 Thanks for giving me this opportunity to say a few words about Basterdicks Holm, who very sadly passed away in January and it came as a great shock to the whole of mathematical community and to everyone who knew him. So I was one of his first students and I would like to say a few words about his work in general, but in particular his work related to the topic of Harry's talk, namely the Androids Conjecture. So Basterdicks worked in many, many subjects, number theory and algebraic geometry on modular curves, modular forms, also some algorithmic problems, and the Androids Conjecture. In fact, he did some seminal work on the problem and his ideas certainly contributed to the eventual resolution of the problem. Buzz was in fact the first person, so the conjecture was formulated first by Yvandre in the late 80s and then by Fransort in greater generality in the mid 90s and Buzz was the first person to come up with a strategy, a general strategy to approach the problem in full generality and in fact his ideas and what his work led to the resolution of this problem under GRH. Well, the eventual resolution is, follows the approach by Pilar and Zanya. However, the ideas of Buzz, for example, the use of monotony and the properties of hypercorrespondences are still crucial elements of the proof. So most importantly, Buzz was the person who isolated the crucial ingredients which is bounding below the degrees of special points and this will be the topic of Harry's talk. And he was actually published this as an open problem and pointed out the connection with Colmey's formula, Conjecture, which for the Foutings height of Cme billion varieties and this is what, this is also the crucial ingredient in the proof. So the proof of the final ingredient and of what you might call Buzz's Conjecture was announced in September and I'm sure he was very pleased to learn it. So yes, I would say that Buzz is passing away, it is a great loss for the community and mathematically, of course, he's family and friends and he'll be a certainly remembered. Well, this is what I wanted to say now over to Harry. All right, thank you, Andre. Yeah, thank you for saying a couple of words. I personally didn't know him that well, but I met him once at the Fields Institute and I was a young student, younger than now. And he was always very friendly and open and open to discussions. So, okay, thank you to everyone who showed up to this talk. I will try to give a brief overview of the history of the Underwater Conjecture and its eventual resolution last year. I mean, I will give my perspective, obviously, which is influenced by my taste. And I would say that the problem or the eventual resolution of the problem had, of course, many ingredients and parts of it come from model theory slash transcendence theory, I would say, and this is more my turf. So I will concentrate on this part of the proof and all the people who contributed to this part. Of course, I will try to mention everyone who made a contribution, but please correct me in the chat or whatever, if I missed something or if I forgot something. As I said, this presentation is heavily influenced by my own tastes, obviously. Yeah, I guess it can't be helped. So I will start with an introduction basically to everyone who maybe has not even heard of the Underwater Conjecture. I mean, I guess there's still a lot of people like that, even some who showed up or maybe students. And then I will explain what the Pila Wilkie counting theorem is. This is perhaps even interesting to people who are not interested in the Underwater Conjecture. We explained the Pila Zania strategy, which applies in many, many situations. And then some ingredients that go into the Pila Zania strategy. And well, particularly, so the sort of the new ingredients come in section six, which I doubt more counting. And at the end, if there's still time, I will talk about recent work of that was that goes perhaps a little bit beyond Andre Ortt towards the pink conjecture or silver pink conjecture and mentioned work of other mathematicians as well. In particular, there are many young mathematicians who work on these kind of problems. Okay, so what is the Underwater Conjecture about? So we start with the lattice in C. So that is its free being group generated by two elements in C that are linearly independent. So one can really imagine this as some sort of lattice in R2. But we think of it as being in the complex numbers. And then given such a lattice, we can sort of study its symmetries, I would call it, or the formal name for it is endomorphisms. And these are all complex numbers that if you multiply the lattice by the complex number, you stay in the lattice. So multiplying by a complex number is really you scale the lattice by a certain number, then you you rotate it a bit. And most of the time, so like in my second example here, if we take generators of the lattice as being one, and i times pi, say, the only endomorphisms that preserve the lattice are the integers. Okay, so you kind of can only multiply, for example, that's by two, and you will still stay in the lattice. But there are there are exceptions. So for example, if you take the square lattice, so generate by one and i, that are sort of perpendicular to each other, then you can also rotate by 90 degrees. So if you multiply by i, and you will still stay in the lattice, I hope that's kind of visual enough. And we say that if this happens, if this endomorphism ring, so this animal is a former ring, is bigger than the integers, then we say that the lattice has complex multiplication. So this is really also where the name comes from, which is a bit weird, why complex multiplication. It's because we can multiply the lattice by something that's not real, not the integers by some complex number, and we still stay in the lattice. Okay. And I think this observation probably goes back to, yeah, the 19th century, I would say, at least. Okay, so what does it have to do with number theory or, I mean, we just looked at lattice or algebraic, sorry, algebraic geometry. So to each lattice, we can attach an elliptic curve. Okay, a curve of genus one with some specified point at infinity. So I've written down sort of a universal family, elliptic curves, the Legendre family here, and which is parameterized by all lambda that sort of by the f i line minus zero and one. Okay. And okay, so each such elliptic curve is isomorphic to see modulo elettis. The isomorphism can be written down explicitly. It's actually given by Weierstrass p functions. So these elliptic functions that were constructed by Weierstrass also very classical subjects. And what what what what comes up, if we look at this particular family of Legendre curve, curves is this particular group gamma two. And I will just explain a second why these are sort of elements in SL2Z. So integer matrices with determinant one. And this particular subject was given by all the matrices that are congruent to the identity model of two. Okay. And these elements, these matrices, they act on the upper half plane by Möbius transformations. So in particular, because they are defined over R, they preserve the imaginary part being positive. So they really act on the upper half plane. And there is a covering map, this lambda function, I think is as it's known, it goes from the upper half plane. So or one could say is a complex analytically the disk to stay without zero one. Okay. And this is kind of, yeah, we'll see this kind of structure a lot in the store. And this lambda function has the property that it is invariant by the action of this group gamma two. And so in particular if you quotient out H by this gamma group, then we get C without zero one. So topology. So really C without zero one is isomorphic to H modulo this arithmetic subgroup of the group of symmetries of H. So all symmetries are given by SL2R. And particularly arithmetic subgroup gives us this curve. Okay. And what's also important is that this function lambda, it in particular if we translate tall by tall plus two, then we get lambda back. So this function lambda has a Fourier expansion. Yeah. And this fully expansion I've written down some, some terms is defined over the integers. So it has some arithmetic nature to it. It actually appears also now in other interesting conjectures, some of them by a Ramanujan, for example. And it is basically given by, this Fourier expansion is defined as basically given by a series in Q and converges on the upper half plane. And this will be important for us later, this particular form of lambda that we can write this lambda function down like this, a sort of an analytic function of an exponential function, but more about that later. Yeah. Okay. So I've written it down. So yeah, as I said, this lambda, it induce an isomorphism H modulo this arithmetic subgroup to, we call it Y2. Y2 is just a projective line minus three points. So let's just, as an aside, this function lambda is also important, for example, to showing like these, I think it's called little Picard's theorem, every entire function that emits at least two values as constant. Okay. And here now we can, the first time this word appears, we can say that this curve Y of two is a Shimura variety, but I will talk about it a bit later. Okay. So because I think this E lambda is isomorphic to C modulo the lattice, each endomorphism of the lattice gives us an endomorphism actually of the elliptic curve. So algebraic endomorphism. And we say that an elliptic curve has complex multiplication. Now, if it's lattice has complex multiplication, and we can write down a specific condition for the lattice for this to hold, namely, if we write the lattice as z plus z tau, which we can always do after scaling, then the lattice or the elliptic curve has complex multiplication, if there is a non-trivial element of SL2Z that fixes tau. Okay. And if you write down an equation for this, then you find quickly that tau has to be an algebraic number in fact. So yeah, algebraic number and is projectic imaginary. Okay. So now, actually, these are the only numbers that are algebraic and whose value by lambda is also an algebraic number. This is a theorem by Schneider. And we already see kind of the connection to transcendence theory, but of course, Schneider was a transcendental number theorist. Okay. Some questions in the chat? All right. So far so good. Now, we come to the first theorem, namely for André, who was mentioned before by André. Okay. So we have, namely, we have these modular polynomials. These are polynomials defined over the integers and two variables. And they have the property that if I have two lambdas that vanish, then the attached elliptic curves to lambda one, lambda two, are isogenes, we say, and they're connected by an isochini whose kernel has cardinality n, say, or even we can even say its kernel is a cyclic group of cardinality n. And, okay, I think at the end of the 90s, at least was published then, André, so Eva André, proved that if you take a curve in the affine plane, and you intersect it with all the invariance lambda one, lambda two, that correspond to CML elliptic curves, and you find infinitely many of those, okay, then your curve has to be special, namely, it has to be a modular curve. So either one coordinate is constant and is equal to a CM, corresponds to a CM elliptic curve, or it defines a modular curve. So actually each point on the curve corresponds to elliptic curve being isogenes. So, so how did he achieve this feat? He used transcendence theory, in fact. So he used very innovatively the theorem of linear forms, logarithms, and he combined it with class field theory. It's a very interesting development. I think it was a bit, well, I probably claimed too much here, but I think it was quite surprising to the community back then that this works. And however, the problem with the proof is, even though people tried, I know some of them, in fact, I haven't tried myself, it doesn't seem to extend to higher dimensions. So it is really restricted to curves. Okay, so this is the first instance I would say of the Andree-Aud conjecture, okay, which I will formulate a bit later. But if some of you are familiar with Man and Mumford, this is really a direct analog of this. The Man and Mumford conjecture says that the sadistic closure of torsion points is a subgroup, is contained a finite union of subgroups, say in the multiplicative group. And this is a very similar statement for CM points, special points. Ari, there's a question in the chat regarding Picard's theorem, a little Picard's theorem. Yeah. Yeah, okay, the question was why it was a little Picard's, yeah, maybe I overdid it a bit with a connection, but if you want to prove a little Picard, one way to prove it is to use the lambda functions that I just introduced, because it gives a, so if you remove from C to points, the covering map is, yeah, is a disk, which is, yeah, compact, which can be put into a compact, or is rather bounded, bounded domain. Okay, great. All right, so now we go to higher dimensions. So we take a product of Y2 now, Y to the n, and we can also define a special point. So here's special point is each coordinate corresponds to a CM elliptic curve. So now we're moving sort of from 2 to n dimensions. Okay. And there's this famous theorem of Pilar, who proved, actually proved a more general statement, but this is kind of what his theorem is known for. He proved that if you take any collection of special points, so that each coordinate corresponds to a CM elliptic curve, you take the Zariski closure, then the Zariski closure is the finite union of special subvarieties. Special subvarieties are just products of special points and modular curves. Okay, maybe I should say what a Zariski closure is. So it basically means that if you take any algebraic variety in Y to the n, and you intersect it with a set of special points, then the intersection can be completely described by special subvarieties. So for example, it could be a finite union of modular curves. Okay. Or it's a finite union of products of modular curves. Okay. Now, what did Pilar use? I mean, this was a real breakthrough because for years, there was only Andres's theorem. People tried to generalize it, and as I will say later, and Andrei mentioned, there were also other approaches, but what Pilar did was he used his counting theorem with Wilkie. So it is stated here. So there are some words that come up that I will explain. So the counting theorem is a very, very general theorem. It says that if you take any what is called definable set, so definable means definable in some fixed or minimal structure in the real numbers, and you count the rational points on this x, then you can bound its number sub-colynomially. So that means if you fix the height, say it here, if you take a rational point on your x, it's just some subset in the real numbers, you take a rational point, you bound its height, which I denoted here, which is the maximum of the numerator and denominator by some t, then you get a bound that is sub-colynomous. You can beat any power of t. What's important here, though, is that you have to remove some set of x. So you have to remove what is called x-alc, and x-alc is sort of the algebraic part of x. So it's kind of surprising we want to do algebraic geometry, but this theorem is important, and you have to sort of throw out all of the algebraic parts. The algebraic part is really everything that is positive dimensional and can be described by algebraic qualities, or inequalities also. So what I would also say, the proof also allows to, here we focus on q where you have the height. You can also put in some fixed number field, and what we'll see later, we can also count algebraic points of bounded degree, and of course you have then to extend the height from q to the algebraic numbers in sort of the usual way, which I hope is familiar to most of you. Now, okay, so more to the pillar, we'll give the counting theorem a bit to its history. It's inspired by a theorem of Pompierian Pila, I think at the end of the 80s, beginning of the 90s, where they proved exactly this theorem, but only for transcendental curves. Transcendental curves are curves in R2 that are not zero sets of some polynomial and two variants. Okay, and I think the idea, I think I heard the idea to get these sub-pollinomial bounds is due to Pompier. But also intermediate results, so from curves to this very general theorem, were previously obtained by Pila, for example, surfaces, they were used by Masa and Zanier to obtain some of their theorems for relative manamumfolk. But I think perhaps one of the great innovations of the theorem was that they introduced this very general theory of hominimality, which comes from logic, into this kind of counting theory. And as I said, I don't know much about this because I'm not a logician, but it was developed more or less, I think, by one of these Pilae, and of course, there are other names like Gabrielov and, as we say later, Wilkie McIntyre, who played a pivotal role in this. Okay, so I will very, very quickly explain how the proof goes. So you have this x, and I'll give some example later, which you can think of sitting in Rn, some, some, some, let's say, analytic set. The first thing that you do is you cover it by smooth charts. So you don't assume axis is definable in hominimist structure. It doesn't mean that it's smooth, but it's sort of piecewise smooth. And you can control the degree of smoothness depending on how many charts you put. So after you sort of parameterize your set, this is the first very difficult step. And I think the first work towards this also goes back to Gromov. Then comes sort of an arithmetic part where you use the determinant method of Pilae that Pilae introduced for the counting. Or, yeah, as I mentioned below later, because it's important later, you can also use Siegel's Lemma to show that all these points of bounded height are contained in an algebraic variety. And then you intersect this algebraic variety with your x. And this is, in fact, where the assumption comes in that you have to put, that you have to take out the algebraic part. Namely, if your x doesn't have any algebraic part, then this intersection will have lower dimension than x. If your x is not completely algebraic, you will cut down the dimension and then you repeat all of the arguments again. So this is roughly the idea. Okay. So now, oh, yeah, I have messed up the pauses here. Okay, then in order to prove a theorem, Pilae used the so-called Pilae-Sanje strategy. So the Pilae-Sanje strategy was developed by both of them to give a new proof, in fact, of the Mandemann-Foth conjecture for a billion varieties. Okay, so what's the idea? The idea is you take a fundamental domain F for lambda as I said, you have the upper half plane. It's not, yeah, it's, and you, but you have sort of many, yeah, lambda is highly non-injective on it. It takes the same value quite often, for example, at tau and tau plus two, the same value. So you have to take a fundamental domain for lambda on H, and it has to be a suitable fundamental domain. This is, you can't take any fundamental domain, but sort of the first one that comes to mind works. And then you can show that for every quadratic imaginary tau, there is some tau in the fundamental domain, such that its height is bounded. Okay. And here M is some number that doesn't really depend on anything. Okay. So this was shown by Pilae in this case. Then there is a theorem of Siegel that says that the number field generated by lambda of tau for a quadratic imaginary is bounded from below by a power of the discriminant. Okay. Now we look at sort of the uniformization map of this product of modular curves by the upper half plane to the end. And this we do is given by each coordinate is just the lambda function. Okay. And then we look at the inverse of an algebraic variety in y to the n, in the upper half plane by pi, and intersect it with the fundamental domain. Okay. So I denote this set by xv. So it's all the toss in the fundamental domain that map into our variety. Okay. What's important here is that because this whole hominimality theory is a real theory, we always sort of identify Sie with the real numbers in the obvious way. Okay. But it's kind of a step that I will skip. So this set xv is very important for the proof. Okay. So first, xv is definable in the structure R1xp. So we have this xv. It's defined by this sort of complicated formula. So you have any algebraic variety. You take sort of all the toss that map into v. And what I'm saying is that even though it's complicated, there is a mathematical formalism that captures it. And it's actually the structure called R1xp. What is R1xp? It's sort of you take the real numbers, you take all polynomials that you, yeah, over the reals. But you also joined sort of a finite number of analytic functions that are defined on some compact on R. Okay. So power series in several variables that are defined on some compact and the exponential function that's in fact unbounded. Okay. Now, if you look at lambda tau, this expansion, you see it's actually an analytic function of q. Okay. You have some, if you think of q as some variable, then it's an analytic function of q and q itself. Well, it's, if you restrict it to a fundamental domain, it sort of consists of the exponential function over the reals and extended a bit. Okay. So in fact, it's not hard to believe that this lambda is in fact in this structure. Okay. But of course, yeah, I mean, I skipped a lot. So if we don't move to the fundamental, if you don't restrict to the fundamental domain, this is not no longer true. It's not definable in this O minimal structure. But yeah, what is an O minimal structure? Indeed. Yeah. So I won't explain it in detail because it can take takes a while, but I will tell you how I think about it. So an O minimal structure is kind of a general, you can think of it as a bit of a generalization of an algebraic variety. So most of what you're looking at are sort of equalities and inequalities of polynomials in analytic functions, at least in practice. Of course, the theory itself, if you want to develop it, is much more complicated. So you allow, for example, to take any zero sets, then project it, and so forth. And you have to, in order to tame it, you have to do a lot of work, which was luckily done by Wilkie. And for this particular thing, the extension of R and X done by van den Driesen, Mechendier and Marker. And as soon as we have their work, we don't have to worry about anything. But to give you some intuition, I will give you sort of two examples of sets that can be defined in an O minimal structure and or not. So the first set X1 is one that you might know from school. It's the graph of the exponential function on the positive reads, or actually, no, on all the reels. So I have not put the empty set here. So I just put an inequality to show that it doesn't really add any magic. So here the graph of the exponential function is definable in O minimal structure, but it's of course a highly transcendental thing. So it's not, it's not given by an algebraic function. But if you take, for example, X minus sine X to the graph of sine X on the positives, this is not defined by an O minimal structure. So why is this if you, for example, fix X1 to be say one half, you get an infinite discrete countable set. Okay. And as soon as you have this, you have sort of a form of defining the integers. And if you can define the integers, then there's really no hope of any tameness. You can think of this, this, this at least is how I think about it. Okay. All right. So there are no questions. Okay. So, so this, this, this was sort of our short primer, our short introduction to O minimal structures. Then how, how do we proceed? Okay. So what I will explain now works is again, as I said, is the structure is, is the strategy of pila insanya. And it was first employed to give sort of alternative proof of money month for really the money month for, for example, proof by the new. Okay. So you take a special point on your variety V in the product of modular curves. Then the first thing that I said was you can then find in the inverse on the in the fundamental domain of the upper half plane, you can find a quadratic imaginary or a tuple of quadratic imaginaries whose height is bounded by the discriminants of these points to some power in. Okay. Never mind that we have a tuple here. It's kind of, it's exactly the same, same principle. And we fix the number few K has been Q adjoined by this toe one up to two N, which is just a number of feet. Okay. Now, if we want to, we can now apply the pilawilki theorem to the inverse of V in this fundamental domain. Okay. So we count the, because we have this height bound here bound by discriminant, we can count these points and we get some sub polynomial bound on the number of points in the inverse. Okay. Bound by the discriminant. So here we can take any epsilon that we want. Right. So we can beat any power of delta. All right. So that's the first part. All right. Then the second one is okay by what I said before, we have a Galois lower bound, a polynomial Galois lower bound for the image of this point in the upper half plane by lambda. So, or you can say for the special point, we have a Galois low bound in terms of the discriminant. Here I've taken one fifth. I could take one over 100. It doesn't really matter. All that matters is that we have some positive power of the discriminant. Okay. Now imagine that your V is defined over Q, some fix now if it doesn't really matter. Each of you can now apply the Galois group, the absolute Galois group over Q on V, and it will permute the special point P. Okay. And fix V as not commute, send P to another special point, but fix V. So that means that for each given special point of some given complexity given by the discriminant, you can find a positive power of the discriminant, many more special points on your variety V. Okay. But this is in contrast to the bound that Pila-Wilke gives you, namely that you have this sub polynomial bound here. Okay. So if we compare the two, the upper bound given by Pila-Wilke and the lower bound given by the Galois bound, so you have just put an M because we have some tuple. Okay. And then if we choose epsilon small enough, we can get a bound for the discriminant delta. All right. And here delta as we have a tuple can be set over the maximum of the discriminants of tau 1 up to N. Okay. But as soon as we bound the discriminant, we are finished. We have only finitely many special points in V. So in particular, its risky closure is of special points is finite, contained in V, excuse me, is finite. Sorry, the intersection of V with the special points is finite. So that finishes the proof. But well, but that can't be the story. That can't be true. Why? Because for example, if V is a product of modular curves, there are infinitely many special points in it. As I explained before, if you have a modular curve, then the coordinates will be isogenist with others. So if one coordinate has complex multiplication, the other coordinate will have complex multiplication as well. Okay. So I've omitted a certain step, and this will be, comes next, is a functional transcendence. Okay. So how do we know that this set that I define the inverse of V intersected with fundamental domain is not completely algebraic. It could be an algebraic variety, a real algebraic variety. Okay. And this is, well, I would, yeah, most say this is probably one of the really significant innovations that in that paper of Pila where he proved the underwater conjecture for products of modular curves is that he showed that if you take some algebraic set in the upper half plane, and you apply the uniformization map, you take the ceristic closure, it will be a product of points and modular curves. So he described sort of the sets that are algebraic on the cover, and whose image is algebraic. Okay. So I think Ulmo, who I've heard many, many fantastic talks on this about, called this bi-algebraic, although I'm not sure who exactly coined this term bi-algebraic. Okay. So in some sense, for the experts, it means that bi-algebraic varieties, the ones that are algebraic on the cover and in the image are weekly special, what it's called. In this context, weekly special means a product of points and modular curves. Okay. So this is what Pila proved. How did he do this? So I think this, this is one of the, one of the, yeah, I heard this back as a student and I was really struck by this proof. So what he did is he, he used the Pila-Wilke counting theorem again to prove this transcendence theorem. And just to make some, some comment on the proof, one of the difficulties of his proof is to show that if you have some algebraic variety that intersects the upper half plane, that intersects many, many fundamental domains in the upper half plane. It's kind of an interesting part of the proof. And it's actually one of the difficulties in where, when the proof of Pila was attempted to be generalized, yeah, which I will talk about later. Okay. So now Pila has, now Pila has obtained, after Pila has obtained the underwater question, why, why two to the end question is, okay, how do we proceed? So the underwater conjecture itself is about Chimura varieties, which the product of modular curves is one of those. But of course, there are much, there are many more. I will not give a definition. I would just say that they are in general given by some Hermitian, Hermitian, sorry, yeah, I think it's Hermitian symmetric domain rather H. And we quotient out by an arithmetic subgroup of its symmetries. And they have the properties, these, these, these objects, they satisfy certain axioms that were rigorously specified by the linear. And these axioms, they ensure that actually this quotient is an algebraic variety, which was actually proven by Bailey Borrell, and yeah, just as an example, above we had the upper half plane is our Hermitian symmetric domain, and the arithmetic subgroup was gamma two. So these are the subgroup, as you said, yeah, and I would just want to say, even, even to show that, that these varieties, firstly, are right varieties, it's kind of very well, and that they have a model of a number fields. I mean, there's lots of work who goes in there by Borrell Boy, for example, and so forth. Okay. All right. But what we always have on S is that we have special points that these are kind of analogs of CM points, points with a lot of symmetry. And they have some attached complexity, like the discriminant of the the endomorphism ring of an elliptic curve. And we have higher dimensional special sub varieties. And these are kind of Shimura varieties, or sub Shimura variety in the fixed Shimura varieties, like for in the example, Y2 to the end, these were modular curves or products of modular curves contained in Y2. And of course, a famous example of a Shimura variety is the modular space of a billion varieties. This is probably the most the best study studied Shimura variety. And of course, there we have the seagull upper half plane and and and it's the theory and seagull seagull already worked. Okay, so one of the I think why people got initially interested in this is also the Torelli locals. So here I just wrote it sort of MG sort of, you look at a modular space or curves of genus G, you can attach its Jacobian to it. And this gives you a point on a G so you can embed sort of the modular space of curves of genus G into a G. This is called the Torelli locals. And one can show that if G is at least four, then MG is not a special sub variety. It's actually not everything after G is equal four. And I think I'm not sure about this. So correct me if I'm wrong if someone knows I think odd especially was especially interested in this kind of questions connected to the Torelli locals. Okay, so what does the underwater conjecture state in general? It states if you take the characteristic closure as any set of special points is a finite union of special sub varieties. Okay, this is a very simple, very elegant conjecture. One consequence is okay, if you should know, if we know in addition that for example, the Torelli locals doesn't contain any special sub varieties, which is not always true. But if say for some G, it is true, then you should have shown that for this G, there is no smooth curve, whose Jacobian has complex multiplication, sorry, there are only finitely many smooth curves, whose Jacobian has complex multiplication. Okay, so there are that sort of direct applications to a right geometry. Okay, so to apply the pilasania strategy in general to so we want to prove now on God conjecture, we need three roughly three ingredients. Okay, the first I would say, yeah, well, no, I wouldn't say anything is polynomial lower bounds for Galois orbits. This was open the longest. The second functional transcendence this I have talked about in length before. And the third is what we had before that you have to show that each special point has a an image in a suitable fundamental domain, a preimage, sorry, whose height is bounded. Okay, so the second part function transcendence was often called functional X Lindemann. I'm sorry if I made it someone here. But I think the first is for for a G, this functional transcendence statement that is needed is was proven by Pilar Zimmermann and then later in India for for for all she move over ties by Klingler, or more in your five using also Pilar strategy. And okay, and right, the problem three, so the height bound for the pre images was completely solved by Donor. Now I should also say before we go further, that actually a proof of complete proof of the underwater conjecture, however, assuming the general Riemann hypothesis was given by Klingler and Yafayev using equity distribution techniques. And as Andre mentioned, there was important input by edexhoven. However, I don't know the history of this exactly, but I think also important input by Ulmo. But as I said, I'm not I'm not an expert on this part. Sorry, we're pushing the chat regarding the complexity. Yeah, it's always defined. Yes, that's a fantastic question. So if you have a CMAB and variety attached to it, then yes, it's sort of, you can think about it as the scrimmage of the endomorphism rings. In general, yeah, I'm not sure that you can actually attach some objects with endomorphisms to it. But yeah, maybe there's some yeah, maybe Andre can answer this. Ah, great, one one one person said one half is possible and thank you others. Yeah, that's true. Okay, I knew I messed up. Okay. All right, so is this okay? Is the so in a G the answer is yes, to the to the last question. Okay, so now we come to yeah, I put a timestamp on it this time because yeah, it's already well seven years almost. So we have a Zimmerman proved the third part. So as I said, we had the second and third part were already done. Now the first part remained for a G and Zimmerman proved it. And yeah, I wrote written down the statement like this. So you have this complexity, which is given as as was mentioned in the case of CMOS is the discriminant, more or less of the endomorphism ring of the, the a beam variety with complex multiplication. And all right, so how does this proof works work. There's two crucial ingredients. One, which I would say is average columnist conjecture, which I explain a second and the second are the social estimates for a beam varieties. Okay, so what what both have in common is that they are completely restricted to a beam varieties. Okay, so you can't go from a G using these two ingredients, you can't go from a G to other she more varieties. Okay, so I would just say the average comments conjecture as Andre, I think Andre mentioned it, it connects the logarithm derivative of some RTL function. So like analytic objects attached, attached to the endomorphism ring of the beam variety to the faulting side of the beam variety. And when it has complex multiplication. And so columnist conjecture is I think is stronger than the average columnist conjecture, but the average columnist conjecture is sort of enough for the application and it was proven by Yuan and Zhang and independently also by Henrietta, Golden Howard and Hera. And I think also the methods of both proofs were really distinct. So they came up with two independent proofs. Okay, so this is number one. And second, isogenous is of course, people are not familiar. This is part of transcendental number three. And for example, I think they were more or less created an attempt to to make parts of the desk conjecture effective. Okay, all right. So as a consequence of the average columnist conjecture, we get a height bound on a special point. So here H stands for some logarithmic while height on a G. So it really a height in the traditional sense, let's say. And it gives you a bound that is subpolynomial and discriminant. Now we want to generalize these two ingredients. Now, so we have the average columnist conjecture with strict AB and varieties, and we have the isosceles estimates that are restricted to AB and varieties. So how do we overcome these restrictions? And the answer for the isosceles estimates is, well, we just do more counting. Now I just want to sidetrack is the Pila-Wilke theorem holds for any o-minimum structure, but Wilke already shortly after they proved it or maybe even before forming a disconjecture that in this particular structure are X. So if you only enjoy the exponential function, a polylock bound should hold. So you should get actually instead of t to the epsilon for the number of rational points, you should get a polynomial in the logarithm of the height. So the observation, so there were some proofs of some special cases by Gerbys Jones, Margaret Thomas, Binyamini, Pila, and but they also, they were mostly concerned with Q or when you attached the number field, the dependence on the number field was actually exponential. And my observation was back then that each time you can sort of prove this, you can prove a polylock bound that is, yeah, that is polynomial in the degree of the number field that you put in. Okay, this comes sort of from how it's proven, running a bit late. So I will do this a bit faster. Just say that it's part two where you intersect your X with an algebraic variety. If you have some control on the intersection, then you can improve on the Pila-Wilke bound. And this is how most of the proofs go. And I should say, excuse me, I have forgotten to say that actually a proof of this was put on the archive by Binyamini, Novikov, and Zak, I think this week. So it's quite, yeah. So you need some end. In fact, what you can do, you can also instead of count points in a fixed number field, you can count algebraic points of a fixed degree over Q, and you still get sort of a polynomial bound. So what's the idea? So the first observation, which when, when, when I saw this, I was quite struck by it, which is if we know on a Shimura variety, this is, we know that given some special point, we can find polynomially many in the discriminant other special points defined over a number field that has the same degree as the number field that is generated by my point. Okay, it's a bit of a complicated statement, but sort of given one point, we can find many. Okay, so this is always true in the case of modular curves. This is already enough to get to get a Galois bound, but in the general case, it's not. And the idea is now to look at the graph of the uniformization map. Okay, so we know that the height of the pre-image is bounded by the theorem of Do and all. And if we knew that the height on the image is also so bounded sub-pollinomially in the discriminant, then we can use, we can use the upper bound coming from the counting traded off with the lower bound coming from this observation to get a lower bound on the degree of the number field generated by P by the special point that is polynomial in the discriminant. Okay, but what we need is we need this log t, this pesky log t needs to be bounded. Okay, so what Binyamini said in the FI we've shown is that we've shown that if you have a special point, then you get a polynomial lower bound for the degree of the number field generated by P if you have a bound on the height. So this replaces the isogenic estimates of Massa and Wüstholz and gives a new proof, in fact, of Andrea Ortt already gave a new proof of Andrea Ortt for AG. But of course, to have the full Andrea Ortt conjecture, you also need a height bound. And this was shortly after then put on archive by Pila Schanker-Zimmern is known in Gröhlich. And they proved the necessary height bound. So you can hear for the HM, you can plug in something that is subpolynomial in the discriminant. So you can bound this, you can beat this M by far, and then divide and you get a polynomial lower bound for your for your field. And yeah, this completes the proof of the Underworld Conjecture. Yeah. So hooray. Okay, I'm not sure, Philip, how much time do I have left? Oh, maybe a few more minutes. Okay. Yeah. All right. So maybe just to finish up. If nobody has a question, what else is there to do for Andrea Ortt? So if you want to go to mixed Shimura varieties, as well, bad news, if you want to do something new, because Gao has shown that this the Underworld Conject for mixed Shimura varieties holds also, but he has kind of reduced it to the Underworld for Piusz Shimura varieties back in the day. So I think five years ago, six years ago. But a big open question is whether we can actually find an effective proof. I want to advertise a bit for that. So can we actually find, given an algebraic variety, find all the special points in special subvarieties contained in that variety. And there is some, there's some progress has been made by many people, but mostly for the case Y2 to the N. And yeah, we just, yeah. And now it's maybe a theorem that they will come out soon where we reduce the effectiveness of the Underworld Conjecture for product of Legendre curves to effectiveness for a product of modular curves. Here we use the awesome theory of Paphion functions and some work of Gareth Jones and myself and I. Okay. And maybe just to finish up, I want to say that there's Pink's Conjecture, which is a mixed Shimura, it's a vast generalization basically, yeah, includes almost all known conjectures in diaphragm geometry, like Modell Conjecture, Manning-Mamford Conjecture. There are similar conjectures by Bombieri, Mars and Zanier, like they were formulated in parallel and also by Zilber. What I want to say is also that Zilber was inspired by model theoretic questions, which I think is quite interesting in itself. And he was interested in a model theoretic approach to Shanwes Conjecture. So there are very interesting connections here. Here's a sort of a formulation of Pink. If you just read it through, you will see that if you replace special sub varieties of codemesh, at least in V plus one, by special points, you get exactly the Underworld Conjecture. And okay. And yeah, I just want to say that there are some cases that were proved, for example, by Baruero, Mars Zanier, Habegar, Pila, who proved certain cases of, in fact, Pink's Conjecture. And yeah, I just want to hear, it's also a formulation of Shanwes Conjecture. And maybe I finish up here now. I just want to say that there are many interesting connections here, especially between this functional transcendence theory and logic. And it has already led to many fruitful collaborations. And yeah, I am hopeful that this will continue. And yeah, thank you very much.