wants to look now at the ring of endomorphism of the curve then it becomes the ring of endomorphism of the lattice in c that is the complex element which preserve the lattice under multiplication okay and so now for an elliptic curve over the complex numbers this curve has a complex multiplication if and only if the ring of endomorphism of the lattice is precisely the quadratic order in the quadratic field is a quadratic order of a quadratic field embedded into c yeah okay so in particular if you have an ideal relative to this quadratic order so I ask the ideal to be a proper ideal so that is that it is not invariant by any bigger quadratic order then the elliptic curves which you obtain by quotienting the complex by okay I should maybe have written okay so a minus one maybe I just just think it's a or it could be a minus one then this elliptic curve has a complex multiplication by this quadratic quadratic order and conversely any elliptic curve with such complex multiplication is isomorphic to a curve of that shape for some proper idea so and in fact what happened is that if I write l of cmd to be the set of isomorphism classes of elliptic curve with complex multiplication by this quadratic order then this set corresponds objectively with the with the picard group of the order that is the the the the as a set the proper ideals modulo homotesis okay this k sorry yeah this k is not the initial k it's not the base field of the elliptic curve it's the really the quadratic field containing the quadratic order sorry for this collision in notation okay so and so here is so this talk will be a lot about duke's equidistribution c or m's and so here is the first of duke's c or m of duke's equidistribution c or m so i so if i called el c to be the set of all elliptic curves over c modulo isomorphism then this this modulus of elliptic curve is naturally identified with the modular curve that is the quotient of the upper half plane by sl2z and this bijection is simply to an element in the upper half plane you associate the lattice and then you associate the torus quotient and if you work out this bijection in the case of cm elliptic curves what you find is that the cm elliptic curves by the order od correspond to the the point in the modular surface of that shape so minus b plus i square root of d over 2a where a b and there is a third number c uh satisfied the discriminant equation b square minus 4ac equal to d and we also ask that a b and c be co-prime so here what we have is that we have a primitive representation of the discriminant d by the vector abc by a primitive by the vector abc so we have this bijection so the set of of cm elliptic curve by od is a finite set of points in the modular surface and the duke's first theorem tells you that when the discriminant grows among the imaginary discriminants of quadratic imaginary quadratic orders the set of cm points which i would call the agner points of discriminant d is equidistributed on the modular curve so equidistribution is with respect to the hyperbolic measure do you see my hand when i move it it works okay never mind okay um so and this is uh what the equidistribution means for any compactly supported function uh continuous on the modular curve the average of the value of the function over this cm point of discriminant big d converge to the measure hyperbolic measure of the function so uh yeah so here is a picture uh of some so agner points of discriminant uh of this uh discriminant can you get back one page yeah what is small d h sub small d oh it's a big d thank you that's an easy question okay so now i want to discuss something else another version of duke's theorem which is the reduction of cm elliptic curves so take cm elliptic curves so so far it is a complex curve over the complex numbers but what happens is that the j invariant of this curve is an algebraic number and this implies that e has an equation uh with coefficient in the algebraic number and the curve is then defined over a number field uh which is the number field generated by this algebraic number over the quadratic field of complex multiplication so it's called ring class field of maybe discriminant d and uh moreover but this this field is is galois over k and the galois group is isomorphic to the to the to the id class group of the quadratic order so this is a form a very explicit form of the reciprocity law for for this extension uh in fact you have more this j invariant is uh is an integer and this means that the elliptic curves has a potential good reduction at every place at every prime and so then what so what you can do is you take a prime p and then you fix a place over the algebraic closure of q which is above p and and then what happens is so then you can reduce your elliptic curve uh modulo that place and by reducing the coefficient uh of the equation of the curve modulo uh says the ideal determined by the place in the in a in any field of definition for the the curve and uh if the prime is inert or ramified in the quadratic field then the elliptic curve e modulo p is uh super singular and i recall you it means that its ring of andomorphism is very big it's a maximal order in a quaternion algebra uh of in the quaternion algebra of so ramified at p and infinity so and okay so and okay and if the prime is uh is a split then the elliptic curve has a as an ordinary reduction but we will uh here we will really focus on a super singular reduction uh okay so now uh what you can do is uh you can fix a prime p a place above p and so and also i uh i consider uh the set of all uh super singular elliptic curves over the algebraic closure of the finite field fp modulo isomorphism and then uh what we have is a reduction modulo p map so i write it uh read p with a little p uh and not uh factor p because uh i have chosen my place so it depends on p with the choice and so i take my elliptic curve i reduce it modulo p and i obtain a super singular elliptic curve and uh also when you have this reduction uh what comes with it is um embedding of the ring of andomorphism so of your quadratic order od into uh the uh the ring of andomorphism of the reduced curve which is uh this op this maximal order uh and is it okay is there any question or okay good so uh here is a version another version of duke uh equidistribution serum uh in fact it's uh it's uh it's a version which maybe can be attributed to uh the joint work of duke and schultzer pillow and uh so duke schultzer pillow equidistribution serum implies the the following so now i take i consider uh imaginary quadratic discriminant d i uh let them grow and uh what i ask is that uh uh the d's are all uh co-prime with this prime number little p i am also asking that uh uh this prime number is inert in the quadratic field uh containing the quadratic order and then you can take all the cm elliptic curves uh by d of discriminant d and reduce them modulo p and so you obtain a multi-set inside the set of all super singular elliptic curves and uh what happens is that when the discriminant d grows uh so you have a multi-set with high multiplicities uh inside this fixed finite set and then this multi-set equidistributes with respect to an explicit measure on the set of a super singular elliptic curves mod p so this measure is the measure which is a probability measure which to any super singular elliptic curves associates uh a mass which is proportional to the inverse of the size of the of the group of units of the of the size of the group of automorphism of the elliptic curve okay so this is a measure for each atom in this finite set and so what does it mean is that when the discriminant grows under these uh conditions which are almost uh which are necessary almost necessary uh then uh the number of elliptic of uh cm elliptic curve with a given super singular reduction uh this number is uh asymptotically proportional to the measure of the elliptic curve up to the size of the whole set of cm elliptic curves okay and maybe if you want to think of uh uh elementary consequence of having a given reduction modulo p so what uh what it says if big uh e reduced to little e modulo p it means and it's equivalent to saying that the g invariant of e is congruent to the g invariant of the super singular elliptic curve modulo the place okay and um okay so i will come back to this later but now i want to to give a c o m and so what we can do is that we can fix a finite number of primes so p one up to p s we will consider discriminants which are co-prime with all these primes and also we assume that we consider only discriminants such that all these select primes are inert in the quadratic field and so when you have this this data so the primes the p i are fixed and now you take a sequence of discriminant with a suitable congruence conditions and co-prime elliptic condition then you have a multi-reduction map so you take your elliptic curve cm elliptic curve and then you can map it first you you can look at it at the complex number in the modular surface so i will call this number red infinity e and this number uh belong to uh so to the this point module belong to the modular surface and then you can take the various reduction of your elliptic curve so red p one of e up to red p s of e and they belong each to one of these factors of super singular elliptic curve module opi and so here is the conjecture is that when the discriminant goes to infinity satisfying the the condition that i have mentioned which are natural then the this set becomes equidistributed on the product relatively to the product of the measure so the hyperbolic measure and the super singular measures okay so that's the conjecture and there is uh there existed one case where this conjecture is true and it it was due to christophe corneux he was inspired by a previous work of of nick vatsal and so corneux established this conjecture when the ah okay there is uh yeah when the discriminants are of the shape the fund and discriminant time uh uh given prime to an even power so the given prime is fixed and it has to be different from all the pis and then you let the the exponent uh uh n go to infinity and you clarify i mean you're inspired by vatsal vatsal used retina of course in in his work a corneux probably used some form of andre haute is that correct that's a special form yeah so corneux he had two proofs of this result so vatsal what he did he did he proved uh an equity distribution serum for uh two uh for uh the same super singular locus el p but he had a shift uh multiplicative shift and he proved the equity distribution and the product and i want to emphasize the input so vatsal used vatsal use retina and then corneux had the proof of this serum using also retina okay but then he had to end the game with another proof using andre haute also yeah okay but i think it was a very special case of andre haute oh yeah sure absolutely even in other words it was known before the proof of andre haute that's what i'm trying to clarify yeah but i'm not sure but uh the andre haute proof gives uh equity distribution i don't remember okay it may just give uh surjectivity or okay but then maybe the the uh the thing to emphasize is retina then yeah so yeah the main the key input in uh i was going to say this later but the key input of the work of corneux and vatsal it was the user of uh retina theory of joining for unibot on flows and yeah absolutely so um so now i want to give another evidence for this so which is uh so this serum that we established with many uh uh manuel and andre haute so uh so it's exactly the same context that before and so what i uh i'm look i take i uh so you need to take two further primes in addition to all the p i so one needs two over primes q1 q2 and then uh what one do is that instead of taking all the discriminant which are naturally uh uh involved in this problem you restrict to the subsequence of these discriminants which are also split at q1 and q2 and then for this sequence of subsequence of discriminant the uh the conjecture hold so you have a equal distribution and so again if you want uh relatively uh concrete consequence you can look at the congruence for the gene variants and uh and then you will uh so what what you get is a kind of asymptotic chinese reminder serum for the for the gene variant of cm elliptic okay so you can reduce the gene variant to anything you want which is natural uh simultaneously for uh uh finitely many uh fixed primes okay and uh i wanted to point out that towards this conjecture there is no uh another uh evidence uh which is the work of uh valentine blommer and pharelle bromelais and uh valentine gave a talk one of the very first talk in the seminar i think it was in uh beginning of uh it was this summer in june maybe or i don't know so it was uh and so what they prove is that if uh you assume that generalized riemann hypothesis hold for a certain number of l function which i will describe later uh what you have is that the conjecture is true without splitting condition modulo q1 q2 but so but you have to restrict to two factors so either factor so either for the product of the modular curve and say one super singular one set of super singular curves or two sets of super singular of distinct super singular curves so and so the method is really very elegant so and uh it it is inspired by some earlier work of uh rady veal and steve lester uh and so use really crucial is a riemann hypothesis so yeah so this conjecture has a number of special cases uh which are which are known i have to describe the riemann hypothesis i assume for automorphical functions so as to get to half integral weight forms uh no i uh they they need it for the all the rank in selberg functions uh that occur in valse purge formula basically not they need really all the rank in selberg with all the characters of the torus uh just to clarify but then through valse purge yeah yeah it's true valse purge and i will come uh yeah i will try to describe the various elements of uh of the proof yeah so um oh excuse me yeah so in this blomber brambley theorem detail mod d tends to infinity along an arbitrary sequence or yes exactly that's the point the sequence of the determinant is arbitrary modulo that it's uh adequate so it means that the d have to be uh inner that uh at p one or p two where whenever there is a p one or a p two but beside this natural compatibility conditions that you do not uh they do not need an extra uh hypothesis for the d is for the quadratic field to be split at q one and q two so do they need a reman hypothesis or just lindelov no they need reman they really need to take the logarithm of the central value okay thank you so yeah it's not yeah it's very subtle so it's um because what in the end what they the saving that they obtain they they take a moment of the absolute value of a product of a function and basically they save a power of log over the trivial bound so it's a very very delicate uh argument so lindelov i don't think it wouldn't it's a two course too yeah and uh yeah it's used the the expansion of the log of the central value so okay um so some just i will just recall some principle of the proofs so uh the the first proof of duke uh c o m it was via theta functions so it's uh the principle is that uh it these c o m were about the problem of representing integers by ternary quadratic form or they all reduce to this so i introduced some notation so you have a three-dimensional quadratic space vq uh like i will call g's its orthogonal group and then you inside your space you take a lattice uh on which the form is uh integrally valued and then uh one the set of representation of an integer by this quadratic form inside the lattice this is just a set of uh element of the lattice taking the value d at uh against the quadratic form you also you may also need the primitive representations okay so it's just a notation and and then uh what this is about is uh so you the the set of representation as natural symmetries which are given by the stabilizer of the lattice inside the orthogonal group and so what one wants to study is either the set of representation or the quotient by this group g of z and uh okay so uh what happens say for the first c o m so the one for the modular surface the situation uh to which is correspond is when the quadratic space is the space of trace zero two by two matrices the quadratic form is minus determinant in that case the orthogonal group is isomorphic to pgl2 and the lattice is this lattice of trace zero matrices so z times the identity plus twice the integral matrices so and uh and then so if you compute and parametrize the the element of this lattice uh what what you see is that the this set of complex of ignore points it corresponds to up to maybe some scaling it corresponds to the set of representation the set of classes of representation of d inside the lattice for this quadratic form and so then so what you prove is that if you scale the the classes of representation it will equidistribute on the corresponding on a corresponding quotient of nipper boloid which is precisely the modular surface so and to prove this duke used the the correspondence of mass to he proved he showed that the vile sum for this equidistribution problem are uh equal so i i write equal if i take proper normalization uh it will be an equality uh to the diff Fourier coefficient of um of uh half integral weight mass form and then uh the objective usually is to show that this vile sum converges to zero if the function of zero mean and uh it it follows from uh showing that this Fourier coefficient uh is bounded by a quantity going to zero with a d again it's all depends on the way you want to normalize the Fourier coefficient is just made to simplify my formula and this non-trivial bound for this Fourier coefficient of half integral weight mass form it was obtained by generalizing a previous work by Henry in the holomorphic case okay so here is a picture uh yeah here is a picture of the equidistribution and this hyperboloid so you see the the fundamental domain being covered by eggner points and okay so uh maybe i will just quickly explain what happens for the reduction the super singular reduction uh in that case the quadratic space uh okay there should have been a zero here the quadratic space uh is the space of trace zero quaternions in the quaternal algebra bp so it's three-dimensional the uh the quadratic form is minus the norm so just like minus the determinant and then the the orthogonal group is the projective group of units of the quadrat of the quaternion algebra and then the lattice so it's z plus twice the maximal order and you take the trace zero element and this lattice is called the gross lattice after a big gross and so for this problem uh what you need to consider is not all the representation of of the discriminant by this quadratic form inside the lattice l but you need to include all the genius classes of this lattice relative to this orthogonal group so this is and this was worked out in by Duke in joint work with Schultz-Pillot okay and maybe just i explain how you pass from elliptic curves to representation in lattices it is the fact that if your cm elliptic curve reduce modulo ix i said it already once i think if it reduces to a super singular elliptic curve e modulo p then this reduction in use also comes with a numbering of the ring of of the ring of endomorphism of the curve inside the ring of endomorphism of the radius curve which is op and then how do you get a representation what you do is you take the image of the element square root of d which has trace zero inside the maximal order and you get a representation and then you want to study the set of all classes of these representation along all the various genus classes and again you have an equity distribution problem to which for which you have to bound the vile sums and these vile sums are Fourier coefficient of holomorphic modular form of integral weights and which so ivaniec provided a non-trivial bond when the for any large enough weight maybe five over two and Duke provided a bond also for the low half integral weights okay so now i want to to pass to another idea for this proof of equity distribution which turned out really to be crucial to the proof of the multiple reduction co m so really the approach through theta function or this theta half integral weight form is not sufficient to obtain multi equity distribution results and one needs something which is more dynamical and the origin of these dynamics is again in the work of gauss so it's the following thing so let's go back to see me cm elliptic curve with complex multiplication so it has a complex torus it's it's it is associated with a lattice with these properties and for it's and the morphism ring and then what you can do is if you take any proper ideal a you can form the lattice of all the multiple of element of lambda by the element of a so it's a lattice and this lattice it satisfies precisely but it's under morphism ring is again od so you have from an elliptic curve with complex multiplication by od you have obtained another elliptic curve by multiplying the lattice by a proper ideal and then in fact this and the new torus up to isomorph it's isomorphism class depend only on the class of the ideal in the ideal class group so in fact what goes approved in so it was not in this language but he really proved this is that the set of cm elliptic curve with a complex multiplication of discriminant d is a is a torsor under the action of the of the ideal class group of the picard group so it's more precise that just being a bijection it's really that you have a group acting on that site simply transitively and okay so yeah and it's 15 minutes 50 oh okay so but then what is important is that the so then the image of the reduction of the cm elliptic curves either infinity or modulo p can have a purely group group theoretic interpretation and which is very important but for this it's really convenient in fact it's an absolute necessity to pass to the adelic language I think and the the point is that the spaces the target spaces we are interested in can be realized that adelic questions uh where the group is the orthogonal group for all these problems and so it's an adelic question with adequate open adequate compact subgroup so possibly with compact at the infinite place and so you have really an identification of these sets with such questions and in this identification you can also realize the set of the image of of the cm elliptic curves so for this you have to consider this algebraic torus so the restriction of scalars of k to q of the the the projective group of this restriction of scalars and if you have this data z of e associated to an elliptic curve or the reduction of these elliptic curves these data they they provide you with an embedding of this torus inside your your group your orthogonal group and then the under this identification your sets can really are identified with an orbit of this embedded group inside the adelic question and so these are various this is important for various things so one of these is a val-spurge formula because this formula so this formula val-spurge will will take which has this shape it tells you that the veil sum under once you have this identification it tells you that the square of the veil sum for the equidistribution problem you want to solve is related to the product of a central value of an L function and a product of local of local integrals and so now if you want to bound the veil sum it is sufficient to bound the central value of the L function and this in that case it was solved this bound was obtained by Duke Friedlander Ivaniek and to bound all the local integrals and then I think it was a close L and you know who were the first to really use the I say so prove a duke theorem in in the pure in adelic language at least that's the approach that they took okay I want to say that there is another formula of val-spurge which here which is more general in which so you have these veil sum but you can in this integral you can also add a character of the torus and you have a similar formula and this time these L functions are ranking cell vertical function and so the general twisted formula it has several applications but one very nice explanation is precisely the work of Blommer and Bromley they use a form of delta symbol method so very neat and but for this they need the GRH for all these L functions so you see it's a case where to have an equity distribution for the product you really need to exploit the the existence of a group structure and here they exploit it in the form of this twisted formula where you where you can also include the characters of the of the torus with the theta function approach you you do not see this the veil sum they come as a block and yeah I don't I don't know how to do it with metaplectic forms but anyway okay so I will pass on this and maybe I will try to explain what happens for our our CRM yeah I wanted to explain Linique but okay so now what I maybe to quickly finish now if we want to look at the multi-reduction equity distribution CRM what you can do is that the product of say the spaces of super singular elliptic curve you can realize it as an adelic quotient where the the group in the adelic quotient is just the product of the various orthogonal group for each factor and then it's a quotient with compact subgroup which will be the product of the compact subgroup for each factor and then if you have an elliptic curve with complex multiplication you have the multiple reduction of your elliptic curve and once you have these multiple reductions it determines you an embedding of your torus inside the product of these groups so and it's a diagonal embedding so t is one dimensional which is embedded in into higher dimensional group and you you also have a torus orbit which is one formed by the adelic data attached to each of the individual torus orbit so there is a slightly non-trivial point still that you need to to see is you need to show that these torus orbit these diagonal torus orbit is indeed the image of of the corresponding to the image of the of your complex multiplication elliptic curve under the multi-reduction embedding and it's not totally obvious in fact to to see this so this is true but to see this what okay it's for this you need an algebraic version of the action of says id class group on the cm elliptic curves so we act this action we we realized it in a elementary way for complex lattices but ser in his short article on complex multiplication he also very quickly introduced what is called the a transform and again it's algebraic version of this action and okay it's slightly incorrect but I will say it like this then what you have is that the reduction mod p of the translate of a curve by an ideal air is the translate by the ideal air of the reduction mod p so of course you have to make sense of this star but it's possible it was done by ser and once you have this you are sure that this orbit indeed represents the diagonal the diagonal reduction map and so now what to conclude the proof what do we have suppose now we have our sequence of discriminant with this primality condition and suppose we give ourselves two extra primes which are in which all these quadratic field split what you obtain from this realization is a sequence of probability measure which are supported along this diagonally embedded along translate of this embedded torus and okay and the splitting condition such that q1 q2 split in this quadratic field it tells you that this measure sequence of measure maybe up to taking sub sequences and by compactness argument you can say that they are invariant under conjugate under fixed conjugates of diagonal elements diagonal matrices of that of that shape so really so q1 the quatern all your quaternion algebra they are split at q1 and q2 because the torus is split at q1 and q2 so they are isomorphic to matrix algebra and then you can your group can contain such diagonal matrices or their projective image and in addition your measure mu d they have by the previous discussion by Duke's theorem basically when you project them on each factor okay their limit the limit of the projection is a measure which have a huge invariance so which is invariant under this group g1 which is the image of the group of quaternions of norm one so the simply connected cover of of of pb and if you have this and this so then you can really invoke so it's an invocation like black magic a theorem of einseidler-lindenstrauss which tells you that so this such sequence of measure as its limits limiting measure which are called the joinings and einseidler-lindenstrauss have classified the joinings for a measure which are invariants by such pairs of elements and so even they prove that the any weak limit is invariant under the product of the simply connected covers and now to finish really the proof it's not so much because to conclude what you have to do is the last function you have to look at are the characters on on this group and then it's really elementary to evaluate these the torus orbits against these characters and because in in your situation the open compact subgroup is big you can you can finish so you don't have a pathological problems which could occur in that case like exceptional subsequence and things like this and so just I want to finish because I wanted to quote Riccardo Menares so I wanted there is a recent very recent work of Sébastien Nérérot, Riccardo Menares and Rouen Rivera-Lotellier so they have studied more refined version of of the reduction mod p of in in this reduction super singular reduction theorem so how can it be more refined so for this you have to know that in fact the set of super singular elliptic curve is just the index index thing set of disjoint union of periodic disk say of radius one and each of these disks what they do is parametrize the deformation of the formal group attached to each of these super singular elliptic curve and if you have elliptic curve with complex multiplication and with given super singular reduction the formal group of that elliptic curve gives you a deformation of the super singular formal group and then you get a point on this big disk so what they did is that using Duke's original method so using theta series they have established that the cm points are equidistributed on this disk so the the answer is more complicated than say a fan say over the modular curve what happens is that the limit depends in fact of the valuation of the discriminant so you take a sequence of discriminant with valuation at p as a given value and their theorem will tell you my set of such cm elliptic curve equidistribute according to a measure which are a certain support in this unit periodic disk or this unit of periodic disk and yeah and you have a different measure for each valuation so and they did it with the theta series approach and as I explained it's I don't know how to to do it to do a multi-reduction theorem refined of their kind so but so we learned a lot with from Ricardo Menares about just already how to formulate this and so with with Ricardo and so many Andreas and Manuel we we could we have implemented this this the torus orbit approach so to be able to to obtain a result for multiple factors at least for discriminant which have a zero valuation at any of the points and so and then we have a quid distribution for the product of measures of the un-ramified measures I would say so it is an elementary consequence of this so you what so for any for any so you fix a finite set of prime s for each of these prime you fix a cm elliptic curve over the complex number say whose discriminant is co-prime with each of these prime of course you need two additional primes for the splitting condition but if you have all this what you can show is that if you take a large amount of discriminant with adequate condition including the splitting condition then you can always find a cm elliptic curve by this discriminant whose j invariant becomes arbitrarily close to the j invariant of the chosen cm elliptic curve okay and so again just to conclude the the proof of this it's okay no I will just stop here okay I should give you a piece okay thank you