 I'd like to thank the chair for the introduction. I'd like to thank all the organizers for having me here, and it is a great honor and a pleasure for me to talk at this conference. In honor of Professor Izu-Zi, let me say a few personal words. The first time that I met Professor Izu-Zi was when he was teaching a course, an algebraic geometry course at Tsinghua University, and I attended the course as a student from undergrad student at Beijing University. And that year was in 2009, almost 12 years ago. And it is, I feel it's kind of like one of the most and the key assistance thing happened to my life, my mathematics career that I managed to learn the modern algebraic geometry from the master of the subjects, and it definitely opens the door of like other asthmatic geometry to me. Even though I did not do my PhD in France, but during the time that I think there was some email correspondence between me and Luke and he was so nice and generous to answer all naive or maybe all kinds of questions, mathematics questions or some life related questions for me and also introduced me to some nice restaurants in the Paris area. I really appreciate all the encouragement and the kindness of Professor Izu-Zi and it's very hard to express all my gratitude towards him. And today I would like to talk about the joint work with Wining-D and in the mental and the racial priests about basic reductions of opinion varieties over totally dual fields. So I would like to start my talk, which actually is like one of the motivation for the project is a theorem of Elkies, which says that if we have a elliptic curve, let me stick with the case when it's defined over the field of rational numbers. And then he proved that there are infinitely many super singular reductions. And let me just make a remark is that it's, I think, and now I would say classical back to the, maybe 80s, as we know based on probably the work of Serum that if we have a elliptic curve over a number of fields, and then the set of ordinary reduction is of density to one, maybe after possibly a finite extension of Q. Okay, so what Elkies theorem said is like, we have a set of density zero and he managed to prove this set is has cardinality to be infinite. And then like to motivate the case that we would like to study, I would like to point out a key input in his proof is that, so we use the moderate space of generalized elliptic curve, which is the compassified J and I, and this one is actually, okay, this is a cost more dry space, but I'll just say it's isomorphic to Q one over Q. Here we have Q one, which is genus zero, and Q is a totally real field. So it explains something about the title. So actually I would say we are grateful to Liangxiao who read some of our previous work and the point out to us is like, there are some more dry space of a billion varieties, which also happens to have this property, and actually we would like to generalize Elkies theorem in this direction. But so before we move on, let me just support the question here. So how about the varieties parameterized by some Schmarck curves? I would like to first make a remark is the odd reversal super singular thing. Let me just stick with the PL type Schmarck curve, although like everything heuristic that I said will still holds for harsh type or a billion type Schmarck curves. Let me stick with the PL type Schmarck curves, and it is the work of Vian and Wethon, who treat so-called Newton's stratification, Newton's stratification for Schmarck curves of PL type, and their works generalize their results to harsh type Schmarck varieties, as we know that, okay Schmarck curve as a generalization of the J line, it has like at modulo a good prime, namely a prime that the Schmarck curve has good reduction, which I will specify later for almost all primes, that we have two Newton's stratum. In this case, you could think about just like using the Newton polygon of the, of the unit variety, the H1 of the unit variety. So you have two Newton's stratum, one is called the mu ordinary, which corresponding to the ordinary one for the J line, and the other is called basic, which corresponding to the super singular case for the J line. So we have two Newton's stratum, this is the open dense one, so which is one dimensional, and this is the finite one, the finite close, and this is the zero dimensional stratum. Okay, so we can replace our question by saying that when we have a unit variety parameterization curve, can we have infinitely basic reductions? Okay, but before that, I would like to continue this remark saying that the, we know, okay, I'll put some name here, but it may not show up in, it's basically essentially using the idea goes back to Sarah, also the idea of cats in the written down in a paper of August, you may also see the work of Pink or reason to work of Fittet. So basically it's a kind of like a standard results you could prove is like, so after a possible finite extension, let me just know this by a unit variety over K is a number field. So finite extension of K, we have density one set of primes with ordinary reduction. Here I just say ordinary when I pass myself to a finite extension, then in the most split primes that the new ordinary Newton-Pyagon is actually the ordinary ones. So which means that the question, so if we look at the set of basic reduction primes A over K is of density zero and our hope is to prove the humanity of this set is equals to infinite. So what is basic reduction point? So basic means that, so when I parametrize, so I have been a variety over K parametrize by certain Schumann curve and then by the theory of Newton strata for Schumann curve, you have two strutters and the one is mu ordinary, one is basic and basic reduction means that when you mod that prime, you hit the basic locals. Yeah, thank you. I'm sending this density zero. Mm-hmm. So I guess is it only after, let's say for your ellipticals, when you look at the singular form of duplication, you can have half of them are ordinary enough. So when you say density zero, you mean after an extension or you mean something else? Okay, so good question. If I just assign to their results directly, I mean something else, I mean after an extension, so which means is like, there are some cases one could prove this part easily by a charcuttar density because without pass to a field extension, the basic reduction may corresponding to a positive density locals, which is exactly happens the case when we talk about opinion varieties parameterized by Schumann curves. It's exactly happened when it has CM, then yes we have, it may or may not be half-half depends on what like CM type it is, but it will be a positive density over the base field K. But I would like to mention the work of, or Simon who treats certain kind of these type of results. So while Simon has a paper about opinion surface and he actually gives you an answer on what is the exact finite extension we need to take? And I expect is like you're using similar ideas, which he used like the set to take type of groups, one should be able to prove is like, if I do not have extra on the morphisms other than the PL structure given the Schumann curve, then we should not need to pass to a field extension of K and hence like this density will be over K. Okay, so and who did you mention? So let me just say, you may look at the work of Will Simon, but he did not treat exact the case that we want here, but I expect the similar ideas will hold. So that's a great question is says like, and these I expect there are specific cases like we won't be able to use a trapezoidal density theorem to prove this type of result. Okay, now go back to the key inputs of Elki's proof. We need a genus zero Schumann curve defined over some total real field. And then here's the candidates. We're grateful to Liang Xiaofu point out these to us like in some of our previous work. Okay, so we want to Schumann curve of genus zero and has some model. Let me emphasize the word has some model over some total real field. So actually we could think about certain Schumann curves arising from so-called cyclic cover mu M covers of P1 ramified at four points. Here I send ramified at four points because I want a one-dimensional family. So when I have four points on P1 then up to equivalence it means it's a one-dimensional thing. And actually in the work of Mounen he gives many examples. So we start from the model space of curve which ramified at four points. It may or may not be Schumann curves and Mounen gives you many examples when a Schumann curve actually coming from these cases. Why did you say four points? So four points gives you like this model space has dimension equals to four minus three equals to one. When I have four points on P1 up to, yeah. Okay, so let me just stick with one example instead of talking about all the families in Mounen's table. So key example for today. And I think our strategy will like generalize to all his families and some other ones. So we would like to think about M equals to five and we think about mu five covers with so-called inertia type 1112. I'm going to explain what inertia type. So let me just allow myself to pass to q adjoint zeta five where zeta five is a fifth unit so that I can actually write down the equation easily. So I would have y to the fifth equals to x, x minus one, x minus t. So this is the parameter t which gives the one-dimensional family and this 111 corresponding should hear 111 and true is the index at infinity. Okay, so we can study these type of curves and it happens that if we take the twirling map, if we take the twirling map we'll have an Abenian family, Abenian varieties of dimension four and actually this is one connected components of the following PL shimmer variety which parametrized so principally four lives of a Bbenian four foot such that we have an endomorphism. So we have z adjoint mu five contained in the endomorphism of A and plus a cold-width condition which coming from the inertial type ramification type. So the cold-width condition is that there exists an embedding of q zeta five into k. So here I write the kind of the Mojai problem overfield but you could also make it over anything away from five. Okay, actually you can make it to be over speckly but let me for simplicity write it for field. So there exists an embedding of this one into k for the such that now I have q zeta five X on the algebra of A over k with characteristics polynomial of zeta five looks like t minus zeta five t minus zeta five square and the t minus zeta five to the fifth sum. So in other words, it's like when I have this is the cold-width condition but let me reformulate it as a unitary shimmer variety. So a unitary shimmer variety corresponding to an emission space V over q zeta five q adjoint root five. So this is the total real subfield of q adjoint to zeta five of dimension two. So this space coming from, so this emission space coming from the h one beta co-homology of the Arbenian variety and the one with the synthetic pairing coming from the polarization but the synthetic pairing is compatible with the anamorphism and thus it makes it into a anti sorry screw some emission form and then we divided by some up to some choice. You can have an emission space of dimension two and this cold-width condition here gives us that such that the signature. So we have two embeddings of q root five into q bar. So for the embedding root five maps to root five we have signature one one corresponding to this one and this one. And when I have root five maps to negative root five I have signature two zero. Okay, so let me just summarize. It's like this is a particular example where we could think about it as a one dimensional family of curves which is indeed one connected component of the following PL shimmer curve which is corresponding to the unitary shimmer variety with an emission space over q root five of dimension two whose signature is one one at one place and the two zero ideas of space. Okay, so this is an example. So let's go back to candidates. So this is an example of a shimmer curve with the modular space to be genus zero and then if you think about this connected component the whole modular description here can make sense. You have a model actually. So first you have a q model and you could like make an integral model defined over z. So this is a situation that we could try to analyze. I'll keep seeing. So now let me just say the expected zero. Okay, so require what I just said is. So unexpected theorem is a zero. It's a zero. Yes, it's a zero. Just I thought expected because the preprint is not on archive yet. Yeah, it's a zero. So I have a linear variety who coming from the Jacobian of this type of particular curve. So where I say C over q zeta five looks like y to the fifth equals to x, x minus one x minus t. And so the title total the real field correspond to exactly these type of total real field coming from the endomorphism. And then we have some technical assumption which, okay, we hope to remove at some point. So we have two technical assumptions. One is that when we reduce, so I have my curve C and then I have my ideal root five. I take the residue field pass to the algebra closure. We assume this one is actually degenerates. Namely is that this is a genus four curve and it degenerates into two genus two curves with z mu five action. What is C or F? So I have the ideal root five and then I take the residue field. I take the algebra closure. Yes. So this is the first technical condition about its assumption at five and the second technical assumption is about its image in the reals. Okay, so now let me draw that. So I have the real points of P one P one is our modular space and we have three distinct points. So using this modular problem, the T parameter is like the lambda parameter for the Legendre family. It's not exactly the J in one. So we actually forget about all the level structure. We have three points with extra automorphisms. One is P is the degeneration one, which means it looks like this one over R. And this is actually a point defined over Q. All these points are defined over Q and this course running to T equals to zero one infinity. And then we have another point and call R. So it's the similar as the J line situation as we have a point corresponding to T equals to negative one or a half or two. This is a point with extra. Here extra means that we have a mu five action in addition to the mu five action. We have an extra plus minus one action. And then we have a third point, Q. We have the extra, I will say mu three action corresponding to the point T equals to plus minus six rows of units. Okay, so we have these three points and you can check is like when I take the J in rent of the T, then these things are defined at least the moduli. So when I say defined over Q, it means we work with the course moduli space and then we talk about the definition of moduli, definition field of moduli. These points are all defined over Q and this actually splits the R line into three segments. And the current in our condition saying that, so our curve C has two embeddings. So C is defined over this real quadratic field. So it has two real embeddings. The two real embeddings, both of them lies on this segment here, okay. So these are the technical assumptions and then we say that A has infinitely. Now T is there. No, T is an element in the, okay. This is the, also this T. This is an image field. T is not in this field. So what is in this field is that, okay. T in this field is okay, but what exactly in this field is like. So the way that you relate to J line was the lambda, the gender lambda. And then you plug in T into that formula. And then as far as that one is over Q root five, that's fine. So yes, so. You need to terminate. Sorry? No, you need to terminate for more. The, what? Sorry. You wrote that C is defined over, I was confused. C was wrote is defined over Q. Oh, I hear, means when you pass, because like I won't be able to necessarily write a curve. Sorry, thank you. Won't necessarily be able to write the curve C in this particular form if I do not pass to this field. But you also need the T. The T lies in this field also or not? So let me just restate this part. So I have a binomial variety coming from a Jacobian for curve C. This curve C, when you pass to, let me want to say, where C over Q bar will look like this particular form. And then the condition of this one defined over Q root five means JT lies in Q root five. This is, you can do it as T square minus T plus one over T square, T square, sorry, T minus one square has infinitely many basic reductions. And I wrote Q zeta five because like in general, if you don't pass to Q zeta five, you may not necessarily be able to write your curve with these properties into this particular form. The mu five action is defined originally on the curve? No, so the mu five action is not necessary to find the original way on the curve. So that's a version of the theorem. And then let me make a remark is like, what could we do probably in a finite amount of time? And then the first thing is like, Modulus some very minor technical difficulties, difficult issues that we haven't checked, but we think we'll be able to do it. So we expect that proof that I'm going to present today will be able to just replace condition one by A has super special reduction at root five. So by super special reduction is like in this case, our Schmuck curve has two active odd strutters at root five and then the super special corresponding to just the zero dimensional locus. And in other words, it's the same as saying that when I take the P torsion of the opinion right, it is isomorphic to the product of the P torsion of super singularity of the curves. And this is, and the particular case there is, so here like when you pass to like product of super singularity of the curve, it may or may not respect the polarization, but here is a decomposition which respect the polarization. Okay, and then second is that we'll be able to show, I think, so this is the PQR picture, we'll be able to show that if our two embeddings of C, not necessarily nice in this segment, nice in these two segments, it doesn't matter where in these two segments, they're nice in this is also okay. So let's replace. Each one lies in another segment. So I view these two segments as one thing and both of them can die on anywhere here. But not on the points PQR and R. PQR is also fine because PQR will be have super singular, sorry, CM of linear varieties and then you can just conclude by Shimura Tan Yama. No, P is the generation, so it's not. Yeah, but it's the product of two CM of linear varieties. Sorry, in this particular case, if I take the Jacobian, it will be isomorphic to the product of two obedient surfaces, CM by Q zeta five. So, for what, for P? For P. And the R is isogenic to P and the Q is the CM of linear four-fold, which is CM by Q adjoint to zeta five and zeta three. But when you write, when you have this curve, x, x minus one, x minus two, and T is zero, one, or infinity, it's not the same. Yeah, so, yeah, so in that case, you see if you look at this particular family, it actually has degenerates and it degenerates into two genus, two curves intersect at one point and then when you take the Jacobian, it's the product. Okay, okay, okay, then you can do it. Yeah, okay. And then to completely remove these two conditions, we need to do a little bit more work than the proof that I present today. And I would like to mention previous work is like in L piece. It has later some generalizations in 89 is that so for elliptic curve over a number of fields, such that there exists an embedding of K into the drills and then also there's the work of Baba and the granus into sun and eight. They formulated as, let me just say they work with small curve associated to the quaternion algebra quaternion algebra B over Q, so quaternion algebra ramified at two and three. They also have like this type of conditioning there, zero. Okay, so let me write down another remark that I was merviling around is that the method, our method generalizes to all curves in one's work, which means those ones coming from Schmura curves coming from cyclic covers or curves. Cyclic covers ramified four points and what we expect is like we expect to generalize to just the genus zero Schmura curve with three points, oh sorry, genus zero Schmura curve over, okay, let me just say with one A model over totally real field and the second is like it has three points with extra automorphisms, but with these three points defined over some totally real fields. So this is what we expect that in general the whole method will work. And here A model doesn't necessarily need to be the canonical model of Schmura curve, especially for this particular Schmura curve that we were talking about. The canonical model is defined over like Q zeta five, but it's actually if you do compute it following the conjecture of long lines and what you work out by Milner and Sheen on like what is the Gaoua conjugates of Schmura curves over like the Gaoua group of the reflex field over Q and you will see this particular one you can actually descent it over Q and so A model over some totally real field. Any further questions about the statement of the results? So for, so I have 20 minutes, I would like to talk about, I'll first just give a very brief sketch of Elki's proof and I'll talk about how we modify each step to actually prove our results. So here we first get a sketch. So first the reduction step, it is enough to show that, so given a finite set S of primes, we can construct find a P, such that E mod P super singular and P not in S. Then we could just search through by contradiction or induction that you can produce more and more primes. So as far as we could, you give a finite set and we can always produce a super singular prime away from that finite set. So we can always produce new ones. Okay. And then how to produce super singular reductions and related to previous question about like the density. It's like when we have Cm, if it curves or Cm of inner varieties, we know exactly by Schrodinger and Yama which the reduction are. So we use Cm points to help us construct super singular reductions. So use Cm points and maybe a little bit more specific here. So we pick up prime D, which is congruent to negative one mod four. And we consider the Higgs cycle or the Cm points Cm by the ring of integers of Q adjoint root negative D. And then I can construct a polynomial. So I get a polynomial PD over Q whose roots are J invariant of all these Cm points. And what we want is like, so we want, so our PD and then we evaluate as the point E and we want this one has a prime factor. Okay, it has a prime factor means that E mod that prime factor P. If it has a prime factor means E modules at prime factor is isomorphic to one of these Cm points. So that's the definition. And so we want a prime factor, but we want a super singular reduction. So we want a prime factor P, which is inert or ramified in Q root negative D over Q. Q root negative D over Q. This is where this Cm point has super singular reduction. When PD of J of E, you want PD of J of E? Yes, oh, sorry. Yes, here, like when I put the bracket, it means, yeah. If I just put bracket, it means I look at this one on the model space. And in order to do this, so the quadratic reciprocity comes in as my, so this condition about P inert or ramified here means that minus D mod P is not a square. And then we want to relate it to a condition about modulo D. So quadratic reciprocity, if we consider P modulo D and take the product of these two general symbols, equals to one. So which means, okay, so here is like, I have, this condition means that I have a negative one here. So by quadratic reciprocity, I just need a negative one here. So in other words, we can study PD J E mod D and we want a prime P not equals to square mod D. So previously we want negative D to be not a square mod P. Now it reduced to that we want a P which is not a square mod D or P equals to zero. And recall as P is a prime factor of this one. So if we know this one is not a square mod D, sorry, if the absolute value of this one is not a square mod D, then we're done. But that is something we don't know how to prove but we can do a little bit more. Is so, the second part is local properties of PD. So first we'll have Lubin-Tate theory. Lubin-Tate theory tells us about kind of like, when I have a point in mod P, how do I lift, do CM lifting of a point mod P? So we use Lubin-Tate theory and the study the CM liftings and we see that the polynomial PD is almost like a square. So times seven X minus 1728 is congruence to a polynomial square mod D. So it means that I always have away from 1728 I always from two liftings at each point mod P. Which in this particular case, you can upside during lifting theorem. I was going to talk a little more later but let me just say Lubin-Tate theory will give you this one. And then we have Archimedean situation. It's like, now we'll just sort of place this one by that one, that's completely fine. And then what is absolute value? So we have the real roots. So PDX has a unique real root which actually goes to negative infinity as D goes to infinity. So this means that, so if X is smaller, sorry, if JE is smaller than 1728, then we have this is negative as D goes to infinity because now this one has two real roots. One is 1728, the other one goes to negative infinity. So if I have something nice in between, then this one is negative. So this also explains why in the main theorem there's a statement about geodesic. Okay, so this would imply is absolute value of JE minus 1728 times PDJE equals to minus one times, which is congruence to minus one times a square mod D, but which is not a square mod D because negative one is not a square. So that's essentially the proof of Elkis. It's like you can produce one and to avoid the set S, you just construct the same cycle which has all the reduction as the set S. Then the new supersingular reductions that we produce definitely will not lie on the S. So you have something which is J minus times D is not a square, but then you take a prime dividing this. Why it is a prime dividing PD and not prime dividing JE? Not this, because one thing that I said, I said this one mod this one is a square, so which means 1728 is actually a root of it. So some more details is this is an odd degree polynomial, so you're never going to be a square and odd degree coming from when you reduce to 17, which is 1728, the CM lifting doesn't show up in pairs. So in order to compensate that, we put a factor here. So this one mod D already has a factor like this. So if it's mod D equals to that, that's fine. You said the choice of P is a P prime dividing. Dividing this, which is the same as dividing this. Because if you go to the factor J minus, okay? Yeah, oh, another word is kind of if you have, let me put this way, if you hit 1728, it also has super single reduction, that's fine. So dividing this one doesn't cause any problem. All the factors dividing this one also gives super single reduction. Now, so for our proof is like we have, from the sketch here we have a couple of things to do is like we construct our CM points we'll study the local properties. So the idea of construction CM point is we want a CM point once we apply Schmarthen-Yama, we have basic reduction at some property which has something to do about ramification of a quadratic, in order to split in a quadratic extension. So let me just directly give the construction as we pick to replace D, we use lambda inside the ring of integers of Q adjoint root five and then lambda totally positive, this is prime. Okay, and then what we do is we consider a CM type coming from so Q zeta five at root negative lambda and the with the CM type coming from this. So we already have like it's behavior, we have the code with condition. So it tells you give some restrictions of the CM type and the CM type particularly, okay. Let me just say there's a up to unique, up to equivalence of CM type, there's a unique CM type which compatible with condition there and using Schmarthen-Yama we'll see that it has basic reduction if P is inert or ramified in Q root five adjoint negative lambda over Q root five. And then let me just mention it's like in there is a quadratic reciprocity for Q root five. So it's talking about P in this extension can be reduced so we reduce the problem. So reduce true study, let me use P lambda and then evaluate at our point. Mod lambda, okay. Now for the second part is local properties and for the local properties we have a couple of questions is the first is moving type still holds about CM liftings with 1728 replaced by P or R. So far we cannot decide which one it is but it's like one of them. They corresponded to that you have trace zero automorphism. So that's 1728, you have a trace zero automorphism and these two points have trace zero automorphism in PDVSO groups. Okay, so the second part is about that. So this gives us the finiteness sorry condition reduce mod D and we also have a real kind of type of condition is we have the odd degree for P lambda and the unique real roots coming from the following is like the unique real roots imply it has odd degree because this is a polynomial defined over P lambda is a polynomial defined over Q root five defined over a real field. And then so upper bound, upper bound coming from so we have some conditions for lambda. So for the quadratic reciprocity to work we need lambda to be congruent to negative one mod four. And for this part to work we have an extra condition is so we want the norm from Q root five over Q of lambda which we already assumed to be a prime to be congruent equals to four mod five. And because how many real roots do we have by CM theory we first account the ideals which gives you a real point and we count the number of polarizations. So we use like some class field theory. So we use some class, I will say class number results some results on class numbers. And it gives upper bound is like we have unique ideal gives real point and the plus unique principal polarization. So although we don't know exactly whether it lies on the connected component coming from the Jacobian curves but we at least know that it says why it's a upper bound it has a unique principal polarization. So we have at most a unique root and the existence is existence coming from actually so we want to produce a real roots and what is the circle that I keep drawing about the real line is we use the apart from the uniformization of Schumer curve is one we have so we use the theory of triangle groups and actually the group we work with a triangle group two, three and a 10. So a subgroup of PSL2R with like automorphism degree two, degree three and degree 10. So like I have this is like P1R with PQR and this is our apart plane, this is the uniformization the theory of triangle groups will give us let me call it tau P, tau Q and tau R and these are geodesics. Okay, so we actually has a very nice description of the real points on the apart plane and these coordinates are explicit and one way to describe points on the apart plane is using its trace zero stabilizer. So if I give a point then up to a constant there's only one element inside positive determined geode to R element which fix the element. So we use the correspondence between trace zero element and the points and then all the points on this is described so the trace zero stabilizer is given by linear combinations. So I have X of gamma Q plus Y of gamma P sorry, gamma R if I'm working with this thing here. And then what is CN point? CN point means that I have a trace zero and the morphism minus lambda. So which means the trace zero stabilizer when I take the determinant equals to lambda. So this is actually a binary quadratic form over Q root five. Okay, so how to construct a point is we've using the theory of triangle groups we get a binary quadratic form and then we use the theory representative theory of binary quadratic forms over Q root five then we'll be able to construct our CN points. And then not just that, sorry, I think I'm running out of time. It's like it not just to construct a point but since this is a quadratic form and we actually could apply Hex theory of equidistribution to actually nicely control where do we want your point to be and hence it's also recover the probability of the real roots goes to negative infinity for Elkin's work. And last remark about avoiding the set S could still work out by assuming that it has mu order reduction at all the other primes but we won't be able to do it at five because at five it's not a good prime but we use Ectel-Ostrator. So we ask it to nice in the, so our CN point will like seeing the generic Ectel-Ostrator the one we construct and thus if we are original point nice in the special Ectel-Ostrator we're never going to hit it by our construction so we avoid the issue of hitting root five and that's why we can produce more and more primes in the end. Sorry for running out of time and that's it. Thank you very much for your attention. I didn't quite, what itself to avoid let us say in Elkin's book to avoid the finite set of primes. Yeah. So he avoid finite set prime by making sure that the CN point he constructed has all the reduction at that given finite set of primes which you can always do. Are you going to take the data? Yes, yes, yes. And then in our case it's like we take our lambda and then for everything away from five for you to have a new order reduction the condition is like has nothing to do with the condition of lambda that we put here. So that's why you will have there compatible you'll be able to like apply travel torque density thing that you will have like positive density set of primes satisfy these conditions making sure they have new order reduction at any given finite set. But away from root five because root five, the strata is slightly different. We need to work with the actual strata instead of the new strata. Are there other questions? And from the web? From the web? So, ladies, thanks, be correct again. Thank you very much.