 Thank you very much for the invitation to the organizers. It's a very nice, like a new format of a seminar. And I'm happy to give a talk. So I'll talk about a point counting for foliations. So the plan of my talk is I'll start by talking about like a tamed topology, tamed geometry, and point counting. So kind of the subject that started with the theorem of Pilla and Wilkie. And I've tried to show how this kind of interplay between tamed topology and the finite geometry has affected some kind of classical problems in arithmetic geometry. And then what are some limitations of this approach? And then I can introduce foliations as a kind of way to maybe try to overcome some of these limitations. OK, so that's the plan. Let me start by talking a little bit about tamed topology. So Gluten Dick in 84, he kind of had a philosophical plan, as Gluten Dick often does, of trying to formalize a type of topology that would be more suitable for the needs of geometry. So I mean, in this document where he outlined this plan, he wrote that after some consideration with hindsight, he feels that general topology was developed by analysts. And it's really kind of geared toward solving some needs of analysis. And it allows things like space-filling curves and some kind of pathologies that you don't really want to deal with when you are dealing with topology or with geometry. And he wanted to kind of axiomatize some kind of topology which would be more suitable for natural problems in geometry. So Ominimality is one kind of a candidate for realizing this goal. I mean, Gluten Dick had many, I mean, his program was very wide. It included things like actions of the absolute Galois group. So it involves also kind of the more arithmetic side. But just thinking about this topological side, Ominimality is a pretty good candidate. So I will not really define Ominimality explicitly in this talk. It's a branch of model theory. And I will not really need the formal definitions. But I'll try to give you a rough idea of what it is. So in Ominimality, you define structures which are kind of collections of sets. And these sets are supposed to have some nice geometric properties. They shouldn't allow things like space filling curves. So one of the nice things, nice properties that these sets have is that they always allow stratification. So dividing a set into smooth subsets, and it allows triangulation. So you can do kind of classical algebraic topology without worrying about pathologies. Now, more recently, it was discovered that Ominimal structures, if you take sufficiently large Ominimal structures, they contain a lot of the natural spaces that come up in algebraic geometry. So for instance, they contain the modularized spaces of algebraic curves, or algebraic varieties, or zeal varieties. And more recently, I mean, OK, so for zeal varieties, it was shown by Petals in the starting quote. And more recently, this was even extended to more general kinds of period maps, for general pure variations of hot structures. So this is work of Bacchier, Wuneberg, Klingler, and Zimmermann. And I will have no time to go into any of these, but I just want to say that this theory of Ominimality really contains a lot of the spaces that we would like to understand when we are studying arithmetic geometry. And finally, there's also in the last 10 years or 15 years, there's been a deep, we discovered a deep connection between tamed topology and arithmetic geometry. So this is based on a theorem of Pila and Wilkie, and then later works by Pila and many other people. So I'll try to show what is this connection and how it affects arithmetic geometry, and then go to what we can hope for next and what are the limitations. So the main motive of my talk is that to paraphrase Gotendic, Ominimality it was developed by real geometers, people like Lauwandendries, and they were thinking about real geometry. They were not really thinking about arithmetic geometry. And this connection to Ominimality, there are some things that we can do with it, but Ominimality is not really completely suitable for doing arithmetic geometry. So there are some limitations to Ominimality, and I will try to kind of show what these limitations are and where we can try to push them. Okay, so let me start with how the story started. Yeah, so what is the connection between tamed geometry and the funded geometry? So it started with the theorem of Pila and Wilkie. So for this theorem, we take a subset of Rn and suppose that it is a nice tamed subset. So it's definable in an Ominimal structure. And I denote by XGH, the set of points that have degree G at most G over Q, so Galois degree at most G, and the height at most H. Okay, so for the height here, you can take, for instance, the normalized V height, but if you're not familiar with this, you can just think about G is equal to one, and then this height will be just the classical height and the maximum of the numerator and denominators. And this is how the theorem is usually formulated first for just G is equal to one. And then the theorem states that if you have a definable set, then for any positive epsilon, the number of points of degree G and height H is smaller than some constant times H to the epsilon. Now I should make some qualifications here. I'm not counting all of the points in the set, A. I'm counting only in the transcendental part. Okay, so that means, first of all, the algebraic part is the union of all the semi-algebraic connected curves in A. Transcendental part is everything else. Now, when you have algebraic curves, this kind of asymptotic will not work. I mean, definitely if you have an algebraic curve, then it can contain many algebraic points of a given height. If you think about X is equal to Y, whenever X is rational, Y is rational of the same height, then you will have like, I don't know, maybe H square rational point of height H. So if you are to prove some kind of bound like this, where you're showing that there are very few rational points, you must throw out the algebraic part. And what the theorem of free language shows is that this is actually all you have to do. Once you throw out any semi-algebraic curves from your set, then you will be left with a set that contains a sub-polynomial number of rational points, sub-polynomial in the height. Okay, so this is the theorem and kind of philosophically it shows that even though all minimally tamed sets, they are only geometrically tamed. So I mean, people were not thinking about arithmetic geometry when they defined this, but still in some sense they are tamed from the point of view of their finite geometry. So there is some kind of control over the number of rational points. So just from the fact that the set is geometrically tamed, we can say something about the finite properties. Okay, so this is the theorem and I want to go immediately to one example that shows how this impacted arithmetic geometry. So there are many examples I'll show maybe the most famous one, which is for the under-old conjecture. So let me start by just introducing this under-old conjecture. So I introduce it only for the case of modular curves, which is the original context in which this was considered. So H will be the upper half space and this curly F will be the fundamental domain. And then the quotient, when you quotient out H by SL2Z, this thing is called the modular curve. So it's the modular space of elliptic curves. Every point in this space represents a lattice. And there is the quotient map going from H to Y1, which is called the modular invariant map. So this is the uniformizing map of this modular curve. Uniformizes the modular curve by H. Now in the modular curve, we have some points that are more special than the generic points. So every point in the modular curve represents an elliptic curve. And we call this elliptic curve special or in this context also CM. If it has endomorphism, if the elliptic curve has endomorphisms other than just multiplication by an integer. So if there are such endomorphisms, it must be just multiplication by some complex number. And then the elliptic curve is called CM. This doesn't always happen. So generically, you will have only multiplication by an integer and sometimes you have some other multiplications. Now in terms of the uniformization, we can understand when the point is special. It's going to be special if and only if a pre-image tau is quadratic, which will be imaginary quadratic because the image is in the upper half space. This is easy to see. Okay, so this is what special points means, both downstairs and upstairs. And the curves can also be special. Some curves are also more special than others. So I don't want to go into the details of how this is defined, but we say that a curve, especially if it is given by one of the modular polynomials, that basically means that the pair of P and Q that are satisfied, that sit on this curve, they are pairs of points that are isogenous. So there is an isogenous, there is some non-trivial group homomorphism between the elliptic curve represented by P and the elliptic curve represented by Q. So in particular, it means that when you have such a special curve, whenever P is special, Q will also be special. Okay, so these are the special curves. If you've never seen the definition, then just consider that there is a collection of modular polynomials with this property that I mentioned now. And now what is the underhold conjecture? So the conjecture states that if V is not special, then it contains only finitely many special pairs. So if it is special, then as I said, whenever P is special, Q is also special, there will be infinitely many of them. But if V is not special, then there isn't an obvious reason why we should expect to have many of these special pairs. And then the claim is that actually, there is only finitely many special pairs. Okay, so this is the case for like a Y1 square of the underhold conjecture. And this is actually was a theorem. This is a theorem that under-approved and then he conjectured some generalizations for Y1 to the n or for more generalization, more varieties. But even though this is a theorem of underhold, let me show another proof, a proof of Jonathan Pila. And this proof actually extends to Cn, the general case and beyond. So, and it is kind of the, this proof will show you how the Pila-Wilke theorem comes into this picture. Okay, so what is the rough idea? So first I'll denote by high the uniformizing map of the modular curve squared. Okay, so we have Y1 squared. I'll just take two copies of A of the fundamental domain and two copies of Y1. And I just coordinate wise, uniformize them. And X will be the pre-image of V. Okay, so remember I have a curve V. I want to study the special pairs on this curve. So I'm going to lift it. Now special pairs are going to correspond to pairs of quadratic numbers upstairs because special points correspond to quadratic numbers. And now the first step is to show that this X contains no semi-algebraic curves. So in fact, this case is not so hard because it's just one dimensional. But I mean, when you do it in arbitrary dimension, this is a really non-trivial step to understand all the algebraic subsets of this X. And this is a big part of Pila's proof, but I would not go in this direction. But I just wanted to mention that kind of this deep step also involves using the Pila-Wilke theorem and has also been extended in many directions, but I just will not have time to discuss it. So let's just assume this for now. So there are no semi-algebraic curves in X. So it means that the Pila-Wilke theorem really counts in X. I mean, we don't have to worry about this transcendental part of X. Okay, so what the Pila-Wilke theorem tells us is if you count points of degree four, I'm taking four here because I want pairs of quadratic numbers, right? I have in this F square, I will have a special pairs, which is two quadratic numbers. So together they will generate a field of degree four. So I count things of degree four and height H. And the Pila-Wilke theorem tells me it's smaller than some constant times H to the epsilon or any epsilon. Okay, so that just says that there are not too many special pairs. It doesn't say that there are finitely many, like the conjecture said. So there has to be some extra ingredients here. So the extra ingredient is as follows. Suppose that you have a special pair PQ and write it as the image of some pair tau one, tau two. So we have PQ, which is the image of tau one, tau two in V and it is special. And denote by H the height of this pair. So well, this height actually turns out to be the same as the discriminant of this dendomorphism ring. I don't want to get into these details, but basically what you can get out of this comparison using class field theory is that the number of Galois conjugates of PQ behaves like some power of H, some positive power of H. So if you have such a point with a big height, it must have a lot of Galois conjugates. And moreover, each of these Galois conjugates are also going to be CM points of the same height. Okay, because this definition of being special, it's a Galois invariant. It just says that you have some endomorphism. So if you apply a Galois conjugation, it will still have endomorphism. So we see that there are many conjugates and they each correspond to a point in this set X for H. And then we just have too many. We have H to the C, but we are supposed to have at most H to the epsilon. So if epsilon is less than C, this will be a contradiction as soon as H is large enough. Okay, so from this, we deduce that actually H should be smaller than some constant. And okay, that means that we have, if you have a bounded height, then you have in particular finitely many such points. Okay, so that's how the proof works for this case. And I want to say something about, okay, I will not mention all the similar applications of this approach, but this approach works for many different kinds of problems, some of which I will mention in my talk. And the main thing that you need in order to make it work is you need to get these lower bounds on the Galois orbits. Just like, for instance, here, I use this important fact that the special points has H to the some positive power Galois conjugate. So this is a Galois lower bound. Lower bounds the size of the Galois orbit in terms of some height. And this is what you need in general. So I just want to mention a couple of examples of such Galois orbit lower bound. So the first case result of Sino-David and also there are other groups by David Master which says that if you have an a billion variety, let's say defined over some number of fields and the point which is an end torsion point, I mean exactly end torsion. So N is the smallest power such that you get the identity. Then the number of Galois conjugates of P grows like some constant times a positive power of this N, of the order of torsion. So again, this is a Galois orbit lower bound. It tells us that if we have a high order torsion, then it means there are many Galois conjugates. So there are some applications in this pillar, this pillar with the spirit of this lower bound. Another type of lower bound generalizes what I, the one that I showed in the previous slide. So if you have P in the modular space of principally polarized the billion varieties. So you have a principally polarized the billion variety and suppose it has a CM. So this kind of generalizes the CM elliptic curves that I was talking about before. Then Timerman proved that the number of Galois conjugates will again be bounded by some constant, bound from below by some constant times power of the discriminant of this ring of endomorphisms of the CM a billion variety. So this is the kind of thing that you need if you hope to use this strategy that you used in the case of under for CM. So Timerman, he proved this result as a consequence, he proved the underword conjecture for AG. So this was kind of the main thing that you had to do the main missing ingredient at that point. Okay, so the point that I want to make is that when you prove these kinds of low orbit lower bounds, they are usually, I mean, in both of these cases, they rely on what are called transcendence methods. So, and these methods are quite similar to the proof of the Pilla-Wilke theorem, but they usually work in a much more rigid context. So, for instance, for this first time or equation three here, you would be using transcendence methods and you would be applying them to a billion functions. Okay, so you have the graph of an a billion function, it's a very nice function, it satisfies functional equations and so on. And then you use transcendence methods and transcendence methods and Pilla-Wilke, what they have in common is that they use auxiliary polynomials to kind of, to find algebraic points on transcendental sets. So this is a very old method that goes back to many Ziegels, Gelfon, many people in transcendence theory and this is also the way that the Pilla-Wilke theorem is proved. So the difference is that when we are doing Pilla-Wilke, we have these general tame sets, definable sets in all minima structures and we cannot say much about how they interact with auxiliary polynomials. So when you're trying to prove something like this equation three or equation four, you are dealing with a much more specialized set and you have much stronger estimates. So you can count how many times a given polynomial vanishes on a given set or you can kind of get your hands on much more finer analysis of these sets than you can for general or minima structures. So that is why it seems that we cannot use like the general or minimal method to prove these kinds of results. We have to first prove them and then we can combine them with the Pilla-Wilke theorem to prove some new things. Okay, so this is basically the, I think the kind of a natural limit of what you can hope to achieve using like purely all minimal methods. And now let me talk a little bit about some conjectures about trying to push these limits. So, well, first of all, this asymptotic, the H-deception asymptotic in Pilla-Wilke, it is optimal in some precise sense that I don't want to formulate, but I mean, basically you cannot really improve upon this asymptotic. So there are examples involving lacunary series where really it contains like nearly H to the epsilon points, rational points of height H. However, there are conjectures. So the main conjecture is by Wilke from this original paper where they prove the Pilla-Wilke theorem. So there are conjectures stating that if your set is definable in some kind of more rigid type of structure, then you should have much stronger estimates. So Wilke conjectured it for the structure R sub X. That means that you start with just algebraic sets and you add exponentiation. So you take any set that you can define with logical formulas involving just arithmetic operations plus just the E to the X. And he conjectured if you have such a thing, then actually you'll get a much stronger bound. So the number of points of the degree G and height H should be polynomial in G and log H. So instead of H to the epsilon, we get log H. I should say that Wilke conjectured this for G is equal to one. And later maybe Pilla conjectured for any constant G. And, but by now we understand that probably it is natural to even want something which is polynomial in G. So, I mean, I still think it's reasonable to attribute this to Alex Wilke, to want this thing to be polynomial in both G and log H. And so this is an open conjecture. There is one structure where it is known, this structure of restricted elementary functions. So that is very similar to R sub X, except instead of adding just the full exponential, we are adding let's say all elementary functions. So the sine function and the exponential function, but we restrict them to compact domains. So there is kind of a technical difference here. I don't want to go into too much details, because it's not really the main direction of my talk, but I just wanted to mention that this conjecture is known for some slightly more restrictive settings. But this result is, I mean, it's not enough for the really interesting applications in the finite geometry for reasons that I will explain. Okay, so why is this connected to the previous slide on the limits of the hominimality? The reason it is connected is an observation of Harry Schmidt that he made a few years ago. And he basically noticed that if this conjecture holds, not just for R X, but for some large enough structure, then Galois orbit lower bounds would follow for a billion varieties. So the first one I showed them for Shimura varieties. The second one I showed them maybe in other contexts. So I mean, this kind of bound, which is polynomial in gene and log H, this is already strong enough to actually get these Galois orbit lower bounds and get like a purely tamed topology proofs where we don't have to rely on transcendence method. So let me just show kind of why this observation is true. In the very simple example of a billion varieties. So suppose that you have an a billion variety A and you have P, which is an end torsion point of degree G. So we want to show that G has to be big. G has to be like some power of N. That's a degree should be big. So denote by gamma the graph of the universal cover. So this is restricted to the fundamental domain. So we have the universal cover of the a billion variety from the fundamental domain to the a billion variety. Now the observation is that on this graph, we get a lot of algebraic points from our point P. So we have the original point P and we have all of its multiples. There will be N of them and multiples. And they are going to be of degree G because the group law is defined over Q or let's say over some number, doesn't matter. So you get a lot of points of degree G. Now on the other hand, downstairs, all of these points have height one or O of one because they're all torsion points or they have just a zero canonical height, no data height and then their way height is going to be just bounded by some constant. And upstairs the height is N. Now here I should specify I'm kind of thinking upstairs in the period coordinates. So I have an a billion variety. There are upstairs there's like lattice corresponding to the billion variety. And I'm going to represent every point upstairs as a combination of the lattice generators. Okay, if you do it like that, then the torsion point is exactly a combination of the lattice generators and with rational coefficients and the order of torsion is going to be exactly the common denominator. So it's going to be N for an N torsion point, at most N for an N torsion point. So now suppose that you have this estimate five that would mean that N, which is the number of points on the graph is smaller than like the, so I'm counting all of the points of N, degree G and height N. So in particular, this contains is N point, but this on the other hand should be polynomial in G and log N, okay. And then it's very easy to see that you can, I mean, this log N is much smaller than N, you can forget about it and you get that N is polynomial in G. So that is a Galois orbit lower bound. If G is smaller than N also has to be smaller. And this works not only for this context, it works also for other context. I will show one more example later. So that is why you would want to prove such things. Okay, so let me go ahead. So in what context am I hoping to prove such things? I'm hoping to prove it in some context that would include all of the, you know, kinds of things that you need in arithmetic geometry. So for instance, the previous result that I mentioned for restricted elementary functions, it doesn't include all of these functions that you need in arithmetic geometry. It does include, for example, elliptic functions and the billion functions, but it does not include modular functions or period maps, those kinds of things. So the goal is to try to find some big structure that contains all of these interesting sets and where we can prove these sharp bounds, like in the witty conjecture. So the context I will work with is the context of foliations defined over number fields. Okay, so I'm fixing some ambient variety in. You can think of it as just an affine space, for example, if you want, it doesn't really matter. And I'm fixing a collection of commuting vector fields. And everything should be defined over some number of fields. Okay, so this collection of commuting vector fields, you can look at their common, like the foliation that they define and this will divide your sub-righty M into a union of n-dimensional sub-manifolds called Leafs, the foliation. Now given a variety, sub-righty in M of co-dimension N, so I have an n-dimensional foliation. So at every point in the space, I have some n-dimensional leaf. And now I also take another algebraic sub-righty that has co-dimension N. So generically you expect that the intersect in isolated points, whenever you have a point in V, there is also a leaf passing through this point and the dimensions are complementary. So they're supposed to intersect just in an isolated point. So I denote by sigma V, the set where this doesn't happen. So where the intersection actually is not proper. There's a whole curve of intersection. So this is what you can think of as an unlikely intersection. So I mean, this is something, some kind of intersection that generically you do not expect to happen at all. And so now let me just fix that kind of some notations so I can state results. So I'll fix a ball in one of these Leafs. And assume for simplicity just to normalize my constants that this is contained in the unit ball in this space M. So it's not so important that it's a ball. Basically I need to fix some compact subset of the Leaf because these Leafs in general, they are transcendental and they can be very wired. So I need to fix some nice subsets. So I just take balls. And I denote by delta V, the maximum of the degree of V and the log height. Well, if you don't know this notion of log height for varieties, then just think that you write V as in terms of equations. And then I take the maximum height of any of the coefficients of the equations, the maximum log height. So this will be the same for my purposes. Formally I use what is called the Chao height. But really it doesn't matter. And I'll also denote similarly by delta C, the maximum of the degrees of these C's and their log heights. Now what I would like to do ideally is to prove Wilkie's conjecture for structures generated by these kinds of sets. So it would take any falliation and any ball and kind of generate a structure using this kind of... So this B is actually a subset of the Leaf. I'm basically thinking that I'm generating structure by taking Leafs of falliations, except that I restrict them to some compact domain. And we would like to prove Wilkie for that kind of thing. That would be very useful for reasons that I would show soon. And there are some kind of ideas for how to do this kind of thing using Nevenlina theory, but we have some technical limitations. So we cannot really prove Wilkie's conjecture for such structures at the moment, but let me show you what we can prove. So the first theorem is about counting intersections. So I'm interested in counting intersections between the Leaf or this compact piece of the Leaf and V. And what I'm saying is that this number is bounded by a polynomial in this delta of V. So that's degree and log high delta of C. And then I get this pesky extra term, the log distance to this unlikely intersection locals. So if you are in the unlikely intersection locals, then you can't expect to count because that really means that if you are meeting this Sigma V, that means that there is a whole curve of intersection between your Leaf and V. So there's like infinitely many intersections that there's no way that you can give a bound. So what we are saying is that if you are not really in this unlikely intersection locals, then the number of zeros grows like logarithmic distance to this bed set. And this is the kind of thing that you somehow, you expect to see in value distribution theory in Nevenlina, this kind of log of distance or log of growth often controls the number of zeros. So I mean, I believe that this should be true even without the log distance. That would be my conjecture, but we don't know how to prove it. However, I will say that when we apply this theorem, we often apply it to not some arbitrary filiations. We apply it to filiations that are actually flat structures of some principle G band. So that means that there is an action of some reductive group G. And this action basically takes, it permutes the Leafs. So all the Leafs are actually the same up to an action of G. And for this reason, usually we have quite a good control over this log distance term. So because all the Leafs are kind of algebraically related, if you understand what happens on one Leaf, you can also understand what happens on all the nearby Leafs, because they're just kind of algebraic images of the original Leafs. And basically in every application that I have tried, you can get away with this. I mean, you can somehow work out this log distance term and bound it. But I mean, sometimes it's quite tricky. So it would be really much nicer if we could get the result without this term. Now, this is about zero counting, but once you can prove something like this, this is kind of like a bazoo theorem, then you can hope to also do point counting of rational points. And what I get in this case, so I'm stating not the most general version, but suppose for example that the intersection between, okay, so now I'm thinking any V. This is not V of co-dimension N, I'm sorry. So now you take any V and intersect it with any L. So this will be, it doesn't have to be pointed. It can be a set of any dimension. If you want, you can even forget about VN. Just think that V is the trivial set, and then you're just looking at L. But suppose that this thing contains no semi-algebraic point for any Leaf. And not just for a fixed Leaf that I'm interested in, but suppose that there are no semi-algebraic curves for any of the Leafs. Then I'm proving the WELD conjecture. So I'm saying that the number of points of degree G and height H is polynomial in G log H. And in fact, it's also polynomial in the degree and log height of V and the degree of log height of the foliation. Okay, so I mean this assumption that I have to make for any Leaf, it exactly corresponds to this problem with the log distance term. But I mean, if some Leafs do have semi-algebraic curves, then this will correspond somewhere in the proof to the fact that you get unlikely intersections. And then you have to control the distance to these unlikely intersections. So well, in fact, I stated a simple version here. I have a more general version where I do allow to have semi-algebraic parts and I need to measure the distance to them. And I also allow to count, not just in sets like this, but also in their images under algebraic maps. So the general statement is much more general, but it would be too complicated to present. So I think this gives kind of the idea of what you can hope to achieve once you have this kind of bound on the number of intersections. Okay, so now I want to show some examples of how you can apply this result to kind of to arithmetic geometry, to the places where you usually would apply the pillow will defer. So I'll start with underhold for modular curves again. But the main point is that the modular Jane variant is uniformizing map. It can be realized as a leaf of affiliation. So it satisfies some differential equations. There are several ways of writing them. So here I write it in terms of the Schwarzian derivative, but it doesn't really matter what the precise equation is. So there is some differential equation whose solutions are the graphs of the J invariant. So that means that we can realize the graph of this square of the J function from H square to C square, we can realize it as the leaf of some affiliation. And now another important observation is that we can replace like in the original proof by pillar that I showed you, we had to count in this fundamental domain F. And this is not a compact set. So we cannot cover it by balls. However, using some, for instance, some Dukes acquisition distribution result or using some height bounds, it is possible in pillars proof to restrict to counting only in some ball. So when we are trying to, I mean, when we apply the pillow will defer and we can assume that we are counting only in some sufficiently big ball. And in that case, we can apply the pillar instead of applying pillow will you, we can apply our results. So we take the leaf L. So the leaf L gives the graph of the J function. We intersect with this equation that defines V. And then this intersection actually gives us this, if you look at the coordinates in H, they will be exactly the coordinates that we want to count. And you can use the previous theorem that I described to count with this Sharper asymptotics. So what we gain by doing this, for instance, in this case is we get polynomial dependence on the degree. Okay, so in general, if you apply the pillow will defer and there is no control on the, how it grows with the degree of PR with anything. I mean, it's just the result about general or minimum structures and you have no control on the constant. But here we know that everything depends polynomial on this delta V. So the degree and the log height of P. So if you kind of trace through the proof, what this means, this would actually mean that you have a polynomial upper bound for the discriminants. Polynomial, I mean polynomial in the degree of P and the log height. I forgot log height here. So if you are given such a polynomial, let's say you are given some pitch representation of it. So kind of the size of your input would be really the degree and the log height. The degree is the number of coefficients and the log height is the number of bits that you need to represent each coefficient. Then the theorem would say that there is a polynomial time algorithm to find all of the special points because we have some explicit upper bound on all of the discriminants. So this is something that you couldn't achieve with the original pillow will defer and because it was, you didn't know how it depends on P. Now I should mention that we don't know what the algorithm is. So I'm just saying that there is a polynomial time algorithm, but we don't know what the algorithm is. And the reason is that the lower bound, so here we make the upper bound fully effective, but the lower bound on the Galois orbit, it involves one universal constants. This constant corresponding to this Z-Gal zeros. And unless you assume that there are no Z-Gal zeros, then you get some constant that you don't know anything about. So this constant has to be kind of hardwired into this algorithm. And that is why the result is not effective, but it does say that the unknown conjecture is decidable in polynomial time. And this extends to CN and probably also to any case where we know the unknown conjecture to AG. Okay, but I want to discuss some other directions. So another direction is to try to use this Schmidt idea, the strategy for proving Galois orbit lower bounds for CM a billion varieties or for general Shimura varieties. So let me talk about the setup for general Shimura varieties. So I have a general Shimura variety and I did not buy pie, the universal cover of the Shimura variety. Then there will be associated to any Shimura variety, some principal G bundle over the Shimura variety. And this has a flat structure and some leaf. Okay, so as I said, this is always kind of what you get. So there is a foliation over this total space P. This foliation is a G equivalent. And there's also a map from P to X hat times S. This is the compact dual. So this is the algebraic closure of the symmetric space X. Nevermind so much. If you don't know the details of Shimura varieties, I'll just show basically how the idea works. So there is such a map fee where the image of the leaf is exactly the graph of the universal cover. Okay, so that means that if we want to count points on this graph of the universal cover, that is the idea that we always want to, that is what we want to do with this Schmidt strategy. We want to count the point on the graph because the graph is the image of a leaf of some foliation, then we can use this previous foliation counting result. So if you apply this and you use this idea of Schmidt, then what you get is that if you have good height bounds, so suppose that we know the height of a CM point P is bounded by the discriminant to epsilon for any epsilon. So suppose that you have sufficiently good height bounds. This is similar to how I said before that when you have torsion points in a billion varieties, the height is always O of one because it is comparable to the canonical height, which is zero. So here we need something like this in order to be able to use this strategy. And what we need is to know that the height is discriminant negligible. And if the height is discriminant negligible, we get a lower bound. By just following exactly the same strategy we get that the lower degree will be bigger than some power of the discriminant. So now this kind of height bound is something that we expect. And the reason we expect is that for AG, we know that it holds. It holds by very, very deep work of Yuan and Zhang and concurrently it was done by Andrei Hatta, Gore and Howard and Madapusi Perra. They proved the average Hermes formula which quite directly implies this height bound. I'd say it's a formula for this height. So in particular, from the formula you can get this height bound. So for AG it holds and Zimmerman used the fact this holds to prove his Galois orbit lower bound. So what we do is we get kind of a new proof of Zimmerman's theorem where the previous proof it was kind of really restricted to AG. So it was using isogenic estimates. It was using the fact that these P's they represent a billion varieties to get the Galois orbit lower bound. And we are saying that if you have this height bound we can get the lower bound for any Shimura variety. So it's kind of one half of the problem is resolved. I mean, the other half is to get these height bounds. And for this, we still need to use average Hermes. So we cannot still prove this Galois orbit lower bound for arbitrary Shimura varieties. But we can prove that the height bound implies the Galois bound for any Shimura variety. So in particular, now if you want to prove under ought you just need to get this height bound. And we have been trying to prove this, but this is apparently a much more difficult problem. Okay, but that is what you can get in this direction of Galois orbit lower bounds using this. Okay, I want to show one other kind of application which was started by Massive and Zanier. So here you consider a family of a billion varieties. So this A is like the total space of a family. Every fiber is an a billion. So I just talk about a billion surfaces. So A is an a billion surface over C lambda. And for every value of lambda, you get an a billion surface. And I'll take C in the total space of this family. It's so irreducible curve. Okay, so think that for every lambda, you get a few points in C, in A lambda. So lambda moves over each lambda. We have the A lambda, which is an a billion surface and a few points in A lambda. And everything is defined over a number of feet. So Massive and Zanier in a paper they considered, or in several papers, they considered the problem of, for which values of lambda do you get torsion points in the fiber? And what they proved is that if C is not contained in a proper subgroup of A, then it contains finitely many torsion points. Okay, so it is an unlikely intersection to get torsion points. You don't expect to get too many torsion points when you have this kind of curve because the torsion points, they form like a dimension zero set in your a billion surface. And here you impose two conditions. So you take a curve inside the total space of A, which is three dimensional. So we expect that this will happen in a zero dimensional set. So in fact, it would happen in a finite set unless there are some good reasons. I mean, if you belong to a subgroup, then there are reasons, geometric reasons why we expect to get infinitely many torsion points. But I mean, if you don't have this geometric reason, then there will be finitely many torsion points. And this was later extended by in many directions by Massive, Zanier, Barroero, Capuano, Mike Schmidt, and probably others. And there are many interesting applications of this result and these kinds of results. So I will show one nice application to the Pell equation. There's also a very deep recent application by Massive and Zanier for elemental integration. So the problem of integrating algebraic forms in elementary terms. And I also learned just yesterday that there is a nice application to billiards. So this was actually in a talk of Bert and Zanier in this seminar a few weeks or months ago. I saw it yesterday in the list and I was very interested. Anyway, but I will focus on only the Pell application just to show what you can get. So if you now use count, foliation counting for this where Massive and Zanier use Pella-Wilke, what you can prove is that not only you get finitely many torsion points, but the order of torsion is bounded by a polynomial in the degree and log height of C and the degree of the field of definition. So this is kind of the natural asymptotic that you would expect. So polynomial in the degrees and the log heights. This looks more like the kind of bounds that you get in the Funtime geometry. In particular, it implies that you can compute all of these torsion points in polynomial time. Because once you know the order of torsion, you have finitely many candidates or these torsion points and you just, you can compute them more easily. And this, I mean, I'm doing it for this sphere, but as you'll see, the method is quite general and probably works for many of these extensions. Okay, so let me just sketch the proof, which is very similar to Pella's proof that I already showed you. So I denote by this lambda the lattice corresponding. So lambda sub lambda is the lattice corresponding to A lambda. And now we want to count the values where, I mean, this points C, which are torsion. So a point will be torsion, as I mentioned before, if and only if the logarithm in C2 is a rational combination of the periods of the generators of this lattice lambda. And again, the order of torsion is roughly the height of the coefficient. This is common denominator of the coefficient. So what the Pella-Wilkie theorem tells us is that the number of torsion values in C, of n torsion points in C, is some constant times n to the epsilon. Okay, and on the other hand, you can get the Galois orbit lower bound. You can also get it using the foliation methods, by the way. But I mean, this part was effective already before in the work of Manson and Zanier. You can get a lower bound. So if you have an n torsion point, then the degree of the field generated by this point is at least n to the d. Okay, so and again, comparing this n to the epsilon to n to the d, if we take epsilon small enough, we get a contradiction for n large enough. And this gives us a bound on the order of torsion. And if you do effective counting, then you can expect to get a good bound on the order of torsion. Now, in order to do this effective counting, you have to realize all of this setup using leaves of variations. So the point is that the generators of this lambda, these are just periods, periods of this family of a billion surfaces. So in this case, they are given by hyper elliptic integrals and they are known to satisfy some kind of system, the Pekan-Fuk system of equations. And the logarithm, you also need the logarithm, right? We want to say that the logarithm of C is a rational combination of the period. So we also need to control the logarithm. And this logarithm is an incomplete a billion integral. So it satisfies a similar kind of equation, an inhomogeneous Pekar-Fuk's type equation. And all of this generalizes to any family. So if you replace Pekar-Fuk's by Gauss-Mannin, then we know that whenever you have periods of an algebraic family, then there are such differential equations. So using these differential equations, you can encode everything using leaves of variations. And you get this, you replace the bound by Masell-Zanier by this polynomial bound in the height and degree of C and the degree and the log n. And then just plugging this into the proof, you end up with the bound that I mentioned. So let me just quickly mention the application to Perl's equation. So this is an application by Masell-Zanier. I'm not saying anything new, it's just very nice. So they consider the Perl's equation for some fixed D, but now over polynomials. So you're looking at an equation like A square minus D, B square equals one, and you're trying to solve it for A and B. So D is fixed and we are looking for solutions in A and B. Now in the integers, this is always solvable unless D is a square. If D is a square, then you can factorize this and there is some obvious obstruction. But if, unless D is a square, then this is solvable. And you can ask, can I also solve it in polynomials? And what Masell-Zanier showed is that actually in polynomials, there are some extra obstructions. So consider, for example, this D, which is given by this family, this is just an example. So actually you can extend it to any D, but let's stick with this one. So this is not a perfect square, but nevertheless, Perl's equation is solvable only for finitely many T. And they do this by translating this problem into a problem on elliptic periods, on the torsion points in the elliptic families. And if you now apply this previous theorem that I showed, this theorem here that gives an effective version and you just translate what you get in their context, then you will get that you can actually effectively compute the set of T's in polynomial time, where D is the input. So everything depends on polynomial degree and log height of D and you can kind of find all of the T's where actually it's solved. And when it is solvable, then their method also shows you how to solve it. Okay, so let me just show this reduction real quick because it's nice. So for any T, we look at this hyper elliptic curve as to Y square is equal to D of T. Okay, so this is a, remember that D was a polynomial in T and X, right? So for every fixed T, we get some polynomial in X here. So this will be a hyper elliptic curve. Now, if you have a polynomial solution, then over this curve, you can factorize this spell equation, right? Because over this hyper elliptic curve, Y, I mean D becomes a special, a perfect square. It is the square of Y. So we can write this kind of factorization. Okay, so A is a polynomial, B is a polynomial and just Y is the coordinate on the hyper elliptic curve. And this implies that A minus YB is a regular function on this hyper elliptic curve, except for the points at infinity. Okay, because A and B, they're just polynomials in X. So of course they are regular except the points at infinity and Y is regular. But moreover, this function will not have any poles or zeros. It doesn't have poles because it's a regular function, but it doesn't have zeros because when you multiply it by another regular function without poles, you get one. Okay, so we find that this function A minus YB has no poles and zeros. So if we look at its divisor in the Jacobian, it will have to be just the two points at infinity for the hyper elliptic curve. And one of them has to be a pole of a certain order and the other has to be a zero, right? The number of poles should be equal to the number of zeros. So we get that the divisor has to be actually M times one of the points minus M times the other points. Okay, and on the other hand, because this is a divisor, then in the Jacobian, it should be just zero. So it means exactly that this difference between the two points at infinity is a torsion point, okay? Because M times this difference is equal to zero in the Jacobian. So we find that there is a torsion point and this is like a section. So you can think for every T, you get some points. The difference between the two points at infinity. And we are asking for which values of T is this going to be a torsion point? And the answer by Maseranzania is that this will happen only finitely many times and this effective version also lets us find the values of T where this happens. Okay, now I see that I'm really out of time. Let me just quickly state the last result to, if it's okay, Philip, I don't know, or I can stop here. It's kind of, so I just want to show what are the limits of this foliation counting method. So one of the things that I'm interested in is the like effectivizing this proof of Andre Hort. And it is really known effectively only in one case. So there's a theorem of law school that appeared in the Annas in 2012. And it was also simultaneously done by Bilou Maseranzania, published here later. And the theorem says that if you have a curve in CN, then you can effectively find all of the special points. Okay, and that is the only case where we know this. So I had before a polynomial time algorithm, but it wasn't effective. I said that there is some constant that we cannot control. So basically the proofs of this fact, they use some things that are very specific to CN. So they use the fact that all of the special points of the same discriminator conjugate. And one of these conjugates is very close to the cast. And then we know very strong transcendence estimates for elliptic logarithms. And kind of all of this goes into kind of effectivizing this proof. And basically none of this works when you go to Averchimua varieties. So it's no longer true that they are all conjugates. So you cannot conjugate them all to a point near the cast. And the transcendence measures we have are not nearly as strong. So it's much more difficult to do anything when you go beyond CN. And sorry, so in CN there are some partial results for positive dimension. I wanted to mention this. There is a national result by Bilu and Kune and I also have some result. But again, it's only for CN where you have this, where you can use this nice properties. So once you go to Averchimua varieties, this completely fails. And we do have a new theorem with David Masler that is in preparation where we show that if you take a curve, a non-compact curve in any Hilbert modular variety, then the same thing is true. So we extend this previous result from CN to any Hilbert modular variety, but now we have to assume that the curve is non-compact. This was kind of present also in the case of CN, but in CN every curve is non-compact. So you didn't have to kind of assume this separately, but we do need this assumption in the Hilbert modular case. And kind of what I wanted to say, just generally the proof strategy is similar. We first, we use g-functions to bound the height of special points in terms of the degree. So this is a result of underlay. And then we can use the Masler-Wussler endomorphism estimates to get the lower bit lower bounds. And then we use a competing pillar-wheel-key bound. So kind of the way that the proof works is similar, although it's much more sophisticated than uses this deep results of underlay on g-functions, but the effectivity of this last step, it already falls outside the realm of everything that I discussed today, because here we have to really work with unrestricted fundamental domains. So in other contexts, it's usually possible somehow to restrict to some compact ball. I was always kind of putting this under the rug, but here you cannot get away with this. I mean, you really must count points on the entire fundamental domain and the fundamental domain is not going to be compact. Anyway, we assume that V is non-compact, but this is really essential for us. So we must be able to count on the entire non-compact domain. And for this, we develop a new approach, which is kind of using much more delicate properties of the differential equations. It's not just for liations. We need for liations that are really given by flat regular singular connections with quasi-unimportant monodromy. So the kind of thing that you get from Gauss-Mannin connection. So I would say that this is kind of the next direction to try to push these results, to really maybe assume more about the falliation and try to cover kind of the non-compact case, which is really needed in the more advanced arithmetic applications. Okay, so sorry for going a bit over time and I'll finish here.