 Last time, I introduced Fortin's height for a billion varieties, defined over a number field. And then I introduced various conjectures about the Fortin's height. The main is about upper bound of these conjectures. And it's obviously a very difficult conjecture. So we hope for some a billion variety, we actually can compute this Fortin's height. As I said that in contrast to the geometric situation, the Fortin's height is basically a cohomology variance or a hardware theory. In a number field of case, the Fortin's height is not. And so there's no direct way to compute that. So we make a restriction and a CM a billion variety. And we hope for CM a billion variety, we can do something. So a CM a billion variety, there is a striking conjecture by Comase. He basically says that the set of the Fortin's height of CM a billion varieties are essentially equal to the derivative of logarithm of CM, out in error function. And this is striking because we actually have no idea even for special values. Then his conjecture about the derivative. And then even this conjecture is hard enough for us. In number theory, we have no idea how to handle out in error function. For example, there's an out in conjecture, for example. Even assume the line of conjecture has become automatic, but we don't have any nice way to compute a special value or derivative. So at most, we can't do a billion case. So this error function, a billion, then that's striking enough for the a billion. If you think about only a billion CM thin, that's still hard. So we do a quadratic one, the simplest one. For a quadratic one, then this is corresponding to the average height of CM a billion variety. When CM field is fixed, you change the type. Then this is what we call the average Comase conjecture. The way there's a two way to prove this conjecture, one is reduce this work to prove this work of yuan, xingyi yuan in 2008, which is extension of the proof of Zagia formula. So in that way, we reduce the computation to Schmurr curve. There's a second way given by the so-called Kudler program. So they reduce the computing of Schmurr, the a billion variety, the height of CM a billion variety to an even bigger Schmurr variety. So there's two ways. One is the increase in dimension two to the g. One is the decrease in dimension g equals one. So you have two different ways, two extremely opposite ways. So today I want to introduce the two different ways. So I give you basic idea. Today's goal is to show that the average of the height of CM a billion variety is essentially equal to the height at the same point in Schmurr curve. So this is a pretty interesting thing. So we have this Sg, the Ziegler modular space, you know, it's h of g sp to g. And this is a big variety. So the CM point at some point there on this variety. And it's average. So basically you fix E over F is fixed, CM, then you consider all the different types. So those are the points. Then here you have a bundle, Haji bundle, right? You do this thing. And the other side, Schmurr curve is completely opposite. You start with, so this is a variety dimension, is a very big g, g plus one divided by two. So that's the one, I mean, presumably you support compute fault in the height here. Then somehow the computation is reduced to a curve. So this is the upper plane modular gamma. The gamma is a discrete subgroup of the quotients. We just compute it in one single point. And this is a process that's very interesting. So how do you, and the boson is a Haji bundle, at x here, omega x. So you have one Haji bundle here and one Haji bundle there. You compute the fault in the height, and there's a big variety. You cut down the compute fault in the height. I mean, the height for one point with respect to this line bundle. Another interesting thing is that this one is something called a PR type, Schmurr variety. This one is not. This one does not parametres anything, even not a Haji structure. You parametre the Haji structure in infinite level, but not in the finite level. The finite level doesn't parametres anything. The reason is that the units of the total real field will have non-trivial action, fixed point at every point. So you cannot parametre anything. So there's something. I mean, in the last year, when I found a proof completely accidentally, even right now, I have no basically intrinsic reason why these two things can be equal. So these can be done by two techniques. There's two ingredients to reduce computation of the high-dimensional variety to the one-dimensional variety. One is the decomposition of Haji bundle. So it's a decomposition fault in the height. The second one, of course, is deformation theory, or piece of a group. The reason is that, as I said, this side has a brilliant variety. This side does not parametre anything. But if you allow it to have infinite level structure at a very, very top, it will parametre the PDSB groups. It's not parametres at the beginning. If you give it's a high level structure so that your units of f act trivially, then there are a parametre of something. And then the comparison between PDSB groups, you can compare the deformation theory of PDSB groups of a brilliant variety of Zika modular space and deformation of PDSB group and Schmura curve. But this one is not trivial. It still needs to be used in. So this will use a hard structure, complex hard structure. So this is usually PRD hard structure. So this is a classical sense. This is a new sense, not completely developed in the last few years by Broglie and Kissing. So it's just a coincidence that both complex hard structure and PRD structure are used essentially for this purpose. But for totally different reasons. So this one give you integral structure. This one give you the complex hard structure. I will not spend too much time talking about this part because this is very technical. So I will spend the time talking about the complex hard structure. Then we are trying to move. Trying to move, how can I move the discussion from Zika modular space to Schmura curve? OK, so this is just the introduction. So the first one, the decomposition of both the heights. So what's going on? Assume A, well, I mean, that's very big. And I don't really need it to A can be any, I mean, a variety of complex number. But I don't want too many notation. So we still have that billion variety over number field. But I mean that this number field here does not play much role. And we assume that A has action, OE action to anamorphism of A, the E is a number field. So this is pretty well known that, or is the E the CM? Either CM or a totally real, because there is a polarization there. You can show this pretty easily. Not an arbitrary number field can be embedded in the morphism ring of A, because this thing has an evolution inside that is a field. It's algebra, you can say, some of them. So then we assume that A, OK, is a narrow model. A narrow model, I mean, there is a canonical integral model of a spec of OK. So that's what's described, I mean, by a narrow. So we can see the differential forms. Differential one form, it defined to be omega A of OK, one form. So this is differential one form. And of course, then there's a hard bundle is essentially determined of this bundle. So we want to compose this thing in one sense. So we have action by O of E x on this bundle. So this is a bundle of rank G over O of K. Well, so then naturally you wanted to compose this action. So assume that K contains all conjugates of E. I mean, what does it mean? This means a homomorphism from E to K. Well, this actually, a cardinality of this algebra is at least equal to the degree K of Q, E of Q. So you have many sense there. OK, so you assume this is true. Then we can compose the same there. We can decompose this so-called eigencomponent. So for each tau, so for each tau, from invariant from E to K, we can form the, I mean, this has action by bundle there. Then we have, we can form this O of E turns the O of K to O of K. So I will still denote this as tau there. But notice that this omega A has action by both this ring and this ring. But for this ring, because an homomorphism, for this ring because I'm of differential one form of this base. So you can use this one to define a component. So let's define the notation W. A tau here defined to be there, this eigencomponent. So it's a formally defined as the tensor product of OE, tensor O of K, go to the O of K of tau, O of K. So this is the O of K bundle. It's still a bundle of O of K. And then we have a homomorphism vector bundle, vector bundle, that omega of A goes to the direct sum of these bundles. So we get this thing. So if you want to take a determinant, you're taking determinant, you will get omega of A to outer the tau determinant omega A of tau. Of course, here is a convention that if omega A tau equals zero, for example, if tau does not appear in eigenvalue at all, then the determinant of this thing is completely trivial. So this is the same there. And this is a morphism of line bundles, the non-trivial morphisms. So this has a metric. And we hope, we put some metric here. So omega A has a metric. I defined it last time just by integration. It's what we usually call the integration, called the Peterson product. Integration of a billion-variety. Because this is the G-form, homomorphic G-form. Homomorphic G-form, which is conjugates integrating a billion-variety, give you a number. But inside, it's just a piece of it. There's a piece of it. We don't have a very piece of product for piece of the one-forms. So this is not G-form. This is part of some small parts. But if your A has a polarist, you do have it. Because once they polarize it, this has a metric. The special one-form has a metric. But I do not put polarizations there. So I have a two choice. Either I work only on the a billion-variety with a polarization. Or usually, I don't like to do that. I just forget about polarization. If I forget a polarization, I have to work in a billion-variety and do a billion-variety simultaneously. That's a typical situation we do representation theory. To create a real number, the only way to do it is take a dual pair of vector space. So what I do is that there is a determinant omega A2 that does not have a nature metric. To solve this problem, we bring A t. So this is a dual-a billion-variety of A in sight. So the dual-a billion-variety. A dual-a billion-variety, then we have the same thing. We have this omega A over t also has a morphism to direct some omega A over t over tau. And the dual-a billion-variety also has action by O over e times O over k. So it's by same there. Again, the same thing. I have omega A over t goes to transfer product omega determinant. I can't choose the omega there. I don't know why I didn't use that t there. Again, we have this stuff. And the interesting thing is that even this one does not have a canonical metric. This one does not have a canonical metric. But they have a pairing together. So that's something we need. So there is a canonical Hermitian pairing between omega of A, omega A over t, and any complex points at any place. Sorry, let me run V inside here. No, it's OK. At any complex place, V over k. So these are going to describe, but this is purely a larger theory. So there's nothing not very difficult to describe. So the idea is the following. Idea is that this omega A of V is equal to, well, it's just omega A of A tends to Vc, as I described there. This is equal to the gamma A V is a holomorphic one form. And the omega A over t come described by the dual space. So this is because we can write down the dual abelian variety using the following parametrization. I mean, a dual abelian variety, if you remember, we will use a Czech homology. It's h1 x gm. You take the real algebra becomes h1, this V is h1. Something's h1 of A, V over A, V in a naive sense. And h1 A, V, Z. And here has something like that. And this part, so you can write this as h1 0 of A, V, right? No, it's a holomorphic 0 1, your 1 0. But this one is a 0 1, let me see, 0 1 dual. This is a dual there. Because this is h1 there, this is a real algebra, it's a dual. So this is by, I mean, these two has a parent, right? One is a complex conjugation of other. So naturally, this is equal to, well, this is a dual of that. So it's a complex conjugation. So this is equal to gamma, omega, A, V conjugation and transpose. You get two of them, right? Your first conjugation, if V is over C, V bar is V tends to C, C by conjugation, just conjugate. And so you have a conjugation plus the dual. So this means that there is a canonical Hermitian pairing between omega, A, V, omega, A, V, T, T complex number. Well, this is a way. But actually, it can be done by a different way. The different way is A, over A, V cross A, V, the dual of A, V, there's a Poincare boundary. A Poincare boundary, there's a truncless 1 1 form. You can use that 1 1 form to get duality. That's another way to think about it, right? So anyway, you have to use this thing. OK, so we get a pairing. So this pairing is functorial. So in a sense, if you have a billion variety, say, if I have a 5 from B to C, this is a billion variety of a complex number, then you will have the, if you take a pairing over B, B really means this pairing. So this one is equal to gamma. Then take a dual, then pull back beta over C. So you have this very nature functorial property. So the functorial property, if I have a billion variety, has an emulsion by O of E, then I can decompose them, right? So yeah, another thing is very important. So another thing is more than, I mean, it's an interesting thing is that, well, this pairing, define a number there. So another thing is interesting. So if you have alpha inside, OK, so if you take, let me try and take a determinant, we also get a pairing omega of A tends to omega L over T to complex number. Here, we have a metric defined when we define a fault in the height, right? Here is similar. So you're wondering, this is a one-dimensional backspace, this is one-dimensional backspace. This has a norm, this has a norm, and a pair also have a norm. So it must be so compatibility. The compatibility is pretty clear as alpha beta equals alpha beta. OK, so that should be naturally true. Otherwise, it's a bit strange. So OK, so that's what we have there. So this is the one thing. Second thing, so the action. So this is a Hermitian pairing. Let me, what I said, OE, O over E acts Hermitian. It's the action of OE on omega of A, omega of T is satisfies the following property, satisfies the usual one, say, I give an element there. Let me, how can I say? Give A times xy will be equal to x A bar of y. This should be something naturally true, right? But this is a pairing there. Because of this reason, then we remember when we decompose this into argon space here, then the above pairing will induce the above pairing. So the second one induces that nature. The pairing, the omega A, V, omega A, V to C is a direct sum of the pairing omega A, V, tau, omega A, V, tau C to complex number. The tau of C equals tau composed of C. Do you remember at the beginning I say my E is either CM or totally real? In a CM case, the C is a complex conjugation. C is a complex conjugation. E is the identity if E is totally real, or the complex conjugation if E is CM. So you get this pairing there. That is great. So finally, we decompose this differential one form into a small piece. A small piece I have is a metric there. OK, so we get this pairing there. So now we defined a one-dimensional backspace norm na tau to be the tensor product of this space determinant omega A over tau tensor determinant A transpose. I mean, the dual-aibillion-variety tau C. And we defined this bundle. And this bundle has a metric. It has a metric and an end of the metric given by the above pairing. So great. So we decompose. We hope to decompose the fault in the height of small piece for a billion-variety. It does not succeed. But what do we do? We do a pair of a billion-variety, A and A dual, then succeed. That's strange. The reason is in a periodic place, we don't need to do that. The periodic space we integral structure automatically give us this one. But at an accumulative place, we have to do that. It's kind of not analog to each other. You have to bring the dual space in. But this narrow model doesn't depend on polarization. That's at whole points. Narrow model is giving you integral structure, but you don't need to use in polarization to define a narrow model. But for height structure, height structure does not have metric. It has a metric. Must come from some calorimetric. So that's something not very balanced. So in this way, we also can define a partial fault in the height as the degree of the two. So twice k over q of degree over the bundle A over tau. I keep a 2 here because I keep telling us that this is the average of two-piece. OK, so I finally succeeded to define fault in the height for small piece. And this definition works for any abelian variety, not necessarily for abelian variety with Cm. This actually is quite important. Because for any abelian variety, because eventually we started Shimura curve, or Pierre type, or Shimura variety, then this decomposes automatically there. And above four Cm abelian variety, but I guess probably the truth. So now we assume that A has Cm of type O of E of phi. So if we assume that, of course, then, of course, the NA tau is trivial with a trivial bundle if tau is not in phi. Because I said that everything's zero. Yeah, zero vector space. Zero vector space determinant defined as a trivial. And so one proposition is important. So for tau inside of phi, we have the following properties. Let me see. So first thing is, this height, A tau depends only on phi and tau. So this is a refinement of Cormac theorem. Cormac theorem says the fault in height depends only on phi. Now it depends on a pair, a Cm type of phi, and one element inside. The second part, of course, so there's a summation. So it is a fault in the height of A if you divide by all this of tau there for tau inside of phi, where I want to compute a difference. You see? My bundle here goes there. I put a metric there, but I cannot guarantee. Even this morphism of a finite place is not necessarily injective isomorphism. So you could have some problem from finite place, periodic place. But this is actually the computable. This is equal to a simple formula. It is equal to negative 4 times k over q, the logarithm d over phi d over phi c. That's a weird thing. So this is the discriminant, partial discriminant. So when you have a number field that can be discriminant, they usually, if it is generalized by one element, they remember it's just the difference of different conjugations multiplied together. So that's called discriminant. And partial discriminant, I only do it in phi. Phi is 50%. So in other words, originally you have how many terms? G squared minus g divided by 2. Now, originally you have 2g squared minus 2g divided by 2. Now you have G squared minus g divided by 2. So it's not half. It's much smaller. So it's just like one of these, you take care of everything here, but now you only take care of this part. Another one, take this part. So it's a really partial discriminant. So the trigger definition, I didn't, I mean, in my paper, I didn't really define it very well. Let me suggest one definition. So I hope it works. I suggest the following definition. I have O over E turns to O over k. So remember, all these type of phi is O inside here. So this phi defines a representation of O over E on a vector space, V over phi, and O over k, with this phi there. Then my O over E, O over k, of course, acts on this V over phi of O over k. This acts on this thing, because there's a vector space there. This acts. So in other words, this has an amorphism of this thing. So I let R be the image of this morphism. So this R has O k inside. My suggest the definition we are following. So this definition will be discriminant at D over phi equal to discriminant of R. I mean, somehow, it's a ratio. So divided by, let me see, discriminant this thing will be discriminant E, discriminant k. Let me try to understand. Discriminant of O over E, O over k, will be the dk. Get a field extension. I forgot some powers. Probably divided by discriminant of O over k. So maybe this is defined by some power of 1 over g. So this probably is a proper definition of this partial discriminant. So this is the order. Once it's the order, of course, you have a trace. So this is discriminant. You have R times R to z by trace map. Then you define discriminant by that one. So this should be the discriminant D over phi. Anyway, it's just a partial discriminant. So this is there. So this has a pretty nice formula. Now, OK, so this is a very interesting thing. Now, I start to introduce another thing. So this is just decomposition. But this decomposition, it gives you some good hint that my fault in the height of a billion-variety is, I mean, even the g-dimensional, I make a one-dimensional. What I mean appears, then you can think of it. This is one-dimensional, maybe something to do with Schmura curve. This is the second thing I want to do. But before I do the Schmura curve, Schmura curve is, I will do Schmura curve here. The type is, it must be parametrized sum of a billion-variety. So the next step I want to do is, I will define the fault in the height of nearby CM types. So what do I really mean? I give a definition of two CM types. Of course, fix the E. Fix the same field. E, two CM types, phi 1, phi 2, over E are called nearby, nearby to each other, for example. Nearby to each other means that they are almost the same type. They differ only by one element to each other. If they differed, they differ by one element. So that's something I want to do. So basically, the cardinality of phi 1, phi 2 equals g. So it's the degree over g of 2g. g minus 1 is equal to g. That is completely the same. No difference. So somehow, I want to study two a billion-varieties. So let tau i equals, well, the phi i takes this intersection out. So we have this h of phi 1 tau 1 plus h phi 2 tau 2. So somehow, I want to add them to the average. Let tau i, well, find the good reason why I want to do that. I define it as the same type. It's a small piece. Now I add two of them to the average. So I give this a name called the h phi 1, phi 2. So I get two same types. So this is a good sign. It's a very strange combination. But suddenly, you can define any two same types. I just define this way. So some effects. So the first one is a pretty clear. This thing, h of phi, all this phi is the average of this thing. Average 2 to the g minus 1. Average of the nearby same type is different. It's slightly different of h phi 1, phi 2 minus 1 quarter log of discriminant. That's very interesting. So you'll get two same types. So one same type will get a nearby same type. The second one is very important. It's this h phi 1, phi 2 is an isogenic variant. So what I really mean is the following. So the second effect is a very interesting thing. So I mean that if a0 is a variant variety of OK, is a variant variety with a variant variety of dimensions 2g with multiplication by o of e of type phi 1 plus phi 2. So let if a0 is isogenic news to a phi 1 cross a phi 2, suppose this is a variant variety already much bigger than our variant variety before, then the interesting part is that the height of phi 1 and phi 2, this originally becomes 1 half the height a phi 1 cross a phi 2 of tau. This actually equal to 1 half h a0 of tau. So this second fact basically means that when you compute a CMI variant variety, usually the isogenic is pretty sensitive. But here actually it's not. If you put a phi 1, phi 2, you can use anyone to compute it. So this gives us a lot of freedom to compute the fault in the height for a nearby same type. So you don't really need to respect the product of two CMI variant variety. You can change it a little bit. So the one thing we're going to use as a following. In a lecture, we're going to use the CMI variant variety as a follows. We're going to use a0 defined as a follows, constructed as a follows. Constructed as a follows. So we do the following way. So what do we do is, so fix, OK? So let tau remember. So once you have a phi 1, if you phi 1, phi 2, of course I just say it give you a tau 1, tau 2. This is the complement. But tau 1 restructed an f equal to tau 2, restructed an f. So these two will be equal because they're both the same types. The same types, when you differ one thing, when restructed on the f, it should be completely same. So I give this name, give a name called tau. So from this tau, we can construct. So I mean, from this tau, we can see the quaternion algebra of B of f, whose ramification state of B, at least at the acumenian place. At the finite place, I don't care. At the acumenian place is all the acumenian place, is all the ramifications state, all the tau sigma divide infinity, this is the place of f, but sigma not equal to tau. So in other words, the ramification state is everything, but not tau. So in other words, the B is definitely the quaternion at all other places except the tau. So we also can see the following thing that we also assume there's an embedding E, embedding to C. Assume there's embedding. Once you assume embedding, then B is naturally have two p's, right as a quaternion, two p's. G of x equals x bar of j. So x inside E. So that's the quaternion. Beautiful thing, the quaternion algebra just like a usual thing. It's a non-so OK. So we will decide, we will compute, we will define B turns the real numbers, right? Equals E turns the real numbers plus, we have E turns the real numbers of j there. We have, this is the thing, direct sum. So here, we have a complex structure phi 1, put a complex structure. So this is isomorphic to c to the g by phi 1, c to the g of phi 2. So I get a two complex structures there, right? So I just simply put a complex structure like this one. Then in this way, I will make a billion variety of complex torus. So this gives us a complex torus of a 0 equal to B turns the real number O over B. So the O over B is any, so O over B, a maximum order over B. But we assume this is contained O over E. So that's the, but these are billion, these are complex torus easy to see from a construction. The complex torus is isogenic to a 1, a phi 1, a 0, is isogenic to a phi 1, direct sum, a phi 2, right? Because this is a phi 1, there's a phi 2 there. So I just construct. The only difference is that I use O over B. That's the only thing changed. So that's the, so this gives you a simple way to relate it to a pair of a billion variety to this thing. So this will give you a good reason why I can say nearby, same type. So this construction brings you one a billion variety with a double the dimension, right? And also with action by OE, it has the same type, phi 1 plus phi 2. So this action a 0 has a phi 1 plus phi 2. It's isogenic to this thing. If I drop this condition, if I say, I all consider all a billion variety of dimension 2G with action by OE of CMOE phi 1 plus phi 2, then this modular space is a Schumann curve. So this gives you the idea that a 0, I mean, that's the funny thing, a 0 is the same point. He say it's a zero modular space 2G. But actually he lives in a Schumann curve. So this is something I want to talk today. So now we want to, so let's, 2.3. So we want to talk about Schumann curve of PR type O of E phi 1 phi 2. So there, so you can think of a Schumann curve as like a same, a billion variety still the same. And what I'm going to do is that basically I will construct the same a billion variety by dropping this thing. But this a 0 has the PR type P means the polarization. So I need it to define a polarization. Anomophism, same thing, is just like that. So let me define this stuff. So I fix. So let V be a B per psi. So B is a quaternion algebra I defined before. Per psi is a synthetic form. So I pick up a psi carefully. Now remember when you, if anybody read Tanyama-Schmura's paper, there's a, when defined as the polarization, you choose one element in E, which is imaginary. So what I'm going to do is, is a synthetic form defined by all the form of the, this way, is a per psi UV equals trace f of q and a trace B of f. So this is the reducer trace of gamma UV bar. The gamma is inside E, right? Cross and the gamma bar equals negative gamma. So this give you a synthetic form. So then I need some positive condition because I couldn't construct a brilliant variety. So I want a synthetic form with a positive type. So this means that I want this per psi U, U i inverse, will be greater than 0 for any U inside this V of r. I'm forget about that. Before I talk about that, I need to find the combless structure, right? So V has a combless, Vr has a combless structure defined as in the same comb structure here, right? Of phi 1 plus phi 2, right? So you define a morphism, V of r. So when the combless structure, I can talk about a modification by i, I define the same. So this is the typically way to find a modular special I mean a variety, you need a vector space. You need to have a combless structure. You need to have a synthetic form. You have everything, right? OK, so now I define a modular problem. I let f prime be the reflexive field of phi 1 plus phi 2. So this means that f prime is generated by of a q trace phi 1 plus phi 2 of x for x inside E, right? So this is the inside C. You define a field f prime. Then so the OB, OB a maximum order, as I before, and the OE contain OB. So this is basically, I mean for uniformity of language, I write lambda equal to OB as a lattice in V. So somehow I mess up their use of V and a B. The reason is the following. When I write a B, I always keep in mind that a B has action by B itself, right? Has a B module. When I view it as a V, I view it as a E vector space. So I will drop the action of B there, right? So I can purely consider it as a V vector space. Conversely, for any Hermitian space of E, if it's type phi 1, phi 2, always come from quaternion algebra. So there's no much difference. It starts with a quaternion algebra or starts with V. So that's a phenomenon in GR2 theory. In GR2 theory, there's many different way to describe Schumann variety. You can use quaternion. You can use Hermitian space. You can use orthogonal space. You guys, the simplest space, of course. But all of them same. So here I dropped action by B there. So make them more precisely viewed as a left E vector space, right? So if I view the left effect space, then multiply by OB at least as a right action. So it's not a bad idea. So you can see in this way OB will axon anamorphism of E of V, so from right action. Maybe it's opposite algebra from right action. OK, so that's basically the data. So now we define a modular problem. So for any, so we define that there, oh, before doing that, I need to describe some space. So once I have this space, I have a beautiful thing. I consider G of U V plus i, right? So this is a unitary thing there. So this is equal to all elements G inside GLE of V, GX, GY, or maybe across GL1 over Q, right? This is over Q. Now what I mean? Q cross there as an algebraic group, right? So I will not going to write on there. So G lambda there equal to lambda X and Y, right? So consider as an algebraic group over Q, right? So it's algebraic group over Q. And this is very easy to do. So the very construction I do here give you the following obstructions. So actually, if I write G prime here, different thing. The G prime is actually isomorphic to at least the Q is isomorphic to B cross and F cross and E cross defined by norm is inside Q. What I really mean? I mean, this is the norm equals inside Q really means that B, E, the reduced norm of B times the norm of E is inside Q cross. And then this is the strangest thing. How this thing acts on this space? The action is by following. So B, if we give you a B, E inside of B cross of E cross, it give you X inside of V, but V is in B, right? It's the same thing. Then the action is given by the following, B, E, X and X. As I said, this is viewed as left, E vector space. So you call it to EXB, just opposite way, right? Because this one, I keep the left action by E free. You put a B this side, then it can't do it anymore, right? So this is giving you a thing. So you see, I mean, I recover, start with the L-commissioned space that we get this in. This also is a very accidental isomorphism that in U11, unitary group of two variables, you always get a quaternion time descent. This is like universal property, right? But not for unitary group of three elements. For two elements, it's the same. So now I define a modular problem. So for any U prime, I use a prime here because later on I use a really not a prime later on. It's open, G prime, Q hat, Q hat, I mean, the hat always means the Q turns the Z hat. Z hat is a product of Zp on open compact. So we define the modular stack called X U prime. Sometimes prime prime here. Sorry, too many prime. It's stupid, yeah, over spec of f prime. So f prime already wrote that somewhere, as follows. So for any f prime scheme, S, there a point in X prime U prime of S is a quadruple of the following subjects, A, yorta, lambda, kappa. You can imagine. So this A of S is Ibelian scheme of relative dimension because 2G, this is the E over Q. Not like before, I have only G, now a double dimension. There's yorta is O of E goes to anamorphism A of S. So such that the trace of yorta of X acts on the algebra of A of S is equal to the trace phi 1 plus phi 2 of X for any X inside O of E. Remember, this makes sense because this thing is inside f prime. This is the inside at O of f prime. So actually, this is for the two property. So lambda A to A du is a polarization which induces whose rossetti evolution induce the complex conjugation, the complex conjugation on O of E. So that's something. There, the last part of the kappa is a polarization. So it's a lambda hat. Lambda, as I remember, is a lattice there. And this thing has a similar form across S to H1 as Z hat. I don't know. I mean there. So these both of them are really a sheaf, at the base. When I write this thing, this doesn't make any sense. But this really means you have to move to there. So it's an orbit. It's not really one morphism. It is U prime orbit of a similar to this of school Hermitian OE forms. OK, this is a lazy way to write it. You first put the full-level n-structure, model n, then it takes limits. So I write orbit. Depends how to interpret the orbit means. Because this doesn't make any sense. I mean, this is at a sheaf. This is a constant sheaf. There's nowhere to be isomorphism. It's just a lazy way. But it will be the way to find it. Yeah, it's just all your pro eter. Or you want to do that way. OK, then descent done. Yeah, you can use a pro eter language. So the whole thing I want to do, spend so much time with this thing, is that, well, A0 defines a point. The whole thing in this x prime, U prime for U prime is a maximum order. So it's a maximum order. We can stabilize it over lambda in G prime of q hat. So it's a maximum order inside. So I construct a Schumann curve. You include this Abellion variety as a one point. So that's the thing. I'm pretty much succeed. So now I want to introduce some additional structure on x prime, U prime. So I'm in some local system, some pitisable group. So for U prime, small, sufficient small, then there is a universal Abellion variety, AU prime to XU prime. OK, so it's actually what I really mean that this XU prime actually is representable by a universal one. And this actually is a curve. This actually is a curve of F prime. And it is not geometrically connected. So this actually, if XU prime of C at one place tau will be finite union of gamma i, some finite union. Once you have this thing, then you can define many beautiful things you want. The first thing will be the Darm homology group. So you have a local system AU prime. This local system will have Darm filtration. This filtration is very important for us. Omega AU prime, then there's omega AU there. So this is typical Darm filtration. And this filtration, everything has action by U of E. You have a type phi 1, phi 2. And the very interesting thing is that one side is phi 1, other side is phi 2. So they actually action by, OK, so we can decompose. We can decompose on the sequence. This turns to F, OK? So don't prime. Don't worry. This is just a vector bundle of some number field. So decomposition is pretty much safe. Not like OF before, decomposition. I mean, you have some problem. You will have something, argon space there, WAU prime of tau C goes to I will decompose this, I will define different name, MAU of tau, WAU prime of tau transpose. So now you understand why I use the W here, because you don't denote the middle as M. M is more like Cartier. Later on, we'll be doing a module. So then we apply the Gauss-Mannig. The middle one is a local system. But this is a local system. This one has a, this is a local system. This is just a vector bundle. So in other words, the middle one has the Gauss-Mannig connection. So there is a Gauss-Mannig connection in the middle. So apply this MAU tau. So we will get the following typical morphism, called the Cordera-Spenser map. I'm going to write everything out. Of tau C goes to MAU prime tau, nubbler, MAU prime tau tensor, omega x, u prime 1. Then goes to, I mean, I continue this morphism, goes to WAU prime tau transpose, omega AU prime. You get this morphism, right? So this composition together sometimes called Gauss-Mannig. So this gets a Cordera-Spenser map. So the composition gives the Cordaria. So it's a WAU prime tau C WAW tau V tensor omega x, u prime. OK? So this is the rank 2 bundle. This is rank over 2, because of my decomposing your phi 1 plus phi 2 structure there. So if you take a determinant, OK, if you take a determinant, so we will get a determinant of AU prime tau C goes to determinant AU prime tau. There's 1, t, which one has a t there? This has a t. There turns omega x, u square. Right, you got this thing. 1 is the transpose, make 1 there. So this actually is isomorphism. Let's give you this actually. I guess even the top one is isomorphism. Sorry. So this is isomorphism there. So this isomorphism. So in this way, if we define AU prime equal to this one times this one tau of C of determinant AU prime tau, then this one gives you isomorphism to omega x, u turns to square. So this will give you the Cordera-Spenser map. OK? That's actually nice. When you keep a bin of variety and the dual together, that's always the case. So in a typical situation, you lift a curve, then usually don't write the transpose here. Actually, sometimes you get confused why suddenly it became like a little curve case. I mean, there's all of the rank 1 bundle. Always got to confuse why this one is a square of other bundle. OK, give you the nice thing. So you remember that the fault in the highs is, I mean, so fault in the highs is easier somehow, is a bundle, is a height, a matrix line bundle. This is shape, right? The degree of metal bundle is a shape. So here, what do we need to do with that? So the fault in the highs we want to compute. So h a0 should be somehow the degree of the nu bar. So this is the Hermitian line bundle on OK to be constructed. Right now, we just have a bundle and a modular curve. Well, first of all, we need to put a metric there. Second thing, we need to put an integral structure there. For the metric, there's no problem. OK, for the metric, we can use this morphism to put an integral structure here. It's the same thing. For the metric, so this is the same thing as degree omega xu bar, that turns to square. For the metric, we can choose the fault in the same. dz equal to twice imaginary of z, if this is for uniformization. If you have h map to xu prime at a place tau, at a place v there, right, the v divided by infinity. You have the bundle. I simply can put this in. But there's no natural way to put an integral structure. Not natural, no good way. So maybe now I think there's a reason there's a paper by kissing and a purpose. They have a different paper. At least when I write this paper, there's no construction. How to extend the modular problem? It's a good way to the OF prime schemes. But now they have, so that's a, but I didn't really check that. The reason is, there's a group G prime I defined. We have some problem at a place, divide the discriminant of the quaternion, and also the place where e is ramified of q. So there's a two problem there. If everything's fine, there is something called a hyperspecial level structure, could be by a quarter width. Then you can define an integral structure. But in general, there's some complexity. There's some problem. So the way, so what we are doing is make a connection to a Schumerer curve, a simpler Schumerer curve. So we want to use a simpler Schumerer curve, xu to study xu prime. The simplest Schumerer curve, actually, the integral structure, xu was already studied by Dringfeld, at least in the carrier. And this simpler Schumerer curve, the structure is much simpler. You see, because there's no ui there. So the simple structure does not involve e at all. But a better thing than a simpler Schumerer curve, it does not parametrize anything. So it's not a peer type. So the carrier started that, but I put a p-divor group in infinity level. At a very infinite level, they put integral structure. Then they can descend it to the dance there. So you get a scheme. The scheme does not parametrize anything. You put a very high level parametrize p-divor group. The carrier started the integral structure of p-divor groups. So they get the integral structure of the Schumerer curve. So I just basically reviewed this Dringfeld, the carrier theory, quote, neonic Schumerer curve, x. So this is a typical, if the x prime is so-called peer type, this x is called a abelian type. So x prime, x has a semi-geometric connected component in the infinite level. But otherwise, they don't have much relation between them. So I want to G be the group, f of q of b cross. It's much simpler. So this is a simpler one. And b of f of quaternion algebra, everything. So we put there hc cross of g of r. So this is a typical Schumerer map. The Schumerer map is defined by z equals x plus yi to, OK, I mean, I put xy next to yx. So let me write this thing. So I make isomorphic this thing to g r2 of a real number plus the quaternions to the, you remember my quaternion algebra split in one place, not split in the other thing. So this is a standard quaternion. So that's a quaternion you learn somewhere. So you get this picture. You get this Schumerer map. Once the Schumerer map, you define the Schumerer curve. So this data allows you to define a Schumerer curve. So this g with h give you a Schumerer curve. x of u, I mean, are all u's curves. And the u is g of q hat open compact. You have this thing. All of the Schumerer curves are defined of f. I mean, it has a canonical of f. As I said, it's a very interesting thing to notice that if you read my paper or my book, I always start with in quaternion algebra. The reason I do that is because when you define a Schumerer curve, you always start with a subfield of a complex number. So a total real number field in banding to complex numbers. I find this one is, of course, it's construction by data. But it's very superficial when we discuss Schumerer variety of other. Because when you start to say my Schumerer curve is defined by the complex in banding, then you have to discuss the complex in band to other place. So I usually start with incoherent one. Incoherent one means I just erase this part. When I erase this part, it means that my quaternion algebra will be totally definite or accumulative place. That's actually very nice. The reason is if a v is a finite place dividing this ramification state, then I get a dream-filled uniformization. So somehow, I find this incoherent quaternion algebra is convenient to define Schumerer variety. There's one reason. Second reason is that in my construction, I will use a very representation. And I use a very representation to start with Schwarz's function over some space. And my incoherent space actually is a much natured way to do that. So there's a good reason studying coherent. But in this lecture, I will not do that. OK, so if I choose, let me define a model here. So if the xu of tau c will be g of q of plus or minus g of q hat over u, there's a standard description. Now I want to put an integral structure. Well, it is a curve. So it's a very nice thing I can have. So there's some facts. So when u is included in 1 plus n o b hat cross for n greater than 3, then genius x of u is greater than 2. Oh, one thing is funny. So this is the geometric component. My xu divided by f, as a curve of f, is always connected. So this genus may be the geometric genus. What? It's f prime, not f. It's a reflection. What? It's f prime. F. No, I don't have a reflex field anymore. Everything is f. So this is the funny thing. So what I really mean is every geometric component, f bar, every geometric component, so what am I going to say? f bar, f bar, so OK. The genus geometric connected component is always greater than 2. So this means that xu has a minimum model. If you have a curve, genus greater than 2 has a minimum model, regular model. x of u is not even 1 over some o f u, o f there. Oh, let me try to understand. OK, I put the same. So xu has a model, has a geometric. I mean, xu map to spec of f. So I said this is connected, but not a geometric connected. So the field of constant on here is spec of f u. So there is a field here. This will be geometrically connected. So maybe my genus really means that over f u. So it has a minimum regular model over o f u. You have something like that. And so usually, if you have a compatible system over curves of generical fiber, each curve has a regular model. This does not imply that your regular model also form a projective system. I mean, you have a rational map from one curve to other one, integral model. But this is not necessarily defined of every point. So miraculously, the proof actually is not trivial. You might spend a lot of time to prove that. So the compatible system defined by this way, actually, extend it to integral models. I mean, I didn't expect such a thing is true. But actually, it's true there. So the fact, so this xu forms inside g of q hat, open compact. And u, I mean, sorry, u is small enough in this sense. Small means that it's included the 1 plus n ob small form projective system. So I mean that you have xu, xu prime u. This inside x of u, xu prime, this is the real morphism. Originally, this is only the rational morphism. It's actually really become a real morphism here. So that's a very interesting thing. O F u prime or O F? Oh, yeah, everything of O F, right? Yeah, of a modular curve, this is actually true because you have a modular interpretation. A similar curve is also true. But this is actually the proof using many other facts. OK, so now I want to define the thing. So one thing you can define is you can use these same models to define the Hajj bundle, L of u. So what I really mean, the L of u under the generative fiber is omega x of u. But when you extend this thing to the special fiber, be really careful. This morphism is projected but not sometimes regular morphism, right? So you have these differentials. If xu prime map x of u is ramified, you pull back your differentials. Let me do this way. But this morph is eta morphism, of course. That differential pullback is differential here. If not eta, your pullback is not. But of course, I mean, every such a morphism is eta almost everywhere. So you use a gluing process to define this system. So such as that, let me give a name. If I write this one as pi u prime, you basically want pi ru ru prime. So in some sense, this L of u is the one in stack sense, right? In stack sense. So it's not a usual differential. These differentials will be compatible with the pullback. So you have a different suggestion. I did not get it. So where does this L of u live? What? What is this L of u? Line bundles. Yeah, you're right. Yeah, it's the peak x of u, right? You have tensor q. You have a lot of some q-query, but you pull back. It means that some powers goes back as a q bundle. OK, so this is just on the generic. No, this is for whole. But you put on x of u. So your extended is a bundle. So I get something. It's called some one-forms in a stack sense, right? It's not in there because of a regular one. If you do regular one, you have trouble. You pull back and go back. You will have ramification. But when u is sufficient and small, the way it is constructed is that you construct part by part, then glue them together. OK, so this is a few things. OK, so now I need to construct some structures. As I said that I don't have a, I mean, this small curve does not parametrize nice thing, but permanent something. OK, so I want to construct some p-duceful group on x u. When u is very small in the x u, when u is inside, it's so small it does not even have a proper p part. It only have o, g, q, p part. Do you see? It's very small because I take a limits. It probably can do slightly big. So what I'm going to do is following the same. I have this g equals b cross there. So this x also x and b, right? v equals b. So I have lambda o, b, this action there. So I have lambda hat there, and I don't want to talk too much. Let me, for simplicity, simply write u equals o, b, p cross. I just define something simple. I don't want to make life too complicated. So I defined h, there's u, up, as a p-duceful group, defined by most of the way you can define. By bp cross o, b, p, it's just a group. Then cross x. So my x is project a limit of x, u. I define it formally. Later I will tell you this formal definition actually makes sense. This thing has action by b cross. So I quote it by o, b, p cross. So this one is a p-duceful group over x over o, b, p cross. So let me write it as this x, up. So I define a p-duceful group in this infinite level structure. So prime to p-level structure, I take a maximo. I don't worry because I study my Schumann variety reduction at a p. So I don't really care about anything away from p. So I make a maximo level structure. Of this one, I define a p-duceful group. Then you worry, oh, come on, this level structure probably doesn't make much sense. But in fact, the remark is a very important saying. So this p-duceful group cannot descend to any curve, a finite level. But for any n, h, up, p to the n, descends to a Schumacher finite level. U p cross U p Schumacher finite level. So in other words, I mean, this peaceful group basically constructed by define the finite group scheme is a finite level. Then I take limits at top. But the limits itself, unfortunately, cannot descend. But each finite level can be descend, OK? This is like a pro et al. So now I fix, let p be a finite place, place of f dividing over p. The beautiful thing is the following, that OK, let k to be u, fp, I take a maximum aromethyl extension. So in other words, this is witter vectors, whatever. Completion, p, completion like that, right? Oh, no, no, no, no, that's stupid. Then times f sub o f, right? Something like that. Now you will know what I mean, just completion. I don't write so much stuff. Make it confusing. Completion, the maximum aromethyl extension, o f, p, OK? So I can see to the x of u, p at a p, to be a limit of, well, it's up x up x of p. I make a big change, it's OK there, right? So I get something, so this is the Shimura curve, defined of OK, where I want to study reductions. So another fact that's very important factor is a factor. So this is a principle group, h up has on integral model over x up at this p, OK? The integral model, let me write x up there. So I extend my integral model, but this is very important because I'm going to study the deformation theory of a point there. So the deformation theory, we'll study that. And not only that, so as usual, the x, let me, I mean, as usual, you see. So the completion. Is it to check where it comes from? OK, so the idea, the idea we use, of course, you have to use the PR type Shimura variety, right? There's nowhere to extend a feasible group to special fiber unless you have something. So what do you do with the following? So I have the Shimura curve. I don't know how to study it. Then I make some best change. Once the best change I can embed into some Shimura curve or PR type. But I pick up this thing, pick up the best change so that it's split at p. So in other words, the integral structure will not be affected. Then I have a universal or a billion variety. Universal or a billion variety has an integral structure. Then I extend this thing using this whole way to do that. And this is not affected by the fact that you are working on the tower on varying this. Is this compatible with different views? Yes. Is this choice of phase change compatible with different views when you go up the tower? Oh, this is even the level. What do you mean? Yes, you are working with different levels, OK? Yes. And what you are describing is on the limit or? Unlimit. Yeah, of course. I don't know how to define. Of course, once that is done, then you can descend to the final level. Each finite subgroup can descend to the final level. But a whole piece of group in the infinite level. OK, so that is great. So I will not talk about that. So the Hartz filtration on the duner crystal may give you induced such an exact sequence. So we get this thing. We're not going to run in this thing, or p prime, goes to duner module HP, goes to duner module FP. Oh, I forgot one thing. Once the integral model of this thing doesn't mean anything. You have to say the formal completion at XUP, p formal completion, so this is the formal completion, at its special fiber, special fiber, is the universal deformation of the HUP respected under the special fiber. You have to say something like that. Otherwise, it's useless. How can you do this thing? You have to say, my differential is, I mean, somewhere I have to use a Kodaira-Spenser map. So I have to say this thing. So this will give you a Kodaira-Spenser map. So this will give you a Kodaira-Spenser map of P that determinant of WPT determinant of WP goes to the X of p square. So remember, this is just error. Let me write it down. Right error, right? My Hatchibondo LUP tensor square, you get a map here. But is the map coming computed? The map is called a negative data B at a P. And so this data B and a P is a divisor. So it's either the trivial if a B, P is split, and then the P itself is not split. So you use P as a group. So you get another one to calculate there. So this will give you integral structure. OK, so I'm almost there. So now, actually, a few words that we're not going to, I don't have much time left. I'll just give you a little bit of idea how this Quaternionic-Shemura curve can be used to study the PR type of Shemura curve. So the idea is a bit of funny. So the group, so for the very fast way, we have this G prime. G prime is inside B cross of E cross of F cross. Then we have this G. G is equal to B cross. So this one has a natural embedding there and also has a compatible. So I can put the h C cross mapped here. When C prime mapped here, if we put everything together, then this h prime here, they also have h mapped to G over R. This h and h prime not compatible. They're different h structure. So if I write it very clearly, this h prime is equal to h cross he. That's the relation. He is C cross mapped to E turns to R cross given by, you remember I have same type, phi 1, phi 2, nearby. So this one is equal to phi 1, says phi 1, for example, C to the G. And this Z goes there to 1, Z, Z. So it missed the first fact. So these are two different h maps, two different Schumerer maps. So even the complex points are so close. As a scheme of F, they're totally different. So this is when you start Schumerer a variety, this is very different because the risk process law is different. They describe a different way of Galov action, completely different way. OK, so the relation is a weird relation. Relation is the following. Relation is, so this action, this G, of course, the first one. So you have the following. You have this Schumerer variety, G prime h mapped to Schumerer variety of prime b cross u cross f cross h prime. According to, by this formula, this means that my h prime, the map to both facts. So this is equal to Schumerer variety of b prime b cross h cross Schumerer variety of u cross he of this morphism. In fact, if I modulate the diagonal action, this is actually isomorphic. So that's the relation between this Schumerer variety and this Schumerer variety. This is x prime. This is x. So xx prime is differed by a twist, a twist of some same type. It's a twist by this thing. So x prime is a twist, a twist of x. Twist of x, that's all of this thing. This has a universal family abelian variety, the PDSO group. This has abelian variety PDSO group. The PDSO group also twist. So what do you mean the PDSO group twist? It means the total module are twist. So the total module of the PDSO group define this one by abelian varieties. And PDSO is defined by this quite a stupid method, the twist between each other. But not twist by a bigger deal, twist by some same type. So PDSO group has no relation to each other, but they are PDSO, but the total module are a bit twisted to each other. So this gives you that, so the A, U, P infinity is a twist of h, U, something like that, a different as above. So in a sense that the total module of P infinity, you call the total module, h, U turns the total module of some i, so i is a PDSO group. It's a one-dimensional representation. So this is a one-dimensional representation. It's a local system, OEP local system defined over. So this product is this product defined over this similar variety, U cross U there. Then here, what do you do? I mean, here, then, what do we care? We don't really care, but PDSO group will care about this exact sequence. There's a hardware filtration. So we need to only figure out, we need to understand how they are doing their crystal related. Unfortunately, this kissing function can result from the color representation to the crystal. So at the end, so this, let me try to see. What? Over which base? Because there, it looks like we have to work over a formal scheme, which is an awful thing. No, this is not a formal scheme. This is just there. Said that we have to complete. What? You said that we need to complete at some point. No, no, no. I said this is a fact. Fact to XU is the integral model of this thing. One fact is that the PDSO group at a special fiber universal deformation is the Schumacher curve of the deformation itself, right? The completion of Schumacher curve of the self. Now, we don't need this fact. So where do we work? What is the base? We work over a scheme? Over a scheme. Which is this projected limit? Yes. And then we apply, break it into this kind of thing? What does it mean? I define a crystal there. During a crystal, this is a Bohm local system. Where do you find the thing? Where are we comparing something? OK, OK, let me try to. What is the base? Just to see that the base is reasonable to apply something, we know. Base is, oh, you mean which scheme I'm talking about. Scheme is here. So I will construct. OK, so I use this integral structure to construct integral structure here. I don't have time, so I try to take a sugar card. So here, I spent a lot of time studying the integral structure here. I get an integral structure there, right? How funtorial is the case in functor? Oh, it's very good, actually. The case in functor, apply case in functor. So we will get that. The dunet module, with the filtration of your infinity, you call the dunet module of h of u times the dunet module of i. So then this one give you the relation between differentials on x and x prime. There is something which I don't really, I'm missing. So here, these kind of statements are relative statements over the base which you. Oh, yeah, relative statement, yeah. Bray cuisine is not a relative theory. As far as I know, it applies only in the absolute case. Sometimes we are still very far from the relative. Oh, right, right, right. You're absolutely right. You're absolutely right. So actually, I applied this at one point. Because first-time I believe I already know extended has a good reduction everywhere. OK, because, yeah, otherwise, it would be extremely surprising. You ask good questions. Yi Chuan and I discussed this thing for a long time. Trying to understand a relative version could not get it. You're right, just be lazy, cheating, you know. OK, maybe I should have finished here. Sorry, it's a bit over time.