 All right, thanks. Thanks for the introduction and thanks for the invitation. I'm really happy to give a talk here with such a large audience. And I should say right from the start that this is joint work with Jerson Carol. He's a PhD student of mine. So this project started as a request. So Jerson wanted to learn a bit about a billion varieties and Russian and points during his first year as a PhD student. That was last year. And we started to meet weekly and the project somehow took a different shape. And this is what I want to present today. Okay. All right. Good. So the first thing is what is this? So first we need to discuss Cavity's theorem. And let me start by saying that C is a smooth projected curve for simplicity. I'm going to discuss things over Q today over the rational numbers. But you can imagine things go through over over a number fields. And when that's important, I'm going to mention it. Okay. So you start with this smooth projected curve over Q and the genus is going to be at least two. And J is going to be the Jacobian of the curve. And Mordor's conjecture from 1922 says that CQ is finite, meaning the rational points of the curve is a finite set. And Chavo T made a breakthrough on this problem in 41. He proved that if the rank of the Jacobian is less than the genus of the curve and of course the genus of the curve is the dimension of the Jacobian. Well, under this condition, then Mordor's conjecture is correct. You get finiteness of the set of rational points. And Falkens proved the full conjecture in 83 along with many other wonderful theorems in the same paper. All right, but nevertheless, you have this more general theorem of Falkens, but we still pay attention to Chavo T's technique. And the reason is that, well, first this rank condition holds quite often. And the other thing is that the proof is way simpler and you can make it explicit in many cases. I don't want to say effective because effective usually is understood as a bound for the height that allows you to search for rational points in a variety. In this case, you get an explicit bound for the number of points, which is slightly different thing, but all right. So here's a sketch of the proof. I'm skipping many meters, but just to give the idea of the proof. You take a rational point to start with if you have any, because if you don't have rational points, you don't need to worry about this, right? So you use this rational point to embed the curve in the Jacobian in the usual way, the Abel Jacobian map, which has this resorted presentation, and r, little r is going to be the rank of the Jacobian. Now, when you look at the closure, the periodic closure of the rational points of the Jacobian, that's going to be a periodic lead group. And the periodic lead groups probably are not in the standard curriculum in undergrad in mathematics, but nevertheless, this is a very classical theory. You can find this in Burbaki, for instance. Well, this periodic lead group, you can bound the dimension using the theory of the logarithm and the exponential map for periodic lead groups. And knowing that this is the closure of an Abelian group of rank r, you get the bound that the dimension of this periodic Abelian lead group is at most r. And therefore, the dimension of this subgroup, the subgroup, the dimension is strictly less than the dimension of the ambient lead group, which is the Jacobian. All right, but then you look at the curve again, and the curve is sitting inside the Jacobian in some way. You look at the periodic points of this curve, and, well, can it be contained completely in gamma in this periodic lead group? Well, no. And the reason is that C generates the Jacobian. Okay, this is standard fact about the Jacobian of the curve. The curve generates the Jacobian as a group, okay, with differences. And therefore, it cannot be contained in a proper subgroup. In particular, it cannot be contained in gamma, even if the subgroup is analytic rather than algebraic. And therefore, you have proper intersections, and there is some compactness argument to conclude that you have finalness. For the periodic points of the curve intersected with this closure of the rational points, but then of course the rational points of the curve are exactly the same as the periodic points of the curve that happen to be rational. Okay, and that's included in the intersection that we just argued that it's finite. Okay, so here's a picture of what's going on. This big ambient here is the Jacobian, the periodic points of the Jacobian. This plane is supposed to be the proper lead group that you obtain as the closure of the rational points that I mentioned strictly less than G, so it's actually a proper lead subgroup. And then this finite curve that's kind of twisted around, that's the periodic points of your curve that's not contained in any proper subgroup. Okay, that's more or less what's going on in a picture. All right, now Coleman, look at this again a couple of years later and well many years later and he realized that you can interpret this intersection of the periodic lead group obtained as the closure of the points. You can intersect this with the periodic points of the curve and reinterpret this as zeros of some analytic functions on the curve. And these analytic functions are obtained by means of periodic integration, Coleman's theory of periodic integration. All right, and without going into details of the proof, let me just say what the theorem says. You take a smooth projective curve over Q you can do this over a number of fields. Okay, no, no problem at all over Q of genius G and the Jacobian is J. And now let P be a prime number. Let us assume a couple of things. I'm going to write this in a long way rather than the short usual version and the long way is just to to make clear what the hypothesis mean. So what is the meaning of this condition. So first, we're going to require that the genius of the curve is at least two. And I want to call this hypothesis as hyperboleicity and I'm going to explain in a moment what this means. Okay, but the genius is at least two. Of course for genius one you cannot have finiteness and sure because you do have elliptic curves of rank bigger than zero. Okay. Now, P good. What is that thing. Well, the curve has to have good reduction of P, then automatically the Jacobian will also have good reduction of P. And P is not too small. And you want P to be larger than two times the genius and all of these conditions you can somehow twist them a little bit and improve here and there under some other hypothesis but The the the theorem of Coleman had had these conditions. Okay, you can somehow improve this on P, but you still need to ask for something else. Okay. And there is this rank condition, the Chabotir rank condition, which as we discussed before is related to the fact that you want to take the closure of the rational points and end up with a properly contained pietic league group. Okay, so you have these four conditions hyperboleicity, the prime is prime of good production, the prime is not too small. And then you have the rank condition under this hypothesis. Then the number of rational points on your curve is at most the number of fp points of your curve which makes sense because you have good reduction and you can just take that model of the curve and look at the p point the model p points. And then you add this number to g minus two. And this number will it involves the genius so it's geometric invariant attach to the curve is not really an arithmetic thing is not going to heights or anything like that. Okay. And the remark here is that these two g minus two is not just some random number. There is a meaning to this. That's the degree of the canonical divisor of the curve, also known as the first turn number of the curve. The first turn number of the curry. And this is important for us because when we go to higher dimensions, we don't have a notion of genius in the way we love to have as for curse. Okay. So the genius somehow gives you a modular space for curves for varieties is not that simple. So it's good to think about this number in more than one way so that we can recognize what's the correct generalization later. Okay. Good. So what is hyperboleicity this thing I just mentioned well. There are many notions of hyperboleicity but when you look at the compact case, most of them are actually the same so I prefer to work with broadly hyperboleicity which is somehow easier to explain. So smooth projective complex variety is going to be broadly hyperboleic if every time you have a holomorphic map from the complex line into the variety, it happens to be constant. That's really hyperboleicity. So here are a couple of examples. If you have a curve hyperboleicity exactly means that the genius is at least two by Picard's theorem. There is a theorem of Picard saying that, okay, fine, you have maps from complex numbers to the room sphere and you have maps from complex numbers to Torah, the bias trust functions, right. But that's it. If you want to go to higher genius, you don't have holomorphic maps defined over the whole complex numbers. Other than the constant maps, of course, okay, that's a theorem of Picard. More generally, you can generalize this thing to sub varieties of a billion varieties. And if you have a sub variety inside some a billion variety. And this sub variety does not contain other positive dimensional a billion varieties up to translation of course, okay. It is hyperboleic or say in a different way when you have holomorphic maps that somehow implies that you have necessarily some a billion varieties sitting inside of M. And if you think this is a generalization because you can put your curve inside its Jacobian. So it's a special case of the second item. And this was a conjecture of blog and then it was proved by green in this special case and then Kawamata in a more general setting. Okay. And, well, then, what if we go back to arithmetic. Why are we discussing this thing about the complex numbers. Well, if your variety happens to be defined over a number field like you, then you can invoke some general There are these conjectures in theory in dimension two and boy time, which gives you more information but everything is, you know, spinning around these principles that lead the conjectures of land on what to expect about rational points in varieties. Okay. The geometry should somehow govern the arithmetic of a direct varieties. And here the conjecture is that if you if your variety is hyperbolic, then the set of rational points has to be finite. Okay. Or as Lang says in his books, he says hyperbolic implies more deli that's that's a conjecture. Okay. And so this hyperbolic is kind of important for us because conjecturally, it implies fineness and fighting this means that the question about counting how many rational points you have makes sense. First of all, okay, it makes sense, because you may have, for instance, in other situations, appropriately contains a variety having infinitely many rational points, but then you cannot count the points. Okay. You should count, for instance, the, the irreducible components of the of the sadistic closure or something like that something fancy. Okay. Now, in the case of some varieties of ability and varieties, this conjecture and more are already proved by far things in a different paper. Okay. And this, this work of far things builds on earlier work by boy time and boy, they gave a different proof of more those conjecture. So this is purely about life and then approximation. Okay. Right. So, why is the chapotee column and bound the chapotee column and bound relevant. Well, there are many reasons, but let me just mention a couple of them. Many people will be left out because I just want to fit this in one slide but I apologize in advance, but just to go quickly about some applications of this ideas and results. Well, first of all, it's used to explicitly compute all the rational points in certain curves. There is, of course, some condition, but when the condition is satisfied, then it works very well and you can compute the points. Either because the bound is sharp and you find all the bounds, sorry, and you find all the points that are allowed by the bound. Or maybe the balance is not sharp, but you can rework explicitly this periodic integrals in Coleman's proof. Okay. Or you can combine the information of different primes in the model they receive and well, there are many ways to apply this ideas. Okay, and that's pretty useful. Also, there is some progress on the Caporazzo-Harris-Maser uniformity conjecture. So this conjectural uniform version of Falking's theorem where the number of points is finite and it can be bounded only in terms of the genus of the curve if you work over Q. So there is some progress by Stoll and Katz and Rabinov and Segre Brown. There are also non-Abelian extensions, first pioneered by Mium Kim, so he worked not in the projected case, but rather the line deleting three points. Okay, and then this approach was extended and generalized to many other settings and in the context of the Chabot-T method, it gives some non-Abelian version where the Jacobian is replaced by some fundamental group. And nowadays the quadratic case, which is the first case after the linear case will be the Jacobian, that's somehow practical in many cases. Okay. And was worked out first by Balakrishnan, Besser Müller, Balakrishnan, and Dogra. Okay. In the case of integral points and rational points. And there are some spectacular recent applications of these non-Abelian generalizations. It's not generalizing for the sake of generalizing. I mean, it can actually solve cases where you cannot apply the usual Chabot-T method. Okay. And especially the curve, and of course Jennifer already gave a talk in this seminar on this wonderful example where all this machinery works like clockwork. Okay, so, excellent. Now, how about going to higher dimension? Okay, can we go to higher dimension? So, well, let's try to understand what's going on here first. Okay. First, if you rework the proof of Chabot-T in a heuristic way, don't attempt to prove the full theorem. Just in a heuristic way, you quickly realize that the correct Chabot-T rank condition here will be that the rank of the Abelian variety, the ambient Abelian variety. This rank added to the dimension of the sub-variety should be at most the dimension of the ambient Abelian variety. Okay, that should be the Chabot-T rank condition. But this, of course, is not enough because your higher-dimensional sub-variety may have, I don't know, some elliptic curve of positive rank sitting inside. Okay, so this rank condition alone should not be the end of the story, but at least it's something that you expect to be part of the Chabot-T method if it ever works in higher dimension. So, so far, this higher-dimensional Chabot-T that people are trying to prove, right, has been explored in the case when the ambient Abelian variety is the Jacobian of a curve and the sub-variety is obtained by adding the curve to itself a couple of times. Okay. Or in a different way, you can think about this as taking some symmetric power of the curve and taking the addition map into the Jacobian. So that will give you another presentation of this sub-variety. So in this special case, there is some work, I mean, not a bound yet. So there is no analog yet of the of the Chabot-T column and bound for this setting, but nonetheless, fineness can be proven and explicitly compute rational points in many cases. Okay. So why people care about this problem? Okay, looks like a very special curve. You have a curve and you have the curve to itself looks a very special setting, but it's really important. The reason it is important is because finding points in this X corresponds to the problem of finding algebraic points. So rational points on X correspond to algebraic points on the curve C of degree D, at most D. So there are there are problems where you really need to compute quadratic points on a curve. Well, then you should look at the symmetric square. So that's why people care about this problem. So Clanson in 93 gave a first attempt and proved fineness on a periodic open set of the symmetric power of the curve, but not the whole thing. And then 6x look at Clanson's approach and we find it in many ways and made it actually practical in some examples. So 6x actually laid the base of a theory about how to compute points on symmetric powers of curves. But still, you don't get a bound. So you don't get a nice close bound like in the Coleman theorem, but it's practical. You can actually run this algorithmically in many cases, not always, but in many cases. And then Jennifer Park in her PhD thesis combined these ideas with with tropical geometry to get some bound. Okay, not another Coleman type of one, but some bound. But unfortunately, apparently there is some additional hypothesis you need to put into the theorem, like the existence of auxiliary differential that when you look at them in a tropical way to say they are not degenerate in terms of series. Okay, and then this was work out in other cases and extended by other authors. Okay, good. So there is some additional hypothesis, but at least this is a step forward in the direction of proving a bound like in Coleman's theorem in at least one example in higher dimension. I mean, well, you can take a curve times itself, but of course we're not thinking about that sort of examples. Okay, we were thinking about some more interesting examples like this one. All right. So, well, how about going beyond curves and that's the answer question. Yes. Yes. I did. Do you Clemson is that is that should I is that Matthew classen. Oh, probably I missed the name. Okay, I'm just not sure. I should go out of there. Probably I should go out of the full screen mode and open the thesis because I have it here. Oh, don't worry. Okay. Maybe at the end, maybe at the end, probably not. Probably not. Okay, so at the end of the talk, we can, we can. But in any case, this is in Samir's paper. So Samir gives a full credit to Clemson to, you know, to be the first person to actually look into this problem. And then he, he develops the theory in a more explicit way. So this is explained in full detail in Samir's papers in introduction. So, all right. So the theorem is this, you take an abelian variety of dimension three. I'm going to state this over cube because the statement over number fields is a bit more tricky. You need to keep track of the ramification of a certain prime. I don't want to stay that it's going to be ugly. Okay, so let me just stay over cubes. All right, so you have your abelian variety of dimension three, and you take a smooth projective surface inside this abelian threefold, we find over cube. You take a prime P. Now the conditions are hyperboleicity. I want the surface to be hyperbolic. In the sense we discussed before, probably hyperbolic if you want. The brand has to be good. Now good. It's a bit more tricky. Okay. So a and X have to have good reduction at P. I'm not saying that a is in the Albanese variety of X or something like that. So I'm asking for a and X to have good reduction of the prime P. And there is one additional reduction condition, which is I want this surface to be hyperbolic mode P. But of course that makes no sense because you don't have a complex maps into something more P. Okay, but still there is a shadow of the condition of this condition which is asking for this surface not to contain elliptic curves. Okay, so that that's actually what I need. Okay, but the surface more P does not contain elliptic curves. Hyperbolic reduction if you want to call it like this. I need P to be not too small, and not too small is not astronomical is just. What is this thing C one square of X is not the square of a number is the square of the first turn class. So this is the first turn number of your surface. Okay, for hyperbolic surfaces turn the first turn number is positive. So this is some positive integers and this positive integer, which is attached to the canonical class of the surface. You take it square. Okay, and then well there is a 15 in front, which is purely technical. I mean, it doesn't really have a deeper meaning this 15. So the print the prime has to be at least 15 times the the chair, the first chair number squared. The table T rank condition where I have a three fold and a surface. So the rank should be at most one, but rank zero is, you know, it's just torsion so I expect one can deal with the torsion in a different way. So we just focus on the rank one case which seems to be the most interesting because you do have infinite many points in this case. So in this setting, then the number of rational points in your hyperbolic surface is at most the number of FP points, which makes sense in this you have you have good reduction you have a way to think about X more P. Plus factor which is of size P. Okay, there is some lower order contribution but it's size P times the first chair number of the surface. Okay, so what is this chair number now explicitly you take the surface take the canonical divisor the canonical class and you self intersect it and you get some integer. Okay, that's it. Now this of course is a reasonable substitute for the genius. When you have a gene the genius in a curve is the first chair number of the curve. In this case, you have the first chair number of surface. It's also a number attached to the canonical class. Hector, excuse me. Yes. One question, please from Carlos. Carlos, would you please ask your question. Can you hear me. Yes, I can. Yes. Yeah, the question was if the condition one implies condition two for P big enough. I don't think so I thought so before. I don't have a contract sample, but Natalia Garcia, she gave me an example of a surface, which is not contained in some ability is just a surface. It's not hyperbolic. But when you reduce more P, there are infirmary primes where you do get rational curves. Okay, so it's not really answering your question, but suggest that it should not be that crazy. If you give me an example of an X which is hyperbolic, but it's not hyperbolic more P for infirmary primes. Okay. I cannot mention you are this. Invert model or surface or something like this now. Well, the examples I mentioned, no, no, no, she constructed something, something different, but probably with Hilbert modular surfaces, you can do something like that. But then you have to think about the singularizations and then you do have some exceptional divisors, I think. Okay. Yeah. I may ask, is this condition to something that can be checked if you have an explicit equations for X and a and you have explicit prime P. The fact that you don't have elliptic curves. Yeah. Okay, well, I'm not sure. I mean, even checking hyperbolicity is not that trivial, but there is a case where it is easy. And I'm going to state this case where it is easy as a corollary in Okay. Okay. So the case where actually you can check this. I mean, this charity business is a theory about having theorems and then finding a situation where actually the theorem can be computed. Okay. It's not, it's not algorithmic. You need a little bit of luck, even for the charity for curse. And in this case, if you give me an ability and three falls over Q rank one and assuming that the endomorphism ring is Z. Okay, of course, on the mobilized space. This is a generic condition. So if you give me a random ability and three fourth of Over C, then this is satisfied for free. Okay. Then you look at a set of primes where the ability and variety is good and absolutely simple. And apparently you can read this from from the set of function. I haven't checked this if this can be done algorithmically, but at least we know that this set of primes has density one in the primes. Okay. And there are examples. I mean, people have computer examples. I cannot tell you for sure. These computations are algorithmic. Maybe you need a little bit of luck. Another question from Peter Sarno. Yes. Go ahead. I'm actually babysitting so I can't. Let me try to read my question. The faulting theorem will give you a bound for the number of points effective. You know what it gives in the setting and how your result compares with what he would give. Yeah, so so the, the theorem of far things. By the Fenton approximation, it gives you a bound for the number of points. But these bound was explicitly worked out by the first time in high dimension by remote. Okay. So there is actually a lot of work to be done there and remote did it. So remote prove a bound and the bound I think is has not been improved since yet but but the shape of the bound is you have an exponential on the base of the exponential. There are two factors. One factor is the degree of the surface. I'm going to state this in the setting. Okay. Will be the degree of the surface. X with respect to a polarization. You need to fix a polarization of the William variety. So that's a geometric term. Okay, geometric terms are fine. Okay. But then there is the findings height of the billion variety, which is on the on the base of the exponential. And then on the exponent, you have a, you have a contribution of the dimensions and the rank. Okay. So the, the bound that you, you get from fighting's theorem depends on the fighting's height. Okay. And that contribution is somehow less uniform in a sense. Okay, that was my question. Thanks. Okay. Yeah. And of course, there are a couple of constants in the bound that make that practical even even in the case of curves. You have this remote bound for curves. Okay, you have it, but it's for computations. It's not practical. You really want to use the tabletic column and approach. Okay. Good. So going back to what I was saying here is that the in the case when you have an absolutely simple ability, then automatically every separate is going to be hyperbolic and when you have absolutely simple reduction. And again, you have this condition on the, on the electric curves, okay, for free. So this is a setting where, where the conditions are immediately satisfied. And of course, you need a little bit of luck to, to, to be in this case, but in any way, I mean, even for the tablet different curves you need a little bit of luck if you want to apply it for some particular But the good thing is that you have plenty of examples here where this theorem satisfied because this endomorphism condition is generic on the modular space. And this run condition, we don't have a theorem, but we expect it to be satisfied a positive proportion of the time. So heuristically, if you start trying a billion varieties of dimension three, the theorem should apply a positive proportion of the time and well here's an example just to tell you that you can find examples that are easy to write down. So you take this, this curve, this genus three curve, the Jacobian is, is suitable for this application. Okay. All right. Good. So about the shape of the bound. Colemans bound is the number of points in the curve is at most counting points more P and then there is the geometric contribution and counting points more P. If you fix the curve and vary P or if you take things randomly in some reasonable way, you expect this to be of size P by the Rima hypothesis. Okay. And then the other one is first turn numbers. The upper bound for surfaces in a billion three foot gives you the number of points more P. This is a surface. So you expect this to be of size P squared. Okay, and then you have this error term contribution and their turn contribution is of size P times the first turn number. So that's why I allow myself to call this thing an error term, but it's not an asymptotic. Of course, it's just an error term for the upper bound, the main part of the upper bound. I expect it to be on the on the point counting aspect. Okay. So in both cases, you have this main term coming from counting points and then geometric contribution coming from the canonical class. So after having two data points, you want to interpolate, you know, to extrapolate sorry to make a conjecture. So, I don't know, I mean it's very tempting that probably there is a general pattern here. If ever someone manages to prove a very general table T theorem in higher dimensions, maybe it's going to look like this. Okay, so what's the key issue in the proof. Well, let me try to explain this, you take the closure of the of the rational points. This gives you an analytic P adic subgroup of what we call a one parameter subgroup. Okay, over the complex numbers. That's what you will do. And then the, the number of rational points that you want to bound you can you can focus not in the rational points but rather the surface of the the addicts intersect with this one parameter subgroup. Okay, and it's a one parameter subgroup because I'm assuming right one. Now the reduction map allows you to cover your ability, but I T periodically with this pediatric open sets called residue this which is just the pre image of a point more P. And on each residue this I'd like to bound this intersection and hopefully it's going to be one most of the time, because if it's one most of the time, then when I vary over the their more key points of the surface. I will get the bound that I was discussing before. Okay, you most of the time you had one and then there is some error term contribution. Okay, that will give you a bound of this form. So that will be the plan. But of course this is not a plan. This is kind of a wish list. You actually need to do something to do something like this. Okay, so here's the picture. Here's a residue disk coming from just one point more P. There is the portion of the surface that lands into the residue disk. And there is this line, which is this one parameter subgroup. I think about this additively because in reality. When you when you look at this pediatric me groups close to the identity, you can linearize everything. Okay. Good. This intersection in one residue disk. Well, you parameterize your pediatric one parameter subgroup gamma you parameterize this with some power series. You need to choose local coordinates, usual business, but there is some power series parameterization. And then you compose with a local equation for the surface. So, gamma will hit the surface precisely when this composition vanishes. And now the question is about counting zeros of this one variable power series, having power series for the for the one parameter group, and then composing with that with equation for for the surface. Okay, and there is some radius of course you need to set up things properly. If at the end you want to end up with a number. Okay, you need to normalize things and All right, so how you count zeros. So you if you start with some one variable power series over the piazza that's convergent on some disk. Let me define n sub zero HR as the number of series counting multiplicity up to radius are some game. And well standard periodic fact one can bound this. Okay, and to bound this you can do it in many ways but something that's convenient will be to have control growth for the coefficients that the conf the coefficients do not grow too fast. That's one thing. And there is some small degree for which you have a large coefficient and large, you know, bigger than or equal to one will do it. If you have these two, then you can use standard periodic analysis to to get a bound reasonable bound for the number of zeros. Now, in our case, this parametrization of the of the one parameter patic subgroup, this parametrization is coming from the exponential map and we have really good bounds for for the coefficients. So item one can be done. Okay, it's not too bad. You just can do it using, you know, suitable choice of local parameters exponential map, lots of time the triangle inequality and then there you go. Okay, the problem is to how how you ensure that you have a small coefficient. Sorry, you have a small degree for which the coefficient is not too small. Okay, and of course that looks very much like non vanishing more P controlling the vanishing order of something more P you reduce the power series more P and you want the zero of this reduction more P is not to be the order of the zero looks like that. Okay. So the key challenge will be to look at this power series and ensure a large coefficient pavically in low degree and large means non zero more P that will do it. I don't need larger than that. So the idea is what I just said you reduce gamma more P and bound the contact order with the reduction of X more P, because you are looking at this composition and F is an equation for F is an equation for X looks wonderful looks very nice except that it makes no sense. Why, because this power series has denominators. Okay. It's coming from an exponential so there is some division by factors there are denominators so you can you cannot really reduce this much. But at least the idea the idea looks fine but it doesn't work. So, the remaining part of the talk, let me just quickly try to explain how to save this idea. And the, the, the approach is what we call the over determined method for infinitesimal infinitesimal omega integrality. The correct version of solving and differential equation will be this omega integrality, which is, if you have two schemes, S and B, and you have a map from B to S, and a differential form on S. Then we say that this map is omega integral meaning heuristically a solution of the differential equation determined by omega is omega integral if the pullback of omega is zero. Because the pullback lifts in the pullback shift, but you don't want to land in the pullback shift you want to land in the differentials of the. So there is some additional work to be done but you define this pullback properly. And that's a condition that the pullback of omega is zero. All right, so we're going to use this on non-reduced schemes in positive characteristic. So the intuition over the complex numbers will not really help too much actually it's not really help place against you. Okay, so the intuition over the complex numbers place against you in this business. All right, so this omega integrality is implicit in classical works by Nakai. He didn't really write this down in this way but it is there. And it's really useful in hyperbolicity. So Bogomolov used this to prove a finite theorem for curves of genus zero and one and certain general type surfaces. So he made some improvements. And Vojta made an explicit version of this business to find curves of genus zero and one in the biggie surface, which is related to some problem in logic that I very much like but I'm not explaining today. Right and then Garcia Fritz. So well Natalia, she worked out a very detailed generalization of Vojta's approach that works not just in one case but in more cases and you can take advantage of verification of higher order. But what is more important for us is that her generalization is purely algebraic. You don't really need any analytic considerations. It's everything algebraic. So what's good for us because we can just take inspiration from this and one or two lemmas from here and there to actually make them work in positive characteristics from non-review schemes. All right, so the remark here is that going from Bogomolov to Vojta is not that obvious because I mean having fineness is not the same as finding the objects, right? It's like punching curves. You have fineness for the points and rational points and curves, but that's a different story if you want to actually find the points. Okay, so that's more or less the step from Bogomolov to Vojta here. Okay, so here's the idea. You take omega one and omega two independent differentials, having nice reduction, more P. Okay, one can do that. And then the condition is that this one parameter subgroup is omega integral. So one has to prove that when you have a one parameter subgroup, first you can make sense of omega integrality because this is somehow it's analytic, but you can think about this as a former scheme. And there is a notion of home integrality there and and the kernel in this space has a mention to in our case. So there are independent differentials. And then you express this gamma as a power series and try to reduce not more P, but more a power of Z, a suitable power of Z and only up to the degree not to be not larger than P so that you don't run into denominators. And then, well, the truncation of this gamma is going to give you an infinitesimal curve. Okay, infinitesimal. So this actually supported on a point. It just has higher order jets but supported by the points. And this is going to be a closed immersion which has to be checked but it's true. Okay, and it's going to be omega integral for two differentials two independent differentials. So nice remote P because the differentials do and the degrees not to be. Okay, and then you reduce and you get a map of similar quality mode P, but now this map exists, although the one parameter subgroup does not. Okay, the one parameter subgroup mode P is no longer there but the map it is. And, alright, so we have this infinitesimal version of the one parameter subgroup mode P. And F is a local equation for X in this residue disk. And the key observation is that if you look at the composition F composed with gamma, the coefficients are practically small. If that happens, you reduce, you reduce more P, not F composed with gamma, but rather F composed with this infinitesimal truncation of gamma. That can be reduced more P. If you get zero, it means that the the PM, which is the reduction of P of this infinitesimal one parameter subgroup is actually a close immersion, not just on the ability and variety but actually on the surface. Okay. That's the key observation. Small coefficients give you a close immersion into the surface more P. And now what happens is that you can take the differentials and reduce more P restrict to the surface. You need to prove they are non zero, but that's a different story. And now you have a surface to differentials and one common solution. And that's over the term that should not exist. Okay, so you, you expect to bound the order to which you can find this infinitesimal solution, you cannot go forever because it just over determined in the sense of differential equations. Okay, so you want to find what is the largest M for which this situation can actually happens and that will give you a bound for what is the largest M for which the coefficients are small. Okay. So here's the state. Here's a lemma. Okay, a lemma to to actually make precise how to bound the order of an infinitesimal solution of an over determined OBE in this setting on a surface. I don't want to read this because I think this is not very elegant, but the important part of the bound is that the order is bounded by a quantity that at the end depends on the geometry of this divisor omega one w one which w two, which is a canonical divisor. So this divisor can be used to bound the order of the over determined solution. Okay. So there is this technicality that some of these summands can be infinity infinity. Okay, that's really, really plain against us. So we need to get rid of this possibility and here's where the hypothesis of not containing elliptic curves shows up. For example, which because of time probably I will not work out here the details but let me show you the picture. Okay. So here you have an omega one DS plus this square D team and the integral curves are these cubics. Okay, these are cubics. So here you have an infinitesimal vertical solution up to order. Two, two, you need to kill the cube and then you have ordered to yes. If you look at this mod Z cube, then you have a vertical solution. And here you have another differential DS plus a square DT. And you have some hyperbolas that move vertically and then this limit curve, which is you can check that s equal to zero is a solution. Now you take the divisor of omega one, which omega two, you can just compute what this is. And the divisor is the diagonal and the anti-diagonal. And you look the diagonal the anti-diagonal they look nothing like this omega integral curves, but nonetheless, one can compute the contribution here of both. And yet, using this ugly formula from before, zero plus one plus zero plus one two, which agrees with the fact that you have a common solution up to order two. So this ugly bound is actually sharp. It's actually sharp in some examples. So in the remaining five minutes, let me just give you a slightly more detailed sketch of proof how you how you bound how to prove a column and bound in this hyperbolic surfaces in a million three-folds. So here's the main result you have an a billion three-fold a hyperbolic surface sitting inside the rank is one and some conditional P that is not going to show up in this argument because I'm skipping all the technicalities. Then the you have that the number of rational points is less than the number of every points plus this error term. That is has it has some geometric origin. We should see the canonical device or contributing to the error term. Right. So the setting is that you you take the closure of the points rank one in place that you have a one parameter subgroup. The reduction map gives you these neighborhoods to cover the the the ability and variety periodically. And on each of these residue this you want to bound the intersection of the periodic points of the surface and the one parameter subgroup coming from the rational points. So if this bound, if it is less than or equal to one most of the time, you're good because then you can add it and get a bound of this form. All right, because when you add over over the p points, you're basically adding one most of the time. So if you want to minimize your your chaotic one parameter subgroup, you compose with the local equation, you get some one valuable power series on a disk and this disk is coming from the rest of the disk. And you want to control the number of zeros because that's an upper bound for this intersection. If you control the number of zeros, of course, if you believe me, we need two things. One is an upper bound for the growth of the of the coefficients and another is an index not to be for which I can find a coefficient, which is actually large. Right. So you take omega one and omega two two differentials on the building variety that kill the one parameter subgroup. And then the as I mentioned before the classical theory tells you that you need these two things controlling the size of the coefficients from above and one coefficient from below. And when you put these things together and do some chaotic analysis, this is the bound that you get P minus one over P minus two times M of X plus one and M of X is coming from this over the terming method. Okay. So it's going to be the largest M with an over the terming close immersion more P supported at the point X and the point X is the point giving you the recipe disk. Our theory of over the term in the omega integral in the characteristic P will give you a bound for this and X, depending on the on the canonical divisor of W one wedge W two. Now this change omega to W is not just funds. Okay. Omega is a differential in characteristic zero on the building variety. So you have the mod P and then you restrict to the surface mod P. That's W. And of course, you need to prove that this wedge is non zero and that's really tricky. Okay, is some sort of weak left chest property in characteristic P that you need to prove. All right. So when you, when your point mod P is not in the support of the divisor, this ugly bound that I mentioned before is actually an empty sum. And it's not just a formality. This situation has to be proof as a separate case in the proof of the ugly bound. So, in this case, you, the, the bound is zero, the, the maximal order of an over the terming solution is going to be just zero. And in this case, therefore, you get the number of points is at most in the intersection of the one parameter subgroup with the surface within the rest of this corresponding to the point X mod P that intersection is at most P minus one over P minus two. Now our prime is larger than 15 eventually. Okay. So certainly this fraction is less than two, but you have an integer less than two. And since the interview is less than two, it is at most one. Okay, this silly thing is actually important when you add up all these contributions. If you don't do it at this point, then you get an additional factor at the end of the bound. What happens when your point mod P is in the in the canonical device or D. Well, in this case, the over the terming bound also works, but it's way more complicated to apply. Okay, there is a lot of things going on there. You need to apply the Rima hypothesis in the for curves, but in the singular case, and furthermore, you need to put some multiplicities to certain points. And you need to do some intersection theory computations. You need to control the similarities of these D because actually these not a nice curve. I cannot assume anything about the these just whatever the differences give you and I have no control on the differentials. Okay. And you need this weak lectures property a blank a bunch of injectivities at the level of comology positive characteristics. It is what it is. Okay. But the important part. I'm not going into these details, but the important part is that at the end of the day, all of this mess depends on the device or D, which is a canonical divisor. So I hope that you you see in this way. How is that this self intersection of the canonical divisor shows up in their return. That's the reason. So here, here's the final bound you add up all the contributions away from the canonical divisor more P you get one. Okay, and when you add one many times you just count. Okay, and that gives you the main term the number of points more P. Then add the canonical divisor you you get some larger bound but you need to do all the work that I hinted before. And you get this error turn. So here's a picture. This is the surface more P the blue lines are the divisor D the canonical divisor D. If the point giving you the residue this if this point is not on the then the intersection may look like this piece of the surface periodically intersected just with the with the one parameter subgroup once, or maybe zero times. Okay, this isn't the one residue disk. And now if your residue this comes from a FP point on the divisor D, then you may have more points in the intersection. Okay, you may have, for instance, this one parameter subgroup meeting the surface of two different points or maybe seven different points within the rescue disk. But that that's basically what's going on. And this is a lower dimensional contribution so at least heuristically I hope you believe me that it contributes to their turn but to actually prove this explicit bound requires some work using this ugly bound for over determined solutions. Okay, and that's all. Thanks for your attention.