 Thank you, thanks for the, thank you for the invitations with my great pleasure to speak here, and today I will be introducing work from my two projects that are joined with Victoria Cantaro-Fafan, Alina Mantovan, Rachel Price, and Yunqing Town, even though there are two projects, but this really started the same problem from two different aspects. So let's start from the concept of ordinary and super singular elliptic curves. Let's go back to our first graduate course on elliptic curves. We were looking at an elliptic curve defined over a finite field. FQ, right? We can even like when we think about the characteristics, not two or three, we can even very concretely just write the representative curve with a short well-stressed equation given by y squared equals to x cubed plus ax plus b, where a, b in the coefficients are in the finite field, FQ. Then we can count the number of FQ points, and this number is given by one plus cubed minus a integer a, and this a is called the trace of Frobenius, is the trace of Frobenius action on the elliptic model of the curve. And then we learned a concept, a thing that we learned a concept that's called super singular, which is a phenomenon that only happens in this positive characteristic case. So the elliptic curve is called super singular if the characteristic p divides this trace of Frobenius, and it's ordinary if p does not divide a. So then in the case of because of the Hasselbaum, we know that this a is bounded between, it's bounded, the absolute value of a is bounded by twice square root of q. So in this case, we know that, well, if the elliptic curve is defined over a prime field at p with p greater than two, then this divisibility and the Hasselbaum together tells you that this elliptic curve is only super singular if and only this trace is equal to zero. So meaning that it exactly has one plus two many points. And then we also learned that if it could curve is super singular thing to bear as many special properties, for example, it has a much bigger endomorphism algebra. And then it had it behaves also like now we know that it's the number of actual points is given by this special value. It has all kinds of special properties, making making the very interest topic of study. On the other hand, then the question, the essential question of today is how many, how many of them are super singular versus how many of them are ordinary. So, since super single as I said have all these very, very special properties, it is, it is makes sense that to us to expect that maybe they're less of them, right. They're really like some some some sort of special ports. So, well, one way to, one way to think about how many of the curves are super singular versus ordinary, we need to put it to curves in a family. So the first way to the first perspective will take is, if I consider all the curves over at p bar, and vary my parameter AB really just in fp bar, and then, but then we get, then we can ask a number of asomorphism classes of super singular the curve versus ordinary to the curve. Right. And we know that over algebraically closed field, if the curves are parametrized by j invariant, right, I can literally just know how many as if I want parametrized as a multiple copies of the curves over algebraically closed field. I just need to pick a j invariant in fp bar, right, so there are usually many of them. So, then, in this case, we see that there are only finitely many among all this you can see many j invariant p bar, only fantasy of them, give you a super singular the curve. And we actually know that on the number of isomorphism classes over p bar is roughly, it's roughly p over 12. It's basically linear in p. And this is this is one way that we feel that there are less super singular super similar to rare phenomenon because, because over the whole modular curve or p bar. There is a open dense locus of repeat curves which are ordinary, and there are only a finite many sets of them are super singular. This is one way to think to ask the question how many to cover super singular how many are ordinary right by literally parameterizing all the curves or p bar. There is another very natural perspective, namely, I can also take anything curve over a global field, and then vary the set of primes is global field, consider the reduction of the curve at these primes. And ask that, at how many primes my reduction are ordinary, and how many primes reduction are super singular. So this is really a very characteristic kind of phenomenon if you're taking the global field to be a number, which is a perspective of taking today. So, let's fix it if you could be divided over a number filled L, then I can consider the set of primes. I'll call them q in L in the sense like you can think about fq as the rescue field for this prime that lies above P, a rational prime P which is a characteristic of your rescue field. And let's consider the places where he has good reduction, which is just all but fancy many places of L. Then, um, then when I when I take the reduction I get to the curve over the residue field, which is a finite field, then we can use the definition ordinary and super singular. I will say that he has already action or at Q, if, if the reduction is literally ordinary to curve or the residue field, and I'll also refer to q at the ordinary prime for same simple super singular. Then when I vary this set of primes. Again, it's a very natural question to ask that, what's the density of the ordinary versus super singular primes, if I take one of the curve or global field and it's a curve, right, and I can see all the primes of this. This number field and constant reduction, how many of them are giving me a super single reduction versus all reduction. Well, let me first tell you one case where we have a complete answer by a complete answer. I mean, not only we know the density will, if you give me a prime and literally can tell you whether the reduction is ordinary or super singular. That's the case when the curve global curve has complex multiplication. So by the work of Shimura Tanyama. If I have a difficult e defined over a number of L, and it has CM by an order in the quadrat imaginary quadratic field q or join squared minus D. Okay, so I have already drawn the field here. So my difficult e is defined over L, right. So if I have a prime q, that's the priming L that lies above the rational prime P. And I want to ask, what kind of reduction, let's assume that that the Q is a good prime of good action for E. Then I want to ask whether my reduction is ordinary or super singular. All I need to do is to look at the behavior P in this quote imaginary quadratic extension, q to q or join squared minus D. So what happens is that, is that P, so P splits in this, in this quadratic extension, if I'm the only if e, e reduction, I'll see more P reduce at P is ordinary. So this tells me that the, so this tells me that whether whether the curve has a, has a ordinary or super singular action at this prime q in L is just given by a Legendre symbol, which is just a minus D over P. And if you think about the other, let's take the other prime or something by quadratic reciprocity. It really gives me a congruent condition, right, in terms of P. So all I need to care about essentially roughly, it just the congruent condition P mod D. Okay, so this tells us that in this case and I said it's a complete answer. Not only I know the density of ordinary primes. I also know, I also know exactly which primes I have ordinary versus super singular reduction, right here. Since the congruent condition. So we know that, we know that the, for, yeah, so so we have explicitly write out the density, okay, by chapter of density theorem. Okay, great. So then how about the non-CM case, which is a more generic behavior for the curve defined over L, right. Okay, so then it is a, it is a theorem there, saying that for any particular E defined over a number filled L, it said ordinary primes always have density one or one half. So one half case. So you really should think about this as ordinary prime has density one, because the one half case only comes from the CM picture that we had. So imagine you're basically the point is if your base field was q, if the curve was defined over q, right, then you should think that half of the primes splitting this extension, and the half of the prime is inert in this extension. So I get a half. And also you can see that if I base change, if my base field contains squared minus D, then when I count primes by norm, I only see the speed primes. So, so, so really, you should think about it as if you have a unique curve E defined over a global number filled L, roughly the set of ordinary primes has density one. Okay, so, so this is one side of the question is, so now we know when we ask how many primes give me order reduction versus single reduction, right. Now we know that density one set of primes, especially for non-CM curve, density one set of primes give me order reduction. So it's so natural to ask how about the super singular, how many primes give me super singular reduction, what's the complimentary set, right. Um, well. So in the, if you think about the modular picture in the first slide, when we think about all the curves depend over fp bar, we know that also almost all the primes, give our ordinary, almost all, sorry, almost all the curves depend of p bar. And then the compliment set is, is finite, it's a finite set, right, but in here, the situation is different. It was, it was in our case thesis, that he proved that for, for lots of number filled L, especially, in one, one special case is when L at the miss at least one row in bedding, then for anything curve defined over L, there, there are usually many primes, where the reduction is super singular. So in this reduction perspective, right, you see that you have a density one set where you have all reduction, but then the super singular set turns out to be infinite compared to the over p bar from the modular picture, where you have an open dense set of gene variants being ordinary versus finally many being super singular. Okay, so, um, well, the takeaway this slide is that if you take a non-seminity curve, which again the generic case or the occurs. Over, let's just take it for simplicity, let's just take it over a basic of q, right, for any non-seminity curve e over q. And then it has all reduction as a density one set of primes, and then had super singular action also have a usually many friends. So this is the takeaway of this slide. And then the goal of my talk is to journalize the work of Sarah and Elkis to a billion varieties of high dimensional a billion varieties. So as an example, I will be talking about the decoding of a curve with the following defining equation as y to the fifth equals x times x minus one times x minus t. So take the smooth projective curve with this often model. And as you can see that this, this curve has generous for. So we will be starting a billion fourfolds. Okay. Okay. So first of all, let me tell you the generalization of Sarah sword in this case. Before, before I tell you just the work we did, let me first tell you the generalization of the notion of ordinary and super singular in the higher dimension of a variety case to set up the problem. Right. Okay. So, well, the generalization of order versus super singular, this look has become much more complicated when the dimension of your ability variety gets gets higher. So let me produce a notion of Newton polygon. So you can pull it out is a notion that you might have already seen in, in a class of algebraic number theory, where you have a polynomial with integer coefficients, and then you can choose a prime P, and, and, and look at the points with. And the draw the points of the, the, the, the, the indices being I which is the, which is the label of the index of the coefficient, and then also the periodic evaluation of the corresponding coefficient, and then draw the lower convex call draw a polygon out of it. So you learned that okay this tells you how this polynomial factors over QP right over the completion, and, and exactly the slope, the slope of this Newton polygon tells you the periodic evaluation of the roots of this polynomial. And very often you can use that to prove that your polynomial is a usable over Q, right. And, and that's it's really generalization of I sense that's criteria, right. So in here, when we see Newton polygon for a billion variety, we mean that if I have a billion variety a defend again or a finance field, then I can again look at the Curtis polynomial for Venus. Acting on the added model of a, and it draws the Newton polygon of this, of this integral polynomial. So, what we do is we will, we will, we will normalize this polynomial to be to be Monic. And also, I will index it in the opposite way meaning that the Curtis polynomial will really be a zero x to the 2g plus a one x to the 2g minus one plus data, and it is symmetric right so the last terms are a one q to the g minus one x plus q to the right which, sorry. Yeah, so it is it is so so so so so if you look at the coefficients right this is my a of 2g, this is my a of 2g minus one, this is symmetric. Okay, so we can draw the Newton polygon of this polynomial. And then because we know the leading coefficient is always one, and then the constant term is always q to the q to the g. What will happen is your Newton polygon will always all the polygons for the operating variety with this fixed dimension will have the same starting point and ending point right. And you can compare whether two Newton polygons lie above each other or not. And, and, and this whether one lies above each other not gives the set of Newton polygons with this fixed g a process structure. What is the work of art, saying that this post that structure turns out to give a stratification of the modular space of principle a polarized operating variety over at people are just like the modular curve case, we've seen that there are two locus is right for you to curve for D equals one one locus is the audio locus which is open sense, and then the super singular locus has a co dimensional one right it's stratification. And let's look at another example, which is, if I have genius to, and I consider a to consider the concept of building surfaces for people are then in this case well first of all the dimension of the modular space is is three, right, and I'm not drawing a three dimensional case so I'm drawing the two dimensional case but I want to remind you that the dimension of a two is three. Okay. So this is what we mean by a stratification. It means that there is an open dense three dimensional locus that is ordinary. Okay, and inside this locus, then there is a one, there is a co dimension one sub locus, which is what we call almost ordinary it has p rank one. And then inside this almost ordinary locus, there is a co dimension to sub locus, which are the super singular up in the surfaces. So this is what we mean by a stratification of a D. Okay, and in this case, again, as you can see that as a generalization with the curve case. So ordinary locus is always the generic locus for for general for general up in a variety inside a G. Okay. And the and it is ordinary if exactly when half of the Frobenius eigenvalues have p added value or q added evaluation zero, and the other half has q added evaluation one so it's the the Newton polygon for the big curve alternative curve is have once look zero and once look one. And then the ability of variety being super singular super singular is always the highest co dimension locus in a G, and it's super singular if all the Frobenius eigenvalues have q added evaluation or half. So if a curve case is it you can also say that say that if that they have both of the problem is eigenvalues will have q added evaluation half. So we're using q added evaluation to normalize it so that the Newton polygon will be a geometrically invariant. So if I enlarge my base field, right, my, my Frobenius will reach the power and by normalizing the slope or the, the, the, the eigenvalue won't have the evaluation the eigenvalue will stay the same. Okay, so that as you can so now this notion of ordinary super singular, but generalizing, which now allows us to ask a similar question, given a higher dimension of any variety defined over a number field, what's the density of ordinary versus super singular primes when I look at the reduction just like the curve case. Right. And the one thing I wanted to note is that in this case, as you can see trace of Frobenius is not enough to determine whether the opening variety is ordinary or not. Unlike the big curve case, right, you actually need to know, like, if you want to know the Newton polygon right really need to know the, the, the, all the valuations of all the eigenvalues really. Okay. So, what, what, what do we expect, what also do we expect from this question. Well, it is a conjecture that was raised by Sarah, saying that for any opinion variety a of dimension G over a number of field L the density of ordinary primes should always be positive. Okay, this is completely general is any opinion variety over any number field. And so that's why we say that it's an ordinary prime. And so that's why it's not an ordinary primes. It's just positive instead of saying it's going to be one. Right. So, if you think about the curve case right as I said for the sea out in the curve, you may have the 31 half but some of the people in politics should be, should be what we should expect always. Well, what do we know about this kind of sure. So first of all cats showed that this kind of sure holds for for our being surfaces. And then later on we are starting with able to explicitly analyze the density of the ordinary primes. So the set of ordinary primes has been to one or one half or or one quarter, and the one half and one quarter case also comes from a building surface with extra endomorphism just like the similar to see a particular case. So there are other more other results in this topic, and I just pick some examples. So one example is that pink showed that this connection holds for certain ability and varieties a with trivial endomorphism, but a particularly small monfortate group by small I really showed the generic monfortate group. And then also very recently, fatigue proved that for also prove this connection for a for a families of a billion varieties, whose endomorphism now the endomorphism is not true anymore, then the morphism will contain some imaginary product field, and then there are some other conditions for for a to satisfy. In this case he was also able to show that the set of ordinary primes has positive density. Okay, so now I'm ready to tell you our work. So join to his contour of a fan mantle when present on this is our current work ongoing work, which hopefully will finish soon. We were able to show that if, well this case is definitely finished is as we are trying to see how much how generalized our result can go. Okay, so for currency. With the current with the following defining equation what's the fifth equals x times x minus one times x minus t defined over in number field L. Let's consider generic is which is when the Jacobian of state does not have CM by CM I really mean so this curve is a generous for. As I said so it's up in Jacobian is up in a full fold by CM here I really mean it does not have CM by a CM field of degree eight a big CM. Okay, if I if I consider that case, then the set of ordinary primes for set of ordinary prime type of positive density. And if I have base change to L join the fifth to unity, or if it is already in your base field L, then this density will be one. Okay, so this is so this is the case for for this particular family we're starting. So I want to make a comment that note that in this. So, for a curve this in this family right over, over, over L joined into five or q joined into five. What happens is this curve has an automorphism, very concretely by mapping x y to x, comma, Zeta five why right, because one way you can view this curve is over over the field L join Zeta five. This is a degree five cyclic cover of P one, and there is this cyclic cover this this degree five cyclic group acting on the curve right, we will model that out to get P one. Okay, so then this automorphism induces a endomorphism of a decoding. And in particular, when the, when the decoding does not have CM really, or for generic member of this family of the opinion varieties. The endomorphism exactly is cured on top of it. And our work, and we expect our result to be for opinion varieties with with this essentially roughly this type of endomorphisms. So for example, one case we currently know how to do is, is that if the endomorphism is just like this case equals to a CM field of degree D, which in this case because this is the degree for CM field. So this is, so I just want to say that our, our general result will be for opinion varieties with certain types of endomorphism. Okay. And another comment I want to make is that as I said, this theorem is about a generic member of my family, mean that Jacobin does not have complex modification. When Jacobin does have complex modification, we can also very explicitly compute the set of ordinary primes using the Schmura-Tanyama formula just like the decode case and give a complete answer like the decode case. Okay, so really the non-CM case is the more difficult and more interesting cases and also generic case. Okay, so then let me tell you a little bit about the Newton polygon of this family and also the, yeah, and why, why this is a family that we chose to study. Okay, so if you consider the family of Jacobins for curves or Q bar, defined by this defining equation, right, then at a prime that is not equal to five, then the reduction is a, so then the reduction is a billion fourfold corresponding to a point in, in the, in, in A4, Fp bar. So note that, so I have a, I have a generous four curve. Of course, Jacobin is a principle formula as a billion fourfold, right. And if I look at all the possible Newton polygons with G equals to four, you can, you can draw them yourself. And there are, there are eight possible Newton polygons, so there are eight Newton stratos. So this is, this is why in general, starting the Newton strata of, of a billion fourfold is difficult because there are so many of them. But, but this is not any opinion variety, right, as I said, the, this opinion variety at the mid endomorphism of Q or joint data five. So because of the actual endomorphism, actually when you look at the Torelli image of, of, of these family curves. So it really like this image, it lies in a PL type Schmura sub variety of A4. And this sub variety S actually is one dimensional. This is really just a Schmura curve looks very similar to a, to the modular curve like the curve case. Okay, so the model of this family looks very similar to the curve. And it is a, yeah. So this, this follows the work of the limostro and the moanen. And then, following the work of Robert pot caught with research, or what, where the home and the, we know that the Newton strata for for you need to reach more variety. So in this case that as I said our S looks like very similar to Jane J line right, and it turns out that for any prime P that is not five. If I consider as a VIP bar. Again, just like the modular curve case. It has two Newton strata. One of them is open events, which we will call new ordinary so you can think it as a similarly to the ordinary locus in the J line of the curve or a P bar. And then there is a closed locus with fancy many points, which we will call it the basic locus. Again, it looks very similar to the super singular locus for the J line recurve, but note that even though this picture, this picture I'm drawing does not depend on P. For any P you have two locus, one of them is one of them is open sense new ordinary while with them is fancy many points basic for any P. But when you change P. So corresponding Newton polygon may change even though that all two locuses, but what's Newton polygon you should look as corresponds, or more concretely, what's the periodic valuation of the problem is again values are for these two locuses will change. Exactly depends on the congruent condition modified. Okay, so when P is one of the five, the new ordinary polygon is ordinary, and the basic polygon has p rank two. And when P is two, three or four more five, then what happens is the new ordinary polygon is not ordinary anymore. And the basic polygon has has p rank zero it is super singular in the actually I mean this is in this case. Okay, so I just want to emphasize that, even though for any P the pictures are the same, but what's Newton polygon, these points corresponds will be different. Okay, so now, after analyzing the modular. So as I said, like, so if you are starting a general operating four fold, there are eight possible you can spread us right, but then now in this case, I know that actually my operating four fold, when I reduce and each fixed P. There are only two possibilities. Okay, so this makes the problem slightly easier to study. Okay, so now let me tell you the general strategy. Not only for for our project, it is a general strategy on how to, how to, how to do these problems, like how to study the sensitive ordinary primes and generalize stairs, stairs work. Okay. Well, so first of all, let's talk about a observation is a very simple observation. Since we are only considering the density of primes, right, where I count primes by norm, or I can say other sweet price, why when I say other primes, who's resting field of prime. So this means that if I think about reduction and I want to start the density, or all I need to think about is on the sweet primes meaning that locally, or at the reduction, or I need to think about are the operating varieties defined over prime field. Okay, which is a simple observation by health. Okay, so now, let's think about it if you curve, if I know let's think about at what times P, my little curve of the maze or reduction. So as we already discussed that if my curve is defined over a prime field. It is ordinary. If I'm the only if, if I'm the only if my tracer for Venus is not a zero. Okay, so because coming from the coming from the hustle bomb. Okay, so this means that my prime my face build is prime. Okay. So this means that this tracer for Venus a. Okay, is a invariant of Frobenius, such that, first of all, it asserts whether if the curve is ordinary or not. Okay, I have one invariant that tells me whether my if you curve is ordinary or not. So second of all, when it's not ordinary. It takes a finitely many integral values, and this integral values are independent of P. So exactly a equals in this case is exactly a equals to zero. Okay, so when that if I have a invariant that tells me whether he is or not. So in particular, when it's not ordinary, it takes a value zero, whichever P it is, it takes a value zero, okay, the integral value that is independent P. And the same thing happens on the surface. Okay, so if you look at the characters for the more Frobenius property surface, as I said, it's symmetric over P is symmetric. It is ordinary. If and only if my middle coefficient, if you think about it in polygon is even only if my middle coefficient is a periodic unit. Okay, it's ordinary if P does not divide a two. So, then, again, if you use Hassabon, you see that a two is bounded by six, six times P, which means that if P is not ordinary, I can start the invariant a two divided by P. Okay. So if it is ordinary, then this invariant a to my over P has to be in this set of finite integral values that is independent of P. Okay, so for opening surface, there's also this very natural natural invariant. Which is a two over P. That also exerts when your ability surface is ordinary, and also takes finitely many integral values for nonordinary services that's independent of P. Okay, so this is the first step for doing this kind of problem is you want you want such invariant. Okay, because this invariant really gives you a function on the attic monotony group, such that the density of ordinary primes is bounded below by the ratio of the connect components of the attic model group, which is function is non constant because because this function turns out that it was on his proof is function actually is only constant when your value is interval. Okay, so by chapter of the city so this will already this will give you a, um, this will give you that they are, they are positive density set of primes, which have already reduction, as long as you could prove that there is at least one kind of component where your function takes a non constant value. Okay, so the first, the takeaway is the first step is to find one such invariant for the ability of writing you want to study. Okay, then the second step is that I said is now you need to, you need to study on the attic monotony group. So what value stuff might be very take. Well, so, which means you need to start the connect components also started to collect components of attic monotony. So for the curve case, this follows from stairs open image theorem. And then in the, in the obedience surface case, Southern, that he was able to explicitly compute the ratio relies on the work of, of the take a lot of Roger Sutherland on the definition of a subtle take groups for our building surfaces. Okay, so this information allows them to explicitly compute the density of ordinary primes for the curves and our building surfaces. Okay, so for our case, because of this cure join data five action, so this is this data five action. So, we were also able to define functions on Frobenius, based on, based on is the image in our war. So, based on whether which Congress class of mod five, the prime has, and then the value of these functions also exactly have the property that we want. It does this when my, when my ability of writing has a new order action. And also, when it's not to me or it takes a bounded integral value of independent OP. And also then the second step, we use the fact that we use the fact that we know, over which field that I think monotony is connected, and the vassals work on them offer a texture for our case. So really, in our case, roughly the analytic monotony group is not that different from the unit for the for the for the for the for the Shimura curve we have. So this is why we're going to conclude really that so our actual results will be that the status of a new order in France for any, for any Jacobi in this family has been to one. And then you recall that we said, ordinary prime has positive density. Well, because, as I already told you that when P is one more five, the new ordinary locus is ordinary. Right. So this means that if my new order has 31, then at the positive density, I am ordinary. And on, moreover, if I can't prime by norm, that if my base field contains data five, then all I can see other prime was characteristic or one more five. And this is why, when the, when data five is in your base field, the density of ordinary prime becomes one. Okay, so this is how you show that there is a positive density of friends that bears ordinary or new order reduction for a bigger variety that you're studying. Okay, so then the next, let's recall our case work started the super singular that the complementary set. Right. Okay, so I'll recall that our case shows that if you have any difficult if you find the work you, or, or more generally a large set of number of fields. There is this infinitely many primes at which the reduction is super singular. And then there are some analogous results for a billion surfaces with Quaternion multiplication. And so these are all of the surfaces case. And you, and now I want to talk about our family curves. These are again up in a full fold. Okay, so, but instead of working on any of them. Now we need more conditions. So let's take C to be a smooth product curve with a defining equation y to the five equals x times x minus one times x minus t. Okay, then I'm going to define a rational function called a J. That is a rational function in key. And I asked this J value to be in Q, and it is between zero over and the 27 or four, I will explain, I will explain what this, these conditions mean in the next slide. And this, so you can think of this as a condition on the Archimedean place for, for my family for my curve. And then, as the prime five I also have a special condition, I asked the reduction of CF file to be singular to be a better doctor. So again notes that this is a family of generous four curve. And my singular fiber actually looks like the intersection of two generous two curves at intersect as a node. So this is what my singular fiber looks like. Okay, I want to reduction of CF five to look like the singular five. So if these two conditions are satisfied, then we were able to show that there exists infinitely many primes on where Jacobian C or the miss basic reduction. So recall that we have previously we proved that at the density one set of primes Jacobian C or the miss order reduction. Right. So now we're we're generating our case in the sense that we are proving that is basically zero set of primes is infinite. So, let me let me next slide let me first explain to you what these conditions really mean. Okay. So, well, what I have this family of curves, some of you may already start to think that this is a defining equation looks very similar to the LeJongere equation for the three curves, where you have y square equals to x times x minus one times x minus lambda, where I can parameterize magic curve law by lambda. Right. But on the other hand, we also know that the information of lambda really gives you a two torsion point the magic curve. So lambda is not just parametrizing the isomorphism classic curve that the actual level structure. So same thing happened here for tea. Of course I can parametrize my curve by varying tea, but then by varying tea is not a one to one correspondence to the isomorphism classes of the curve in a family. So, just like the lambda to J map, which is a six to one map, I can just model this ambiguity coming from tea, just by defining what we call the generalized J function, which is this degree six rational function in terms of tea. And then just in just the same way as a correspondence between lambda and J, this is even the same function because you are modding out with the six, the six possible lambdas or possible teeth corresponds to a J. Okay, so then this J will uniquely determine the isomorphism class over Q bar and FP bar for my curve or for Jacob Jacobin for any P that is not equal to five. So, so J is a parameter for the course modular space S, which we already, we already showed up in the previous, in the previous slides of the family. And from this, this parameter we actually obtain a Q model for the for the Schmura curve, even though that's the reflex field for the Schmura curve is Q drawn data five. Okay, but it actually has a Q model, which is actually very important crucial for our work. So the theorem, when we ask this generalized J value to be in Q, it means that the field of modularity is Q. So you can, that's why you can think it as we are really generalizing LK theorem for the curve divider were Q. Okay, so, and then, well, what happens for five. Okay, apart away from any prime away from the prime five, the Schmura curve S has good reduction, and we've also already discussed that. In this case, if I consider S over P bar, then there are two Newton stratus, one of them is new order and one of them is basic. But as the prime five, S has a bad reduction, and there is only one Newton locals there and it is for singular, meaning that if you take any of your variety in my family and reduce more five, it'll be super singular. And this is essentially why we will need to impose the condition at five, I asked my seat to be a bad reduction at five. So, the condition of bad reduction can be replaced by some other condition, but in our proof, because of this fact of the five, we will need some condition five for our C, which we all explained in the proof, why we will need some condition. But here I can tell you that the reason we need a condition at five is exactly because the modular, the Schmura curve has a bad reduction at five. Okay, so one thing I really like about these projects is that I give you one a billion variety, like a handy one a billion variety defined over a number of years, and I want to start this reduction at different price. And it turns out that instead of starting this one a billion variety, I'm really I need to know which family is a billion variety behaves. And then I need to know properties of the modular space that this ability that parameterized the family was a billion right it is like this ability of writing even though it just one single I've been right it remembers which family belongs. Okay. So, and this is a prospecting to pick. Okay, so now I explain to you what the means for j to be in queue, and, and essentially where why we would need a condition five. Let me explain the weird, the weird interval zero and 27 or four where that comes from. Well, to do that, we need to think about the, let me tell you a little bit again more about the family where it's a very variety belongs. Let's think about the complex points of the Schmura curve. This S over C is a compact in a curve, as I said with reflex your cure is a five more over it. It is a, it is a triangle modular curve for the triangle group delta 2310. Okay, so it looks very similar to the, to the modular curve where we know that the modular curve X one is H as upper half plane mod S or to the right here, our curve over C is given by upper half plane modulo this triangle group. Then the fundamental domain of my S is two copies of a hyperbolic triangle which I draw here. So, so because our S is really just a P one right it's paralyzed by J this one function so it's a P one. And you can think that you take two triangles, a thing that looks the same and the glues them together along the side this gives you a sphere, which is a complex P one right. Okay, so what does the fundamental triangle looks like. Well, the fundamental triangle it has three vertices. Each, each corresponds to a curve or I've been a variety in my family with extra automorphism. Okay, so the vertex P correspond to a degenerate curve as I said it is two copies of a generous two curve. But note that my Schmura curve is actually compact. Okay, so because the generator coping for this genius for for this curve is still a billion fourfold. So, my, so, so this is not a cost. I want to emphasize this is not a cost. Okay, even though the curve it corresponds is singular but the ability in variety is good actually. It's good. Okay, so it corresponds to actually smooth up in a variety, not a semi up in a variety. Okay, so. So the point is because it's too coffee genius to curve extra extra automorphism coming from it I can take automorphism or one copy and fix the other copy. So that other smooth curves don't have this automorphism. And then there's a point q with a extra automorphism of order three with J value zero. And then there's another point are with extra automorphism of order two, with J value 27 or four. So you can think of this as very similar to the busy curves with CM by by cure to I and the cure to the three. Okay, so then, when I asked the J value for my curve I started to be between zero and the 27 or four means that I'm asking my curve the motive I point to be on this side. between this q and r. Okay, so the sides of this triangle is actually all the real points on my smaller curve. And, and these are geodesics in the upper half plane. Okay, so this condition is restricts our motive point to be on the side qr. And the reason here just a technical reason just in our current work. We know much better. We know more information about this geodesic this side that the other two. So we will, we will hope to work to get rid of this, this, this restriction, this is our current current restriction from our pool. Okay, so this is how I explained the serum. Let me tell you a little bit about the strategy. Again, let's think about the strategy of elk is cool. Because when you when we ignore all the details roughly, that's the same strategy, just like the case for the ordinary right is the same strategy always. So the proof for this for this serum is, is like the proof of there exists evenly many primes, right, how do you prove they're using many primes. Well, I assume there's a finite set of primes, and then I construct a new one. So it's similar here. So the point is another way to visualize it I always think it as like a fishing game, where there's an ocean of primes, and I'm building some fishing nets to catch some primes with a special property. Okay. Okay, so well what's the property do I want is exactly the property that my fix it because this is my fix me. I want to start it. Mod p is in the super singular locus of my motor a curve. Okay, so what are my fishing nets, what my fishing nets will be see Emily curves. The reason is the following. The reason is that I won't start it where he has super singular action, right, let me start with some of the curves where I know exactly where they have super interaction. And as I explained the CM was whether they have ordinary super direction comes from in the jungle symbol that we discussed before. Right. Okay, so things that I know I build some fishing nets which are standing curves. And what are the primes will catch me. Well, it exactly catches a prime where E, and it's super singular, sorry, this CME isomorphic modus p so this exactly when the intersect with the respect in the section right. So, so every CMD curve will catch me some primes by intersecting my E. Okay. So, okay, so I want, I want to catch infinitely many friends where I have super single reaction. This means that I want the prime that catch to be a super singular. So what do I need to do. Well, I need to guarantee that each fishing that will intersect a super singular locus. Okay, okay, every time I saw that this fishing actually doing something, right. But on the other hand, I want to catch infinitely many. Right, this means I need to make sure I have this infinite set of super singular prime, a CMD curve, I need to make sure they don't intersect at the same single point. So what I'm going to do is as I said, I assume the finance that I just need to make sure that the next one. I'm sorry, I just need to make sure that the next one. The next one will not intersect the same super singular prime as the previous one, right, then, then I'm done because I have infinitely many each will each one will catch me at least one super singular prime. And once I've been caught by this net is not the ones that have already been caught. Right. Okay, this is a general strategy, and this is also a general strategy for our work, which I don't have time to discuss. So in the last minute, let me explain one thing, which is the condition at five. Right. So the point here there is, how do I guarantee the new fishing that will not catch me a previous prime. The easiest way is, I want to say that for the order for all the primes that I already know of super singular action, I can construct a CMD curve, which has all the reduction at all the known primes, so that I can guarantee the new one I guess is actually new right. So this would work for any prime that is not at five. Because remember, for all the opening varieties in my family, they all have super singular action at five. So I can't, I can't use a Newton's strata to guarantee that the CM curve I construct, not intersect my original curve at five. So I need to use some other discrete invariant to tell me to guarantee that even though I have infinitely many fishing that they are not just all catching five. Right. So the current condition we impose is smooth versus singular. But we will also work on trying to, to use other discrete conditions instead of that one to be able to generalize to make our system more general. But this is behind why we need a condition at five. So I'm running out of time. So I don't have time to tell you the schedule approved. So let me just finish there and thanks for your attention.