 Thank you very much for this kind invitation. I've been following this seminar since the beginning. And well, I cannot always be at the same time, but watching the recorded lectures had it's a great possibility. OK, so today I will talk about joint work with Sebastian Herrero and Juan Riera Litalier and the title is Distribution of Seven Points, but the added one. So let me start with the complex case just to have the motivation for this work. So I will denote Y of C, the modular space of elliptic curves over complex numbers, up to complex isomorphism. And as it is well known, this can be uniformized by the upper half plane. It's a Gaussian H over SL2Z. Now we pick a discriminant. So by definition, a negative integer, which is either 0 or 1 mod 4. And this integer determines a unique order of the discriminant D inside the quadratic field Q square root of D. So the main actors today will be the elements in the set called HD, which is the set of complex elliptic curves with endomorphism ring isomorphic to OD. So this HD is a finite set. The number of elements can be, well, it's a set that can be put into one-to-one correspondence with the car group of the order. This is the very classical CM theory. And in this talk, a CM point will be an element of HD for some D. So the basic theorem is a theorem proved first by Duke and then extended to full generality by Closet and Ulmo, stating that the sets HD equidistribute on this complex modular curve as D goes to minus infinity according to the hyperbolic measure. So more concretely, this means that for any test function, say continuous and compactly support, when you take the average, when you average the function on the set HD, so you evaluate at every point in HD and divide by the number of elements, then as D goes to minus infinity, this will converge to the integral of f according to this measure, which is dx dy over y squared, which is the hyperbolic measure on the upper half plane. And there is this constant 3 over pi to account for the fact that you will have a probability measure in the limit, because all of these measures are probability measures. OK, so Duke proved this theorem in the particular case where the sequence of discriminants is a sequence of fundamental discriminants. And then Closet and Ulmo extended this for arbitrary discriminants. So in this talk, we will look at the periodic situation. So we fix prime number p. We will denote by Cp the completion of the algebraic closure of Qp bar. So Cp by definition a complete and algebraic closed field. Throughout the talk, we will fix an embedding of Q bar inside the Cp. And once we do that, we can think of the elements of HD since we know they are actually defined over number fields. So we can think of them as elliptic curves defined over Q bar. So you can take the Weistras equations with algebraic coefficients. And using the embedding, you can see this as a subset of the periodic points of the modular curve. So I haven't defined Y of Cp, but it's by definition the set of Cp elliptic curves up to isomorphism. Cp isomorphism. So now you have these Cm points. You can see them also as points inside the biotics place. So the main question we want to address today is that if you take a sequence of discriminants going to minus infinity, so how are the sets HDN distributed inside this biotic modular curve as M goes to infinity? And I will explain during the talk that the answer really depends on the sequence of discriminants you consider. So you see in the, I will come back to the complex case. In this theorem, there is no sequence. And it's for any discriminant going to minus infinity. But in the periodic situation, there is a difference. And you have to classify some how sequences in order to state any distribution here. So OK, so I will start. There will be a few cases. So I will start with what we call the transient case. I will explain what I mean by transient. So definition, the distribution of the sets HDN is transient if for every point in the periodic modular curve, there is a neighborhood U such that the proportion of CN points landing on U tends to zero as N goes to infinity. Yeah, so transient distribution means that these points are not concentrated in any way. OK, so the first theorem will characterize the sequences giving rise to a transient distribution. And for that, I need to introduce a definition, another definition. If you take any point inside HD, then we know that it has potential good reduction. So call it E tilde reduction and it take curve over FP bar. And this reduction can be ordinary or super singular. So we will define this symbol, the periodic P super singular valuation of D. This will be by definition infinity if the reduction is ordinary. And if it's super singular, then it will be the periodic valuation, the periodic order of D. So the exponent of P inside D. OK, so the transient theorem tells you that the distribution of the sets HDN is transient if and only if the limit when N goes to infinity of this P super singular valuation at the end goes to infinity or is infinity. So you see this limit condition will be satisfied by definition if you take a sequence such that all reductions are ordinary because we define this symbol to be plus infinity. So OK, so you will have this condition. So it's telling you that when the reduction is always ordinary, CN points are not concentrating anywhere. The situation is transient. The other possibility is that the reduction is super singular, but the exponent of P inside the N goes to infinity. OK, so that's our first case. Now I will explain what happens in the other cases. So what are the other cases? So we will concentrate on the non-transient situation. So that means that we will assume from the other cases that we will assume from now on that every elliptic curve in HDN has a super singular reduction and also that the periodic valuation of the N remains bounded as N varies, because otherwise we know that there's transient institutions. So we don't want that, so we assume it's bounded. A first remark is that the fact that the reduction is super singular can be read inside the order the ODN. And it's a given to say that the completion of P of ODN is an order in a quadratic extension of QB. Well, since we are assuming that the periodic valuation is bounded, we will fix it. We will fix the periodic valuation, so we assume it is fixed. And remark that that can also be read in terms of the order, because the valuation is fixed if and only if the quadratic order you obtain by completion is fixed. This quadratic order only depends on the periodic valuation of the N. So if we fix this periodic valuation, we fix the order. So with that in mind, it makes sense to fix. We fix an order in a quadratic extension of QB. And look at all elliptic curves, such that the completion at P of the N. Isomorphic to this O we fixed. And moreover, we will take the periodic closure of this set. So that will be what we call lambda O. So if these elliptic curves are going to equidistribute, the limit measure will have to be support on this set, lambda O. So any possible limit measure must be supported on this set. If we go along a sequence, the N, such that the completion is O. So with that in mind, the theorem, the supersingular situation is the following. First of all, the closure of this set of CM points is a compact set. That's our first point. A second point is that if you take two different orders, so you have two different sets lambda, then they are actually disjoint. So these orders, they somehow, they sit inside the modular curve, but they do not touch each other. So you have compact sets which are disjoints. OK, now in order to state the distribution theorem, I will make an assumption which is not really necessary. It's just for simplicity. So assume P is 1 mod 4. Under this assumption, we can show that there exists a measure called new O, which support equal to this lambda O, such that when you run HDNs where the completion of ODN at P is fixed, isomorphic to O, then the sequence of sets equidistributes as the discriminant goes to minus infinity with respect to this measure new O. So that's the situation. And as I said, that if P is 2 or P is 3 mod 4, there is also an equidistribution theorem, but it is likely more complicated to state. So in this talk, I will stick to the case 1 mod 4 just for simplicity. But you can also write down an equidistribution statement. OK, so this is our main theorem, I will say. So now, OK, this is fine, but I would like to show an application, a diophantine application of this result. So let me remind you that the J invariant of a CM elliptic curve is usually called a singular modulus. And these are known to be algebraic integers. They play a very important role in Hilbert's first problem. So what we can prove is the following. Fix finite set S of prime numbers and singular modulus J0. Then the set of singular modulus J such that J minus J0 is an S unit. This is a finite set. The statement is that this set displayed here is finite. Just a reminder, so to be an S unit, that means that, well, these are algebraic integers. So when you take the absolute norm, you get an integer, a rational integer. To be an S unit means that the prime factors, they are all inside P. So if you constrain the prime factors to belong to a fixed set, then you only get finite the main differences of singular modulus. Well, this statement has a bit of a history, so I will explain. For us, the starting point was theorem of Philip Habeghe. He treats the situation when the set S is empty. So being an empty unit means to be an algebraic unit. So an algebraic integer such that the inverse is also an algebraic integer. So he proved that the set of singular moduli, which are algebraic units, is finite. And his method used the complex distribution of cn points. So what we do essentially is to take Philip's method and inject our biadic results to extend this statement for S units. Well, this first result of Habeghe and also our result is non-effective. We don't give any estimate on the number of elements of this set nor on their height. But there are a number of effective results. So below Habeghe and Cune, they prove that the set of algebraic units is actually singular moduli, which are algebraic units, is actually empty. Not only finite, but empty. And the same if you plug here 1728 instead of 0. Later, in Cungli, he proved that in full generality for every cn, for every singular modulus j0, this set is empty. So that means that for any pair of singular moduli, the difference is always divisible by some prime number. And there is also a result by Campania that he shows that the set of S units related to 0, so the set of singular moduli, which are S unit, is empty when S is what I call S ordinary, which is the set of primes of ordinary reduction for the elliptic curve with j invariant equal to 0. So this is an infinite set, but only concerns ordinary reduction points. And he has a similar result for 1728. OK. But well, this is what we had. But at some point, Philip, he asked about other Hab module. What can you say about only small corrections? So this bilu, Habegger, and CUNY, they did not prove that this set is empty, but it follows from least work. Sorry about that. OK, so let me continue, so we realize after some emails exchanged with Philip that we can generalize a bit our statement in this form. So take F to be a Hab module for a G0 subgroup of GL2 plus of Q. So by definition, a Hab module is an element generating the function field of a modular curve of G0. Assume that this function F and the j invariant are algebraically dependent over Q bar. So this is a sort of arithmetic normalization of F. Now the same result follows. So if you let S to be a finite set of prime numbers and if you fix an element in the upper half plane tau 0, which is a quadratic imaginary algebraic number, then the set of say CM value, so F evaluated at tau, where tau is quadratic imaginary, and such that the difference F of tau minus F of tau 0 is an unit, this set is always finite. OK, so some examples, well, an example is what I already showed, the case of the j invariant, so take F equal to j, that's the previous theorem. But this also applies to the so-called lambda invariance, which is a Hab module for a modular curve of level 2. It also applies to Weber functions, which are Hab models for, well, a group which is a bit difficult to describe, but it's a classical function used in the CM theory to generate class fields. And another interesting example is, well, a family of interesting examples is the so-called Myk Thompson series attached to elements of the monster group. So the monster group, the theory, the representation theory of the monster group is also a source of interesting Hab models. And this theorem applies to them. OK, so these are the results I wanted to explain. And now I will explain the main ideas behind this theorem of a good distribution in the super singular logs. So I will explain how we prove this theorem. So the first remark is that once you fix P, there are only finitely many super singular elliptic curves over Fp bar. Let's call them E1 up to EK. The number K can be computed in terms of P. That's very classical, you know, but here we will not use it. Let's call DEJA will be the set of piatic elliptic curves, such that the reduction mod P is isomorphic to EG. So this is a disk. It's a residue disk, set of points reducing to some element over a finite field. So the mental picture is that you have at the bottom all these finitely many super singular elliptic curves, and above every one of them, you have a disk, which when you perform reduction mod P, entirely collapses to the elliptic curve. So what we will do is to fix one of these super singular elliptic curves and seek to understand the asymptotic distribution of the set's hdn landing inside the disk corresponding to E. So these cm points will be somehow distributed in the super singular logos, and we will see what happens at the level of one disk. So say that one. OK, so for now, we fix one super singular elliptic curve. And yeah, I will also assume some simplifying hypotheses, which are not really necessary, but in order to explain the main ideas in a simpler way, I will simplify a bit the setting. So OK, first of all, as we are in the super singular case, we will assume that all elements in hdn have super singular reduction. That's the first thing. We fix an order inside the quadratic extension of QB and elliptic curve, super singular elliptic curve E. So we will assume that O is a maximal order, which is more or less like if we are in the complex situation, it's like assuming that we are dealing with fundamental discriminants. And we will actually assume that as well. We will assume that dn is a fundamental discriminant for all n, like in Duke's theorem. Also, we will suppose that dn is not divisible by p. Well, p equals 1 mod 4, because that's the statement I gave. And another simplifying hypothesis is that the super singular elliptic curve we fixed has known a trivial automorphism. So this hypothesis will tell you that dn is not a square mod p. So p is inert in the quadratic field of QS square root of dn. And the completion of p of this ODN are equal to the maximal order of the unique non-ramified quadratic extension of QB. You have when p is at least 3, you have only 3 quadratic extensions of QB. One of them is non-ramified. So this is we are working on that case. Now, before I read all this slide, I will tell you the goal. The goal is to uniformize the set lambda o, which, let's remember, lambda o is just the closure of all cm elliptic curves, such that the completion of p of the automorphism ring is o. So we need some description of this space that will allow us to prove a QS tuition. What we will do is to uniformize this space by some quaternion algebra. So I will explain that. So we start with the local division quaternion algebra, so the quaternion algebra over QB, which is a division algebra, called OB, the maximal order, which there is a unique maximal order. This algebra is relevant because the completion of p of the endomorphism ring of E is isomorphic to OB. And using that, and the deformation theory of the formal group attached to E, and those the E with an action of the units inside OB. So this is where we are using the simplifying hypothesis that E has no non-trivial automorphism. OK, now fix discriminant of inside O. So a small remark, since O is a periodic order, there's no unique discriminant. You have many discriminants. So fix one, which is by definition you take a basis and then you change the basis by the Galo action and you take the determinant of a 2 by 2 matrix and you get a periodic number. That's one determinant. But it's not unique. It's unique up to a square. OK, so you fix one or choose one. And we will define the set SL, which is by definition the set of elements in the order, the maximal order, having trace 0 and reduce norm equal to minus L. So this is a subset of OB star, because since the norm is minus L and L is not divisible by P. Well, I didn't say it, but L is not divisible by P because it is a discriminant and we are in the un-ramified case. So that's why discriminants are not divisible by P. So these are units, in particular the elements of SL act on the disk DE. OK, so the uniformization we use for the set lambda O is the following. We show that for every element in SL, every element in SL acts with a unique fixed point. And such fixed point, well, will be an element of lambda O. And moreover, the map going from SL to lambda O that to any element that attaches the fixed point is continuous by ejection if we mod out SL by plus or minus 1. So we will think of this map as a uniformization of lambda O. So when you try to predict the distribution, it is useful that your space is a homogeneous space. But it turns out that the super-similar locus of the modular curve is not really a homogeneous space. But what we show here is that this lambda O, they are. And this is how we will exploit that to prove a equestria. OK, so still at the philosophical level, so we want to prove something here. We will construct a measure on the left-hand side on SL. And we have to somehow describe the CM points in lambda O in terms of points in SL. So now we'll explain two things, one, how to construct measures on SL, and two, how to read CM points inside SL. So let's start with the measure. So well, this set is a compact set, and I'm sorry. And it is invariant under conjugation by B star. Because conjugation by B star will not affect the trace, nor the norm. So this set has an action by B star by conjugation. So by definition, we will denote by mu L, the uniformation SL, which is the unique probability measure on SL, which is invariant under this action by conjugation. Now, this perfectly defines a measure on SL, but it can be written down more concretely. Let me explain that. So fixed an integer R, bigger than or equal to 1. And take a reduction map from the set of quaternions with trace 0 to Z over P to the R Z cubed. Because the OB is a ZP model of rank 4. But if you impose the trace to be 0, the rank drops by 1. So you have a rank 3 ZP model. And so by choosing an integral basis, you can construct a reduction map to the set to the right. OK. So fix one reduction map. Then now, if you run through the elements of Z over P to the R Z cubed and look at the inverse image, these sets will cover the SL, this periodic sphere. So let's call M sub R. Yeah. M sub R will be the number of non-MP sets that are covering this sphere. OK. So with all this in mind, so if you take any one of these sets, so the uniform measure will be really uniform. So it will attach to the set the measure, which is 1 over the total number of such sets. So in other words, it's like you fix an element, or you look at the set of elements in SL, which reduce to the same point. That will give you a set. This set, if you move it by conjugation, will cover all of the cell. And since you want your measure to be invariant, then you can move it to the other set. You can do no other thing, but you really have that mu L of this set has to be 1 over M R. So that's why this is a uniform measure. OK. Now I have to explain to you how you read CM points on SL on the dashboard. So for that, let me introduce this set, which I will call VG. Here, this little D, you have to think of it as a discriminant, a fundamental discriminant. So VD will be, by definition, the set of endomorphism of E of the elliptic curve. There's no completion here. That's important. So the set of endomorphism of E, it's an order in a global quaternion algebra. So this is a Z-module of rank 4. So you take all the elements there with trace 0 and norm equal to minus V. Again, this set can be sent to OB star by completion. If you complete at P, then, well, if you assume at least that P does not divide the discriminant, you get an invertible element in the local quaternion algebra. OK. So what we show is the following is that if you take a fundamental discriminant with such that P does not divide D, and also it's not a square mod P. So this is telling you that the reduction of the corresponding elliptic curves is super singular. Then every G inside the set VD has a unique fixed point in DE. And moreover, the set of CM elliptic curves with discriminant D landing on the disk DE is just the union over all VD of the fixed points of G. OK. So this is how you can read in the quaternion algebra the CM points. So they will be fixed points of some special elements that come from a global quaternion algebra. OK. So now we have all the tools. So I will explain how the argument runs. OK. So we take a sequence Dn of fundamental discriminants with P does not divide Dn, and such that Dn is not a square mod P, so super singular reduction. For every discriminant, we will consider the set VDn. So global endomorphisms, which is 0 and norm Dn minus Dn, that now we know they parametrize CM points. And remember that, as I said before, that the elements in the local quaternion algebra with trace 0, they form a ZP module of rank 3. And these sets VDn, they sit inside a global quaternion algebra, so they lived inside the Z3. OK. An artifact of the proof, it's not really necessary, but in order to simplify, we can assume that this sequence of discriminants converges piadically to some piadic number, some L, which is invertible. By taking a subsequence, we can always assume that. Then when Dn is close enough to L, they will differ by a square. So we write Dn as L times the square of some piadic number, An. So if we do that, then use this An to scale the set VDn. So you divide the elements in VDn by An, and if you do that, then they will land inside this piadic ball, SL, because the norm will be minus Vn divided by An square, so that will be minus L. And this is the key point is that the elements to the left, they are like integral points, because they live in a global space. So these are integral points. And this SL can be thought as a piadic ball of radius L. So this reminds of a linux problem. So to understand the distribution of Hdn intersect with the E, what we have to do is to understand the distribution of these integer points, 1 over N VDn, inside the piadic ball S of SL. So this is like the classical linux problem of integral points inside a real sphere. Now we are in a very similar situation. We have integral points, but inside that piadic sphere, well, the sphere is defined by the norm in the quaternion algebra. And we can solve this linux problem. We can really prove that the sets 1 over A VDn are equidistributed according to the uniform measure on S of L. Remember the uniform measure that we discussed a few slides before. So just, well, I won't go into the proof of this theorem, even though it's important for our result. But we follow, we use the classical bounds by Ivaniek, Duke, and also more recent refinement by Blohmer, on Fourier coefficients of half integral weight model forms. In particular, for this problem, we have to deal with a model form of weight 3 over 2. And we need good bounds for the Fourier coefficients of that. And these bounds are available thanks to this works. OK. So now the conclusion will be as follows. So first of all, the fixed point formula tells you that when you look at the fixed points of the elements of VDn, then you land exactly at the same points inside this DE. So we deduce that if mu L is the uniform measure on S L, then the same points will be equidistributed according to the push forward measure. We can use the fixed point map to push the measure to the disk. And then one has to check, but that's really very formal, that this push forward measure does not really depend on this L, auxiliary L that we chose. Remember that we had this sequence of discriminants. And we say, well, we can take a subsequence which converges. The point is that if you took another sequence that converts to another L, then in the end, that doesn't matter. The measure is the same. The measure only depends on the maximal order you have. OK, so this proves the equidistribution statement. So by this, you put together on the information that you have a global equidistribution. OK, now perhaps I still have some time. So I will make some comments about this simplifying hypothesis, because I said they were not essential, and they are not. But I will give an idea or some idea of how you get rid of all these simplifying assumptions. So the first assumption was that this local order O was maximal order, which amounts to work with the fundamental discriminants. So in order to pass to the general case, what we did was to use the Katz theory of the canonical group. Yes, somehow, well, if you know this theory, then somehow when you start with an elliptic curve with such that the completion of the endomorphism ring is certain quadratic order, if you mod out by the canonical group, you get a new elliptic curve. The order has a peak on doctor, which is one less. So you decrease the peak on doctor by applying the canonical group. So you can somehow use that fact to promote results for maximal orders to arbitrary orders by somehow applying the inverse map attached to modding out by the canonical group. OK, so it's not entirely, it's not really so simple, but you have to do some theory, but this is how it works. So it's mainly due to Katz theory. Another assumption we did was that p does not divide the discriminants, so that amounts to say that the order on ramified. When the order is ramified, what will change in the uniformization of the lambda O set is that the points inside SL will have two fixed points. So you have to work with two fixed points and somehow average over both. But you get an equation statement anyway. There is also this assumption p equals 1 mod 4. When this assumption is not fulfilled, what happens is that the set lambda O is split into two subsets, which are also compact. And you can prove that same points of fundamental discriminants, they will all land inside one of the components. And if you modify the discriminant by a conductor, which is a square mod p, then you will stay in this part. But if the conductor is not a square mod p, you will jump to the other part. So the conductors will somehow force you to move from one part to the other. So you will have one equation statement for conductors, which are a square mod p and another equation statement for conductors, which are not a square mod p. So it's a bit of a mess to write it down, but it has the same structure as the theorem I already explained. Yeah. This hypothesis, well, if you really use deformation theory, then you see it's not difficult to get rid of this. The deformation space is actually a covering of the E. It's not really the E, but a covering. So you work in the covering. You do all I said, but in the covering. And at the very end, you push the results of the covering to the disk. And that's how you get rid of this simplifying hypothesis. OK. Yeah, well, I'm reaching the end of the talk. So I will just comment that we have current ongoing work to treat the same problem, but on more general shimura varieties. That's joint work with P.L. Goen, Sebastian Herrero, Paimon Kasei, and Juan Rivera. And there's another direction, which is the joint distribution on products of periodic model occurs for different primes. So there was a talk by Philip Michelin in this seminar about this. And well, this is joint work with Manny Aka, Manuel Lliuati, Philip Michel, and Andreas Wieser. OK, well, I think that's all I have to say. Thank you.