Log Drinfeld modules and moduli spaces
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Transcript: English(auto-generated)
00:00
Each week, I stay in the studio in Residence Omae, and this studio is the studio in
00:26
which I stayed for one year, from 1980 to 1981, and all the memories return to me.
00:40
During he was here that time, and he was always very kind and always smiling. And Christopher was also very kind to me, and thank you very much. Your help was so great for me that time. And I could not speak English, I could not speak French, so thank you very much.
01:03
And then, Ufa came from Israel. After Ufa came, Duini, every day Duini said, Ufa approved purity, Ufa approved that, Ufa approved reason. And Ufa gave lectures in the lecture room there, and in the lecture he said,
01:26
For many times, I can prove that, I can prove that. And each time he says, I can prove that, Duini became like this. And one time, Duini could not stay on the chair.
01:44
He was too much impressed by the result of war, and he dropped from the chair and started to swing, or float on the floor. So I wrote a letter to Shuji Saito in Japan.
02:05
He was still a student, and to explain that a great young mathematician appeared. And I put in the letter my drawing that Ufa is saying, I can prove that.
02:21
And Duini was singing on the floor. Smiling. And I, through some period, I shared the office with Ufa, and I thought that it was better to talk with him.
02:41
And so I explained the fact that I was studying to him. Then, next day, I heard his, I can prove that. And that was the end of my study.
03:00
I could not understand his method, so my study finished, but I could not see how it was finished. Then many people came to our office to ask questions to Ufa. And they sometimes made a sequence, and then the first person entered and discussed with Ufa,
03:28
and then became happy, saying, oh, I see, oh, I see, and then left the room. And then the next one came and became happy. One day, I saw a body laser in the sequence of people,
03:44
and I think that time he was proving, it was our main conjecture with Ang Lee Wiles, and probably he had some difficulty. And then he entered, and then, of course, he became very happy,
04:02
saying that Ufa gave a perfect answer before he explained his question well. So the memory of Ufa at that time is so strong for me, and he's getting now, he's getting greater and greater now, and so this is very nice.
04:26
And I took about the total compactification of Greenfield modules, and if a function is in one variable over finite field fq,
04:56
and infinity is a fixed place of a variable,
05:06
and a is an element in fq, integral outside fq, infinity outside infinity.
05:23
And then, so N is an element of A. N has, I assume that N has at least two prime, two prime divisors.
05:53
And then, let, let, oh, two, oh, yeah, yeah, of course, yeah, yeah.
06:03
Yes, yeah, yeah, yeah. And then, the, then another thing, Nd, Nd is the homogeneous space of, space of, of, of Greenfield modules is over, over, over,
06:28
and with, with, through, through n-level structure. That is the, is the sense of the derivative, the derivative n-level structure.
06:42
I, I don't, I don't explain to you. This depends only on the idea, and it doesn't have to be principle. Oh, I, I know that, that, yes, but today, I, I, I, I assume that, I just consider that principle idea.
07:01
Yeah, yeah, yeah, yeah. J is by N, yeah, yeah. And so, so then, then, then the principle has the property that it is regular, and, and, and that is smooth over, over, over fq, over fq,
07:22
and also the, after inverting the, the Nd, then, then, then this is, this is smooth over, over, over fq.
07:47
Yeah, I forgot to be in the, in the description of the big reaction of Drini. I, I felt that Drini was very happy to see a younger mathematician who was as excellent as him, yeah.
08:02
That was my strong impression which I had that time. So, so, so, so, and then, and then, so then, now, we, we, we construct,
08:25
but we construct the, the, the M&D, the enlargement of M&D in sigma. This is a cone decomposition explained later, explained later,
08:47
but, but in the theorem I assume, I assume today that, that is, is just the fq t. The general case, general, I, I, we believe that, this is a joint, joint work with,
09:08
with, joint work with, with, with, and, and, and we believe that the, the case generally,
09:26
A can be, case of general A is okay. But then the formulation of the result is a little different from the, for general A, and then for A equal fq t, then the, the, and also the, we already,
09:44
almost completed the paper for A equals fq t. And, but for general A, then the method is to reduce the problem to the case A equals fq t, but, but the, the manuscript is not yet, not yet written so much,
10:02
and so I, I am afraid to say some, any wrong things about the general case. So, so today I, I, I am also sure that the, the paper is completed soon for, for this A equals fq t case, and the, and the paper will, I hope to submit the paper to the archiver in one week, one month,
10:23
I hope, yeah, and so, so I, I think in around 1994 there was a lecture of Primck in those days, on Smutskoportifica, but I don't know exactly what he did because it was not in the lecture, and then he told me that there is some idea I don't really know about.
10:43
What's the relation? Oh, yeah, yeah, yeah, actually, I have not, yeah, actually he wrote a note, a short note, and so I will soon say about it, the, the, the, yeah, the, that I will soon say, okay, okay, I will say that the, concerning this, the Sateke compactification was,
11:08
this is some minimal compactification of, of, of, of MN, MNDs. This was written in, this was by Kaplan for A equals fq t case,
11:28
and, and, and Pink for A, for A gender, and for, for, through that compactification about which I am, I am talking about the, the, the, Pink wrote a summary,
11:50
a summary of the, in, this is, you can see this in,
12:02
yeah, this is published in, in, in the, in some, some, some series of lectures, something about the research institute in Kyoto University, and that can be read. But details were not yet published using results of, results of Kuziwara.
12:28
And I, actually, I, we believe that what he wrote in this note is correct, and almost, essentially sufficient to, to develop the details theory,
12:43
and so I, I'm sorry I have not yet contacted Pink, so I, I, I don't know how, how he, he did not write the details. And, but, I think his ideas are nice and great and, and, yeah.
13:02
So, so, and we, we add some log, log context, log thing, and we did some hard computations, actually, which are not written in that note. So I am not sure if I, the details were not published,
13:20
I have to contact him, yeah, yeah. So, so, then, yeah, so then, then, yeah, then, so the theorem, the theorem is that, that, I, I state the theorem without, without showing the,
13:40
what is the thing, what is the thing, yeah, yeah. And what is this modular space, and what is the modular problem, I, I don't explain now. But this is, this is a log regular, and this has log structure and log regular,
14:00
as a, so that, that is a natural generalized log, log regular. And, and, and the proper, proper by, and so that is, ah, yeah, this is open, yeah, open set, yeah. And then, then, then so open set.
14:21
And, and, and, and, and these is, is, is, is, you know, outer tensor a, this then, then this is, this is logarithm over, logarithm over, over, over a.
14:45
Yeah, just, just, this is what, putting log, yeah. So, and then, and this is, and after there are some sigma, prime, sub-division, finite rational sub-division of this condition, sub-division of, of, of sigma,
15:09
such that, for which given sigma, then we have such, such that, ah, m n, m l d, m, sigma is, is regular, and, and, and this is smooth over, over, over, over,
15:27
over, over, smooth over, over. That is a resolution of the singularity can be done in this, in this, yeah, smooth over a. And, and, and, and the boundary here is the normal crossing divisor and, ah, thank you very much, thank you very much.
15:51
Yeah, yeah. This is, and the boundary is normal crossing, simple normal crossing divisor here, and the, all reducible components of, of, of the boundary are smooth,
16:05
and there are any intersections are also smooth over, over this. So, yeah, such, such a nice resolution of the singularity exists here. So when you say that your space, so you claim that the space is regular,
16:20
but it's a-minutes, it's log-regular, but let us say in the regular case even, so you, it maps to the curve, spec a, and what is the local structure, is it, is it in some sense log-regularity or puts enough, is it log-smooth when you put enough, divide enough when you put enough structure, or?
16:46
Oh, yeah, because in the log-regular case, this is just exactly the place where the log structure is trivial. So, if you have this log-regular, and, and, and then, and S, U, U is a place here,
17:03
where the, the log structure of, this is log structure of S is, is trivial, then, then, then the log structure, this is log, log structure of M is equal to, if this is J, is, is equal to,
17:24
yeah, so then, that is, and in this case, this is, this is, this is exactly the place where the, the log structure is trivial, so, so the log structure of this is given by, by, by that, by, by just by those schemes, yeah, yeah, yeah.
17:45
This is not, my question was, you said it's log-smooth over A after inverting N, and if you don't invert N, but suppose I add the, the divide of N to the log structure.
18:01
Oh, I don't put the log structure in A, yeah, just, just, I, I, I am not putting the log structure in A, yeah. So, so then, then, if I don't invert N, then, yeah, we can say something that is, that is,
18:24
there is some, yeah, look, look, yeah, maybe I may say about it, the data, if I have time, the, the, the formal completion of, of this is, I mean, formal, if you, you, you,
18:45
some formal completion of this is regular and, and, and times, through some, locally, locally, locally, locally, you have a smooth map to, to, to, to, to, oh yes, to, to, to, yeah.
19:06
I don't know the, I have not yet studied the, the, the, what happens on the divisors of N. Sorry, sorry, yeah, yeah, so, yeah, yeah, maybe I have, I can say something, but today I don't prepare anything about the fiber at N.
19:23
Yeah, sorry, sorry, yeah, so, so, that is my, my, my, yeah, yeah, yeah, yeah. And so, so, so, so, then, this is the theorem, but I have to be, say, assume that there is LQT, sorry, sorry,
19:40
otherwise the statement is, is, should be the same, but, but my script is not yet written, so, so, so, not written here, so, so I, I have to, and so, so then, then I hope to introduce a log-during-felt module, yeah, yeah, so, so, first, recall that, recall that
20:18
drink-felt module of rank d and module of, of rank d and generalized drink-felt module, drink-felt module, module
20:40
over S, so S is a scheme over A. Is a, is a, is a, this is recipe, yeah, recipe, is a pair, pair L theta, theta, theta, phi, L phi, sorry,
21:02
phi, here, L is a line bundle on S and, and phi is the, the action of, action of A on, on the additive group scheme, L,
21:27
satisfying, satisfying one and two, one and two, here, this put two prime.
21:40
So, yeah, so the, so, so the one is, one is that, one is that, one is that the, the, the, the action of this phi A is, is A,
22:02
is a multiplication by A. The second is, is, is that the, that's why locally I take isomorphism, take isomorphism L with O S. Then the, the phi A X is written as, as, so X is now in, in L, but it is in O S.
22:23
Then the, then this is written by a polynomial and the first term is A X because by this. And then the, the here, the last term is unit times, unit times X to the A to the B.
22:42
Here, here A is, this is absolute value, standard absolute value of, value of, of, of F in T. And the, and the, on the other hand, the two, two star, two, two prime for the, for the, for the, generalize means that, that then we don't put such assumption.
23:06
That is, then this is, this is, just for any, any S, then the phi of this, this is the, it is the, is, is the,
23:22
module of R for some, module of R for some, R for some, R for some R. R, R, R on this. So here D, D, D should be nonzero. So that is the, that, that, polynomial phi A should not be, be a linear function.
23:50
That is the, some higher degree should appear. Yeah, yeah, yeah. So, so then, and then the remark is that, roughly speaking, the, speaking the, the, the, the, the, the, the Sateke compactification,
24:10
Fitchi-pink compactification, Fitchi-pink, Fitchi-pink constructed is the modular space
24:28
of generalized bring back to modules. Does it follow that these colors in A,
24:42
like the base with FQ, acts in the usual way? Oh, yeah, yeah, that follows from the definition. Yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah. And so then, to have a toroidal compactification,
25:01
we should put some logarithmic thing, yeah. And so then, I will do, I will do. So then, the next, we talk about log-drink at modules. So the, now for, then by log scheme,
25:22
I, here, today, this is so-called FES log scheme, log scheme, what we find on the, on the Zariski site is the, for simplicity. Yeah, et al. site is okay also, but just to,
25:43
because I prepared, we are writing the manuscript to using Zariski site, so, but I think et al. site is also okay, yeah. And then, now, for et al. be a line bundle on the, on log scheme S.
26:05
Then, we define L bar to be the union of L and MS inverse L cross. Here, the L cross is a basis of L inside L.
26:29
And MS is a, MS is a log structure of S. And this, so I have to make this more precise,
26:42
but the feeling is that this is an enlargement of L, putting poles which belong to MS. And then, that is, so MS inverse is the F inverse, if MS inside the group, group, group, group,
27:04
group, MS group. And the MS inverse L cross means the, this twist of MS inverse by always cross the, this L cross, sorry, L cross is the,
27:22
is the tosser of always cross, and so, yeah. So then, we have such a thing, yeah. And this union is, this is the union, union is the union in which we identify L cross here and L cross here, yeah, yeah. And actually, L is not a strange thing.
27:44
This is just a shift of morphisms of, that's a shift of morphisms to P of L plus O, which contains L as a dense subspace.
28:01
And putting, but this has S morphisms. More S, this morphisms, and this is projective space associated to the beta band wave plus O. And what we put a log structure here. This is just pullback log, pullback log of S,
28:24
pullback log structure of S and plus the log by cultured divisor, log at infinity. You have the, complement is a cultured divisor, so you can put the, so you add the log structure
28:44
coming from that cultured divisor to infinity. So then, then this is, then the, yeah, this part is treating the outside,
29:00
this space outside, yeah, yeah, yeah. No, no, so then this is not a strange thing, yeah. Then, now, so, then.
29:22
It's called the pitch, this is as a space of line, it's a classical projective lines or the space of hyper-plants. Oh, maybe here it's an equivalent, it goes like this, so. Ah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah. The vector band doesn't mean anything. Oh, yeah, you can make a vector band version of this.
29:44
And my big hope is that this notion is useful for the log-driven modules, then the vector band version may be useful for the log-stilke or something like that. Log-stilke, stilke is a vector band, so I hope that such a thing can be also.
30:04
Yeah, then if you put vector band, you replace that L by V, and then you have, you can have some vector band version of this, yeah, yeah, yeah. And then I, now, now, by log-driven module,
30:23
I build module of rank D with in-level structure, structure of S, this is a log scheme over, over, over, over, yeah, yeah,
30:42
is a, is a, is such a thing. Here, L theta is a generalized Greenfield module, Greenfield module,
31:02
module over S. And the iota is a, is a homomorphism, not homomorphism, map of Sibs to Elba, the, the, the map, homomorphism of Sibs, just Sibs,
31:22
we, just map, map, map of Sibs. And, and because Elba is not a group, so we cannot say homomorphism. Elba is, yeah, so, so then it's satisfying the following, find the, the, the,
31:40
the maybe following this sharp, yeah. And I write the condition, yeah. The, the, yes, so, so, so I write the condition, the, the, so if, so if in the case,
32:06
in the case, S is log, log-regular, then I, then it is like the, the, the, and the U, U is a,
32:22
is a, the part is three, the, the, the part open set of S on which the log structure is trivial, and then you have zero, zero in this. Then, then, or maybe, I don't know, then the, then the restriction of this to U is a,
32:44
is a during-helter module of, module of, of, of rank D. And, and the restriction of this Eota to U is a, is a during-helter N-level structure,
33:05
structure of, of L U, L U, no, L phi is restricted to U, yeah, yeah. So this is a condition, yeah, that is. So in this case, you have this, yeah.
33:20
So, so, and in general, in the general case, the, for general S, then, then the condition is that, condition is the following. So for, for, for locally on this, on this, there is some, some,
33:41
some S, S prime, and L prime, phi prime, Eota prime, and the, and the, and the F from S to S prime, such that, such that, such that O by S, O by S A,
34:01
such that, such that, this S prime is log-regular. This is, yeah, log-regular. And, and, and the L prime, phi prime, Eota prime is a, is a, is a log,
34:21
during-helter module, during-helter module. So maybe I, I call the log, log, log, log, log, log, log during-helter module of, of rank, rank D and D, D, D, D, N. This is a log during-helter module of rank D and N level structure,
34:42
is prime in the sense, in this, in the sense which we have, we have just defined. And the, and the, and the, and the L prime, L phi, phi, Eota is a product of, of L, L prime,
35:03
phi prime, Eota prime. So here F is just a mouth of log schemes? Yeah, yeah, yeah, log schemes, yeah, yeah. Yeah, yeah, log schemes, yeah, yeah, yeah. You choose it to be straight? Oh, no, no, no, no, any, any, anything is fine. Yeah, anything is fine. Locally, locally, for the less people, for the greater people, for the greater people?
35:21
I am using, sorry, the, both of the log structure for Zariski to produce locally. Yeah, yeah, anything is, Zariski is okay, and Litter is okay, not, at the end, these are not, they don't give a difference.
35:41
Why, at the end, the group is okay, because, because the module space is log-regular, so the, anything like this comes from the module space, so I actually, the, yeah, yeah, so we don't need to be careful about this locally, but to study the localities better, and so I take Zariski locally, yeah, yeah, yeah, yeah.
36:03
So then, now, so, so then, then, then, now, I have some remarks on this subject, remarks, yeah.
36:20
But now, I have 20 minutes left, okay, okay, okay. So the, sorry, the, ah, so, so, so, thank you very much, thank you. So the, the remarks are,
36:43
well, so, four, four, four, four, in the cases, log-regular, then, then the, the, the four, four, log-regular, two-module, log-rank, D, and level N is equivalent to the, just the, such that, here, here, if I, if I,
37:01
you, ah, ah, ah, this is a generalized, generalized, during delta, during delta module over, over, this, and this is a, this is, and such that L phi U is over, during delta of rank D, and, and this is a,
37:21
ah, ah, and, but this one is now, ah, ah, the level, level structure, in-level structure, ah, structure, on, on, on, on L phi, phi U. So just in-level structure, then, then, this is just, you, you get by restriction to U, ah,
37:46
but here, you can, the, the unique extension of Yota, the, the, such Yota extends uniquely, ah, ah, to the, to the map on, on this, ah, ah,
38:01
from, from, over, over, L phi to there, yeah, to L t, yeah, yeah, so that, that is, ah, ah, so, so that, that is the first remark, and the second remark is, is, is that the, the, if, if L, L, ah, so that is, I am not defining log-rank filter module,
38:22
I am just defining the log-rank module with rank D, with in-level structure, so, ah, without in-level structure, I can, that, I, I don't tell you what was, what was, what is log-rank module, yeah, yeah, yeah. I, and, so, the, the, this,
38:43
this, if, this is log, log, log-rank filter module, rank, rank D and in-level structure, then, ah, then, ah, you have, you have, for, for any, for any a, b, a, b in, in, in this thing, ah,
39:07
then, ah, for any, then, ah, you have, ah, you have locally on this, ah, on this, you have, you have, we have either, either 4a device,
39:20
4eota a device, 4eota, I will explain 4, yeah, or 4eota b device, 4eota a, that is some totally, total ordered structure appears, and what is 4, yeah, 4 is that, 4 is that,
39:41
4 is that L bar, from L bar to, to Ms over S, you have, you have, so L, L goes to one, and, and, and then, ah, if we inverse e, here f is in Ms and ee in L cross goes to, to F,
40:00
that is how, how 4 is b, yeah, yeah, the, the F mod modulo 4 is cross, yeah, yeah, so, so that is the 4, yeah, so the 4 is always, ah, one device this, one device this means that for, for, for, x divides y in, it means that y over x is in S, here,
40:26
Ms in, in, in S cross, ah, in, in, in Ms always divided by, by group, in the group, yeah, the quotient is in the group, but, but the x divided y means it is inside, inside here,
40:41
yeah, yeah, and then the third remark is that now, hi, hi, and then, okay, so, third remark is that, now, now I assume that
41:01
is fqt, so then as a story becomes special and simple as a, otherwise the, there are some very complicated things, yeah, so, so, ah, so, so the property of the first device, the other, these photos, these photos form,
41:20
one is to prove it using the, the proof is not trivial, and so, you have to use the structure of the local ring of regular scheme, yeah, that is not so trivial, yeah, yeah, yeah, yeah, so,
41:41
and remark three is that, that, ah, for you assume that now is fqt, then, ah, then, ah, then, ah, four, l, phi, ah, phi, ah,
42:01
iota, ah, log, log derivative module of, of, ah, assume, ah, of, of rank d and table n, then, ah, over s, over, over s, then, then, ah, the, then, locally on this, ah,
42:23
on this, there is, ah, basis ei, ah, so I, I like to, to, to make a index like this, ah, a over ei basis of, ah, of, ah, n1, y1, a, a to the d, ah,
42:45
such that, ah, such that, ah, such that, ah, ah, ah, four over ei, four over i, ah, four over ei, device four over a, ah, iota, iota ei, iota ei device, ah,
43:03
four over iota a, ah, ah, for any a, ah, ah, ah, inside, inside, in this, ah, minus, ah, ah, ah,
43:22
first i equals zero, i minus one, j equals zero, i minus one, in ei. So in your definition, l bar was the union of, of, of l and something else, and you defined the pole as on the second part. Oh yeah, yeah, yeah, second part,
43:40
the first part goes to one, yeah, yeah, yeah. The first part goes to one, ah, okay. Yeah, yeah, yeah, yeah, yeah, I am just, yeah, so, so, we are worrying about the pole, so, and then, then, ah, this remark is important, three is important, ah, for, for, for a whole case a equals lqt. Then, then, ah, remark is that
44:01
for, for any four, four, four ei, ah, as in, as in remark, one, three, then, ah, the pole of ei is always equal to, is independent of the such choice, of such, such ei, so,
44:23
this pole is, is determined. And, and furthermore, and, ah, pole e0 is, pole e0 is always one, yeah, yeah. Such a thing is, can be proved. And, so I, I hope to use this such thing
44:41
to formulate the moduli functor, yeah. So, so, ah, so the, now, ah,
45:08
now, ah, moduli functor is, in term of functor, mn from log scheme, scheme over, over a to, to set, ah,
45:23
ah, this is just, just the, the, the isomorphism L phi is, is go, mn, s goes to, to the, the logarithm at the module of rank d, and then, over s, ah, ah, with, with module isomorphism.
45:43
And, and, and, ah, mn, mn bar d is the sigma. Now, ah, we will use sigma. Sigma is, ah, let the c, d be the, the, the cone, ah,
46:01
s, i, i, one, two, i, d minus one, ah, such that s, i are real numbers, and then you have, you have this, we consider this cone, yeah, yeah, yeah. And then, ah, let the sigma be, ah, ah,
46:21
finite, ah, finite rational cone decomposition, ah, position of, of, of c, d. Then, ah, the, then you have mn bar sigma. So, ah, this sends s to be, to, ah, ah,
46:43
to the, ah, that is, this is, ah, mn, mn in sigma, s is, is, ah, is, ah, elements l, pi, yota, ah, yota, ah, plus, plus of this, ah,
47:02
inside mn, mn, dn, s, such that locally on s, ah, ah, on s, ah, there is, ah, ah, basis ei, as in, as in remark three, ah, remark three, yeah, yeah.
47:20
And such that the, that, such that, ah, such that, and the, and the, and the sigma in the, in the, some cone in sigma, yeah. Such that, such that, such that, ah, the, the formula is, ah,
47:42
that the, ah, i equals one to d minus one, ah, for, for yota ei, ah, bi is in ms of os cross, ah, ah, for any, for any, ah, bi in z to zt minus i, one, ah,
48:05
such that, such that, such that, ah, bi si, i equals one minus dy minus y, is, you know, negative for any s in sigma.
48:21
That is, ah, that is, ah, something like, ah, the poles of those position points, ah, ah, ah, satisfying the, the, the, we don't belong to, to sigma or something, something like that, yeah. Yeah, the, the, the, the, the,
48:41
the stations of poles are in sigma, yeah, yeah, yeah. So you want s d minus one, not sb? Ah, because, ah, I am here, oh, sorry, sorry, this is d minus one, so, sorry, sorry. I am neglecting the first e zero, because you thought e zero is always one,
49:00
so I am just neglecting, so, sorry. So this is d minus one, yeah, yeah, yeah. So, so that, that, this is, this is inside, inside r d minus one, and you have a positive element, yeah, yeah, yeah, yeah. So, thank you very much, thank you very much, thank you very much. And so the, the, then the theorem, theorem is that this one is representable,
49:23
and, and, now, now, the theorem here. The theorem, so as you, theorem is that assume, assume f is, is, is equal to t, and then, then the, the, the, the, functor m n, m n t,
49:41
theorem is represented by a log scheme, log scheme of m bar d n, theorem satisfying the, satisfying this, this,
50:03
this theorem, yeah, yeah, yeah, yeah. So, so, so, so, now I think I have only 10 minutes, so then it is rather hard to explain the details of the, the key points of the proof.
50:20
But, but the, yeah, but the, shortly speaking the, the, the, the, the case m t is, so let sigma one be the, the trivial decomposition,
50:42
trivial, ah, the, the, the proof, proof, that I'm, that I will speak, that I will sketch of the proof, trivial, are, can decomposition then, of, of, of, of c d, that is the all face, consisting of all faces of c d,
51:02
then, then, and if you take m t, t, this is just a, a growing up of, some growing up of, of p, p, a, a, d minus one.
51:23
And some, some growing up, I, I don't, I have no time to, to explain. And for, for them, for any n of them, or just this is growing up, for, yeah, but, but for, for any sigma,
51:40
this is a growing up of this, t, t, you, you, you are t t, t is just t, t of, of, of f q t, yeah. Ah, sorry, sorry. So the, if you have one, only one divisor, then if you invert that n,
52:01
then you can get a, a modular space, yeah, yeah. So the n b n, for, for n b n or n b n b sigma exist, exist, that's the, this one exist only, only, only this one exist, and without this localization, that, that can, is not, it is hard to define to the,
52:20
maybe in the world of stacks, then they may exist, but in the usual sense, the, you, you have to, after, after, after a, and if, if, if, if, if n has only one prime divisor, yeah.
52:45
Yeah, so, so then this one is defined, and then if you have, for a general n, then, general n, then the, the n can be written as n one times n two, and, and n one and n two are one, and, and then, then the, and the,
53:04
they are not constant, they are not constant. Then n, n, n sigma tensor, you, it is, n, n, sigma is obtained as, as, as this,
53:21
and this one is obtained as a growing up of, of the, the, the integral closure of, closure of, of n, n, n one, n one,
53:40
sigma one tensor a one, in, in, in, yeah, d, d, d, so d. And then, then, finally, so that is, so this is first remark, what the,
54:01
second remark is this, and then, but finally, the, the, this one, this one, n, n, and finally, the, the, m, n, m, n bar sigma one is the,
54:26
is the, the normalization of the closure, of the closure of, closure of, of m, n, n. I mean, after, after tensor, this is the closure
54:41
of this in, in m, d, m, d, k, t. Here, we use the map if q, t goes to, if q, n, if q, t, sending t to n. Then, if you have a, doing it to m, n bar sigma one.
55:01
Other, if, if you have a, so the, this gives a, a map from m, n, m, d, n to, to m, m, d, k, t, where, where k is the degree of, the polynomial.
55:21
That is the absolute value of n is q to the k, yeah, yeah. The, the, that is, if you have a doing it to module with f, q, t coefficient, and then, you restrict the coefficient to this thing, then the rank becomes, is multiplied to, by k,
55:42
degree k, and you have a doing it to module over this former f, q, t, by, with the higher rank. And then, you have a embedding of, this is a embedding on this, you can have a m, n, d, k, t,
56:01
but this sigma one is now a sigma one of, of, of c, c, d, q, c, d, k, yeah, yeah. And then, you, you can take a, you can take a closure of this inside here. Then, you get, you get that one,
56:21
and this one, this one. And then, then you again take the integral closure, and then, and you take, you have to take a growing up, some growing up. And I think that the process is so simple, but the most important part, difficult part is to prove the log-regularity
56:43
of the, of the such, such thing. And so, that then, for, for to have a, I, I have to mean something like that. So, to prove the log-regularity of the,
57:00
of the such thing, we have to use the, the formal theory, formal completion. And then, I have just one remark, I can give only one remark, that we are able to study some, some, something like state uniformization
57:21
in the formal completion, just like a theory of Halting's chai in the mojilai of abelian varieties. So, for example, if an elliptic curve degenerates, then, then it is, it is a state curve is like this. And, and then, I, then, if you,
57:42
if the, the local mojilai space is, space is, is, is this, spec of this, or SPF of this, are the, the formal mojilai space.
58:06
It, it, so that is, the Q is the, this Q. And then, you, if you, by this local theory, you can see that the space is log-regular, log-regular, in this case it is regular. And in general, log-regular.
58:21
And after inverting N, it is smooth over, over Z, yeah. So, the, the, such thing can be proved by using such formal, local study, by completing the story. And then, just as this, then, the dream vector mojilai has something like
58:40
theta-uniformization. But it is, we need a iterated theta-uniformization. That is different from the usual theta-uniformization. You have to divide something like gm, a good reduction.
59:02
And, and then, you have to divide x0 by gamma i. And then, you get xi plus one. And then, the xm is the, the, the, the, the, the, the. So, from, to get L5, what we get E, then,
59:21
you can, here, you have only once, you can get E, the elliptic curve E, just dividing gm, which has good reduction, dividing by it, by some lattice. But here, you have to, you cannot get the dream het mojilai from earlier.
59:42
This is log dream het mojilai. Log dream het mojilai. Log dream het mojilai from earlier, by, by just, by one rank, dividing by, once by rank, by lattice, lattice. You, you, you have to divide the, the.
01:00:00
lattice for several, you divide it for several, several times from starting from object of good, during the measure of good reaction of lower rank, yeah. So the iterated, theta, theta, sorry. Iterated theta informization is necessary,
01:00:20
but once you do this, then such a local, local coordinate, you can find a local coordinate using the generator of such lattice. So then you can see that the space has nice, nice coordinate functions, and space is regular, or space is smooth, log-smooth, yeah, yeah, yeah.
01:00:41
This is the main part of the story. But already in Pink's note then, such a thing is illustrated using the result of your hand, so I have to contact Pink why he has not published the,
01:01:04
yeah, yeah, yeah, yeah. This will be in the end of the story, yeah, yeah. A lot more questions.
01:01:24
I think you mentioned a couple years ago, you had some applications in mind for this theory? Oh, yeah, yeah, yeah, yeah, so much. The high IOs, so much, here in the collaborator, Sharihi is in the collaborator.
01:01:41
Why we are studying this is that we are interested in Sharihi conjecture, and it is the modular symbol of G.L.N.
01:02:02
One is related to your theory of the modular forms of G.L. N minus one is our project. And the case N equals two as stated in Sharihi's paper in analysis of mathematics.
01:02:23
And now, but we wanted to consider N equals to F equals Q. N equals to F equals Q, these are the Sharihi conjecture, original Sharihi conjecture.
01:02:42
And we are interested in the case, if it's a function field case. And then the theory of modular symbols, and yeah, so the, yeah, so the, your theory means that your theory of modular forms of G.L.N. minus one.
01:03:02
And then our plan is to, this comes to so-called Kondo-Yasuda element, element in the K group of the modular,
01:03:22
K in of the modular variety, during filter modular variety. And MDN, and MDN. But if we compactify that, we can consider this at the boundary.
01:03:40
The boundary is the world of G.L.N minus one, so this is related to, this is our plan. And so we started to, this thing is to study, we wanted to get a boundary, yeah.
01:04:02
And, but, yeah, but we spent three, still, yeah, I think you had my talk three years ago or something like that. But we needed three years to complete the paper. And I think the paper is soon completed,
01:04:21
at least for equals FQT. Yeah, yeah, yeah, yeah, yeah.
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