 thanks for the introduction and the invitation. And today I will talk about the sparsity of rational and algebraic points. I'll start with an Asian question. We know that in mathematics it's a fundamental question to solve equations like PDE or ODE, but in this talk I will focus on one type of equations and finding certain type of solution. This is this example that I wrote. So we take a polynomial in two variables with coefficients in Q. The question is what can we say about the Q solutions to f x, y equals zero? And this kind of question is very ancient question. It dates back to the ancient great time and it's called different time problems. And here's the book where these questions were first formulated. And here are some examples. The first example is x square plus y square minus one. And for this example, we have infinitely many Q solutions corresponding to the Pythagorean triples like 3 over 5, 4 over 5, 5 over 13, 12 over 13, etc. In the second example, we have a cubic equation y square minus x cube minus 3. And we only have finally many Q solutions, three of them. The third example is similar to the second example in the sense that the equation is still cubic. But in this example, we have infinitely many solutions. The fourth example is a polynomial of higher degree. And again, we have only finally many solutions. So what's the difference between them? Well, here we see that the degree of the polynomials are very different. The first one is quadratic and we have infinitely many, both the second one and third one are cubic, but one of them has finally many, one of them has infinitely many Q solutions. And the last one has a degree higher than three. The way to understand one or at least one way to understand this phenomenon is to write down to consider all the complex solutions and then try to draw the, try to draw the Riemann surface or the algebraic curve associated with these equations. We consider all the complex solutions and draw them down in C square and then compactify them and then do the desingularization. What we get is the so-called algebraic curve. And in the first, and the difference between these algebraic curves is that they have different number of genuses, a genar, or the number of holes are different. In the first example, it's just a sphere. In the second example, we have one hole. A third example is also one hole. And the fourth example, we have two holes. These holes, if you know them, they're called the genus of the curve. In the first example, it's a genus zero curve. And both the second and third example are genus one curves. And the fourth example is a genus two curve. Okay. And now from now on, I will talk about these equations according to their genus. And in the whole talk, I will fix a number G and a number D, which are integers. G is negative, D is positive. And I will fix the number field of degree D, which I call it K. And C will denote a irreducible smooth projective curve of genus G defined over this number field. That means if we write down the system of polynomial equations dividing C, we can choose one system of polynomials with coefficients in K. And usually we denote by CK, the set of K points on C, or the set of K solutions to the system of polynomials. And the first case when G is zero, this is relatively simple. We have two cases, either CK, either C doesn't have any rational point, K point, or C has at least one rational point, at least one K point, in which case C is isomorphic to P1 over this K. And then we have like local global principles. And the next case is when G equals one. This is a very interesting case. In this case, either again, C doesn't have any K point, or C had at least one K point, in which case CK, this set of rational points, it has a structure of a building groups with an identity element. And we call this pair C with this distinguished point, an elliptic curve defined over K. So both the curve and the point are defined over K. A famous theorem of Modell-Way asserts that EK is a finely generated Abellen group. So first of all, see, EK, it has a structure of Abellen group. So that means it has its torsion free part, which is Z to the power of rho, with some rho, which a priori might be infinite. And it has the torsion part, which also a priori might be infinite. But the theory of Modell-Way says that no, rho must be finite, and the torsion part is also finite. So it's finely generated. And then to understand this EK, we have two parts to understand. The torsion free part is rho and torsion part. For the rho, in general, there is no effective method to calculate it, although in some particular cases, one can. But the biggest conjecture in this area is the Birch and Schrodinger die conjecture. It asserts that rho equals the order of some L function associated with E using some, again, local global principles. And here is a list of people who contributed to this conjecture, according to Wikipedia. Yeah, I'm not an expert on that. And if you don't agree with this list, just feel free to edit the Wikipedia page. In practice, sometimes we are okay to just find an upper bound on this row in particular cases. And there's a very explicit upper bound by O and top in 1989. So here it depends on the conductor of E and with two constants depending on k in an explicit way, meaning that they depend on the degree and the discriminant of the field. A bigger question is, is rho bounded for a fixed k? Let's just look at the case when k equals q, is rho bounded for all, is rho bounded for all the curves to find over q? Well, there are the original opinions on this. For example, recently there's this characteristic of Parc-Cooner, Lloyd and Woot, which suggests that there might be only finding many elliptic curves to find over q with what we ranked with this row larger than 21. On the other hand, we do already have examples of elliptic curve to find over q with this row bigger than 21. For example, Elkies in 2006 showed constructing an elliptic curve, a very explicit one with more than we ranked this row at least 28, and it is proved to be 28 under GRH. Okay, so this is the torsion-free part, and for the torsion part, Barry Mazer proved a falling result. So the cardinality of this torsion part is uniformly bounded for all E, and this result was generalized to arbitrary number field by Kamyani, Mazer, and finally by Morel for arbitrary number field. And actually when proving this result, which is about curves of genus 1, Mazer and Morel and Kamyani and Morel, what they actually proved is a result for curves of higher genus. So this is what is proved in that paper in the paper of Mazer. He showed that if n is large enough, then the only q points of some other curves, which are called modular curves, are the only q points are the rational cuffs. So in other words, for these kind of particular curves, the set of q points can be determined very explicitly when n is large enough. And these curves are often of genus at least 2. So to study curves of genus 1, we also need results on genus at least 2. So in the rest of the talk, I will pass to curves of genus at least 2. So this is the general statement for this case is the Modell conjecture made about 100 years ago, and it was proved by Fulton's in 83. He showed that if g is at least 2, then c has only finally many k points. So before moving on, let's digest what this theorem says. First of all, this theorem is a very, very strong theorem because the hypothesis is short, it's topological, it's just this weak topological hypothesis on the genus is at least 2. That's all. But the conclusion is a very strong arithmetic one. It says that the system of polynomial equations has only finally many rational solutions. So this is a very strong conclusion. If we look at Meiser's result, which was 77 before Fulton's theorem, it says that for this particular kind of curve whose genus is easy to calculate in general, if n is at least 16, then it has only finally many q points, finally many rational points. So this is really a strong result. The particular applies to Meiser's result. But on the other hand, this proof is not constructive, meaning that given the particular, given the particular curve, even if it's over q, we cannot determine the set of rational points of this curve by the proof or by any known proof of this theorem. So for example, in Meiser's result, Meiser really determined this set. Well, Fulton's theorem doesn't. At least the proofs up to today, they don't. So this is the first digest of the theorem. Another phenomenon which is maybe more famous about this theorem, Fulton's theorem, versus a particular concrete example is Fermat's Laugh theorem. Here, let's fix n to be an integer. I will write down at least four and fn to be the curve defined by this equation. One can show that the genus is at least two. Then applying Fulton's theorem, one gets only finally many rational solutions to this equation. And this is Fermat's Laugh theorem. Well, the equation showing up in Fermat's Laugh theorem. So just Fulton's already proved that this equation has only finally many rational solutions. But for this example, more expected, we want to know that the only two solutions to this to this polynomial are the trivial ones, meaning that one of x and y must be zero. And this is indeed proved by Welles and Taylor Welles in 95, known as the Fermat's Laugh theorem. Okay. So again, in this example, just by Fulton's theorem, we get the finalness, but we need some other tools. Or actually, the proof of Fermat's Laugh theorem doesn't really depend on Fulton's theorem at all. So yeah, it shows that it determines the exact set of Q solutions to this particular polynomial. Okay. But this example also suggests that it's extremely hard to compute the set of rational points on the given curve, even just for Q. Instead, here is a more achievable but still fundamental question. Is there an easy upper bound for the cardinality of the set of rational points? And how do these K points distribute on C? So this is a question which we can ask and probably it's easier to answer. And for this question, it's back really to at least to model, we have different grades to understand and to analyze this question. We have finalness, upper bound, uniformity of bounds on this cardinality and effective model. So in the rest of my talk, I will explain this question and what is known on this question according to this grace. The first one finalist was proved by Fulton's. And here is the proof. This is extracted from Simeon Archer de Bensou Arrhythmétique as a risk 127 by Lucien Spiro. This picture is part of the picture. What I want to talk or say about this picture is that this is the proof of Fulton's, this is the proof by Fulton's when they'll conjecture. And actually what this proof really showed is a result on integral points on modular spaces. It really established the Chavarage conjecture. And then using the Kodava partial construction, this result on integral points of modular space implies a result on rational points on curves. So what Fulton's really proved is a result on integral points of modular space. And now recently there is a new proof to this fact by Lawrence and Venkatesh by modifying some cards in this picture. Okay, so this is the first proof of Fulton's of the model conjecture. Today I will focus on the second proof of model conjecture, which is different. It is by Vojta. This is really a different proof and it's by Dufontaine method and some article of geometry. And in this proof we also see some descriptions of this distribution points on C. At this leads to explicit upper bound on the cardinality of CK. Fulton's first proof gives an upper bound but then ad hoc one. This gives an explicit one and the proof is simplified by Bombieri generalized by Fulton's to some high dimensional cases. So the starting point of Vojta's proof also of Fulton's first proof is that I assume this CK is not empty. So in this case we take one particular rational point and then we see C as a curve in this Jacobin via the Abel Jacobin embedding based at this point. And then if we base that a rational point then the set of rational points is a subset of the set of rational points of this Jacobin. And JK is an Abelian variety. It has more structures. It is an Abelian group and the more structures we have in general the more tools we have the easier we are able to attack this the problems. So we have this embedding first and then on JK we have more tools and here is one particular tool that we're going to use. So on the Jacobin we have a normalized height function. It is a function from the set of algebraic points on J on the Jacobin to the set of negative real numbers and it vanishes precisely on the torsion points. We restrict this height function to the set of rational points and then tensor it with R. The good one good property of this height function is that it extends linearly really linearly to a function like this and we get this function after the extension is still quadratic and positive definite. Also pay attention in this process also the torsion points are killed. So the only points killed by this function are precisely the torsion points. And now we have on the left-hand side we have a real vector space which is finite dimensional by model weight theorem and we have a quadratic positive definite function on it. If we take the square root then it becomes a normed Euclidean space. So here we have an Euclidean space on the left-hand side and Jk it is the lattice inside this inside this Euclidean space. So here it is the dotted points. Some dotted points are points on Ck which are in this picture the red points and the blue points. They are the points in Ck. And if we have on a finite dimensional real vector space if we have a norm then we can define an inner product and the angle between each two points. For example if we have one point here and one point here we can define the angle of these two points. Okay there's a definition of angle of these two points. Okay now let's see. A starting point is the following observation which by by by Man's fault which was in the 60s actually it is called Man's fault's fault meter. What I'm going to show here is the consequence of this fault meter. For two distinct points on C really really on C not just not on J we have the following inequality. So here we have a quadratic form so p square plus q square minus 2g pq the inner product. This is a quadratic form okay this is a leading term which is a quadratic form and then we have a linear term here which is not a leading term. The sum of them is always non-negative if p and q are non-zero. So from this formula or inequality we also see the difference between the case g equals 01 and the case when g is at least two. That is when g is at least two then this quadratic form is indefinite and we know that an indefinite quadratic form real quadratic form a priori it could take any real value it could be as negative as possible but here this this inequality tells us that no if we consider if we study all the rational or the algebraic points on C or in particular rational points on C for different points then this leading term cannot be too negative if the norms are large enough so that already gives a strong constraint on the paired pq if they are distinct so that is what that only happens when g is at least two well of course when g equals one then this leading term is already positive definite so yeah so this this formula doesn't give much more information as it does for g at least two so this already shows that there are already things that algebraic points on curves of genius at least two they are very smarts in some way so this is already Manfred's observation in the 60s a Manfred's formula and the proof of this formula is somehow really geometric and here is what for here is another phenomenon to prove model conjecture and this phenomenon was proved by Vojta and I will I hear in this theorem I merge Manfred's result and Vojta's result in the same theorem just just this is a statement of theorem but let's really understand the statement by this picture by this beautiful sunshine here the theorem says so Manfred Vojta says the following thing inside this Euclidean space we can we are able to find a capital r just so that if that we draw this ball of radius r center and zero we draw this ball of radius r center at zero in this Euclidean space and now let's only consider points outside this ball outside this ball and there exists a number three over four such that if we divide this Euclidean space into cones according to this angle condition because this thing it really says something about the angle in their product over the product of the norms so this is like the angle of p and q if we have two points whose angle is small that is that means it's smaller than this number here then these two points they cannot be too close to each other what sorry what I took it someone just unmuted by accident ah okay so I will um okay I continue so here um yeah in this Euclidean space we draw this ball of capital r radius capital r we only consider points outside this ball and then we divide this um Euclidean space into cones according to this angle condition and let's consider points lying in the same cone outside this ball um Manfield claims that each two distinct points here in this cone outside this ball they cannot be too close to each other that is here if we if we fix a point p here then the other point q it must lie somewhere outside this large this area uh Vojta on the other hand proved that well they cannot be too far from each other either so q must lie somewhere here in in this stripe here um and this is a very strong condition um what can we do now now let's consider all the points lying in the same cone as p we first of all number them um well a priori we don't know that there are only finally many of them but let's just take n of them so p1 to pn are distinct points in the same cone in this cone um and we uh order them in non uh in in non decreasing norms then Manfield's inequality Manfield's result says that okay then the norm of pn must be at least twice the norm of pn minus one and then two to the two two square the norm of p um two square the norm of pn minus two etc so in the end it's the norm is at least two to the power of n times the norm of p here this is this part is Manfield on the other hand Vojta claims that if we just look at pn pn then this holds true the norm of pn cannot be greater than kappa times the norm of p and and then in this inequality the norm of p cancels out and we get a bound on uh n that means in this cone we can't have more than log two kappa plus one large points in this cone and also we know how many cones there are because there is there there is this angle condition we can show that there are at most seven to the power of the dimension of space such cones according to the angle condition and then we do have the fineness not only the fineness but also we have a uh an explicit upper bound of a large point of the lattice points outside this ball which are called large points they are the number is bounded by this it's very explicit kappa can be made explicit uh and so on and now let's uh further summarize this so in the previous slide we saw that uh Vojta proved that there is a there's a capital r so that we can this that we can divide the points into small points which are the points inside this ball and large points which are points outside this ball and then most Manfield and Vojta proved the results about large points which gives uh fineness but also it gives something more it gives a an upper bound or a commonality of large points and this are actually it can be chosen in a in a further explicit way that is attached to each curve there's a cannoli defined number which is called the faulting height uh it showed up in the first proof of model conjecture by faultings it this this height measures the complexity of the coefficients of the equations defining the curve c in my talk i will abuse the notation i will use this number to denote um the maximum of one and this faulting height so i normalize this height so that it's almost at least one and um Bombieri the Diego uh actually Bombieri and the Diego they show that um i think it's really Bombieri uh this capital r can be chosen to be linear in terms of the square root of the faulting height okay um and in the previous slides we showed that the number of large points here it is bounded above by a number which depend only on kappa only on g times seven to the power of the model way rank but here but actually levin alpagay in 2018 he improved his seven to a number smaller than two um so anyways we do have a nice upper bound on the cardinality of large points it is very nice uh it here you see only the genus here and here you see the model way rank here we do have a nice upper bound but then that's it of course but then that's it uh for the small points temporarily for the finiteness we don't need to do anything because small points are lattice points inside a ball and then automatically they are only finding many of them so prove to prove the finiteness of rational points we don't need to do anything about the small points just results on large points is enough but of course we do want to have some kind of bound and a an explicit and explicit bound was indeed attained by david filimone and kimon so they proved that the cardinality of this rational point the set of rational points on c is bounded above in terms of four numbers we have the genus we have the model way rank as is shown in the large points we have also another number which is very um uh natural that is a degree of the number field i mean anyways the model way rank it depends on the degree of number field and then this fault in height which is the which is some number measuring the complexity of the the equation defining the curve to coefficients so this there is an upper bound so up to now for the classical result uh here we have the finiteness on the degrees uh we do have the finalness we do have an upper bound and on this side regarding the sparsity of the algebraic points first we do have manfalls result which things which suggest that algebraic points are sparse on these kind of curves and uh we also have manifold and void has inequality which are results on the sparsity of large points they have uh yeah they have consequence on large points then we have then the next step is to ask do we have some kind of uniformity on the on the bounds of the cardinality and do we have effective model so here uh on this side to describe the sparsity of algebraic points and uh rational points here we can ask are there other descriptions which can say at least something about the small points uh so that we have so that here we have some some some other phenomenon and here we have some better bound better more uniform bounds uh do we have something like that and um before trying to prove any kind of uniform bounds on this cardinality let's see how uniform such a bound could be first of all this cardinality it must be depend on the genus and here's an example of high periodic curves uh defined by this equation um it has this genus uh one one thousand and twelve and it had at least uh two thousand twenty five different rational points if we consider uh also the points in the infinity or actually two two thousand twenty six um um the thing is when you write further terms the genus got higher higher and the number of different rational points at least the the simple ones we have more and more and the cardinality also must depend on the degree of the field this is also very reasonable because if you if our field get larger of course you get more rational points so these two numbers must appear in any um reasonable upper bound of this of this cardinality in some way um then here's a very ambitious bound that is the question is are these two numbers enough so is this possible to find a number depending on depending only on the genus and the degree such that the the set of cardinality the set of rational points on c has cardinality bounded by this capital b that's such a number exist well this question has an affirmative answer if we assume a wide open conjecture of unbiary long on rational points on the right of general type uh this was proved by cabrasso harris and mazer and then improved by bachelli in the in 97 so after this proof there are two divergent opinions either one believes this this bound or one doesn't believe that the very long conjecture is the current statement is is true um instead here is a um a bound a rather uniform bound which is proved uh this this bound was conjectured by mazer it's called mazer's conjecture b um this is in a joint work with dimitrov and habegger we proved that when g is at least two we have an upper bound of the on the cardinality which is in terms of the genus and the degree of the number field these two things appears but also depend on the uh jacobin the the rank of the jacob model the model we rank of the jacobin and we also show that this this uh this constant here grows at most polynomially in the degree of course if you only care about the case where k equals q then this then this k doesn't then this degree doesn't show up here um well this is mazer's conjecture b and compared with classical result the height of c is no longer involved in the bound um and later on quina improved this result he removed the dependence on the degree in this uh in this constant but still i want to emphasize here that because we anyways have in also in quina's result we anyways have this model they rank it must be depend on the the field in a very serious way so you you somehow see this con see this degree in some way so um it's it's that that's that explains why in in quina's result you don't it's seemingly you don't see this degree here so this result is still reasonable because you have this model we rank okay and for example if k equals q and then you believe the heuristic uh such that the model we rank is bounded then then the the cardinality is bounded uh just in terms of g this is but if you don't believe the uniform bound uh then the model we rank might be unbounded okay so this is the result that we proved as a rather uniform one and before our proof uh there were already results in this direction by different time methods and based on void has proved by david filimon and david magna camay and filibon in some cases and levin alpagay he proved that when k equals q and g equals two the average number of the of the cardinality of rational numbers on on c is a finite number is a finite number he proved this and um by the shabotekolman method which is another powerful method to determine the set of rational solutions um michan still proved this result for small rank curves a hyper elliptic curves and the hyper elliptic condition was removed removed by kanstrebgenab and schreibbrock but still small rank so each this method each each of these this methods are very different at least currently they are very different but they have their own advantages like the different time methods we don't depend on the model we rank we just it works automatically for all curves of any rank but the kalman shabotekolman method uh we we need some because uh condition on the model we rank currently only applies to curves of small rank but whenever it applies their bound is very sharp it's so sharp that one can sometimes determine the set of rational solutions rational points but also currently uh yeah yeah so these are the disadvantages of each method okay here's an example um this is a family of genus two hyper elliptic curves so y square equals x times x minus one x minus two x minus three x minus four and x minus s so we have a parameter we have a parameter s it's a one parameter family in this family of curves something more particular happens that in regards of rational two points on the jacobin which i'm not going to talk about but what i want to say is in this example the model we rank has a better bound in this way so that we can by our result the number of rational points on this point on this curve c s in this one parameter family uh grows sub polynomial okay sub polynomial again in this particular example i want to emphasize that it's particular regard of rational two points rational two torsion on the jacobin okay so how did we prove this result well actually we give a further description of of the of this distribution of this rational points on the curves which also said something about the small points and this is the theorem that we prove i will merge our result with quina's result uh we show the following thing um on such a curve like this around each rational points the number of around each algebraic point uh the number of other algebraic points which are not far from this point in this euclidean space that we defined before it is uniformly bounded in terms of the genus and again what do we mean by not far well not far means that the distance is smaller or equal to c one times uh to some some constant times the fault inside will actually square root if we talk about the distance well first of all this is like a more cognitive version of the bogomol of conjecture proved by yu mo and shou wu zhang they showed that they showed this result uh but with c one and c two unexplicit uh they are they are not explicit at all it's just two two numbers depending only on the c uh in in the unknown way our result is more explicit for these for for for the constants and this is another phenomenon of sparsity because it really says that algebraic points they they are far from each other in a very quantitative way in a very you know quantitative way um let's see how it implies that bound that rather uniform bound well again this is the picture here we already showed what the classical result we don't need to work with the large point now uh in contrast to model conjecture now let's look at the points inside this ball of of radius capital r and now we want to cover this large ball by by by some small balls of radius small r okay and the small r is uh the square root of c one times the fall inside so this is capital r this is small r and we want to cover the large ball by the small balls and so first of all how many small balls do we need to cover the large ball well this is the number this is yeah this is by some simple uh packing argument we know that them the number of small balls we need to cover this large ball is this number the ratio of the radius so because if you see the the the ratio of the radius to the power of the dimension and in each small ball the number of algebra or rational points is uniformly bounded by c2 so combining these two results the number of small rational points is at most c2 times capital r over small r rank of jk and capital r of small r from here we see it's c0 over c1 rank of jk because the fall inside here cancels out in this ratio and all these numbers depends only on g so we have this result and this is what we desire desire we're done with the random uniform bound as soon as we have this new gap principle yeah and actually we proved this result with Dimitri Habegar we proved this result for curves of large heights and Qina proved this result for curves of small heights yeah um so now for Gina so now we do have uh some kind of bound more uniform bound on the kindality of ck but the fall inside doesn't show up in this bound um so here the the the final uniformity um is true subject to the model way rank subject to the model way rank and here on the sparsity part we do have another description of the sparsity of algebra points by this new gap principle and next step is effective model um okay so so here's the summary so Manthold and Vojta's inequality describe how the large algebraic points are sparse and we give another description of how all algebraic points are sparse in c in terms of distance we don't talk about angles we don't we don't have uh yeah we don't talk about angles in this new gap principle um the next step is of course uh effective model it is a conjectural statement which describes where to find the rational points and at least temporarily the slogan here is there are no large rational points if we take this capital r good enough the slogan is no large rational points and what does it mean um i will show it in two minutes uh but here is uh some questions related to our results um which i don't think i have time to talk about so i'll skip it um so here is the conjectural effective model it says that there exists an effectively computable constant depending only on g the degree of the number field and it is the discriminant such that there are no large rational points if we take this c in the capital in the definition of capital r and little is known about the effective model uh there are some cases known following the many of the meanco method by uh gagali venetiano and viada uh again there are some rank conditions and another approach is the colman uh shaboti colman key method by obtaining sharp bounds and currently it is known in a case of k equals q and the rank most g uh the traditional one called colman shaboti for rank of smaller than g and recently the quadratic shaboti for some curves of rank g with various publications of janitor balakrishnan in collaboration with besser muradogra etc okay so this is for rational points on curves of at least two another aspect is about algebra torsion points so we know that on a on a on a curve any of any genus the number of algebra points must be infinite for many many reasons um because otherwise it won't be a curve um we still want to have some fineness and there's one and there's a natural way to divide these algebra points on a curve into equivalent classes they appear in a natural way algebra geometry and number theory um they are called the torsion packets and here this is the definition that algebra geometry definition of torsion packets and me as a number theorist i take another definition that is consider a point p inside this curve the algebra points on this curve and the torsion packet containing this p is the set of the set of algebra torsion points on the jacobian intersect with the the image of this curve based uh under the arojacobi embedding base at p oh so this is the set of torsion packet uh on c uh containing this p the theorem of many the theorem we know of you know known as many mental conjectures show is that when g is at least two then each torsion packet is a finite set he showed the fineness in some kind of uniformity was proved by uh baker and poonan it says that most torsion packets has cardinality uh have cardinality and most two so the in this torsion packet you have p and then and most one more point and this kind of sort of uniformity is that the number of torsion packet of points uh with torsion packets greater than two is bounded above by a number depending on the curve again this number depends on the curve in some unknown way okay and also all these theorems they can well the the many mental conjecture is also made in made into uniform eventual uniform by larce kunan he showed that the torsion packets the cardinality it's bounded above you know by a constant depending only on the genus and prior to kunan's proof uh the marco krigan here proved this result for genus two bi-liptic curves using asthmatic dynamics and kunan's proof uses distribution theorem in addition to techniques and results from dipping chocow and harvegger um uh and and a second proof is proved by yuan using adelic line bundles and a third proof is just posted on archive last month by harvegger and myself uh which does not use equidistribution we use pilazanier method and um so then uh i will finish my talk with the the last slide here um of course now we have this uniformity a national question to ask is that can we compute this constant do we have explicit formula for this well our function field yes by lupur serum and wombs they prove that the c prime g can be taken to be quadratic very explicit over function fields of any characteristic for any non-isotrivial curve so this is a very very nice bound it's quadratic uh the proofs of junan and yuan unfortunately they are currently not effective because this equidistribution result is not effective uh our new proof uh is in principle effective subject to bianmini's effective pilawilki constant result however one can prove a bound maybe one can prove a bound by our new proof but the the bound is too large to be practical um so to get an explicit bound like lupur serum and wombs really one needs more things and other other ideas but here is another question which may be more interesting so can we make bacon pull this result more uniform that is in their bound they said they proved that the number of points p with large torsion packet um the cardinality is bounded above in a number depending only on c but the curve but on c in an explicit way but here we can ask whether we can make this bound depend only on the genus and here we change this two to six uh the number six is suggested by by our proof uh actually michelle stow also in in some paper one can also read off this six by in some previous paper of michelle stow and in the end i want to mention there are also some exciting results uh for as dramatic dynamics by the marco maraki uh maraki schmidt and go tia vini and etc i think my time is up so i'll stop here thank you