Shintani generating class and the p-adic polylogarithm for totally real fields
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Transcript: English(auto-generated)
00:17
So welcome to Paris Peking Tokyo Seminar.
00:21
So it's my great pleasure to introduce the last speaker from Tokyo, Kenichi Banai, at Keio University on the weekend. So he will give us today a syntax generating class and the periodic pluralism for total real fees. So please start. Thank you very much for the introduction, and thanks for this opportunity
00:41
to give a talk at the seminar. Yes, I think everyone probably says this, but it's my first time giving a seminar. I assume. So a little bit not know how to do this, but today I'd like to talk about Shintani generating class and periodic fault logarithms. It's a series of work that I've been working on with Kei Hagihara, Kazuki Yamada,
01:02
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 this. 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 Q,
01:21
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 diplo values. So there's one very good generating function in this case.
01:42
And in the case of imaginary quadratic fields, there's the Robert's theta 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,
02:00
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
02:21
is so consider the case of the rational field. So this GT is a rational function on GM. And so there's this classical theorem, which says that if you take the logarithmic derivative
02:42
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.
03:01
So Zn is just nth power of the root of unity. And this function is known to have another continuation to the whole complex plane. And the relation to this layer theta function to the Dirichlet L values is that you have this lemma.
03:23
So if you let C chi be this value here, then the Dirichlet character 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 chi n can be written as some sum of these disease.
03:43
And then if you sum over n equals one to infinity, then you can prove that L chi S is of this form. And then you see that this is exactly where theta function. So yeah, so if you know the layer theta function, then you know all the degree L values
04:02
or the Dirichlet L function. So this layer theta function is very useful. And what the theorem that I just gave said is that g t, t over one minus t, it's one rational function. It's this one simple rational function. But this function knows all the values
04:23
of the layer theta values for all zs 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.
04:43
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,
05:04
but more precisely it's prove its theta 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?
05:21
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 generative class, which we construct to generate these layer theta values.
05:42
So that is the main theme of today. And of course, there's been lots of people who have been studying theta function of totally real fields. So our results based on old work by Shintani, it's sort of a reformulation of his work with some inputs from Barsky, Casanogas
06:02
on construction of Heckel L functions for totally real fields, and also observation by cuts using algebraic torus. And our results, I think, are also related somewhat to the works on Eisenstein co-cycles, Chekhov cycles, and Shintani co-cycles by mainly Dasgupta and his collaborators.
06:23
And I think their work is more group theoretical in some sense. And there's also work by Bellingson, Levy, and Kings on the topological polylogarithms for totally real field case. And that work is a little bit more on the topological side. And I think our construction,
06:41
I would say it's more algebraic. It arose from the study of polylogarithms 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.
07:02
So first, I want to talk about layer state of function for totally real fields. So I said that in the classical case, the layer state of function was very, very important. But in the totally real field case, it's not clear what layer state of function should be. So there's no,
07:21
I don't think there's been a definition of layer state of 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 rank of integers. And we denote by F plus cross
07:42
the group of totally positive elements in F. And we let I be the group of non-zero fractional ideals of F and IG, those which are prime to G. And then we denote by CLF plus the cross group, the narrow cross group of F with conductor G,
08:01
which is defined as such. And then a finite hectic character is just a finite character on the narrow cross 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
08:22
sufficiently greater than one. And then it has analytic continuation and all the good properties that you want. And so we want to write these values in term of something, the layer theta values. So how can we do this?
08:40
So we can write it like this, because there's a one-to-one correspondence between ideals and OF equivalent to A inverse and I inverse alpha, where alpha is an element in the totally positive part of R divided by the units,
09:03
because if you multiply alpha by units, 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 units
09:21
and you have to divide by delta, which gives lots of trouble. And so we have this function here, this character here on each off loss. So we denote this by chi off, which is given like this. So it gives a character on divided by gare off.
09:43
And then if we replace this by this notation, then we have this. So how should the layer state of function in this case look like? So copying the case of the rational field, we take c chi xi to be the finite Fourier transport
10:01
like this for some xi, which is an additive character on R divided by gare off. Then finite Fourier transport gives this formula here, very similar to the case of the rationals. And so a naive definition of the layer state of function would be is that one could try to define
10:23
layer state of function as the sum over these gazie alpha. But the problem we face with this naive definition is that so you want to sum alpha over all plus, but divided by the units.
10:40
But the problem is this additive character gazie, this is not well defined module of the units. So we have a problem here. And how to avoid this problem is actually very simple. So what we do is because gazie module delta is not so good, we take the delta orbit of gazie
11:04
and add all the gazie epsilon. So the units act non trivially on the characters. So we add over all the orbits of gazie and then consider this function. And then this function itself is well defined.
11:22
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.
11:42
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 r it was defined by the hectic character
12:02
that'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 Heckel functions, then we have this.
12:21
We just plugged 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 layer state of function. So you get a formula like this. So what we just proved is the following theorem,
12:43
or what we just proved is the following. So if we define that layer state of function to be of this form, then you can write all the finite Heckel characters in terms of some of these functions. So what I wanted to say here was, is that this layer state of function
13:02
is very, very important that if you know all these functions, then you all know all the Heckel functions. So in order to find the canonical generating function of the special values of these functions of finite Heckel characters, then it's sufficient to find a generating function
13:20
for the layer state of function here. And so that is the first section of my talk. So are there any questions up until here? Or is it okay? Yeah, too fast or okay?
13:42
Yeah, it seems okay. Okay, yeah, yeah, yeah, yeah, yeah, yeah. Okay, so this is just a notation that maybe you were, can you go back several pages? Of the thing modulo G.
14:03
So before, yeah, so, so when you work with prime to G, I suppose this should be effective because you work with same congruent to one or G.
14:22
So it should be, the Gothic G is effective, is a. What do you mean by effective? No, no, the Gothic G. When you write a prime to Gothic G. Right, right, right. This is a, you say G first it was a fractionary ideal,
14:41
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 data function? There is the data function, yes.
15:01
On just the previous page also? This? Yeah, okay, okay. And your convention for Dirichlet characters is that you consider them, you consider primitive Dirichlet character? That is? Yes, yeah, in most of the formulas, yes, yeah.
15:23
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.
15:43
Okay, yeah. Okay. Yeah, please go ahead. Okay, okay, okay. So go back to the, please. Okay, so that was the definition of large data function
16:05
and we think this is very important because it knows about finite Hecke characters and the Heckel functions. And yeah, so I mean, one thing we realized when doing this research
16:21
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 Zeta function
16:41
and the generating function. So the Larry Zeta function itself, it doesn't a priori have a good generating function, but what we want to use is the generating function for Shintani Zeta function studied by Shintani. So in the next section, I want to talk about Shintani Zeta function
17:02
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.
17:22
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
17:41
and we define the 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 define a cone to be a G dimensional, F rational simplicial closed polyhedral cone.
18:03
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 D union zero of the form this. So it's a closed cone for some basis alpha one,
18:21
alpha G and F plus D linearly independent over R. And in this case, we say that alpha is the generator of sigma alpha. So I wrote F plus D, but usually, yeah, usually one takes some fractional ideal here
18:43
and then take a basis as a fractional ideal. So the Shintani zeta function, what it is is if you take a fractional ideal R and for a cone and some torsion element
19:03
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,
19:22
and then you sum over alpha in this place. And this is the intersection of R 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
19:43
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,
20:02
so the cone is something in R2, so a closed cone is something like this. And what the upper closure is, 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.
20:21
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,
20:42
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
21:01
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
21:22
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 exists a set phi of G dimensional cones,
21:42
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 a finite, yeah, it can be represented by a finite number of cones, module the action of delta,
22:01
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, this area into very nice parts. And using this decomposition,
22:20
what is important about this decomposition is that if you take a torsion point in G, 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,
22:44
you can write it in terms of the Shintani theta function of the cone and phi. So you wanted to, we wanted to sum over our R plus module delta, but this phi divided by delta gives the fundamental area of R plus divided by delta.
23:05
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,
23:20
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 the 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
23:41
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.
24:02
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 multiplicity additive or additive characters, yeah, in here.
24:22
And these objects have a underlying just scheme. So in the case of GM, it's home to that GM and for this, it's just home a GM, which written as an affine scheme is of this form.
24:41
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 natural multiplication here and it's a fine scheme. And in the case of the classical case, it was T over one minus T.
25:00
So we want to see what comes over here. And a little bit of preparation, we say that an element and an ideal I is primitive if for any N, if you divide alpha by N, then it's no longer in the ideal.
25:20
And we denote by script A odd, the set of primitive elements in R. Then if we take a basis G like this and the 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
25:41
that defined by these spaces. And then half is just the upper closure. So this is a finite sum. So this is a simple polynomial, a rational function on TA. And we let U alpha R be T R minus the divisor, T alpha equals one.
26:02
Then Shintani proved the following theorem. So for any integer K one to K 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.
26:23
So Shintani proves 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.
26:42
Actually, if you take the complex valued points of the algebraic torus, then this really corresponds to the direction of differentiating with respect to the embedding of the real in that direction. But it's also an algebraic differential operator
27:02
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,
27:22
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.
27:43
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.
28:03
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 you can prove that it exists. But there are many, many ways to take Shintani decomposition. And so this function is not so canonical.
28:22
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 our Shintani generating class.
28:40
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.
29:03
So if you take any positive element F plus cross, then if you multiply by X, then you get trivially multiplication by X gives an isomorphism of O F modules R and X R. And this gives an isomorphism of algebraic tori,
29:23
T X R congruent to T R. And on the C-valued points, you can really see it explicitly. A C-valued 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 C X, which maps R to C cross.
29:42
But the definition of C 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 an unit, then if you multiply a unit by an ideal, then you get the same ideal.
30:01
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.
30:23
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.
30:41
And so we have an action of group on the torus. So we want to say a little bit about equivariant sheaves and cohomology of this object. So actually, we don't work on T, but we want to work on U, which is you take out all the units from each of the components. And then U also has an action of F plus cross
31:02
induced from the action of T. And an equivariant sheaf is so something a sheaf on U, which is has good properties with respect to the group action. So the precise definition is this. So for each X, you have an isomorphism,
31:21
which is compatible with the composition. Compatible with the composition means that this diagram is commutative. So an equivariant sheaf is just a set of family of sheaves on each of the components of U R, which behaves well with respect to the group action. So you have an isomorphism here
31:43
for each element of the group X. And then you can define the equivariant cohomology by just the right derived functor of this and M derived functor of taking the global section and then taking the group invariant part.
32:01
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 U alpha R be take out the divisor T alpha
32:22
equals one for each alpha. Then this gives an open covering, a fine open covering of U, because you're taking out just the divisor T alpha equals one. And if you take the sum of all of it,
32:42
you're just, only the unit is removed from each R component. Then again, this F plus cross acts on the indexes and the open sets. And then we can define the equivariant check complex simply by, so if you have a F plus cross
33:03
equivariant chief on U, then just define the complex as this. So R is for each component and for each index alpha. So Q plus one component, then it's just a usual check complex,
33:21
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, differential. Then what we can do is we can prove that actually this complex
33:41
calculates the equivariant cohomology of U with respect to the action of F plus cross. So yeah, this is a very good complex. So calculate this cohomology. And so what we do is we fix once
34:01
and for our 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.
34:22
And then if we take the generating function, there should be an R here somewhere, R here. But if you take the Shintani generating function and multiply it by sign alpha, then actually this, all of this defines an element in here.
34:44
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 cohomology here.
35:03
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.
35:23
And if you take all of it, then all of them together form a single canonical cohomology class. So this is what we call the Shintani generating class. Excuse me, what is the sign of matrix?
35:41
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 took the determinant. Yeah, okay, yes. Sign of the determinant, I'm sorry, yes, yes, exactly. Thank you. Yeah, it is to make the check co-cycle
36:02
cancel out properly, yes. 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,
36:22
if you multiply all the delta tau's, then you get the differential delta. And this is a differential, but actually this induces a homomorphism on the equivariant cohomology because it induces a map on the complex. And so delta is all the delta tau together. So if you delta the t alpha,
36:42
then it's the norm of alpha times t alpha. So it's a differential kick in like this. And what we could prove is as follows. So for any integer k greater than or zero, and any torsion point z in t r for any r.
37:00
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 that point z, then you can specialize. So it gives an element to the cohomology
37:20
of an equivariant point. And it's the g minus first cohomology. And on a point, it may seem like it disappears. But fortunately, delta is isomorphic to z g minus one. So it has rank t minus one. And the cohomology of this is just a group cohomology of this. So you have one dimension surviving.
37:44
So actually, it's a cohomology class, but you can evaluate this cohomology 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
38:03
the layer theta values in this case. And this works for any integer k, positive integer k, and any torsion point z, which is different from one. So this means that this g t knows all the layer theta
38:21
values for all non positive values, non positive integers and all characters. So this is 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
38:41
and all torsion points. So that is our main theorem. So do you have questions up until here? Okay. Yeah, it looks okay.
39:02
Okay. Yeah. So now I want to talk about p-adic polylogarithms because I mean, I like to study polylogarithms and this research started originally from trying to figure out the polylogarithm in this case.
39:22
And so our observation now allows us to define the p-adic polylogarithms very clearly in this case. But we want to talk about the p-adic 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
39:42
containing the Galois 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 tr
40:04
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 p-adic affinoid spaces. So what we're doing is, so we're not just removing the point t alpha equals one,
40:22
but we're removing the residue disc around t alpha equals one, but still it's some p-adic analytic space. And then, you know, then uk hat i, you just define to be the sum of all these. You take residue discs out,
40:43
then you take all the union and then u hat k is this union over all the fraction ideals. So you have this p-adic analytic ring. And then what we could prove is that for any fractional ideal i of f
41:04
and for any integer k and 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.
41:21
We remove, we consider only the alpha in a tensor zp star. And a tensor zp star is the set of generators of the 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
41:41
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.
42:00
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.
42:21
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 p-adic help functions. And I just show, sort of explain this.
42:41
So this is again, like a homology 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 t minus one,
43:00
there is one dimensional left. So you can think about the value of this. So this point is well defined. So using these values of the p-adic polylogarithm, we want to relate it to p-adic help function. And so what is the p-adic help functions in this case? So the p-adic help function for totally real field case was,
43:24
it's a old result by Barsky, Casanogus, and Lien-Ribet. And Lien and Ribet, they used modular forms or Hilbert modular forms. And the one which is closer to ours is probably the one by Casanogus. And, but the p-adic help function,
43:42
it's certain function on zp, that's on zp, which interpolates all the Hecke L values of the finite Hecke characters. So because it's a p-adic interpolation, you have this a little bit tight in the lower character coming in, but in any case, and you have to remove some p Euler factors,
44:03
but in any case you have 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 r B it's in r for any fractional ideal r,
44:22
which means that this inclusion defines a map from t r to t r B. 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
44:46
and take the primitive part, this 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.
45:01
And what is interesting about this is that this row B actually gives an action of the ideal class group on t zero g, which is simply transitive. So this looks very much like complex multiplication theory
45:23
where if you have a 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 our torus also, this class group acts on the G torsion point.
45:40
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.
46:00
And so if you have a element G torsion point z in t i g, then we denote by z B the image by row B like this. And so the result concerning p-adic of function is as follows. So suppose G does not divide any power of p.
46:23
So G is an integral ideal, as Ofer 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 p-adic L function at any integer
46:41
can be written as a sum of a Gauss sum, some Gauss sum and the points of the p-adic 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 p-adic polylogarithm function
47:01
can be used to write the Kubota-L-pool p-adic L function, but using our p-adic L function or using our p-adic polylogarithm, we could prove a very similar result in this case.
47:21
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.
47:44
So this is the result concerning p-adic L functions. So are there any questions until here? Is everything okay? Yeah, it's okay. Okay, okay. It's okay? Okay, okay.
48:00
So yeah, so maybe I rushed through too fast, but yeah. So that is basically what I wanted to talk about today. Wait a second, Ola has a question. No, I just wanted to say again the formula where you add something with primitive points acting,
48:22
I didn't quite catch the, what is this? Just all the notation and so on. So this is what, A runs over... The fractional ideals of F. All the non-zero fraction are ideal?
48:41
Yes, yes, yes. Okay, and so this then will permute. Ah, okay. And then Ola gives an action on this, which is simply transitive. Okay, this is not difficult to see. Okay, thank you.
49:01
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.
49:22
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 state of function for the real fields. And so what we did was we did not just take the point,
49:42
but we took the delta orbit of the point to define it, and it gave a very nice function. And then we constructed the Shindani generative class as a canonical class and cohomology of a certain algebraic torus with an action of F plus cross.
50:02
And then we were able to prove that it generates all the non-positable special values of layer state of 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,
50:22
definition of the p-adic polarizm became very, very natural. So one did not want to find a p-adic polarizm function, but one wanted a p-adic polarizm class. And then it's very natural, you just remove the p part and it gives a p-adic polarizm function.
50:40
And then one can prove that it's related to special value of p-adic health p-adic health 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
51:02
or we wanted to understand with my colleagues, basically the theory by Neckover and Scholl on plectic structures in the fully real field case. And we wanted to find good examples of varieties with plectic structures that we could work on.
51:21
And it seems that this algebraic torus seemed to be a very good candidate. And so we yet don't have a good equivariant plectic polylogarithms theory or Pott's theory or things like this. But because of our calculations, we sort of conjecture that the specialization
51:42
to torsion points of the equivariant plectic, I should say Hodge. The real Hodge realization of the equivariant plectic polylogarithm for T should be related to positive values of our layer state of functions. So in the case when f equals q or g equals one,
52:04
the layer state of functions are in fact the special values of polylogarithm functions. And I guess this should be, the natural generalization would be is that the polylogarithm function specialized to torsion points should give layer state of functions,
52:21
special values of layer state of function. And also in the Hodge case, we have no idea how to do the calculations. In the suntomic realization case, we are not able to make the theory work just yet, but we have lots of p-adic differential equations and calculations.
52:41
And it seems to be that the suntomic realization of the equivariant plectic polylogarithm for T may be expressed using our p-adic polylogarithms. So today I just introduced the version with just one integer k. But in fact, one can do a plectic version
53:01
where you have g weights. And then these g weighted polylogarithm function seems to describe the suntomic 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.
53:24
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 detorion point doesn't make sense
53:40
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 Gallo action. So I don't know what is a good way to think about this.
54:01
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 Gallo action structure and make it compatible with the action of the ideal class group, then can one do something with,
54:24
so yeah, Kronecker's union term or how to, 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.
54:42
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 this was, can one imagine that you're lucky
55:05
that the function can be some constant term of something? Probably, yes, yes. I haven't really thought about this, but I don't know what the right framework is.
55:20
Some maybe Hilbert module thing, and then you degenerate to the cusps, then you get these torus. And yeah, I don't know, but I would imagine something like this. Thank you.
55:40
So are there more questions? I think yes, yeah. Of raise his hand and I think also another participant should raise his hand. Okay. How do I see to, who gets to see who is raising his hand? Let's see, probably we only can see. So maybe offer and then.
56:01
Yeah. Okay, so again, I asked to show me again, the point where you discussed the computation using certain kind of check or cycles. I can hope. Divine cohomology. So this, I want just to again, the notation.
56:23
So you take the torus modulo for any, for all you mean, for all,
56:41
you mean t alpha equal one for any means for all? No. This is the notation, here, here, yeah, here for all, for all, yes. No, no, but what is, ah, okay, this is for, okay, okay, no. This is just a notation for one and then this is the covering.
57:02
Then can you show the definition of A, A? A is the primitive element in all, so. Ah, okay. And then, okay. And then, okay, this is all in on zero.
57:24
And then, okay, so there is no fix, so everything, all the action is free, so there is no, because you repeat? Right, right, right. Okay, the action is free, so there is no problem, okay.
57:42
Right, exactly, exactly, that's how you prove, yes. Okay, so, thank you, I was just, I was confused, I thought that you had some, okay, it was not, okay. Yeah, thank you. Mm-hmm, yeah, okay, ah, yes, yeah.
58:08
So, Mladim, you can ask. Yeah, Mladim, yeah. Can you hear me? Yeah, go ahead. Thank you for, I just have a question, it's a little bit of a follow-up of the question.
58:22
So, you have this interpolation formula, which is the usual formula for a PRL function and a delivery-bed L function. So, sometimes you can have an exceptional zero, if you can show the formula.
58:44
Yeah, so for, well, in your normalization, that's for, yeah, k equals zero. So, for example, if the character is trivial at many gothic piece, it has a multiple zero, or this gross conjecture,
59:00
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 does this give some insight, like the fact that the high power of p would divide the value at,
59:23
because you have an actual zero at k equals zero, so this means that 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
59:43
gives additional insight as 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.
01:00:00
like, can you avoid this with some other point? Yeah, I haven't thought about this. So I don't know exactly. Yeah. Thank you. Thank you for the question. So yeah, so what, what kind of problems are interesting in this
01:00:24
direction? So Well, what's interesting to me is, as I've been, this, this empirical function is a constant term of the right time series. Right, right, right, right, right, right, right, right. The rules are related to some, for example, interesting
01:00:41
phenomenon on the Eigen curve and things like this. Right, right, right. But you're out. Yeah. I see. Okay. Thank you very much. Okay, so please go ahead.
01:01:04
I want to know, are there any kummer congruence satisfied by the new analytic function LP in your story?
01:01:21
So, so, so the periodic polynomial functions, it's sort of defined via kummer congruence in some case, in some sense. So, you know, when one tries, when I said that this power series converges as a, as a limit of
01:01:43
polynomials, that is sort of the kummer 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 kummer congruence that's everywhere. I
01:02:00
think in the, 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 everything looks like, yeah. Thank you.
01:02:24
Okay, are there more questions? Okay, if not, thank you very much for the speaker. Thank you very much.