 So, welcome to Paripekin Tokyo seminar. So, it's my great pleasure to introduce the last speaker from Tokyo, Kenichi Bannai at Keio University and Rikken. So, he created us today, Sintai generating class and the PRD for your students for totally our fees. So, please start. Thank you very much for the introduction and thanks for this opportunity to give a talk at the seminar. Yes, I think everyone probably says this, but it's my first time giving a seminar so a little bit not know how to do this, but today I'd like to talk about Sintai Generating Class and Piatic Fall Algorithms. It's a series of work that I've been working on with Kei Hagi-Hara, Kazuki Yamada and Shuji Yamamoto and with lots of other people and some preference are on the archive. So, if you're interested then please take a look at those. And so, what the general story that I want to talk about today is really a very simple question. In the case of the rational field two, there was this rational function one minus T which is related to cyclotomic units. And if you take the logarithmic derivative, you get T over one minus T, which is known to be related to deeply other values. So, there's one very good generating function in this case. And in the case of imaginary quadratic fields, there's the Robirth data function, which is related to elliptic units. And if you take the logarithmic derivative, then you get this theta dash T over theta T, which is known to be related to Heckel values. So, the simple question is, what do we have in the totally real field case? So, do we have something that we can do? And what we want to talk about today is some generating function in this case which gives Heckel values. And we want this to be very, very canonical. And so, what I mean by very, very canonical is, so consider the case of the rational field. So, this gT is a rational function on gN. And so, there's this classical theorem which says that if you take the logarithmic derivative of this function k times and specialize at roots of unity, which is not one, because at one you have folds, then you get the special value of layer theta function. And so, what is the layer theta function? It is a function given by this form where z is some root of unity. So, zN is just nth power of the root of unity. And this function is known to have another continuation to that whole complex plane. And the relation to this layer theta function to the Dirichlet L values is that you have this lemma. So, if you let C chi be this value here, then the Dirichlet Calcure can be written in terms of the layer theta function like this. And the proof is very simple. It's just the finite Fourier transform of finite characters. So, chiN can be written as some sum of these disease and then if you sum over n equals one to infinity, then you can prove that l chiS is of this form. And then you see that this is exactly layer theta function. So, yeah. So, if you know the layer theta function, then you know all the Dirichlet L values or the Dirichlet L function. So, these layer theta function is very, very useful. And what the theorem that I just gave said is that the gT, T over one minus T, it's one rational function. It's this one simple rational function. But this function knows all the values of the layer theta values for all gS and all non-positive kS or minus kS. So, yeah. So, in case what is amazing in this case is that you have just one very canonical function that knows all the twists and all the weights. So, that is what we mean by a very canonical function. So, if we go back to our picture, when I said that this knows Dirichlet L values, more precisely I meant that this knows layer theta values. And in the case of imaginary quadratic fields, this robust data function, I said it was Heckel L values, but more precisely it's prove its data values. And because of the functional equation, it's also layer theta values in this case. So, we have a very good theory in the rational field case and the imaginary quadratic field case. So, what can we do today? Is there a canonical function that knows Heckel L values? And then it's not really Heckel L values, but layer theta values that we want. So, I want to talk about a very canonical class which we call the Shintani generating class which we construct to generate these layer theta values. So, that is the main theme of today. And of course, there's been lots of people who have been studying data function of totally real field. So, our results based on old work by Shintani, it's sort of a reformulation of its work with some inputs from Varsky-Casanoogis on construction of Heckel L functions for totally real field. And also observation by cuts using algebraic torus. And our results, I think, are also related somewhat to the works on Eisenstein-Cocycles, Czech-Cocycles and Shintani-Cocycles by mainly Daskupta and his collaborators. And I think their work is more group theoretical in some sense. And there's also work by Bernice and Levine-Kings on the topological polar algorithms for totally real field case. And that work is a little bit more on the topological side. And I think our construction, I would say, it's more algebraic. It arose from the study of polar algorithms in the totally real field case. And the construction is very algebraic. And hopefully it's a little bit, it's very simple to understand. So, that is what I want to talk about today. So, first I want to talk about layer theta function for totally real field. So, I said that in the classical case, the layer theta function was very, very important. But in the totally real field case, it's not clear what layer theta function should be. So, there's no, I don't think there has been a definition of layer theta function exactly of this form up until now. So, what we want to investigate is finite hectic characters for totally real field. So, we prepare some notations. So, F is a totally real field with degree G and ring of integers. And we denote by F plus cross, the group of totally positive elements in F. And we let I be the group of non-zero fractional ideals of F and I G, those which are prime to G. And then we denote by CLF plus the class group, the neural class group of F with conductor G, which is defined as such. And then a finite hectic character is just a finite character on the neural class group. And then by extending this hectic character by zero to all the fractional ideals, then one can define the heckel function as this, which converges absolutely for the real part sufficiently greater than one. And then it has analytic continuation and other good properties that you want. And so, we want to write these values in terms of something, the layer heckel values. So, how can we do this? So, we can write it like this, because there's a one-to-one correspondence between ideals and OF equivalent to a inverse and a inverse alpha, where alpha is an element in the totally positive part of R, divided by the unit, because if you multiply alpha by unit, then it gives the same ideals. So, dividing by this unit is really the difficulty in the higher dimensional case. Yeah, so this delta is the totally positive unit and you have to divide by delta, which gives lots of trouble. And so, we have this function here, this character here on each plus. So, we denote this by chi-a, which is given like this. So, it gives a character divided by ga-a. And then, if we replace this by this notation, then we have this. So, how should the layer heckel function in this case look like? So, copying the case of the rational field, we take C chi-z to be the finite Fourier transport like this for some gz, which is an additive character on a divided by ga-a. Then finite Fourier transport gives this formula here, very similar to the case of the rationals. And so, a naive definition of the layer heckel function would be is that one could try to define layer heckel function as the sum over these gz alpha. But the problem we face with this naive definition is that you want to sum alpha over r plus, but divided by the units. But the problem is this additive character gz, this is not well-defined modulo, the units. So, we have a problem here. And how to avoid this problem is actually very simple. So, what we do is because gz modulo delta is not so good, we take the delta orbit of gz and add all the gz epsilon. So, the units act non-trivially on the characters. So, we add over all the orbits of gz and then consider this function. And then this function itself is well-defined. So, we define the layer state of function to be of this form like this. And this is a very simple trivial point, but actually this has a very good interpretation in terms of geometry. So, I want to emphasize what we're doing here. And then the finite Fourier transform becomes, so it was without dividing by delta before, without dividing by delta before, but if we divide by delta, then add all the delta orbits here, push it in here. Then because chi-a, it was defined by the Hecchi character. It's independent of multiplication by units. So, these finite Fourier coefficient is independent of the action of the unit. So, you can put it in here and you have this. And then, so if we use this formula to write out our Hecchi-a functions, then we have this. We just plugged it this in here to get this formula. And if you change the order of these, then you get this. And so, this part here is exactly the Larry's data function. So, you get a formula like this. So, what we just proved is the following theorem, or what we just proved is the following. So, if we define the Larry's data function to be of this form, then you can write all the finite Hecchi characters in terms of some of these functions. So, what I wanted to say here was is that this Larry's data function is very important, that if you know all these functions, then you all know all the Hecchi-L functions. So, in order to find the canonical generating function of the special values of these functions of finite Hecchi characters, then it's sufficient to find generating function for the Larry's data function here. And so, that is the first section of my talk. So, are there any questions up until here? Or is it okay? Too fast? Or okay? Yeah, it's okay. Ah, okay, okay. Yeah, yeah, yeah, yeah, yeah, yeah, yeah, okay. Okay, Joe, this is just a notation that maybe you were, can you go back several pages? Okay. The definition of the sing-moderu G, so before, yeah, so, so when you work with prime to G, I suppose this should be effective because you work with sing congruent to one or G, so it should be, the gothic G is effective, what do you mean by effective? No, no, the gothic G, when you write a prime to gothic G. Right, right, right. This is, you say G versus was a fractional ideal, but here it should be actually. Right, right, right, right, yes, yes, yes, okay, yes, okay, yes, yeah, it's an integral ideal, yes. Okay, okay. Okay, thank you. Can you go back to the definition in the classical case of the large death function? There is a state of function, yes, mm-hmm. Just the previous page also. This, this, mm-hmm. Okay, and your convention for Dirichlet characters is that you consider them, you consider primitive Dirichlet character. That is. Yes, yeah, and most of the formulas, yes, yeah. Okay, okay. Yeah, I mean, some parts, it doesn't have to be primitive, but in most of the important formulas, we assume primitive, yes. Yeah, so it's the usual for, okay. Because sometimes it gives different, okay. Okay, mm-hmm, yeah. Okay. Okay, mm-hmm. Yeah, please go ahead. Okay, okay, mm-hmm. So go back to the, please, mm-hmm. Okay, so that was, so that was the definition of large data function. And we think this is very important because it knows about finite HECI characters and the HECI-L functions. And yeah, so I mean, one thing we realize when doing this research is that when one tries to use the formula for large data function, then many formulas become very simple and easy to understand. So it's just a minor tweak, but I think it helps to streamline the theory. So next I want to talk about Shintani's data function and the generating function. So the large data function itself, it doesn't a priori have a good generating function, but what we want to use is the generating function for Shintani's data function studied by Shintani. So in the next section, I want to talk about Shintani's data function and the generating function for those functions. And so we let I be the embeddings of F into the reals. And since the degree of F is G, so we have G embeddings. And for the sake of order, we're just going to number them. So we fix an order, which means that F tensor R is isomorphic to RG, the G dimensional real space. And for any alpha tensor one, you just embed into RG by each of the embeddings here. And then in Shintani theory, we have to think about cones. And we define a cone in R plus GU zero as follows. This plus means the positive part of R. So it does not include zero. So you have to include zero here. And we define a cone in this talk, we say we define a cone to be a G dimensional, a rational, simple, shall close polyhedral cone. And it's very long. So I'm just going to say just cone in this talk, but it's any subset of R plus the union zero of the form this. So it's a closed cone for some basis alpha one, alpha G and F plus G linearly independent over R. And in this case, we say that alpha is the generator of sigma alpha. So I wrote F plus G, but usually one takes some fractional ideal here and then take a basis as a fractional ideal. So the Shintani zeta function, what it is is if you take a fractional ideal, ah, and for a cone and some torsion element of this additive character, we define the Shintani zeta function by this formula. So you have this cone here for this torsion point Z or torsion point, I mean additive character, and then G variable complex numbers. And then you sum over alpha in this place. And this is the intersection of all with the upper closure of the cone. And what is the upper closure of the cone? It is basically, so you add some points on the cone and maybe if you're smart, then this notation is sufficient for you. But yeah, when Yamamoto-san first told me this, I couldn't follow what he was saying. So what the upper numbering is, is for the case when the dimension is two, so a cone is something in R2. So a closed cone is something like this. And what the upper closure is, you always take one direction to be up and you include one side to be closed and all the other ones to be open. So the definition is, if you move a little bit downwards in one direction and if it's in sigma, then you include it. So this point here is in the upper closure because if you move down a little bit, it's included. But this border here is not because if you move down, then it goes outside. So it's because when you want to paste together the cones, then you want to count the boundaries without redundancy. So you have to put in some system of how to put in the borders and that is just for this. So the upper closure is something like this. And then, so the Shintani-Zeta function is you just take the upper closure of a cone, intersection with the integral ideal and you just sum over all of them. So this is a G variable complex function and it is known to have an analytic continuation to the whole complex plane. And then why these cones are important is because of the Shintani decomposition which was first proved by Shintani. And this version, upper closure version was proved by my colleague, Yamamoto-san. And so what they proved is that there is a set phi of G dimensional cones which is stable under the action of delta such that if you divide it by delta, then it's a finite set. So you have a finite representation or finite, yeah, it can be represented by a finite number of cones, module to action of delta. And then all the r plus G can be written as the sum of the upper closures of the cone and phi. So it's a way to break down this area into very nice parts. And using this decomposition, what is important about this decomposition is that if you take a torsion point in GZ times some additive character which is not equal to one, then you can divide this. So this is the definition of the layer theta function. But then because of this summation here, you can write it in terms of the Shintani theta function of the cone and phi. So we wanted to sum over r, r plus module delta, but this phi divided by delta gives the fundamental area of r plus divided by delta. So that's how we use the Shintani decomposition to calculate the layer theta function in terms of the Shintani theta function. And because Shintani theta functions, you had to add over all the orbits of the GZ. We also take all the orbits over GZ of the Shintani theta function and then we have this nice formula. Yeah, and so what is good about this Shintani theta function is that it has a generating function. So Shintani proved that there is a generating function for this function. So we want to now talk about the generating function of the Shintani theta function. And so where does the generating functions live? So in the very classical case, it was T over T minus one, which was a rational function on GM. And in the totally real field case, we've already seen, I've already introduced this notation. So this is the homomorphism additive home from R to C star. So R is additive and C star is C crosses multiplicative. So it is characters multiplicative additive or additive characters, yeah, and here. And these objects have a underlying just scheme. So in the case of GM, it's home to the GM. And for this, it's just home AGM, which written as an affine scheme is of this form. So this T alpha is, if you have T to the alpha and T to the alpha dash, then the multiply, you get T to the alpha plus alpha dash. So you have this national multiplication here and it's a affine scheme. And in the case of the classical case, it was T over one minus T. So we want to see what comes over here. And a little bit of preparation, we say that an element in an ideal I is primitive if for any N, if you divide alpha by N then it's no longer in the ideal. And we denote by script AR, the set of primitive elements in R. Then if we take a basis G like this and a cone generated by this, it should be alpha, I'm sorry. And then define the function, rational function like this, where P alpha is like the fundamental parallel pipe that defined by these spaces. And then hat is just the upper closure. So this is a finite sum. So this is a simple polynomial, a rational function on T. And we let U alpha R be T R minus the divisor T alpha equals one. Then Shintani proved the following theorem. So for any integer K1 to KG greater than equals zero and torsion pointing here, if you take the derivative of this function here and plug in T equals theta, then you get the Shintani theta values. So Shintani proved that for Shintani theta functions there's a very, very nice rational function. And here delta tau is the differential satisfying delta tau T alpha is alpha tau T alpha. So this is a differential and for any embedding. Actually, if you take the complex value points of the algebraic torus, then this really corresponds to the direction of differentiating between with respect to the embedding of the real and that direction. But it's also an algebraic differential operator and it is defined as such. So we have this very nice generating function. So we have this generating function and we had this formula connecting their theta values and Shintani theta functions. So if we just take the sum, because it's the sum like this, I think I forgot maybe this, but if we take the sum like this, then yeah, we're just summing and this is the generating function for this, then we get a nice formula like this. So this is a generating function for a value like this. So this looks very nice. And I think up until now in many theory, this function was used many, many times. But our criticism of this function is that, first, this function depends on z, the point that you want to investigate. And the second criticism is that this function depends on the choice of the Shintani decomposition. So Shintani decomposition is a very nice decomposition and it can prove that it exists. But there are many, many ways to take Shintani decomposition and so this function is not so canonical. So these two things are something that we wanted to avoid in our research. So how can we create a canonical generating function? That was the question that we really thought about. And the answer is, is our Shintani generating class. And so what we do is we use these functions to create a very canonical generating class, which really knows the values of all the layer data functions. And so, but to start, I want to define some actions on our torus, algebraic torus, t, r. So if you take any positive elements f plus cross, then if you multiply by x, then you get trivially multiplication by x gives an isomorphism of f modules, r and x, r. And this gives an isomorphism of algebraic tori, t, x, r, congruent to t, r. And on the c-valid points, you can really see it explicitly. A c-valid point of this is a character from x, r to c star, c cross. And then if you map this by x, then you get to see x, which maps r to c cross. But the definition, because the x is just, you take multiplication by x on the inside. So it's a very natural map. And so if epsilon, so if x is a unit, then if you multiply a unit by an ideal, then you get the same ideal. So you get an isomorphism from t r to t r. So you have an action of delta on t r. But more generally, we can take all the sum over all the t r for all fractional ideals, then you can multiply by x and get an isomorphism like this. So you get an action of f plus cross on t. And one of our idea, especially when working on the case when the cost number is greater than one, is that this t is really a nice guide to work with. And so we have an action of group on the tourist. So we want to say a little bit about equivalent sheafs and cohomology of this object. So actually we don't work on t, but we want to work on you, which is you take out all the units from each of the components. And then you also have an action of f plus cross induced from the action of t. And an equivalent sheaf is, so something a sheaf on you, which has good properties with respect to the group action. So the precise definition is this. So for each x, you have an isomorphism, which is compatible with the composition. Compatible with the composition means that this diagram is commutative. So an equivalent sheaf is just a set of family of sheafs on each of the components of you are, which behaves though with respect to the group action. So you have an isomorphism here for each group element of a group x. And then you can define the equivalent cohomology by just the right drive function of this and drive function of taking the global section and then taking the group invariant part. And so this is a very abstract definition, but what we do is we construct explicit complex to calculate this cohomology, especially in this case. So we want, we give an explicit complex to calculate this. So let your alpha R be, you take out the divisor t alpha equals one for each alpha. Then this gives an open covering, a fine open covering of you because you're taking out just the divisor t alpha equals one. And if you take the sum of all of it, you're just only the unit is removed from each component. Then again, this F plus cross acts on the indexes and the open sets. And then we can define the equivalent check complex simply by, so if you have a F plus cross and covariant shift on you, then just define the complex as this. So R is for each component and for each index alpha. So Q plus one components, then it's just the usual check complex, but with the invariant part here. And one can make this into a different, into a complex by taking the differential to be the usual check differential. Then what we can do is we can prove that actually this complex calculates the equal variant kohomoji of you with respect to the action of F plus cross. So yeah, this is a very good complex to calculate this kohomoji. And so what we do is we fix once and for a numbering of embeddings, then for any alpha, so alpha g elements in here define the sign of alpha to be the sign of the matrix of each component with respect to the each embedding. So it is plus one or minus one. And then if we take the generating function, other should be on R here somewhere, R here. But if you take the Shinkani generating function and multiply it by sign alpha, then actually this, all of this defines an element in here. And what we could prove is that actually this element defines a co-cycle. And then because it forms a co-cycle, it forms a very canonical class. It forms a single canonical class in the kohomoji here. So our question up until now was, we have lots and lots of generating function and lots and lots of cones. And the natural question we asked at first was, how to take a canonical choice? But actually the answer is the best is not to choose. You take all of it. And if you take all of it, then all of them together form a single canonical kohomoji class. So this is what we call the Shinkani generating class. Excuse me, what is the sign of matrix? Sign of the matrix is, so the determinant is positive or negative. So if it's positive, then it's plus one and negative, it's minus one. So you take the determinant? Yeah, okay, yes. Sign of the determinant, I'm sorry, yes. Yes, exactly. Yeah. Thank you, yes. Yeah, it is to make the check cycle cancel out properly, yes? Yeah. So yeah, the generating function pays together to form a single canonical class. And so we were very happy with this observation. But what can we do with this class? So the differential delta, if you multiply all the delta tiles, then you get the differential delta. And this is a differential, but actually this induces a homomorphism on the equivalent kohomoji because it induces a map on the complex. And so delta is all the delta tile together. So if you delta the t alpha, then it's the norm of alpha times t alpha. So it's a differential given by this. And what we could prove is as follows. So for any integer k greater than or zero and any torsion point gz in tr for any r. So the Shintani generating class lived here, but if you take the differential k times, then this guy also lives in here. And if you specialize this point at this point gz, then you can specialize. So it gives an element of the kohomoji of an equilibrium point. And it's the g minus first kohomoji and on a point it may seem like it disappears, but fortunately delta is isomorphic to zg minus one. So it has rank t minus one. And the kohomoji of this is just a group kohomoji of this. So you have one dimension surviving. So actually it's a kohomoji class, but you can evaluate this kohomoji class at the point because the point with this action of delta is one dimensional. And when we evaluate our Shintani generating class, then what we can prove is that it actually gives the layer theta values in this case. And this works for any integer k, positive integer k, and any torsion point gz, which is different from one. So this means that this gt knows all the layer theta values for all non-positive values, non-positive integers and all characters. So this formula here, this is a very clear generalization of the case when g equals one. I mean, it gives all the values for positive integers and all torsion points. So that is our main theorem. So that, do you have questions up until here? Okay. Yeah, it looks okay. Yeah. Okay. Yeah. So now I want to talk about periodic polylogarithms because I mean, I like to study polylogarithms and this research started originally from trying to figure out the polylogarithm in this case. And so our observation now allows us to define the periodic polylogarithms very clearly in this case. But we want to talk about the periodic case. So we fix an embedding of q bar into c and q bar into cp, the usual embeddings. And we let k be a finite extension of qp containing the galore closure of that. And so this is a little bit complicated, but what we're doing is just thinking about the rigid analytic spaces associated to what we just used so far up until now. So a is just the okay version of the ring of t r then you take the affinoid space attached to this. And then you remove t alpha minus one and take the affinoid space. So these are all periodic affinoid spaces. So what we're doing is, so we're not just removing the point t alpha equals one but we're removing the residue disk around t alpha equals one, but still it's some periodic analytic space. And then, yeah, then UK had I, you just defined to be the sum of all these. You take residue disks out then you take all the union and then you had to take this union over all the fractional ideals. So you have this periodic analytic ring. And then what we could prove is that for any fractional ideal r of f and for any integer k and con sigma, if we define this polynomial by this sum, it is very similar to the sum that we use for the generating function, but we have this here. We remove, we consider only the alpha in a tensor zp star and a tensor zp star is the set of generators of the OF tensor zp module, a tensor zp. So we sort of remove the parts divisible by p in some sense. Then what we can prove is that this itself is a formal power series, but one can prove that in fact, this is a limit of polynomials and hence it defines a rigid analytic function in here. And we have this for r and sigma like before. So what we could prove is that when we bring in the sign again, if you attach the sign to these polynomials, then you get an element in here again. And then we could prove that this defines a co-cycle. So it forms a canonical class again in here. So you have rigid analytic spaces this time instead of algebraic ones, but still you have something in here. And actually what we could do with this function is that we are able to prove that it's related to special values of piatic alt functions. And I just sort of explain this. So this is again like a homologic class, but you can specialize this to points again. So this is a class which lives in here, but if you specialize, I'm sorry, specialize, then the point is again, because delta has a rank g minus one, there is one dimensional left. So you can think about the value of this. So this point as well defined. So using these values of the piatic poly-algorithm, we want to relate it to piatic alt function. And so what is the piatic alt functions in this case? So the piatic alt function for totally real field case was, it's a old result by Barski, Kassel-Nugus, and Lien-Ribet. And Lien-Ribet, they used modular forms or Hilbert modular forms. And the one which is closer to R is probably the one by Kassel-Nugus. And but the piatic alt function, it's certain function on Zp, as on Zp, which interpolates all the heck yeah L values of the finite hectic characters. So because it's a piatic interpolation, you have this a little bit of a tiny little character coming in, but in any case, and you have to remove some P Euler factors, but in the case of this interpolation. And before going into the details, just one more thing about these torus T, which is interesting. So if you have an integral ideal B, then Rb is in R for any fractional ideal R, which means that this inclusion defines a map from TR to TRB. So, and if you sum over all the fractional ideals, you get a map like this. And actually what can prove that this action is compatible with the action of F plus R star. So if you look at the G torsion points of each torus and take the primitive part, the zero means the primitive part and sum together and divide by F plus cross, then you get sort of the G torsion point of the quotient stack in some sense. And what is interesting about this is that this Rb actually gives an action of the ideal class group on T0G which is simply transitive. So this looks very much like complex multiplication theory where if you have an elliptic curve with complex multiplication, then because of the Galois action, the ideal class group acts on the torsion points of the elliptic curve, for R torus also this class group acts on the G torsion point. However, in this case, these are all defined over Z or Q. So unfortunately this doesn't have any Galois action of the totally real field. So I'm a little bit lost what to think about this, but this action of this ideal class group on this, which is simply transitive is something very interesting. And so if you have a element G torsion point Z in TIG, then we denote by Zb the image by Rb like this. And so the result concerning periodic L function is as follows. So suppose G does not divide any power of P. So G is an integral ideal as over said, which does not divide any power of P and let Z be an arbitrary primitive G torsion point. And here then for any integer K, we have this formula. So the value of periodic L function at any integer can be written as a sum of a Gauss sum, and the points of the periodic L functions here. So this is a very natural generalization of a result by Coleman. In the case when F equals Q, he proved that the classical periodic polar logarithm function can be used to write the Kubota-Leopold periodic L function, but using our periodic L function or using our periodic polar logarithm, we could prove a very similar result in this case. I mean, so maybe it's not, I don't go into the detail of the proof, but the existence of the row B, the action of the ideal class group makes this formula very, very simple. And so I mean, I want to understand what that is doing, but I don't yet have a feel of how things should be. So this is the result concerning periodic L function. So are there any questions up until here? Is everything okay? It's okay. Okay, okay. It's okay. Okay, okay, so yeah, so maybe I rushed too fast, but yeah, so that is basically what I wanted to talk about today. Wait a second. Who has the question? No, I just wanted to see again the formula where you had something with primitive points acting. I didn't quite catch the, what is this? Just the old notation and so on. So this is, A runs over... The fractional ideals, okay. All the non-zero fractional ideal. Yes, yes, yes. Okay, and so this then will permute, ah, okay, and then gives an action on this, which is simply transitive. Oh, okay, this is not difficult to see. Okay, thank you. No. Okay, okay. Right. Yeah, so yeah, this object has an action of this. Yeah, it's not difficult to see. You're right, yeah. Okay, thank you. Mm-hmm, mm-hmm, mm-hmm, mm-hmm, mm-hmm, mm-hmm. Yeah, okay. So yeah, so the conclusion is so what, so I guess I talked about what I wanted to talk about. So what we did was we newly defined the layer-zeta function for the Ligubial fields. And so what we did was we did not just take the point, but we took the delta orbit of the points to define it and it gave a very nice function. And then we constructed the Shintani-Generic class as a canonical class and cohomology of a certain algebraic torus with an action of F plus cross, and then we were able to prove that it generates all the non-positive special values of layer-zeta functions for all non-trivial finite characters. So we were able to prove an analog of the case when g equals one. Then using this idea of thinking not about functions about cohomology classes definition of the periodic polar algorithm became very, very natural. So one did not want to find a periodic polar algorithm function, but one wanted a periodic polar algorithm class. And then it's very naturally just remove the P part and it gives a periodic polar algorithm function. And then one can prove that it's related to special value of periodic health-tech functions for totally real fields. So that generalizes the result by Coleman. And so the conjectures and questions that we have is this research originally started because I wanted to understand, or we wanted it to understand with my colleagues, basically the theory by Neckover and Scholl on plectic structures in the Pallori real field case. And we wanted to find good examples of varieties with plectic structures that we could work on. And it seems that this algebraic torus seemed to be a very good candidate. And so we yet don't have a good equilibrium plectic polar algorithms theory or hot theory or things like this. But because of our calculations, we sort of conjecture that the specialization to torsion points of the equilibrium plectic, I should say hot, hot, hot realization of the equilibrium plectic pallori algorithm for T should be related to positive values of our layer of data functions. So in the case when F equals Q or G equals one, the layer of data functions are in fact the special values of poly logarithm functions. And I guess this should be, the natural generalization would be is that the poly logarithm function specialized to torsion points should give layer of data functions, special values of the data function. And also, in the hot case, we have no idea how to do the calculations. In the symptomatic realization case, we are not able to make the theory work just yet, but we have lots of chaotic differential equations and calculations. And it seems to be that the symptomatic realization of the equivalent plectic poly logarithm for T may be expressed using our periodic poly logarithms. So today I just introduced the version with just one integer K, but in fact, one can do a plectic version where you have G weights. And then these G weighted poly logarithm function seems to describe the symptomatic realization. So in level of calculations, I think we have some results, but we are not able to fit it into a formalism just yet. So I'm curious how things should go. And finally, this stack. So taking T to be all the sum over all the fractional ideals then dividing by F plus star, the G torsion points of this stack. So maybe a G torsion point doesn't make sense, but if you sum over all the G torsion points, then it gives something with a natural action of the ideal class group. So this sort of looks like complex multiplication theory, but without Galois action. So I don't know what is a good way to think about this. If one could define some F structure on this, I don't know if this makes sense on this object. And then if there is some way to put in a natural Galois action structure and make it compatible with the action of the ideal class group, then can one do something with, so, yeah, Kronecker's union tromb, or how to, yeah, yeah. So yeah, I don't have any ideas, but I'm curious to see how this can go. So I think that's what I wanted to say today. So thank you very much for your attention. Yeah, thank you very much. So you discussed on this L value. So this L value can be regarded as the constant term of Einstein series. So you can imagine that your Lahi data function can be some constant term of something. Probably, yes, yes. Yeah, I haven't really thought about this, but I don't know what the right framework is. Some maybe Hilbert-Modger or thing, and then you degenerate to the cusps, then you get these torus and, yeah, yeah, I don't know, but I would imagine something like this. Oh, that would be interesting. Thank you. So are there more questions? I think yes, yeah. Offer raises his hand, and I think also another participant, who should now raise his hand. How do I see who gets to see who is raising his hand? You cannot see, probably we only can see. So maybe Offer and then... Yeah, no. Okay, so again I asked to show me again the point where you discussed the computation using certain kind of cheque or cycles. I know, mm-hmm. Revalent cohomology. So this... Maybe this, yeah. I want just to, again, the notation. So you take the torus, modulo, for any, for all you mean, for all, you mean tr for equal one for any means for all? No. This is the notation, and here, here, yeah, here for all, for all, yes. No, no, no, but what is... Ah, ah, okay, this is for some, okay, so, okay, though. This is just a notation for one, and then this is the covering. And can you show the definition of a, a? A is the primitive element in off, so. Ah, okay. Mm-hmm. Where it was. And then, mm-hmm. Okay, and then, mm-hmm. Okay, this is all in on zero. And then, okay, so there is no fixed, so everything, all the action is free, so there is no, because you repeat. Right, right, right. The action is free, so there is no problem, okay. Right, exactly, exactly. That's how you prove, yes. Okay, so, thank you for just, I was confused. I thought that you, you had some, okay, it was not, okay. Yeah, thank you. Yeah, okay, ah, yes. Yeah. Some, let me, I'm gonna ask... Yeah, grab them, yeah. Any query? Yeah, go ahead. Thank you for, I just have a question. It's a little bit of a follow-up of the question. So, you have this interpolation formula, which is the usual formula for get L function of, didn't re-bed L function. So, sometimes you can have an exceptional zero if you can show the formula. Mm-hmm. Oh, yeah. So, for, well, in your normalization, that's for, yeah, K equals zero. So, for example, if the character is trivial, it's P, at many gothic P's, it has a multiple zero, that's gross conjecture. These derivatives at zero has been studied, but I mean, because your formula later is true for all K, right? So, if your K is positive, but like very periodically close to zero, then can, does this give some insight, like the fact that the high power of P would divide the value at, because you have an actual zero at K equals zero. So, this means that since your formula is true for all K, like if K is very positive, but periodically close to zero, so does this give some insight? I don't know. So, I don't know if our formulation gives additional insight with respect to what was already known, because, would that be right? Yeah, because, yeah, because, yeah. No, I don't mean additional insight for zero, but like, can you explain this with some other point, say something? Yeah, I haven't thought about this. So, I don't know exactly, yeah. Thanks. Yeah, thank you for the question. So, what kind of problems are interesting in this direction? So... Well, what's interesting to me is related to application act, as I mean, this PIDKL function is a constant term of analysis time series. Right, right, right, right, right. Zero's related to some, for example, interesting phenomenon on the eigen curve and things like this. Right, right, right. But you're, oh, yeah. Mm-hmm. I see, I see, mm-hmm. Okay. Thank you very much. Okay, so, please go ahead. How many of you made it? I want to know, are there any kumar congruence satisfied by the new analytic function, Lp, in your talk? So, the Piatec Paul logarithm function, it's sort of defined via kumar congruence in some sense. So, yeah, when I said that this power series converges as a limit of polynomials, that is sort of the kumar type congruence that is used. Yeah. I see, I start with this, okay. Yeah, yeah. So, that everything fits together means that there's lots of kumar congruences everywhere. I think in the totally real field case, it's very different from the imaginary quadratic where if the prime is super singular, then it does lots of bad things. And the totally real case, yeah, it's very flat. So, every thing looks like, yeah, it's very multiplicative. Yeah. Thank you. Thank you. Okay, are there more questions? Okay, if not, thank you very much for the speaker. Thank you very much.