 Okay, thank you. I changed the title because I think this title goes better with the day because all my talk will be around the conjecture of Mazer and also a collaborator of mine is a recent student of Alexander Gontroff, Mr. Dimitrov. Okay. So first of all, the motivation is, let's start with Fulton's theorem. In my whole talk, I'll fix G to be an integer, at least two, and C is a smooth, projective irreducible curve defined over Q bar. The motivation is, first, Fulton's proved in 1983 that if C has genus at least two, defined over a number few k of degree B, C has finally many rational points. So this is the model conjecture. This proof also gives an ad hoc upper bound on this number. It's on G and D and the Fulton's height and about Jacobi rank and so on, but the bound is really not explicit and ad hoc one. Around 90s, Voita gave a new proof to Fulton's theorem using our club geometry and more in the sense of Dufontine method which was simplified by Fulton's and further simplified by Bombieri purely in terms of Dufontine geometry. It's a new proof of the same result, but also at the same time gives a rather good bound on it. So the bound is in terms of the genus, the degree of the number field, the Fulton's height of the Jacobian of this curve and the model weight rank. And the model weight rank. All these numbers are explicit in this bound and later on, the bound was improved by David Philippon and Heimann and so on but it remains in this term. So this is in theory, this kind of a computable expression. Yes. It doesn't involve the other and the height of the points is not known to be. No, no, no, no, that is the effective version but no. Mayser asked the following question. First of all, in 96, this is a conjecture. Later on, it was simplified. I will state it in this most general form, maybe here because this is the conjecture I'm going to study. Let's take any algebraic point P0 in C and let's consider the Abel Jacobi embedding via this point. We take any finite rank subgroup of JQR say of rank R. We hope that the cardinality of the following points, the points on C intersect this finite rank subgroup. It has a sort of not really uniform but it's not uniform only in terms of rank. So this bound does not depend on the choice of the point, first of all, and also it does not depend on the finite rank subgroup as long as the rank is fixed. This was the original conjecture made by Mayser. Afterwards, there were some simplified version. I'll talk about some. My main result today is this is a forthcoming paper with Vesselland Dimitro and Philip Habegger, is that we can prove this inequality for large curves, large in the sense that for any fixed G there exists a number depending only on G such that as long as the height of the curve is larger than delta, then this inequality holds. High of the Jacobi and all the curve. Some people also define it as the height of the curve. Some immediate application of this theorem, first of all, particular cases, if we take P0 to be a rational point and gamma to be JK, which is a finally generated subgroup, then our bound here will say that the number of rational points is uniformly bounded in terms of G, D, and the rank. Here, the dependence on D is really artificial. It is there because we have this restriction of large curves. And D is here only because if we fix D, then there are only finally many curves with small heights. So the dependence on D is only here. And this is actually another version of Mayser's conjecture. Secondly, if we take P0 to be, say, any q bar point, but gamma to be the subgroup of torsion points, right here, we only take finite rank subgroup. And this is a rank 0 subgroup. Then we have a bound that I'll just use Cq bar tall to denote it. But this really depends also on the P0 we chose at the beginning. So this is the size of the torsion packets. If you know the terminology, it depends uniformly on G and D. Again, this dependence on D is really artificial. And of course, it shouldn't even be there, especially for the second one. And currently, no results for the first one. In this direction, David and Philip Bohm proved it for the case when J is a subrarity of copies of the same elliptic curve. And they also find some examples that David Nakamaya and Philip Bohm also produced some examples of families satisfying this. And for the second one, there is a recent paper of holy Laura DeMarco. And yeah, they prove this for G equals 2 by elliptic. But their result is stronger than ours in this case, because they have no dependence on D. They really prove this. Otherwise, there are some results in these directions using the Shabotikol man method. But all of them, of course, there is some rank condition on the curve. Now let me quickly go through the method of Umberi faultings and Vojta to see some input we need. And our proof is based on the second proof of model conjecture. Recall that if we have J, the Jacobian of C, on J, we fix a symmetric ample line bundle. Then this L defines a function from the q bar points to non-negative real numbers. It's called the Nehontate height function. It satisfies two properties. First of all, this function defines a quadratic form. Simply in this term, if we take the multiplication by n of the height, it is n square of the height of the original point. And secondly, it vanishes precisely on the torsion points. Now, if we want to fix our finite rank subgroup gamma in Jq bar, we can always do a map from gamma to gamma tensor r. And here in this step, this is almost injective except that we lose all the information for the torsion. But here, we also lose all the information for the torsion. So actually, in terms of the height function and this mapping to gamma tensor r, actually we are losing the same kind of information. So we can use it to study this. And it is known that, first of all, this is just rr. And hl hat square root defines a norm on it just by the linear extension of this function. All right, now we divide gamma into two sets. For this first simplicity, I'll just present a case where gamma is finally generated. For example, the rational point case, just for simplicity, gamma finally generated. Then in this case, when we embed gamma into gamma tensor r, we obtain a Euclidean space here. And gamma will become a lattice in this Euclidean space, normally Euclidean space. And the division and Bombardier-Foltings' Vojta and their method, they divided the points in gamma into two sets, which we call the set of small points and large points in the following way. They found a c0, I'll explain later this number, and divide it into small points, which are the points with height smaller or equal to c0. And the set of large points, so you may guess, is the points with height larger than this number. OK, now we have, say, this is the origin. Say, this is the origin. And c0 is here, the radius. Now, the set of small points is here. The set of large points is outside this ball. Well, for modell, it is immediate that the set of small points is a finite set, if I restrict to finite generated group, just because it's a lattice and contained in a closed ball. So it's automatically finite. So for the sake of modell conjecture, we don't need to do anything for these small points. But then, bounding the number of large points is what we need. And this is what is done in the method. So theorem, this is Vojta faultings from Biele. And then, it was later on strengthened by Davis-Fiery-Boum and Heimann that the number of large points is uniformly bounded in terms of, I think it's 7, I'll just write it as c prime g1 plus r, c prime g r, sorry. The number of large points is already uniformly bounded. Actually, this number is g times 7 to the power of r. But yeah, let me state it in this way, every term is. And Davis-Fiery-Boum and Heimann they also proved this result for finite rank subgroup so that this part actually works also for finite rank subgroup. And they prove it using the Manthold gap principle plus the so-called Vojta inequality. So this is a review of the Bonviere faultings Vojta method. It tells us that now, in order to study measures conjecture, we only need to study the set of small points, which for modell is the automatic thing, but now it's the hard thing. And they found this c-node. So fact, the property of c-node is that it is linearly in terms of the faultings height of the Jacobian. This is how they constructed the c-node. The whole point is to find a good c-node such that this Vojta inequality holds. All right. OK, now we see that we want to study this set of small points with height bounded linearly in terms of the Hfj. And we want to bound the number of these points like independent of this Hfj. So what we do is, so it suffices to prove the following thing. Small points are far from each other. Or in mass terms, if we have any two small points in gamma, actually, we will prove it for any points in j cubed r. Then, their distance defined by the Nehontet height called Nehontet distance is uniformly bounded in terms of g and Hfj. And we cannot prove it for the moment, but we can prove it plus some number. Let's say devil not g. And this number is why we are restricted to the large curves. I'm sorry, but is it a male number? It's just a number, depending only on g. This is what we can prove. But actually, if we can prove this term, then we prove the four measures can get her in it. You were right to write it clearly. What we can do is this. Everything is positive here, but not negative. Everything is positive. But yeah, everything is positive. This may be negative. This is the problem. This is the reason why you are limited. Otherwise, you will have a bound. Yes. OK, so. Just height can be zero points. It can be zero. So that's why this bound. You mean different small points? Difference of small points. He is different from you, I suppose. It could be the same. If it's the same, then this is zero here. And it just says that the height of the curve is small. Because this is a negative number, I'm sorry. If this is zero here, then we will have zero greater or equal to a positive number times hfj plus a negative number. Yes. So that means the height hfj here is small. I think you really do want to take different points. Otherwise, you'll just conclude by taking p equals q that's behind us. Oh, OK. I see your point. Yeah, sorry. But what I want to say probably is that p minus q could be torsion. p minus q could be torsion. Yeah, thank you. Yeah. Yeah. But actually, what we proved is not just for gamma, but really for Jq bar with this term, so that we're done. Let me write down exactly what we do. We proved the falling height inequality for one parameter families. And this is a theorem. Yeah, because it's a project, it concerns several papers. So I will make it precise. Which papers are published? Which are preprints and which one is forthcoming? This one is a published one this year. It basically shows that this is true for one parameter families. So let S be a curve against smooth but not projective curve defined over q bar. Let A to S be an abelian scheme of relative dimension G, and L be a symmetric, relatively ampulline bundle on it, so that now we have a fiber-wise defined mayhotate height. Let X be a irreducible sub-variety, dominant or not. It doesn't matter. Close irreducible sub-variety. Then, OK, this is the setting I may want to use. You said you expect this to hold without the extra term delta not G? Yeah, but under certain conditions. But then it means that there is no difference which is a non-zero torsion point, or it seems too strong. Yeah, maybe I shouldn't say that, because when we put everything in a family, we see exactly what we expect from us. When we put everything in a family, we see more clearly what is expected. But here, I want to explain the idea of what should be proved to get this result. So you don't claim that you think that that's true? No, no, no. I don't claim that. I don't claim that. I just say that if that could be proved, then the conjecture is solved. All right. Now, we assume that this X satisfies some condition which I will call non-degenerate. I'll explain what it means later in my talk, non-degenerate. Then we have two conclusions. First of all is if X is non-degenerate, then we sort of can compare two height functions. The first height function is fiber-wise defined. The second height function is completely on the base. Write it. There exists a number depending only on X, and it's RST open non-empty subset of X such that the following inequality holds if we consider the two height functions. First of all, let me first of all finish this and then explain what I mean. First of all, my small s is just a point on the base curve. So this function, it actually means hf as, but I write it in this way to emphasize the fact that this function is completely on the base. It does not really depend on the fiber. And this point P lies in this RST open dense subset, but it's also on the fiber over s. So this height function is really the fiber-wise defined Nehontate height function I talked about, which is above. So the left-hand side is something depending purely on the fiber. The right-hand side is something depending purely on the base. After all, there's no relation between them. We can't expect to have a bound. But here, what we proved is that if we assume, if we restrict to a sub-right, he's satisfying some property, then these two heights can be bounded. And of course, in order to make this useful, we need to say that what is non-degenerate? There is an original definition, but it's ad hoc. So here, we also prove that we also give a criterion of this non-degeneracy. X is degenerate if and only if it's very rigid from geometry. It's simply the translate of an abelian sub-scheme translated by a torsion section done by, I would say, something constant. This means that the abelian scheme may have a constant part, it's a trivial part, and then the other part of X is just a trivial product up to a finite covering. So in practice, we prove that in practice, we can often check this condition to determine whether X is degenerate or not. And then if it's not, then we can apply this. I'll give you an example. X is a dominant over S. Yes, it's better to assume it, but otherwise these whole things work because this is just one number. Let me assume this. But it is not necessarily absolutely reducible, only irreducible? Absolutely irreducible. Everything is defined over Q bar, so it's absolutely irreducible. No, in the total space, in general fiber it is... No, it's not as well. So when you add torsion... This also may not be irreducible. I say abelian sub-scheme, but of course I mean up to a finite covering with base change, then it's abelian sub-scheme. I mean the finite cover of S. So up to a finite cover of S is... Yes, yes, yes. Okay, up to a finite covering of S. So what happens if A to S is just a product and it would seem that the two sides are independent? Then if X is not a constant, then they are not independent. Sorry? If X is not constant, then they are not independent. Then there's height in what it holds, if and only if this X is not a constant thing. So this result, I priori, also applies to constant abelian scheme. Yes, it does. The example is this is a preprint with three authors. We proved this conjecture that the theorem for one parameter families. Let's have a look. Now suppose we have a one parameter family of curves of genius at least two. First simplicity, I'll assume G is at least three, just for some reasons we will see. Then we consider its relative Jacobian. Well, there's no canonical way to embed C into J. But because actually here, we don't care about the actual heights of those points, but the distance of those points. So the height of the difference of two points. So we only need to care about this subrarity, C minus C in J, which is well-defined because it can be defined as the image of this fiber product to J, where this is fiberwise defined by sending PQ to P minus Q fiberwise. So we don't need to even fix a section for the base of the abel Jacobian belly. Okay, let's call this one our subrarity X. And of course, this will be the A over S. Then the assumption of G at least three, this will imply that X is also a non-associative family. Then this assumption will say that X is non-degenerate, just by computation by hand. X will be non-degenerate by this geometric criterion. Now we can apply this height inequality. Then if we take two points on the same fiber of C over S, it's equal to this C depending only on X, so S, so C, and so on, times the faulting height of the base point. Of course, for PQ satisfying those conditions. But this is an open condition for the rest of points. For the rest of points, we use those packaging arguments. But at least now we see that because this number, it depends only on the family, not each individual fiber. So we can see that now we really have this result. So the height of the distance of any two points, algebraic points, are uniformly bounded below linearly in terms of the faulting height. And this is what we want to prove this. Yes, P is different from Q again, thank you. So this tells you that if the difference is torsion, then you'll get still some consequence. Yes, if it's torsion, then the height of the curve itself is bounded above. And the P and Q are small, no, they are small. Then the curve is small, say. But you don't say that if there is a small point, the curve is, you should only say there is a torsion point. The torsion difference of small point, the curve is small. Yeah, well, if here P minus Q is a torsion, then the curve is small. Yeah, so that it can be negative. But I mean, we can take different normalizations to make it positive. There are different normalizations. Some convenience of normalizing so that this is positive. Actually, yeah, you might talk about things how it will always be positive. But the thing is, actually, we prove this purely in terms of height machine. So it's OK to add a number to this, or 1 to this height function. And this basically proves the theorem for one parameter families up to some details for the packaging argument. OK, now, I hope I convinced you that if we can generalize this height inequality for arbitrary families of a billion varieties, then we can prove the theorem. And then next comes the generalization. And to do that, I will need to explain what is this net degenerate. You will see that it comes from geometry. To do this, because we want to solve this conjecture, we can just work with the universal curve. And this is the modular space. This is the universal curve. It does not have a section, but still, it's a universal curve. Here, I'm taking some level structure. And again, I have the universal Jacobian. And I can take this map to construct a CG minus CG inside J, which is well-defined, which is well-defined. And furthermore, I would like to embed everything in the universal a billion varieties, using the Torelli embedding to embed it into the modular space of a billion varieties, principally polarized modular space of a billion varieties. And here, there's one to curly AG, which is the universal family. So in my talk, I will always work with this system. I have the universal Jacobian, and this is the universal a billion varieties. And this is the map pi. So you fix a level structure? Yes, yes. Because actually, in my talk, everything is for height machine, height inequality, and so on. So taking a finite covering, it's always allowed. It doesn't do it. Yes, yes, yes. So this is the object we will work on. And then, whatever sub-variety we take will be a sub-variety of the universal a billion varieties, but it might not be dominant. Most of the time, it's not dominant to the base. Reducible sub-variety. Again, geometrically reducible sub-variety, defined over a Q bar. And furthermore, we actually have another object in this picture that is the totological line bundle. It exists just because the modular space parameter tries not only a billion varieties, but each a billion variety is equipped with a polarization, a principle polarization. So this is the totological line bundle on the universal a billion variety. We can take the first Shen class of this line bundle. This non-degenerate condition, well, this is not the definition we take, but it's really the geometric meaning behind the definition we will take in a few minutes, is non-degenerate if x is called non-degenerate. If we take this first Shen class, we restrict it to x, and we take its correct wedge power. It is not always zero. It is not always zero. Well, one thing about this form is that this first Shen class is always non-negative. This can be computed in explicit or coordinates, and this is done by, I would say, Mach, 9 in Mach. So this is always non-negative. So saying that this is not always zero means that it's positive somewhere. At one point, that's enough. Yeah, because it's a one-one form. We have not come to the boundary, so it's really a form. Of course, it's hard. The geometric meaning of this will be then, well, if x is non-degenerate, then sort of if we want to integrate this thing over x, we will get a positive number. So that is almost equivalent to say that this lg restricted to x is a big line bundle, except that now nothing is projective. So it's really the geometric meaning, the geometric background behind. But of course, in order to get something, we need to be more precise. But this is our idea. Our description is as follows. We use the so-called Betty map, introduced by Daniel Bertrand, when they studied other kind of different time problems and like the intersection problems. The Betty map is defined as follows. Here we take the universal identity. I want to take its universal covering. Well, we know that we take the Ziegler path space. It uniformizes in the complex category, the modular space. And here on this AG, we want a similar thing. We want a similar thing. And it can be defined as follows, universal covering AG. I'll call this one, I was in very bad notation. I'll temporarily call it yg, y2g. First of all, it's defined as follows. First, as set, or as real algebraic object, it is r to the 2g times the Ziegler path space. And then we want to give it a complex structure. The complex structure encodes the complex structure of every abelian variety over this modular space. So it is actually given in the following way. Here we take any vector, AB, both in RG and a matrix, G by G matrix. We send it to A plus CBZ. And this is, of course, a bijection. And it's also automorphism as real varieties, real algebraic varieties. But the complex structure should be this one. Then this is the universal covering. I'll use this one. Now we have y2g covers AG, which maps to HG plus. This is just a trivial product with some twisted complex structure. All right. Now X is in AG. I'll take an irreducible component, irreducible in the sense of analytic geometry, component of u inverse X. Then this is going to be in this uniformizing space. This is going to be in this uniformizing space. And the Betty map is something very, very simple. It's just y2g when we write it in this way. Actually, it projects the first factor and the Betty map, essentially, is the composition. This is the Betty map. Well, this is something very, very simple, but in close, for example, we can really read off this condition completely using this very simple map. So I'll give the actual definition which we have. X is non-degenerate if and only if, or if and only if. This is definition lemma, maybe. B, restricted to X2, has full rank. The generic rank of this map is twice the dimension of X. So this is because this 1, 1 form, it defines a point-wise, a symplectic pairing. And the kernel of the symplectic pairing are precisely the fibers of this map. This can be computed explicitly. And then saying that this is non-zero precisely means that there should be no direction along X not vanishing along some positive dimension of the fiber. So this is precisely the definition. And also from this new definition, we do know what kind of sub-right is degenerate. So from this new definition, we have that X is degenerate if and only if for any X2 in this X, the big X2 component, it lies in a fiber. So the dimension of B inverse BX intersect X2 is positive. The dimension of that is positive. By the way, the fiber, the map is just real analytic. It's just real analytic. It's not complex. The fibers are complex analytic. And this is used in the verification. Yes, because it's pure. It varies homomorphically. Thank you. And then this says that now we have actually a curve in HG plus, a complex analytic curve such that if we take BX times this curve, this is inside R2G times HG plus, it is contained in X2. It's containing that for each X2. And then it simply means that X2 is covered by such things. So sitting there just a local curve is a... Yeah, but it's complex analytic. You can use analytic extension somehow. Yeah, but it just defines a small neighborhood. Yes, yes, yes. Our priority is only a small neighborhood. Yes, yes. And here, I want to say that now then this hopefully implies that if X is degenerate, then it's the union of such things, but I can also take the diracic... Oh, sorry. So X, now when we pass to the universal abelian variety, because X is algebraic, so I can take his diracic closure. So that now X, if X is degenerate, then it is this ad-hoc union. It is this ad-hoc union. Actually, the other direction also holds by some... a lot simpler reasons, but maybe I... If I have time, I'll explain it, but the other direction is the simple... it also holds. All right. Now we want to study these things now. Okay, now the question is to study these diracic closures. What properties should they have? And this is the next part of my talk, maybe number three, functional transients. I'll start with the weak X-chanu statement. This is actually a sort of statement. It says the following thing. If we have two varieties, two algebraic varieties, and this is more or less... this is subjection. This is usually the... the usualizing map linked by a trans... an analytic map. So both omega and s are supposed to have some algebraic structure, but Q itself is very far from being algebraic. It's highly transcendental, but it's only analytic. Then weak X-chanu says that things here and things here, they should intersect transversally, except if they... except if they actually intersect in a smaller base, meaning that if we have z inside omega irreducible, complex analytic, again irreducible in the sense of complex geometry, then the dimension of z bar inside omega plus the dimension of Qz bar inside s is greater or equal to the dimension of z plus the dimension of this Qz Bzar. Let me explain this term. So we have a bioalgebraic system here, and we say something is bioalgebraic if it is algebraic here and here. That is the Bzar thing. So this is sort of the bioalgebraic closure. weak X-chanu says that this thing, it is in omega, an algebraic thing in omega, and this Qz-zar is an algebraic thing in s, and the dimension of their sum is at least the dimension of the z itself plus the Qz-Bzar thing. And it means that these two things, if there is a proper bioalgebraic thing containing both, then they don't intersect transversally. Otherwise, they always intersect transversally. So this says that if we have some sort of unlike intersection, then there must be a reason. And this reason, we can write it down. This is the result we're going to use. This kind of result was first approved by ARCS, of course, to study the shadow. It can be a functional analog of x-chanu's conjecture, but it got interest again by the study of the Andriot conjecture. And it's another, it's a weaker form of a weaker form, Ax-Linderman, was proven by Klinger-Rilmore and the FIF. This is for Shimura variety. This is for Andriot. But this general form recently is proven by Mark-Pilan Zimmermann for the modular space. And this paper is published. It has two extensions, two directions of the extensions. First direction is by Baker and Zimmermann for variation of pure hard structures. This is also published, and this is used in the recent proof of Lawrence-Tanban-Kentash proof of model conjecture. Actually, it's used for the high-dimensional, high-dimensional case. Another extension is done by me for this universal Abelian variety. It's a mixed Shimura variety. It's not pure anymore, but this is a preprint. And in studying this question, studying this y, it is this form of use. Now to study that y over there, we see what condition it satisfies. We apply the weak Ax-chanu to... Here, z will be r times C-tutor. It is also complex analytic, as we said here. And of course, to this system, to the system y2g2AgU. Well, then first of all, the first term is dimension z-bar, which is the dimension of r times C-tutor zar. This is the first term. The second term is precisely y. And the third term is 1. Well, because it's a curve. The third term is 1. And the last term is y-bizarre. Now, in this inequality, actually, we know many things. This is what we want to know, so we don't need to do any operation on it. This is a number. And this is a biogereg thing. Well, in biogereg geometry, usually these kind of things have very good geometric interpretation, so we don't need to do much about it. So the only thing we need to study is the first term. Well, a first step to do is that this is our C-closure. Although this is not trivial, but it's not very hard, actually certain things commute. We can first of all do the C-closure of C-tutor completely in Hg, and then do this product. They are actually the same. We can show that. Then the equality becomes something purely in the base plus dimension of y is greater or equal to 1 plus dimension y C-bar. Okay, now let's study this. Well, by the choice of C, we know that y is r times C-tutor and take the zar C-closure. So of course, this thing is smaller or equal to dimension of pi y bizarre. This is smaller or equal to pi y bizarre. So now we have the equality being dimension of y is greater than, I removed this one, but then replace this by strictly larger than dimension y bizarre minus dimension pi y bizarre. Okay, this is actually the eventual form we want because both things have very good geometric interpretation. I can also tell you what is this difference. This difference has an even better geometric interpretation. Say here we have our universal Abellion variety over the modular space and this is our pi y bizarre. The y bizarre is something over that and it is part of it. This part, it is precisely up to a finite coloring. An Abellion sub-scheme translated by a torsion section then by a constant section or isoconstant section. So in some sense it is the smallest Abellion sub-scheme containing this y. It is something like that. So now y is something here. Say this is y and if we restrict the orange part to the yellow part then y will be contained in no smaller Abellion sub-scheme in this whole thing. Then dimension y being larger than this difference means that the dimension of y is larger than the relative dimension of this Abellion scheme and if we look at the definition of the Betty map, of course the rank of the Betty map cannot exceed twice the dimension of the original variety. So that's why as long as y satisfies this then of course y is degenerate and if x is the union of such things then x must be degenerate. This is why I said that the other direction is rather easy to prove. You say it's in a billion sub-scheme and sub-scheme over what? Over a sub-variety of A.G.? Yes, over its base. And this glass section? The torsion section over its base and the constant section over its base. Now I'm restricted everything in the base. The base is fixed. It's translated by torsion section of this restriction and then by constant section of course when I talk about this I always mean up to a finite covering. So the torsion section is actually constant after the finite cover? Yes, it is. I just want to separate torsion and constant but otherwise I can say just constant. It's fine, it's fine. But because in studying this mixture or whatever this stuff, torsion and constant, they are really very different things so I like to separate them. One of them is special, the other one is wicked special and so on. So, okay. Now using the mixed-axe channel we can show that X is really the union of something and still an ad hoc union but each member of this union have a better interpretation. It satisfies some condition which is more or less standard in electrical intersection theory. And then next step is really do electrical intersection theory. So next step do electrical intersection theory. Again, this is basically says that the maximum Y is satisfying this condition they come from finally many of families. This actually goes back to even Bogomolov for a billion varieties and then Emmanuel generalizes to Shimura varieties for weekly special sub varieties and on the other hand when studying those electrical intersection Zuberpin conjecture Habegger and Pila introduced this weekly optimal thing and Dao and Ren proved it for Shimura varieties I proved it for mixed Shimura varieties so we can show that those maximum Ys come from finally many families. So eventually this ad hoc union becomes a finite union so that for any X this union will be ZRC closed and also we can prove the criteria in one this ZRC close-up variety is the whole thing doing this electrical intersection stuff I don't think I have time for it but just back to our counting problem back to Meisler's question of course as I said we want CG minus CG to be non-degenerate unfortunately it's not true for stupid reasons because the dimension of this is too large compared to G so this is not the correct X we take the correct X we take there are two ways I'll take the easy way it's just a fiber product with piece I think here I can write 3 but I really should write 3G minus 2 okay for some reasons for some induction reasons and of course here assuming that G is at least 3 if we take this power to be 3G minus 2 then X is non-degenerate so that we can prove this height inequality it says that okay we cannot prove that each pairs of different algebraic points are far from each other but we can show that around each algebraic point are at most 3G minus 2 points which are not far from it and for the purpose of counting that suffices so for each algebraic point at most 3G minus 2 points close to it close means that the distance is smaller than the linear function on the Fulton's height then for the purpose of counting we are done just multiply everything by this number which still is explicit on G I think I will stop here thank you well thank you are there any questions could you repeat why you have to rule out the case G equals 2 oh actually it's just because when G equals 2 this will be the whole thing whatever power you take it will still be degenerate the way we yeah yeah yeah the way we solve this problem actually this is not the actual thing we take we take I know this the map we take this is K so this is K we take is P1 minus P0 the map we take is actually this one for this as long as K is at least 2 or 3 then G equals 2 will also be solved at the same time for one parameter families it's almost equivalent to take this one and this one doesn't matter but actually when we work on this even for G at least 3 it's simpler to take this one just for the proof with Joe Aris and Luceca Forazzo he has been even more optimistic about rational point optimistic I don't know for that thing I really don't know because it's a rank stuff I can I can do nothing with the ranking but I have a question about it so you do it for finite rank subgroups of J, Q bar and of course also J torsion what about a sum of finite rank and J torsion I think it's a finite rank right what it's a finite finite rank Q rank oh no no I don't mean that I mean the visibility of if I allow myself to divide finite rank subgroup by integers and I get a oh that's actually what I mean by finite rank it's Q yeah yeah yeah it's Q rank yeah yeah yeah finally generated I mean that it's a Z module but finite rank is really over Q yeah yeah yeah because in the proof actually for the large points they already solved this problem and for the heightened quality we don't see we actually don't see this gamma thank you