 And thanks to Mike and Philip and Alina for putting the series together. It's a great pleasure to speak in the series. What I'd like to talk about today is some methods for determining rational points on curves and I'd like to illustrate each of the methods with A prototypical curve that might want to consider. And so all these techniques sharing common some chaotic methods and I'll kind of start from the ground up and build our way to some interesting curves that we'll look at. So the first question comes to us from Question about rational triangles. So we say that a rational triangle is one with slides that are all rational. So we can ask then if there exists a rational right triangle and a rationalized sauce least triangle that have the same perimeter and the same area. And this might seem like a question that the ancient Greeks looked at maybe one that We've known the answer to for quite a long time, but it turns out that this was just looked at very recently by two graduate students in Japan, my KO university here, Kawa and Matsumura in 2018. So this will lead us to a question about a particular genius to curve. I thought we'd take a look at their solution. So we'll assume that there is such a pair of triangles a rational right triangle and a rationalized sauce least triangle. And so we rescale and we can assume that their lengths look like the following. So for the rational right triangle to KT K times one minus T squared K times one plus T squared Where K is a positive rational and T is a rational between zero and one and for the rationalized sauce least triangle one plus you squared one plus you squared and for you for you between zero and one rational. Alright, so we've set up our two triangles. And now the question was, can they have the same area and the same perimeter. So we compute And we compare their parameters and areas and now we have two equations and three unknowns and KT and you and then we do a small change of coordinates and find ourselves looking at a genius to curve. So the question now about our rational triangles is about rational points on this genius to curve given by the following equation. Why squared equals this cubic squared minus 8x to the sixth. So here is our first curve that we'd like to look at. So what can we say about rational points on this curve. All right. So something very special happens and the shadow tea Coleman bound kicks in and it tells us that this particular genius to curve has at most 10 rational points. So this is incredible and I'll tell you where this comes from and why it applies in a moment. But we'll hold on to that for now. And since we know that this curve has at most 10 rational points. Let's try to find some rational points in this curve. So, you know, we search in a box and maybe write a little script to try to find some points of small height. And when we do that. Well, there are a number of rational points and in fact there are 10 rational points. So this is great. We've determined the set of rational points on this curve. So from the point of view of this question we've now found what we can say about our triangles. And if we trace through the different coordinates. We find that the one point with x coordinate 12 over 11 and positive y coordinate gives rise to a pair of triangles. So, we've answered this question about triangles using the shadow tea Coleman bound on this genius to curve. And this is the theorem of here a column at some of the that up to similitude, there exists a unique pair of a rational right triangle and a rationalized sauce least triangle that have the same parameter and the same area. All right. So, you might be wondering about the shadow tea Coleman method, which is what allowed us to compute the set of rational points in the previous example. So that's what I'd like to talk about next. So it implied for this curve that there were at most 10 rational points. And behind the scenes crucially, we satisfied an inequality between the genius of the curve, which is to, and the rank of the mortal V group of its Jacobian, which we were able to compute and find out that it was one. All right. And since that applied, we were able to apply some work of shadow tea that was later be interpreted by Coleman to get an effective bound that turned out to be sharp. And that's a bit of luck that's involved for this particular curve. So, there's some choice of prime. This is a periodic method and there's some choice of prime that that you're using behind the scenes. I'll tell you more about that a little bit. All right. So maybe to just give a little bit more context about why we care about questions like this. So, faultings proved more dels conjecture and so we know that we have a smooth project of curve X over the rationals of genius at least to find out that the set of rational points is finite, but faultings proof doesn't give us an algorithm to determine the set of rational points. Now, there's another proof of finiteness due to Voida, but it also doesn't give us an algorithm for finding set of rational points. There is recent work of Lawrence and bank attached that also gives another proof of finiteness. And here remains to be seen that this can be made algorithmic. But what we do have that works in some cases is the method of shabushian Coleman. So it only applies to curve satisfying certain restrictive rank bound. But then when it does apply, it works very well in practice for determining rational points possibly combined with other methods. So that's our motivating problem to look at the method of shabushy Coleman or possibly other refinements of this to give an explicit version of faultings theorem. So if I give you a curve to find over the rationals. I know that it's genius is the least to compute explicitly the set of its rational points. All right. So here is one more example one more curve. Smooth plain quartic reasonably nice coefficients. So maybe this is some random curve that we'd like to look at, but maybe not. So it's not so random, but we would like to look at it. This is the split carton modular curve of level 13. And there are some interesting consequences for knowing its rational points. And I'll tell you more about that a little bit. This will be the third curve that we look at. What we do know about this is that Galbraith did some searching and found seven rational points. And of course, that then leads to the natural question. If that's everything the set of the seven points precisely the set of rational points or were there more just kind of beyond the range that he considered, and we'll answer this at the very end. So maybe just a little bit of setup for working with higher genus curves. So for curves over the rationals of genius at least to the set of rational points is just a set. So we need to have a little bit more structure to say something about rational points. And so what we do is we associate to our curve other geometric objects that do have extra structure. So with all of this, I'll assume that our set of rational points is not empty, and I'll fix one point I'll fix the base point and use that to embed the curve into its Jacobian. I'll do that using this obel Jacobi map Iota that sends a point P to the class of the divisor P minus B. So the more del V theorem tells us that the group of rational points and the Jacobian, it's finally generated. So you know it's isomorphic to our copies of Z plus some torsions and finally a billion group. And here, the rank this is very important for us often will need to know what it is before we can even start, but I just want to emphasize that this is also something that's very hard to compute. Typically we get it through some Oracle, like magma. But in general, this is also something that's really difficult to compute and we don't have an algorithm that will compute this in general, but in all of what we do we kind of assume that we're handed the rank and that we can proceed. All right. So the idea, the main idea behind all the methods that I'll talk about is to associate either one or maybe a collection of other geometric objects to our curve. And then use that to explicitly compute a slightly larger, but very importantly, finite set of chaotic points containing the set of rational points. And then from that finite set of chaotic points, use that to extract the set of rational points. And this is what the Shabbat-T Coleman method does. And in subsequent years, there are various refinements of this method that allow us to consider other curves of slightly higher rank. So by associating covering collections or maybe maps elliptic curves over higher degree number field something that is elliptic Shabbat-T. There's also a very large generalization of all this. This is a program initiated by Min Young Kim, where instead of just considering the Jacobian and Abelian geometric objects, he considers non Abelian geometric objects associated to the curve. And this leads to a non Abelian version of the Shabbat-T method and non Abelian Shabbat-T program. What I hope is that this will allow us to understand rational points on curves whose Jacobians have higher rank. So beyond the frontier what's possible with the Shabbat-T Coleman method. All right. So let's start with Shabbat-T's theorem. So Shabbat-T proved the following special case of Mordell's conjecture in the 1940s. So that if X is a curve of genus at least two over the rationals, and the Mordell Bay rank is less than G, then set of rational points is finite. And in the 1980s Coleman gave a theory of periodic line integration and a number of spectacular applications of this theory. And one application that he gave was a reinterpretation of the method of Shabbat-T. He gave an effective version of Shabbat-T's result by reinterpreting everything in terms of periodic line integrals. He constructed a certain annihilating regular one form and computed its integral and show that this really annihilated rational points. So the rational points had to occur as solutions to this integral of an annihilating regular one form set equal to zero. He did more than that. He was able to count the number of zeros, this periodic integral. And in doing so, he gave this bound that for a good prime greater than 2G, the number of rational points is bounded above by the number of Fp rational points plus 2G minus 2. So this bound comes from studying the number of zeros of a periodic power series, but also using some global information about this differential that he constructed. All right, so here's a little bit more about the method of Shabbat-T and Coleman. So we'll fix the prime of good reduction for our curve. We'll assume that is greater than two. And then remember we have the obel Jacobi map, iota that embedded our curve x into its Jacobian j. And so that induces a map between the spaces of regular one forms in the Jacobian and the curve. And in fact, this is an isomorphism of g-dimensional piatic vector spaces. So we'll suppose that we have a regular one form on the Jacobian omega j and that it restricts to regular one form omega on the curve. All right, now we have a piatic integral on the Jacobian. The Jacobian over Qp is a piatic lead group, and so naturally carries piatic integrals. And we'll use that to define a piatic integral on the curve. So I want to define the piatic integral on the curve between points Q and Q prime on the curve in terms of this piatic integral on the Jacobian. So we'll define it to be the integral between zero and the class of the divisor Q prime minus Q. And then if the rank of the Jacobian is less than g, the claim is that there exists a regular one form such that the integral between our base point and any rational point of this differential omega vanishes on all rational points. All right, so why, why should this be true? Well, the Jacobian, we said this is the genus G curve. And so the space of regular one forms is g-dimensional. And if the Jacobian has rank r, essentially we have r independent linear conditions. And so if g is greater than r, then we have some positive co-dimensions in the space that we're trying to cut out using conditions coming from the rank. And so if r is less than g, then we can construct at least one annihilating differential. And so just using the linear algebra, using the values of the Coleman integrals coming from each piece of rank, we can then compute the kernel of the corresponding matrix and then compute a basis for the space of annihilating differential. So this really is now linear algebra problem if we're able to compute values of these Coleman integrals on each of the independent points in the Jacobian. So if we're able to do that, then we can construct a basis for the space of annihilating differentials and then study the corresponding zeros of those integrals and then find a finite set of chaotic points containing the set of rational points on the curve. All right, so just to kind of recap what's happened so far. So we're going to have our curve of genus at least two. We're going to embed it inside its Jacobian. And if the rank of the Jacobian is less than g, then we're going to use a Shabu-T Coleman method to compute a regular one form whose Coleman integral vanishes on rational points. Coleman computed an upper bound for the number of zeros of this chaotic integral and showed that if the prime is greater than 2g and if the prime is good, then the number of rational points is bounded above by the number of fp points plus 2g minus 2. So this is first a local computation in each residue disk and then globally using lemur rock, we get this bound. Now, in practice, this can be sharp. So in the triangle example, the genus was two, the rank was one, and behind the scenes we picked five, and that gave us eight f5 rational points and putting that all together we got the separate bound of 10. But you know that there's some dependence here on the prime p, right? And we kind of understand how the number of fp rational points on our curve grows as p grows, is linear in p. So we want to pick this p to be as small as possible, essentially, so that we have the best shot at getting a good upper bound. In practice, it isn't always going to be a sharp upper bound. So you compute whatever you can on the right hand side and you do a search on the left hand side for the points of bounded height, but you might have some discrepancy, right? And it might just be because genuinely there is some difference between the size of the set of chaotic points cut out by these intervals and the number of rational points you have. Yeah. So it's a question in the audience, from the audience. Rasima Kumar, please and ask away. No, in the second point, the number of points, the fp rational points and x, is it for a specific p or is it for all p? What is the p? So this is the p that you choose. p has to be a prime of good reduction. And also it has to be larger than two times the genus of the curve. But then this is true for any such p. You get to pick. Okay, thank you. Are there any other questions? No, I don't think so. You can continue. Okay, thanks. All right. So when there is some discrepancy between an upper bound that you compute and kind of the size of the known rational points that you have, it might seem that all hope is lost, but in practice, there is a good method of sitting so you kind of do a little bit more work and then throw all this into the more divisive. So you compute Shabu-T Coleman data at a number of primes, you carry out the Shabu-T Coleman method for a collection of favorable primes, and then use conditions on the Jacobian at these various primes to then sieve out the fake points, the points that are not honestly rational. And in practice, this works quite well to then give you the set of rational points. All right, so what we're trying to compute in the Shabu-T Coleman method is the following. So we construct an annihilating differential omega. And then we want to compute all chaotic points in the curve Z for which the integral vanishes. And we're going to call that set of chaotic points X QP one. So we want to do two things. We want to compute this annihilating differential omega, or maybe a collection of annihilating differentials, and then the finite set of chaotic points X QP one. So I thought I'd try to make this as explicit as possible to kind of convince you that we can compute these objects. So this is the case of genus two and rank one. And we'll assume that we have at least one rational point on our curve and we'll fix the base point V. So genus two, we have two dimensional space of regular one forms, call the basis elements omega naught and omega one. And since the rank is one, which is less than two by setup, we know that we can compute this set X QP one is the zero set of the periodic integral. So the next point isn't essential, but it just kind of helps simplify some of the computations. So if we know one more rational point P for which the divisor P minus B is not torsion and the Jacobian said some gives us a point of infinite order in the Jacobian. Then we can compute these two integrals, the Coleman integral between P and B of omega naught and the Coleman integral between P and B of omega one that gives us two periodic numbers and B, and then that gives us the constraints that we need to construct our annihilating differential. So taking, for instance, B omega naught minus a omega one gives us an annihilating differential. And then that is what we want to integrate and find all points Z for which that vanishes. And so maybe I should say a little bit about computing these intervals. So for hyperliptic curves, this is something that you can use sage to compute these numbers and B. And for more general curves. There is a package that young Topman and I put on GitHub to compute these values of Coleman integrals. So if you need to compute Coleman integrals either for this or for other purposes I encourage you to check out those two software libraries and let me know, especially if you find any bugs. So, in any case if you're able to compute these numbers, then you're able to compute the set of chaotic points for which the integral vanishes. And then you're able to compute the sex QP one. So, from that you might have to do a little bit more work to compute the set of rational points, but hopefully things like the more divisive will kick in and then you can determine the set of rational points. All right, so we've talked a little bit about finding rational points on curves. Maybe there's another question of which curves we actually want to do the sort of thing for. And, while sometimes we like to look at curves that have been mysterious for a number of years. And so one particular curve that was difficult for many years was a curve that appeared in the Earth Medical. So, DiFantis posed the following question. So, to find three squares, which when added to give a square and such that the first one is the side or the square root of the second, and the second is the side of the third. So, if we put this all together, DiFantis is asking if we can find positive rational x and y, such that the equation y squared equals x to the eighth plus x to the fourth plus x squared is satisfied. And DiFantis found a solution. So, he gave x is one half and y is nine sixteenths. So, from DiFantis's point of view, he was done, he solved this problem. But from our point of view, we'd like to see if we can find all the rational points on this curve. So, let's remove the singularity of the curve at zero zero. So, this amounts to determining the set of all rational points on our next curve now this genus two curve with affine model y squared equals x to the sixth plus x squared plus one. The reason this curve is very nice is because it's the only higher genus curve that's considered in the 10 known books of the Arithmetica. Another reason that this is nice is that it's just beyond the frontier of what we can do using shepherds Coleman strictly speaking. Why is that? Well, this is the genus two curve, but it's Jacobian has more del V rank two. So, just one more than what we can use shepherds Coleman for. Nevertheless, Joe weather all in his thesis show that the set of rational points in this curve is precisely the following. And let me tell you how weather all did this. So, this curve is a little special so it's a genius to curve, but you'll notice if you look at the equation here so y squared equals x to the sixth plus x squared plus one. There is an extra automorphism, right so at least one. So you look at x going to minus x in addition to the hyperliptic evolution, you have more symmetry than you would in a generic hyperliptic curve. Indeed, this is a bi elliptic genus to curve, and the Jacobian of this genus to curve is isogenous to a product of two elliptic curves. So what weather all did was to consider a collection of covering curves and apply which he Coleman on the covers, and then deduced the result about his curve about di fantasies curve. Alright, so by a cover of a curve I mean a surjective map from a curve F on to see. And whether all idea was to look at a cover or really a collection of covering curves, such that every rational point on his curve came from a rational point upstairs or on one of the curves in the covering collection. Then we can compute the rational points on our curve by computing the various sets of rational points in the covering collection. Now because the Jacobian is isogenous to the product of two elliptic curves in this case, they're very nice covers that you can construct this genius to curve. And whether all found some genius five curves G1 and G2 covering this genius to curve. And then he was able to further quotient out by some automorphisms, which then gave genius three curves that were hyperliptic. And moreover, those hyperliptic Jacobians had ranks zero and one. And so then Shabbatee Coleman kicks in and he carried out Shabbatee Coleman on those genius three curves and use that to determine the set of rational points on his genius to curve. So Shabbatee Coleman plus a little bit extra geometry looking at this collection of covering curves, and then carrying out Shabbatee Coleman upstairs is the idea that allowed whether all to tackle the second curve in our trilogy. All right, so taking this question a bit further for which curves do we care about and do we want to actually compute rational points on. So there are a number of interesting questions that come from modular curves. And so maybe one very nice example of this is this beautiful theorem of major that tells us if we have an elliptic curve over the rationals and P rational point of finite order and then N has to be one of the following values one through 10 or 12. And the idea behind this is to find the rational points on the modular curve X one event. So non hospital rational points and X one event correspond to elliptic curves over the rationals with a rational point P of order n. So Mazer's theorem is equivalent to the fact that the set of rational points on X one event consists only of cus if n is 11 or n is greater than equal to 13. So here is another modular problem. So we'll look at elliptic curves of the rationals and L of prime number. And then we have a natural Galois action on the L torsion points. And if we fix the basis of our L torsion, this is isomorphic to Z model C model. We have the following residual Galois representation row bar EL. And we can try to understand its image and Sarah proved that if he does not have complex multiplication, then row bar EL is surjective for L sufficiently large. So here went on to ask if this can be made uniform. Regardless of E, does there exist an absolute constant L not such that row bar EL is surjective for every non same elliptic curve E over the rationals and every prime L greater than L not. And the folklore conjecture is that taking L not to be 37 should work. Alright. If we're trying to understand the image of this model gal representation. Well, to show that it's surjective we want to show that its image is not contained in a maximal subgroup of GL to FL. So what are these maximal subgroups, we have the borrel subgroups that were triangular ones, exceptional subgroups, those with projective image a four s four or a five. These are the normalizers of the split carton subgroups. So these are the ones that can be diagonalized over FL. And then the normalizers of the non split carton subgroups. And the idea and doing this analysis is that for a maximal subgroup G. There is a corresponding modular curve XG over the rationals, for which the non hospital rational points correspond to elliptic curves with the image of this gal representation containing G. So now we've translated this question about looking at these images into, well, we've translated this question about looking at the possible subgroups into question about the possible rational points on these modular curves. So what do we know about Sarah uniformity. So in the borrel case. This is handled by major his work on rationalisogenes of prime degree, the exceptional subgroups were handled by Sarah. The normalizers of the split carton subgroups. This was work of below and parent. And the non split carton subgroups are much harder and essentially, not much is known. So this is pretty wide open. What I'd like to focus on is the work of billy parent and then billy parent rebella. So they showed that the split carton curves, their rational points are costs and CM points for L greater than or equal to 11, but they couldn't say much about 13. Alright, so what went wrong at 13. So a number of things went wrong at 13. In fact, they refer to 13 as a cursed level. And one thing that they use is Mazers method for integrality of noncustital rational points, which uses the following decomposition the Jacobian. So the Jacobian of the split carton curve of level L is isomorphic to the Jacobian of X not of all squared plus, which in turn is isomorphic to the Jacobian of X not of L cross the Jacobian of X non split of L. And Mazers method applies whenever the Jacobian of X not of L. This is what J not of L is whenever this is not trivial. And this is true for L equals 11 and L greater than or equal to 17. And this is the first thing that goes wrong for 13 J not of 13 is trivial. So the first curse of the cursed curve, and there are quite a few is that the Jacobian of X split of 13 is actually isogenous to the Jacobian of X non split of 13. And there are not any nice factors that we can take out and analyze the Jacobian of X not X split of 13 is absolutely simple. All right, so we said earlier, the Jacobian or the non split carton curves are kind of more difficult to deal with and so the fact the Jacobian of the split carton curve is isogenous to the Jacobian of the non split carton curve level 13 doesn't bow ball for us kind of suggest that maybe things are somehow more difficult level 13, but it gets worse. So, virtually on did some very nice work on the non split carton curve level 13 and the split carton curve level 13 she computed explicit smooth playing quartic models and showed that actually the curves are isomorphic over Q. So, the curve at level 13 that split carton curve level 13 actually turns out to be isomorphic to the non split carton curve level 13 not just that their Jacobians are isogenous, but the curves themselves are isomorphic over Q. And there's no real good reason for this there's no modular interpretation of this exceptional isomorphism. And again we said that the non split curves are much harder and so now we find ourselves dealing with the non split curve of level 13. So, here is Farron's model for X split of 13. So this is that curve that I showed you earlier and so now we can maybe try to analyze this curve and see what we can say about its rational points. So of course the first thing that might want to try is the Shabbat C Coleman method. So, what can we say about the rank of this, first of all, that's our third curse. So the, the rank of this Jacobian is at least three. Beyond Shabbat C Coleman. And there aren't any nice covers that we construct, at least not obviously to kind of use the second idea of whether all. So we're really out of luck here, and we need to do something else. But this is exactly what Kim's method proposes. This is kind of the right sort of example to consider in Kim's method. One can possibly hope to carry out Shabbat C Coleman ideas when the rank is greater than equal to the genus using some non-Abelian geometric objects and corresponding iterated Coleman integrals. So, in this situation, or kind of beyond the frontier of Shabbat C Coleman, we would like to construct some non-Abelian geometric objects associated to our curve. So this is what's often referred to as looking at Selma varieties. So these are cut out using homology and some local conditions. And so this should kind of feel like the construction of Selma groups and setting elliptic curves. And these summer varieties give rise to a sequence of sets. So XQP1 was the first Shabbat C Coleman set that we were looking at. But there is a depth 2 set XQP2 that's cut out by these double Coleman integrals and further endfold iterated integrals give rise to further refined sets XQPN. So the hope is that maybe we can compute the equations that these iterated Coleman integrals have to satisfy, and then we can use these now to cut out these finite sets of piatic points and then extract rational points from these sets. That's the idea. Alright, so Kim has conjectured that for n sufficiently large, the set XQPN is finite. And this is implied by standard conjectures like block HATO. So what do we know now what's what's known in practice. Well, one of the first nice results in this area to do coats and Kim for curves whose Jacobians have CM Jacobian. So for n sufficiently large the set XQPN is finite. And Ellenberg and Haast showed that that can be extended to give a new proof of faulting theorem again. So for n sufficiently large we have some finiteness and then that applies to curves whose curves that can be written a solvable Galois covers of P1. So both of the first two results are showing this asymptotic finiteness that there's some controlled dimension of Selma varieties as n gets large, and that gives you the finite result that you need. In a different direction together with Dogra, we showed that for curves X over the rationals with genus at least two and the rank less than G plus the rank of Neuron severity of the Jacobian minus one, then the set XQP2 is finite. And this XQP2 now is kind of this next set cut out by double integrals and computing XQP2 is what we call quadratic shepherty. So we would like to compute the set XQP2 because it turns out that the curse curve satisfies this bound that the rank is less than G plus the rank of Neuron severity minus one. And so if we can compute the set XQP2, then maybe run business and we can compute the set of rational points. All right, so more broadly, we can hope to use quadratic shepherty to compute XQP2. In the case when the rank is equal to the genus and the rank of Neuron severity is greater than one. But there's some difficulty making all this effective and implementing this on a computer to get some sensible answers at the end. So the functions, you know, where do they come from? Well, it turns out that they can be expressed in terms of piatic heights. And the idea is to move from the linear relations among these abelian integrals and the shepherty Coleman method to bilinear relations among these integrals. And these piatic heights have a natural interpretation using piatic differential equations and the constants can be computed using a collection of known rational points. So there is also still a bit of luck here involved as well that you kind of need enough rational points on your curve to start off with. So not only do you need to know the rank of the Jacobian and to have some explicit work on the Jacobian as well, but your curve naturally to have a collection of rational points and sufficiently many to be able to start this process. Right, so just a little dictionary between the two methods classical and quadratic shepherty. So the Jacobian now is replaced by this non abelian object the Selma variety. The integrals are now replaced by some iterated integrals, and there's some bilinear algebra of piatic heights replacing linear algebra computing kernels. So to reframe what happened in classical shepherty Coleman, instead of just using the linear relations, we're going to try to use some bilinear relations as well. So let me just remind you about the linear relations so these are among the functionals integral from B to X for omega regular one form in the case when the rank is less than the genus. So this gives us some obel Jacobi map with base point AJB. And what we like to do now is to generalize this to consider now the bilinear relations when the rank is equal to the genus. So this brings us to the definition of a quadratic shepherty function. So this is a function theta from QP points on the curve to QP, such that on each residue disc the map obel Jacobi base point and the quadratic shepherty function this is given by pietic power series. And moreover, we have an amorphism E, a functional C and a bilinear form B, such that for all rational points, the quadratic shepherty function minus the bilinear form on obel Jacobi base point and then slightly translated version of this, the and we can show that a quadratic shepherty function induces a function that vanishes on rational points and has finally many zeros. So this is how we'll compute the sex QP to I'm sorry to interrupt. There is a question please from Joel Rosenberg. Joel, will you please ask your question. Sure. So, earlier in the talk, there was an obstruction to using classical shabbatee in G equals three and R equals two. Does what what is that obstruction does quadratic shabbatee solve this case and is there a corresponding obstruction to quadratic shabbatee. I might have misheard the question. So you were saying there is an obstruction to genus three and rank two. I thought that was the problem in diaphantesis. I thought that was the setting of diaphantesis problem. Oh, okay. So yeah, and in diaphantesis problem is actually a genus two curve with rank two and so indeed there is an obstruction to using classical but you're absolutely right there is an obstruction to using classical shabbatee Coleman and genus to rank to whether all didn't use quadratic shabbatee. He had this covering collection, but you're totally right. You can apply quadratic shabbatee in that scenario as well. So this condition about now and so very ranks works in that case as well. And in fact, my student Francesca Bianchi worked out an example using quadratic shabbatee to solve diaphantesis problem again, and to show that it has just the rational points that whether all found. Thank you, Jennifer. Are there any other questions. Okay, great. All right, so in quadratic shabbatee we need to construct this function so we need to have some way of having this endomorphism is functional, and we need to solve for the bilinear relationship. Let me just say for this last thing solving for the bilinear relationship is similar to solving for the linear relations and shabbatee Coleman using linear algebra. And one natural source of bilinear functions for us comes really through the theory of chaotic heights. So the fact that this is quadratic shabbatee suggests that maybe there should be some quadratic functions, maybe taking patic values and patic heights are precisely the sort of tool that does that for you. So that's that's how we're going to find some of these quadratic functions. Okay, so to warm up. Before we get to the curse curve I thought I just say a little bit about how this can be used to find integral points on hyperliptic curves. In the case when the rank is equal to the genius and then how we can extend this then to rational points on more general curves. So the little bit of theory that you need now here is the Coleman gross patic height, which is a symmetric bilinear pairing on degree zero divisors on the curve. This takes values in QP and the height on D with a principal divisor this is zero so this is a well defined pairing on the Jacobian. So global height has a decomposition is a finite sum of local heights over the finite primes fee. And then how do you compute each of these local heights HP. Well, there's some distinguished crime P that you're picking this is a patic height so at P, there is something honestly patic that's happening. So there's a normalized differential that you construct with respect to splitting the hodge filtration on each one drop. And then you compute the Coleman integral of this differential the third kind away from P. This is more common tutorial, and you compute intersection multiple cities of your device is extended to a regular model. Roughly speaking, what is happening in quadratic Chevy T is to use this decomposition of the global height h is a sum of local heights. So, local height at P is something really honestly patic it's the solution to a patic differential equation. So local heights away from P, take on finitely many values on rational points, but in certain favorable cases, these are all trivial, and there are no local heights away from P to consider. And that's kind of a situation that we'll want to look at. So in the case of looking at integral points on curves, the main idea is to first compute this local height HP is a double Coleman integral, and then to control the finite number of values that the local heights take on integral points. All right, once we have those two pieces well the other thing that's going on is on the left hand side, the global height, we said was a quadratic form. So let's take down a basis for the space of quadratic forms using again Coleman integration so we can rewrite the global height h in terms of a basis for space of quadratic forms, and then everything on the left is a Coleman integral, and most of what's on the right is a Coleman integral so we move things around so that we have a Coleman integral set equal to finitely many values. And then, just like shepherty Coleman, it mounts to this pietic calculation of looking at pietic power series and understanding when they can take on each of these values. Right, so that's the theorem with Bessar and Mueller, then the case of a hyperliptic curve this can be made very explicit. When the rank is equal to the genus. And you know that these linear functionals f by the intervals of the regular one forms a mega I are linearly independent. So we have an explicitly computable finite set of values s and explicitly computable constants alpha ij such that the quadratic shepherty function theta, which is a double Coleman integral the integral of omega i omega i bar, where omega i bar is the dual undercut product pairing such that theta minus this combination of products fi fj this takes values in this finite set s on integral points. So this theta minus alpha ij fi fj. This gives us the analog of that function that we looked at the integral the nightly differential on the shepherty Coleman method. And then we just systematically set it equal to each of the values in this finite set s, look out all the possible pietic solutions and then take this across the whole set us. So this now gives us the sex QP to the case of integral points. So we'd like to do something like this for rational points. But there's one main problem here is that controlling the local heights away from P is quite difficult when X is rational but not integral. So in the Coleman growth setting these local heights away from P are essentially measuring denominators. And so if you want to study your rational points well you need to have some sense about the denominators look like, but this is, this is the whole thing right this is the entire question we don't know some absolute So this is tricky. How do we control the local heights away from P. So the workaround is to look at a quadratic shepherty function that comes to us from a slightly different construction of pietic heights one due to neck of our, and here instead of pairing divisors we're going to look at the pietic height on an associated gal representation. For every rational point we can construct a pietic gal representation a of X. This depends on having this data of a correspondence, a nice correspondence Z. And we have such a correspondence when the rank of Neuron Severe the Jacobian is at least two. And then we're going to make the following extra assumption that our curve has everywhere potential good reduction, because in this case, the neck of our height has the following very nice property that the local height each V of this representation is trivial. So then there are no local heights that we need to compute. And so we're essentially done here we just need to compute the local height at P. And this we can do using some pietic hot theory. So just like with the Coleman gross decomposition the neck of our pietic height has a local decomposition as a sum local height of local height at P of the representation plus the local heights away from P. The local height at P is a locally analytic function. This will give us our quadratic quadratic shepherty function theta. And by assumption since we're now going to put ourselves in the position that our curve has everywhere potential good reduction, the local heights away from peer trivial. And this will give us a quadratic shepherty function whose bilinear pairing is now this global height H and the anamorphism is induced by this correspondence Z. So with all the assumptions now that we've kind of stacked on. So if the rank of the Jacobian is equal to the genus, the rank of Neurons of very is greater than one. The pietic closure of the Jacobian has finite index and jqp access everywhere potential good reduction, and that we know enough rational points on our curve. So if we can solve the following two problems, we're in business and we can compute the quadratic shepherty set. So we need to be able to compute the local pietic height as a pietic power series, and then evaluate the global height. And so we can compute the known rational points, but really the second problem, computing the global height on each of the known rational points is essentially the first problem since we have everywhere potential good reduction, computing the global height is just computing the local height at P. Right. And so the cursed curve. There are many things that go wrong with the curse curve but some nice things about the curse curve well it satisfies all these conditions. So we can check and compute the rank of Neurons of very and so on, we can compute the rank of the mortal V group the Jacobian. And we can show the rank, the Jacobian is three. It turns out to have enough rational points so Galbraith found seven rational points and we can do a small change of coordinates to work with this model of the curve so that we can put five rational points in each of two affine patches covering the curve. Since the rank of Neurons of various three, there are two independent non trivial nice correspondences that actually give us to 17 at a kite so we work with 17, but there are two height functions that we can write down. And so it's a little bit better than just having one independent function so having two functions now we want to look at the intersection of the solution sets, and then to see what happens if this is precisely on the seven known rational points, or rational points. All right, so the theorem. This is joint work with Doug Romulo Taubman and Bonk is that the split carton curve level 13 has just the seven rational points that Galbraith found earlier. So this completes the classification of rational points on the split carton curves that was started by Billy Pajón Rebolleto. And then thanks to the work of Bertrand Barron. Since we know that the split carton curve level 13 is isomorphic to the non split carton curve level 13. We get for free that we know the rational points and the non split carton curve of level 13 as well. There's seven rational points there. And I think I'll stop there. Thank you very much for your attention.