Function field analogues of compactifications of period domains
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Kato, Kazuya
For a function field in one variable over a finite field, we will consider analogues of compactifications of period domains.
 Thank you very much. I am thankful to the organizers for the invitation. And I am very happy to talk in this conference for Professor August. The book notes on Crystalline Co-Homology by Bertel August was a Bible for me in learning Crystalline Co-Homology. And I brought that book every fair thing I was young. And he is now completing a book on big book on logarithmic geometry and I hope that that book will become also a Bible for young people. And my talk is related to log structure. This is a joint work with T. V. Fukaya and R. Romi Yosharihi. So, first I compare the characteristic zero situation and characteristic p situation. So, here is the modular space of the brilliant varieties and then there are many compactifications. And then there is a Borrel cell compactification. This is O.B.C. in the case of O.B.C. And here also are some deductive Borrel cell compactification. And here the so called troidal compactification. And here is the satake-B.B. Borrel compactification. And we consider similar thing for the function field case. Here is the modular space of the drift effect modules. Which appears for function field case. And here the so here already there is a work of Kaplanoff and pink compactification. This is and this is Kaplanoff in my understanding Kaplanoff did the case if he calls function field is equal to this rational function field case. And pink constructed this in general. And this is similar to that the above satake-B.B. Borrel. And now here the so that for why we I wonder for what is happening there. And we are with these collaborators. They are like wrote a paper on this analog of the deductive Borrel cell compactification. And this paper will be posted to the archive soon. Just maybe in 10 days or something like that. But we are trying to do that construct the troidal compactification. But still there are several things to solve. So I hope we can succeed. But I cannot say perfectly that we did such a compactification. And this is the map to the some analog of deductive Borrel cell compactification. And then here is the troidal compactification. But as I said, this is not yet written that paper is still not yet perfect situation. So that is a small question should be here. But this is maybe too early to talk about this. But these are related closely. And so this one is some preparation for the troidal compactification. And so I need to talk a little about this one because I hope to present the total picture. And concerning the troidal compactification, pink published paper, summary paper already in 1994. And to my knowledge, the details are not yet written. Probably our method is a little different from the method described in the short article. And so log appears because the troidal compactification for the Arbelian Varities case is a modular space of log Arbelian varieties. And this troidal compactification is a modular space of log Drinfeld modules. So the log appears. And these three, the collaborators and I are studying Iwasawa theory. And the reason why we need this troidal compactification is that we are studying the Sharifi conjecture, which was formulated in the paper of Sharifi in the analysis of mathematics in 2012, in 2011. And that is a new theory in Iwasawa theory, which uses the boundary of the modular curve. And these three are collaborating to generalize his conjecture to more general situations of similar varieties or these varieties, and not from GL2 theory to GLN theory. And then such boundary becomes necessary. So that is why we need this for the new study of Sharifi conjecture for function field case for GLN. And then so first I introduce omega and x. So we consider the three situations. So here f is q and f is q here, and f is a function field in one variable over some finite field. And then we consider the finite dimensional vector space over f. What is the difference between A and B? That will appear. Yes, yes. That will appear here. And here the dimension v is dDd and here the dimension v is 2g and v is endowed with a pairing, f, f, it is qc. that is non-degenerate and antisymmetric. And so here D is also dimension. And I need a place infinity. So for A and B, then infinity is the Archimedean place. And for C, then we fix our place infinity of A. This is for this. And then I first define omega. So when we talk about omega, then we assume that we are in B or C. And B then we simply assume that base is already fixed. Base is fixed so that it is identified with this thing. And this is the standard one. And in C then, B is equal to G. And then in B then omega is G. Then maybe C is maybe most important for us. I first define this. This is... I also define C. In the case of C, then C implies that C is the completion of the R. So infinity is the local field at infinity. So C is the completion of the closure of infinity. So infinity is the local field at infinity. So that is equal to R for A and B. And then this is... This is G1 to GD, a linearly independent over infinity. And in the case of B, then omega is the Z-gelapa half space of degree G. And A is not important for us, but I discuss about A. I talk about A for comparison with other things. And then... Next, so then roughly speaking, this is a modular space of the Dlinfeld modules of rank D. And this is a modular space of polarized Abellian varieties of dimension G. And so then now next I define X. So X is if X be infinity and X be infinity and this, in the case A and B, there are... that is... C and this is for B. So this is the... Norm's zone on V infinity, if infinity turns to V, modular homotocity, that is... This is a norm... I will explain this. And then this is a norm zone on V infinity, compatible with this. And divide this. Here the... this norm is... Norm means that the norm is in the case of A and B then it is a function from V infinity to R. Such that for some basis, then... I don't know, for AC. Yeah, yeah, yeah. Races then... then... OK, OK, A, B and C. Yeah, yeah. Such that... for some basis such that... such that the mu, mu A1, E1 plus A and E, N is... E, D is... Yeah, maybe, maybe. This is A and C maybe. This is equal to A1 square plus this one, 4444. This is 4A. And for function field case, this is the max of A1, A, A, D, R. This is just absolute value at infinity. So here, A, A, R in infinity. Oh, yeah, yeah, A, A, A, in infinity. So in the case C, you are working with the analog, what you call C, which is F infinity by roof. But the value group is not the... is less than the real. So this means you don't include norms that... to the value group. You multiply maybe AY by constant. Oh, no, no, no, sorry, sorry. That... that is a problem though. No, in the analog communication, you multiply absolute values by some real constants. If you want to get the world space of periodic norms, it is the space which is sometimes considered... depends on what you want to consider, but... Sorry, ah... You want to consider, but some people consider like... in the whole, like, I think, Goldman and Deva. Ah, sorry, this is a real value to normal. Yeah, but in coordinates should be maximum... some real constants multiply by absolute value. Oh, maximum constant, but we... there is some basis, and so that is OK. No, no, no, no, no, no is the... is this standard... has a standard form for some basis. Yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, just the... for some basis then the... the... the function has this nice shape. That is the definition of the norm. Yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah. Yeah, and in the case of B then the... the... the norm is compatible with... with... with... with... this means that there is some basis... basis EI... of V... here, now, antisymmetric pairing is given, and for which is... for which is the... the... the... this pairing is a standard one. The... and... and the... and the mu mu mu mu mu A1, A1 plus AAD, A2G... E2G is... is again of... A1 square plus A2G... square. Yeah, yeah. Oh, sorry. In... in non-artimidate case, you can see... it's only a countable set of un-countable... It is uncountable. That is at the political space. No, no, no. You can see the set. It's a loss of countables. You don't have to get countables. Countable space. Countable set. No, you can't do that. You can't do that. You can't do that. But on f infinity, yeah. Yeah. Yeah, yeah. So this is a countable, no maybe no. You can make a convergence. That is like a real vector space because you are considering the values in real numbers and then the countable, no, no, no. Oh, yeah. You are right. Perfectly right. Yeah. Sorry, sorry. There is a... Yeah. Yeah, yeah, yeah, yeah. Sorry, sorry. This is... I am very sorry for this. Yeah. Yeah, yeah, yeah, yeah, but that is yes, yes, yes. I am sorry, I am sorry. I am sorry. Yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah. Yeah, yeah, yeah, yeah, yeah, yeah. Very sorry for this stupid thing. Oh, yeah. We need this. Yeah. Yeah, I am sorry for that. I'm sorry because... So, then here you need such square. Yeah, yeah, yeah, yeah. You need... Yeah, sorry, sorry. Yes, so this is a... Yeah, so that is die. We can make it diagonal by taking the basis. Yes. So, then... Yes. So, then now... So, then... Ah, this is important. So, I will not... Not... Not erase this one. But, then... Yes. Oh, yes. Maybe... Yeah, yeah, yeah, yeah. Okay. This should be here, but I think the most important thing should be... Maybe... I think the convention here is that maybe the most important source. Ah, sorry, I didn't... Okay, okay, yeah. So, then... Then... Yeah. So, then... Sorry. Ah, sorry. Then, ah... This is just a multiplication by... multiplication by... by some element, positive. Then, we will have a class of those. Yeah, yeah. And then... Then, ah... So, in the case of A, and then... Then, X... X is... I can modify it to SLD. R divided by SOD. Ah, sorry. Sorry. Sorry. Sorry. And B, in the biggest B, I... X is isometric to ASPG, R... R divided by a maximum factor. And in the case of C, then X is a standard geometric realization of the... realization of the... of the Bvialfitz building... building... for... for... Pgrv infinity, yeah, for this algebraic group. And then we have a map. We have also in the case of B and C, then we have a, if they are there, we have omega. And we have a map from omega to x, a continuous map. This is in the case of C, it is defined as z, d goes to mu, mu is r. Mu is mu A1, Ad is A1, z1 plus Ad, zd. So this is inside the C, but because z1 to zd are linearly independent of infinity, so this becomes a norm. And so this is a Krasov. And then for the case of B, then it is z goes to mu, mu is a Hojimetric, the Krasov is a Krasov, and the Krasov is a Krasov. So this is the Krasov. And then we have a map from omega to x, and then we have a map from omega to x, and polarized by this. And then in the case of B, then this is the Homme-Omori-Shizun. And so next we consider the section 2 is concerning the, first we consider the compactification of this x, and then the total compactification is related to how to extend omega. And so first we consider the extension, enlargement of x bar of x and x bar flat of x. So then the, so first x bar, this is the case A and C. C is C means G, this seems that I deleted something. Sorry. This is the, the first case is G, just the, yes. Second, B is the for the case with pairing, and A and C are similar, but the fields are number q, and this is function fields case. And here then, then this is x bar is the search pair, where p is the f parabolic subgroup, subgroup of p, g l, g l v. And, and mu is, and then p correspond to a filtration, v minus 1 v 0, v m v, parabolic subgroup is a, isotropy group of such a flag. And then mu is mu, mu i 0, i m fair, mu i is in x v i divided by v i minus 1. And then the, and in the case of B then, then some, some modification is necessary. And B is, in the case of B then, x bar is again p mu, fair, fair, p is a f parabolic subgroup of, subgroup of g c s p, v o i s p v, that's the same thing, essentially. And mu is, mu i, here, here, so the p correspond to a, some, some such, such that the, the, the annihilator of v i for the sparing is v minus 1 minus i, for all i. And, and mu, mu, mu 0, mu 0 is, is in x v 0, v minus 1, with, with, with, sorry, I forgot the notation infinity there. And here, here, we can have a pairing for the restriction of the original pairing. And mu i, mu i for, for i, not 0 is, is, is in, in x v i divided by v minus i infinity. Here, there is no, no pairings, we just take the space of norm's, modulo, modetti. And then, the main, now, the theorem is, then we have some topology on this x bar. The definition of the topology is rather involved and I'm afraid that I don't have time. That is the, the, some kind of topology. And I, or, or, or some, or, or some, or, or, or, or, or, or, but, but I, I, I, I am afraid that I don't have time to, to, to, to explain. Then here is the theorem. So that is, for a, b then this is well done. It should be so that our parabolic, which I think relates to the grandian. Since we are symphlectic, you can also have a parabolic. No, no, no, I think. There are some, some exceptions. No, no, do you get all the parabolic subgroups in this way or? Maybe, I, I, I believe so, yeah, but if, if it is not so, then please take this, this, this, this such thing corresponding to such thing. Yeah. Otherwise, the definition doesn't work. So, so I, yeah, yeah, yeah, yeah. So, and this, ah, and we don't know. And, and see the, the, the, this is a recent paper of, joint paper of, of, of, of, of, of, of, of, of, of, of, of, of, of, people including, including me. So, so that is probably new, the, the, the, but, because of, we wrote a paper on, on this and that, that paper will appear, posted, you will be posted in archive soon. But, but, and we are not perfectly sure if this such thing is, ah, I will write a, ah, theorem, the gamma X by the compact household, ah, for, for, for, for, for gamma, ah, commensurable, ah, to, to, to, to GLDZ, ah, if, if these, if, if, if the case A and G, GSP or SP, maybe SL, SL, SL, ah, the, no, SP, SPD, GZ in the case B and SL, SLD, OF in the case C. Now, OF is, OF is a, is a element in F such that, ah, A integral outside, outside infinity. Yeah, and so, ah, so for A and B then, then, ah, the A and B then gamma, this, this thing is, ah, so called reductive Borrel cell, ah, Borrel cell compactification of, ah, and so, such a standard of Jikutan. So, this is some, ah, we, we, we, we imagine that some person already studied C, but, but, ah, we could not find the reference. And there is some similar subject, ah, ah, similar subject, ah, is, ah, X is inside, X bar is inside X, ah, X is inside X bar by, by taking the P to be the, the, the, the whole, whole, whole, parabolic subgroup, ah, whole group, yeah, and then, ah, yeah, so, ah, ah, this is, this is compact and this has a sataketology and not the induced topology. And here, this is defined by using, ah, infinity, parabolic subgroup, ah, ah, in place of, in, in, if, if, if parabolic subgroup. Ah, sorry, yeah, in the case C, C, C, yeah, this is, this is ABC, ABC, ABC and, and then, ah, there is such a story and so, so, ah, in the case, for example, in the case A and, and B maybe, and G is 1, then, ah, the X bar is, is a upper half plane, ah, and P1Q, that is a boundary point, ah, and X2 bar is, is, ah, so, that, that is in this case, then, you have a homomorphism from the, the, this H2X, this is, this is H is, I think, 2, 2, 2X by, as I, I, I, I wrote some here, yeah, and, and then, this is H and P1R and so, ah, parabolic corresponds to, to such, such thing. And so, ah, this is compact, this is compact for the natural topology, ah, but, but this one has a, it's usually considered with, with Satake topology, not in this topology, so, so, then, after, after dividing by gamma, then you have haha space, yeah. And this one, this one for CX bar for C was, is a, is a so-called So, the general compactification of X are studied by already, in the old works of some people, I'm sorry, I, yeah, here, and, and, and, yeah. And so, such similar things are already studied and so, it is strange that the theorem for C is not known, but, but, maybe known, but, and, but the proof is, is just a, proof of the theorem for C is just the imitation of the other, other cases, A and B, but, but still some, some strange things are happening there and so, so the, this paper by three people have, has 50 pages, not so, so simple, that was not so simple, yeah, that is strange thing, but, but, and then, then I think, I, I also talk about X, X bar flat, X bar flat is, is, is, is, is, in this A and B then, in C, yeah, in C then X bar, X bar flat is, is a quotient of X bar and X bar is, is W here, here, here, W is a subspace, a subspace and, and, and, and, and, and, and, and, and, and, and, and, and, and this is, and B in the case of B, B then X, B X bar flat is, is, again, W and, mu, yeah, and then, this is, W is in V and, and then, such that W, W, the manipulator of W is contained in W and, and then, mu is in, in X, W, W, this and, and, and, and this, yeah, and then the map from X bar to X, B is, is a, just a, a, P, P mu goes to V zero, mu restricted by, just mu zero and, and, and, and, yes, this is the definition, I know the B, this, this is for, for A and C and for B then, it is P mu goes to, to, to, to V zero, to, to, to V zero, B minus, mu, mu, mu zero and, yes, no, no, no, no, no, that is, fine, yeah, yeah, yeah, yeah, that's same, same thing, yeah, yeah, yes also, yeah, this is same, yeah, for, for, for all A, B, C, yeah, yeah. And so, that if a subjection and, and so the X bar flat has a coefficient topology of X bar, or one This is another. Also, we can define the sataketopology, which may not be the image of the sataketopology of X. The search complicates things. But I don't discuss such too much details. And again, Seorem is that again, this is how it's looked compact. And for A and B, then this is a result of sataketopology. This is proved by him in 1964, I think, in his paper in the analysis of mathematics. And so C may be new. And we can have, again, X bar, flat. This is also a compact space. This is compact, first rule. And this has a sataketopology, which is different from this space. And this is so-called compactification by Uena of the pre-war-techs building. In the case of C. And A and B, in the case of AB, it is already studied by sataketopology. In another paper, his paper in 1964, something like that. This paper also appeared in analysis of mathematics. So when he was young, sataketopology, two papers in analysis of mathematics. One is this one, and the other is this one. Yeah, yeah, so then these stories are actually, and we have nothing to return down, but maybe the results are maybe generalized to all. All reductive groups are not only GLD or PTSD. And the problem should be the compactification of the understanding of the building of the space of norms may be not so good for generalization. And then we have to directly consider the building, I think. But similar arguments may be OK. It can be down there. And also the version of SU. Of course, in the case of number fields, then those are already known, essentially. And for X bar, X2 bar, then people already studied all reductive groups. Yeah, and so the version for S, S are finite set of primes, S are arithmetic subgroups. S for S are finite set of primes, set of places, including all arachnidian places. Then X is now the XV for all V. And then we can define X bar and for X bar, S are, and then the S arithmetic subgroup group is has no compact. This is in written. This one is written in that paper, which will be submitted in archive soon. So there is some such S version. So these are remarks. But I did not introduce this generalized version because the story becomes too much complicated. So I just considered the case of just one place. And then the final thing which I will talk about is the story of colloidal compact fiction. But as I said, already the paper is not yet written. Just we are starting to write. And so here can happen as usual that some terrible problems appear in the writing and then searching. At present, I hope that the things can be solved. But I'm not so perfect sure. But I write the picture section three. The first is we only consider B and C. In the case of GLD of the Arachnidian number field, then there is no analytic structure for the search space feature appears here. And so we consider B and C. And then the MKB, in the case of B, this is a modular space of, so K, G is GLV. And this is a B case, GSP. And this is GLV. And K is an open compact subgroup of G, AF, finite part. And the finite part means just this is without infinity. And then we can define such modular space. And modular space, MK is the modular space of polarised Arbillian varieties with K-level structure. This is for B. And this is the modular space of the dream-field modules of dimension G. And this is modular space of dream-field modules of rank E with K-level structure. This is B and this is C. And then, as is well known, you have a presentation of MKC. So I talk about only analytic theories. We hope also the algebraic theories can be done, but today, and only I present a very rough story. MKC is, it is well known that MKC is written as K. So this is for B and C. The omega was, in the case of C, then omega was dream-field upper half-plane, upper space. And then, this is then let MKB be flat, MK flat B. This is modular space, the compactification of MK by Bayley-Borrell, in the case of B. And the capra pink in the case of C. And then, and also MK bar, bar-troll, the toroidal compactification. In the case of B, then it is constructed by Manfred and other collaborators. And in the case of C, then I believe that we can construct this, but the paper is not yet well. So, but I hope to present the picture. The picture is that the, so this is not unique, not unique. And there's no, and here this is a standard one, unique, unique one exists, standard one exists. And then MK bar, bar be the inductive limit of all toroidal compactifications, all MK bar, MK bar, there are many. And so then, then we have inverse limit. And then we cover diagram, cumulative diagram. MK bar, bar, bar-troll, MK bar, bar flat. And here you have, you have X bar GF, Gaf, FF, divided by K. And you have GF, X bar B times Gaf, F, flat K. So, that is, then, not that we have a map from this to GF, X times. So we have a map from monogat to X, so you have a map already here. And then what I am saying is that this map may be phi. Phi can be, phi is extended to continuous maps. Phi is extended to continuous maps. And this one, this one is in the case B, case B, that is a national number, this is case, this is our Avalian variety case, this is a homeomorphism. That is, this is the, this is a Sadake compactification. So this is just a political space, but this is an analytic space. And so that is Bailey and Borrell put a complex analytic structure. So maybe C should be here. Complex analytic structure on the topological space feature, Sadake constructed. And so this is the picture. And then what I have five minutes left, but I, in any case, the paper is not written, so I can give only a love story. So then how, how the, so concerning this then, I have one, so just small thing. So that, that is, so the love story of log dream belt modules. But I am hoping to describe this Barl version. Without Barl then the, the, without Barl, the Trojan compactuation should be defined as the quotient of this Barl version. So then I describe this in short, so, well, but if it's module then. So let S be a log smooth, rigid analytic space. I am talking about the analytic situation. So log smooth, rigid analytic space over, so we are considering C, C, C, yeah, yeah, analytic space over C. Then, and the log smooth then we have S Barl that is a limit of, of, of blowing up of, of S along log structure. If the log is given this way then we, we blow up, then you have this and then we blow up this, this is intersection then, then you have this is S and this is limit of, limit of this is S Barl is the limit of, of, of, of such blowing up. And then the, the log dream belt module is a, is a, log dream belt module, module over S infinity over S, S, S Barl is, so that is, that with, with cable level structure. So that K, K is assumed to have, we are, assume, assume the, for simplicity the, the G, GL, the OV, other, that is, yeah, then, then is a, is a such thing is, is a, is a pair of, pair of V and L and V and lambda, here L is a, L is a line bundle on S Barl and, and so the S, inside S, the, you have open set U, the, this is a part where log is trivial, the part log is trivial. And then V, V is a, V is a, and then you have, you have then here you, the blowing up doesn't change U, so then you have J here and then, then V is a, V is a inside here and, and then this is a, this is a, this is a D dimensional F space, C for V, C for V, D dimensional F space and lambda is a, is a level structure, K level structure, the, the, the A, F, F, T, but, cosidate module, K, the, the isomalism is, the only, J, J, oh yeah, thank you very much, thank you very much, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, and then, this should satisfy the, the, the, the, the, the conditions are, conditions are, there is some, for example, there is some condition is the following, yeah, so for all such that, so which topology is it, is it? Ah, yeah, yeah, yeah, yeah, that is, I think there's a rigid, rigid topology, oh such thing, yeah, this is, The limit of the rigid topos, Oh yeah, yeah, I think so, yeah, yeah, such that, such that the, all T in this, but then, and all S in this, we have the following, so maybe just T is enough, yeah, all T then, then we have, we have, we have, this is, suppose, stock at T, V1, in the stock of V then, we have this, and, and, and, such that, satisfying having the following property, that property is the first, V0 is the, the inverse limit of, you know, inverse image of, image of, of, of OS, VAR inside j star OU, ah, ah, ah, ah, ah, intersection, intersection, sorry, sorry, V is V, V, sorry, V, V, and intersection OS, OS VAR inside, inside j star OU, yeah, yeah, and, and furthermore, the condition is that, then, from OS VAR to, to, to, to see at T, the, you have a, is, is, then, then induces, induces, ah, injection, this, this is injection, so that is the, the, the, we have a norm of, for this, so then, so from this then, ah, norm 1, on, on, V0 appears, ah, V0 infinity, ah, this is V0 infinity, appears, and the second condition, so that we are, we are, how, how I am actually showing how we get such V, V, V, V, V, V, the, the point of the, of the, ah, X bar, yeah, yeah, and then, then, ah, the, the, then, another condition is that, second one is that, ah, sorry, I need to, to, to, to, to, thank you very much, I am already, I need to, ah, sorry, sorry, ah, so then, then, ah, another is that, so, there is a, you have a, you have, you have, let VIO, FBR, VR, the, the, then, you have here, by, in this here, you have, or V, and then, by taking the inverse image here, ah, you can, you can have a, this defines the V of the inverse image, the integral structure is defined, and then, ah, then you have VIO, then, then for all I, ah, ah, ah, yeah, the, you have a, VIOF divided by VIO minus 1, VIOF, ah, from here, then, you have a, JOU, divided by VIO minus 1, VIOF, and then, you have a exponential of VIO minus 1, VIOF, ah, this is a Dlingel exponential map, ah, and that is the, this is the additive map, ah, so, ah, then you have J star of U, and, and the condition is that, by, by, the, by, by, by, by, ah, then, the image of, image of, VIOF minus, minus, minus 0 is in, in m minus 1 inside, inside J star of U, this is the, m is the log structure, ah, that is the, ah, CF feature, ah, yeah, that is the log structure, and then, that is the inverse of the log structure, and then, then, ah, the last, last thing is that, the, another condition is that, the, the further condition is that, condition is that, there is some homomorphism from, from m of, of, of, of, of, plus, this is the log structure, S, S, S bar, T, 2, 2, 2, 2, R, additive, such that, the, the, the composition, lambda, OF minus, lambda of minus 1, OF minus 0, ah, M inverse, and then, ah, this is by, exponential, ah, yeah, then, then, M, F, close to F inverse, H, all, all are D and T, yeah, yeah, the, this composition extends to, to a, to a norm on, on, on, on V infinity, V infinity is the, I don't know, V i, V i minus 1 infinity, it is, it is a, F, F infinity times lambda i, OF divided by lambda i minus 1, OF, so, ah, so, ah, so the norm, also, norm is defined on, on, on, on, on this, so, ah, then, by this then, ah, the, the, that, X bar and, and the, the, X bar and, ah, the, the, this, toroidal compactification is related, and so, the, the, the, this, Mk bar, bar, tol, bar is, is a, is a modular space of such, such object, so then, if we have such thing, then, you have a morphism, extending S, Soda, yeah, yeah, Soda, that is a fine modular space of, of, of, of, in, in, in this sense, yeah, so, so then, ah, yeah, sorry, sorry, the last part is, the, the, the, two-love story, ah, finish here, yeah, yeah, yeah. Thank you. Question. What is the relation between the curly end and the rest of the data? Ah, ah, ah, sorry, sorry, sorry, sorry, very, very sorry, that was, this, this, this should be eres addicted to you, sorry, the, that is, I was just misunderstand the L's, oh, you, the, the, the, the, the locale, locale exists oh yeah so, I was oh, oh ah, oh ah, yes, I was sorry, sorry, sorry, so this, that is, so, ah, here, this is locale, locale is, localeおおowany, I was, I was thinking of locale sorry, yeah, yeah, yeah. KAVARTY Is capital lambda the capital V over there? Capital lambda I, O, F? Capital lambda I, O, F, sorry. What is that? Oh, V, V, V, V. Sorry, sorry, V. Yeah, yeah, I was confused. Sorry, sorry. This is V, yeah, sorry, sorry. V, yeah, that's V, yeah. Yeah, yeah, sorry, sorry, sorry. Oh, this is just, so I want to have some fun questions on the end. So when you consider J-Law style, if you do it in rigid geometry, then you could consider sections with possibly essential singularities or just morphic. Yeah. So what do you mean here? Is it... Yeah, terrible singularity of the functions appear. But the condition is that the M is reasonable functions, so like this is just something like generator of the device, generated by... If you go... We went to... Yeah, yeah, we went to the inverse limit, but in the middle, then this one is reasonable thing. Here, this is just reasonable thing. And this is invertible element of this and generated by invertible elements and some device, generators of the device. So the condition is that the here, we have a reasonable object. Having not so terrible singularity, just they have poles. They have just, at worst, they have poles there. Yeah, this is having poles. But in general, it has essential singularity. Yeah, yeah, so that is... Every terrible singularity appears, but the condition is... This is the condition. Yeah, yeah, the assumption, the condition is... This is the condition, the big condition. The assumption, this is condition one is this. And the condition two is this. And the conditions that you put is relative to a C-valued point of the inverse limit space. So does it imply the same condition for other points for the Berkowitz point? Or do you just consider C-points? C-points, so I only... C-points, yeah. C-points only. So this is also VI. And then this is... Here is what we have. Yeah. If there are no other questions, let's thank Mr. Casano again. Thank you.
 Thank you very much.  I am thankful to the organizers for the invitation.  And I am very happy to talk in this conference for Professor August.  The book notes on Crystalline Co-Homology by Bertel August was a Bible for me in learning Crystalline Co-Homology.  And I brought that book every fair thing I was young.  And he is now completing a book on big book on logarithmic geometry and I hope that that book will become also a Bible for young people.  And my talk is related to log structure.  This is a joint work with T. V. Fukaya and R. Romi Yosharihi.  So, first I compare the characteristic zero situation and characteristic p situation.  So, here is the modular space of the brilliant varieties and then there are many compactifications.  And then there is a Borrel cell compactification.  This is O.B.C. in the case of O.B.C.  And here also are some deductive Borrel cell compactification.  And here the so called troidal compactification.  And here is the satake-B.B. Borrel compactification.  And we consider similar thing for the function field case.  Here is the modular space of the drift effect modules.  Which appears for function field case.  And here the so here already there is a work of Kaplanoff and pink compactification.  This is and this is Kaplanoff in my understanding Kaplanoff did the case if he calls function field is equal to this rational function field case.  And pink constructed this in general.  And this is similar to that the above satake-B.B. Borrel.  And now here the so that for why we I wonder for what is happening there.  And we are with these collaborators.  They are like wrote a paper on this analog of the deductive Borrel cell compactification.  And this paper will be posted to the archive soon.  Just maybe in 10 days or something like that.  But we are trying to do that construct the troidal compactification.  But still there are several things to solve.  So I hope we can succeed.  But I cannot say perfectly that we did such a compactification.  And this is the map to the some analog of deductive Borrel cell compactification.  And then here is the troidal compactification.  But as I said, this is not yet written that paper is still not yet perfect situation.  So that is a small question should be here.  But this is maybe too early to talk about this.  But these are related closely.  And so this one is some preparation for the troidal compactification.  And so I need to talk a little about this one because I hope to present the total picture.  And concerning the troidal compactification, pink published paper, summary paper already in 1994.  And to my knowledge, the details are not yet written.  Probably our method is a little different from the method described in the short article.  And so log appears because the troidal compactification for the Arbelian Varities case is a modular space of log Arbelian varieties.  And this troidal compactification is a modular space of log Drinfeld modules.  So the log appears.  And these three, the collaborators and I are studying Iwasawa theory.  And the reason why we need this troidal compactification is that we are studying the Sharifi conjecture, which was formulated in the paper of Sharifi in the analysis of mathematics in 2012, in 2011.  And that is a new theory in Iwasawa theory, which uses the boundary of the modular curve.  And these three are collaborating to generalize his conjecture to more general situations of similar varieties or these varieties, and not from GL2 theory to GLN theory.  And then such boundary becomes necessary.  So that is why we need this for the new study of Sharifi conjecture for function field case for GLN.  And then so first I introduce omega and x.  So we consider the three situations. So here f is q and f is q here, and f is a function field in one variable over some finite field.  And then we consider the finite dimensional vector space over f.  What is the difference between A and B?  That will appear. Yes, yes.  That will appear here.  And here the dimension v is dDd and here the dimension v is 2g and v is endowed with a pairing, f, f, it is qc.  that is non-degenerate and antisymmetric.  And so here D is also dimension.  And I need a place infinity.  So for A and B, then infinity is the Archimedean place.  And for C, then we fix our place infinity of A.  This is for this.  And then I first define omega.  So when we talk about omega, then we assume that we are in B or C.  And B then we simply assume that base is already fixed.  Base is fixed so that it is identified with this thing.  And this is the standard one.  And in C then, B is equal to G.  And then in B then omega is G.  Then maybe C is maybe most important for us.  I first define this.  This is...  I also define C.  In the case of C, then C implies that C is the completion of the R.  So infinity is the local field at infinity.  So C is the completion of the closure of infinity.  So infinity is the local field at infinity.  So that is equal to R for A and B.  And then this is...  This is G1 to GD, a linearly independent over infinity.  And in the case of B, then omega is the Z-gelapa half space of degree G.  And A is not important for us, but I discuss about A.  I talk about A for comparison with other things.  And then...  Next, so then roughly speaking, this is a modular space of the Dlinfeld modules of rank D.  And this is a modular space of polarized Abellian varieties of dimension G.  And so then now next I define X.  So X is if X be infinity and X be infinity and this, in the case A and B,  there are... that is... C and this is for B.  So this is the...  Norm's zone on V infinity, if infinity turns to V, modular homotocity, that is...  This is a norm... I will explain this.  And then this is a norm zone on V infinity, compatible with this.  And divide this.  Here the... this norm is...  Norm means that the norm is in the case of A and B then it is a function from V infinity to R.  Such that for some basis, then... I don't know, for AC.  Yeah, yeah, yeah. Races then... then... OK, OK, A, B and C.  Yeah, yeah. Such that... for some basis such that... such that the mu, mu A1, E1 plus A and E, N is...  E, D is...  Yeah, maybe, maybe.  This is A and C maybe.  This is equal to A1 square plus this one, 4444.  This is 4A. And for function field case, this is the max of A1, A, A, D, R.  This is just absolute value at infinity.  So here, A, A, R in infinity.  Oh, yeah, yeah, A, A, A, in infinity.  So in the case C, you are working with the analog, what you call C, which is F infinity by roof.  But the value group is not the... is less than the real.  So this means you don't include norms that... to the value group.  You multiply maybe AY by constant.  Oh, no, no, no, sorry, sorry. That... that is a problem though.  No, in the analog communication, you multiply absolute values by some real constants.  If you want to get the world space of periodic norms, it is the space which is sometimes considered...  depends on what you want to consider, but...  Sorry, ah...  You want to consider, but some people consider like... in the whole, like, I think, Goldman and Deva.  Ah, sorry, this is a real value to normal.  Yeah, but in coordinates should be maximum... some real constants multiply by absolute value.  Oh, maximum constant, but we... there is some basis, and so that is OK.  No, no, no, no, no, no is the... is this standard... has a standard form for some basis.  Yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, just the...  for some basis then the... the... the function has this nice shape.  That is the definition of the norm.  Yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah.  Yeah, and in the case of B then the... the... the norm is compatible with... with...  with... with... this means that there is some basis... basis EI...  of V... here, now, antisymmetric pairing is given, and for which is...  for which is the... the... the... this pairing is a standard one.  The... and... and the... and the mu mu mu mu mu A1, A1 plus AAD, A2G...  E2G is... is again of... A1 square plus A2G... square.  Yeah, yeah.  Oh, sorry. In... in non-artimidate case, you can see...  it's only a countable set of un-countable...  It is uncountable.  That is at the political space.  No, no, no.  You can see the set.  It's a loss of countables.  You don't have to get countables.  Countable space.  Countable set.  No, you can't do that.  You can't do that.  You can't do that.  But on f infinity, yeah.  Yeah.  Yeah, yeah.  So this is a countable, no maybe no.  You can make a convergence.  That is like a real vector space because you are considering the values in real numbers  and then the countable, no, no, no.  Oh, yeah.  You are right.  Perfectly right.  Yeah.  Sorry, sorry.  There is a...  Yeah.  Yeah, yeah, yeah, yeah.  Sorry, sorry.  This is...  I am very sorry for this.  Yeah.  Yeah, yeah, yeah, yeah, but that is yes, yes, yes.  I am sorry, I am sorry.  I am sorry.  Yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah.  Yeah, yeah, yeah, yeah, yeah, yeah.  Very sorry for this stupid thing.  Oh, yeah.  We need this.  Yeah.  Yeah, I am sorry for that.  I'm sorry because...  So, then here you need such square.  Yeah, yeah, yeah, yeah.  You need...  Yeah, sorry, sorry.  Yes, so this is a...  Yeah, so that is die.  We can make it diagonal by taking the basis.  Yes.  So, then...  Yes.  So, then now...  So, then...  Ah, this is important.  So, I will not...  Not...  Not erase this one.  But, then...  Yes.  Oh, yes.  Maybe...  Yeah, yeah, yeah, yeah.  Okay.  This should be here, but I think the most important thing should be...  Maybe...  I think the convention here is that maybe the most important source.  Ah, sorry, I didn't...  Okay, okay, yeah.  So, then...  Then...  Yeah.  So, then...  Sorry.  Ah, sorry.  Then, ah...  This is just a multiplication by...  multiplication by...  by some element, positive.  Then, we will have a class of those.  Yeah, yeah.  And then...  Then, ah...  So, in the case of A, and then...  Then, X...  X is...  I can modify it to SLD.  R divided by SOD.  Ah, sorry.  Sorry.  Sorry.  Sorry.  And B, in the biggest B, I...  X is isometric to ASPG, R...  R divided by a maximum factor.  And in the case of C, then X is a standard geometric realization of the...  realization of the...  of the Bvialfitz building...  building...  for...  for...  Pgrv infinity, yeah, for this algebraic group.  And then we have a map.  We have also in the case of B and C, then we have a,  if they are there, we have omega.  And we have a map from omega to x, a continuous map.  This is in the case of C, it is defined as z,  d goes to mu, mu is r.  Mu is mu A1, Ad is A1, z1 plus Ad, zd.  So this is inside the C, but because  z1 to zd are linearly independent of infinity,  so this becomes a norm.  And so this is a Krasov.  And then for the case of B, then it is z goes to mu,  mu is a Hojimetric,  the Krasov is a Krasov,  and the Krasov is a Krasov.  So this is the Krasov.  And then we have a map from omega to x,  and then we have a map from omega to x,  and polarized by this.  And then in the case of B, then this is the Homme-Omori-Shizun.  And so next we consider the section 2 is concerning the,  first we consider the compactification of this x,  and then the total compactification is related to how to extend omega.  And so first we consider the extension,  enlargement of x bar of x and x bar flat of x.  So then the, so first x bar,  this is the case A and C. C is C means G,  this seems that I deleted something.  Sorry. This is the, the first case is G,  just the, yes.  Second, B is the for the case with pairing,  and A and C are similar, but the fields are number q,  and this is function fields case.  And here then, then this is x bar is the search pair,  where p is the f parabolic subgroup,  subgroup of p, g l, g l v.  And, and mu is, and then p correspond to a filtration,  v minus 1 v 0, v m v,  parabolic subgroup is a, isotropy group of such a flag.  And then mu is mu, mu i 0, i m fair,  mu i is in x v i divided by v i minus 1.  And then the, and in the case of B then,  then some, some modification is necessary.  And B is, in the case of B then,  x bar is again p mu, fair, fair, p is a f parabolic subgroup of,  subgroup of g c s p, v o i s p v,  that's the same thing, essentially.  And mu is, mu i, here, here, so the p correspond to a, some,  some such, such that the, the, the annihilator of v i for the  sparing is v minus 1 minus i, for all i.  And, and mu, mu, mu 0, mu 0 is, is in x v 0, v minus 1,  with, with, with, sorry, I forgot the notation infinity there.  And here, here, we can have a pairing for the restriction of the  original pairing. And mu i, mu i for, for i, not 0 is, is, is in, in x v i  divided by v minus i infinity. Here, there is no, no pairings,  we just take the space of norm's, modulo, modetti.  And then, the main, now, the theorem is, then we have some topology  on this x bar. The definition of the topology is rather  involved and I'm afraid that I don't have time. That is the, the, some kind  of topology. And I, or, or, or some, or, or some, or, or, or, or, or, or,  but, but I, I, I, I am afraid that I don't have time to, to, to, to explain.  Then here is the theorem. So that is, for a, b then this is well done.  It should be so that our parabolic, which I think relates to the  grandian. Since we are symphlectic, you can also have a parabolic.  No, no, no, I think.  There are some, some exceptions.  No, no, do you get all the parabolic subgroups in this way or?  Maybe, I, I, I believe so, yeah, but if, if it is not so, then please take  this, this, this, this such thing corresponding to such thing.  Yeah. Otherwise, the definition doesn't work. So, so I, yeah, yeah, yeah, yeah.  So, and this, ah, and we don't know. And, and see the, the, the, this is a  recent paper of, joint paper of, of, of, of, of, of, of, of, of, of, of, of, of, of,  people including, including me. So, so that is probably new, the, the, the,  but, because of, we wrote a paper on, on this and that, that paper will appear,  posted, you will be posted in archive soon. But, but, and we are not perfectly  sure if this such thing is, ah, I will write a, ah, theorem, the gamma X  by the compact household, ah, for, for, for, for, for gamma, ah, commensurable, ah,  to, to, to, to GLDZ, ah, if, if these, if, if, if the case A and G, GSP or SP,  maybe SL, SL, SL, ah, the, no, SP, SPD, GZ in the case B and SL, SLD, OF in the  case C. Now, OF is, OF is a, is a element in F such that, ah, A integral  outside, outside infinity. Yeah, and so, ah, so for A and B then, then, ah, the A  and B then gamma, this, this thing is, ah, so called reductive Borrel cell, ah, Borrel  cell compactification of, ah, and so, such a standard of Jikutan. So, this is some, ah,  we, we, we, we imagine that some person already studied C, but, but, ah, we could not find  the reference. And there is some similar subject, ah, ah, similar subject, ah, is, ah,  X is inside, X bar is inside X, ah, X is inside X bar by, by taking the P to be the,  the, the, the whole, whole, whole, parabolic subgroup, ah, whole group, yeah, and then,  ah, yeah, so, ah, ah, this is, this is compact and this has a sataketology and not the  induced topology. And here, this is defined by using, ah, infinity, parabolic subgroup,  ah, ah, in place of, in, in, if, if, if parabolic subgroup.  Ah, sorry, yeah, in the case C, C, C, yeah, this is, this is ABC, ABC, ABC and, and then,  ah, there is such a story and so, so, ah, in the case, for example, in the case A and,  and B maybe, and G is 1, then, ah, the X bar is, is a upper half plane, ah, and P1Q, that  is a boundary point, ah, and X2 bar is, is, ah, so, that, that is in this case, then,  you have a homomorphism from the, the, this H2X, this is, this is H is, I think, 2, 2,  2X by, as I, I, I, I wrote some here, yeah, and, and then, this is H and P1R and so,  ah, parabolic corresponds to, to such, such thing. And so, ah, this is compact, this is  compact for the natural topology, ah, but, but this one has a, it's usually considered  with, with Satake topology, not in this topology, so, so, then, after, after dividing by gamma,  then you have haha space, yeah. And this one, this one for CX bar for C was, is a, is a so-called  So, the general compactification of X are studied by already, in the old works of some  people, I'm sorry, I, yeah, here, and, and, and, yeah. And so, such similar things are already  studied and so, it is strange that the theorem for C is not known, but, but, maybe known,  but, and, but the proof is, is just a, proof of the theorem for C is just the imitation  of the other, other cases, A and B, but, but still some, some strange things are happening  there and so, so the, this paper by three people have, has 50 pages, not so, so simple,  that was not so simple, yeah, that is strange thing, but, but, and then, then I think, I,  I also talk about X, X bar flat, X bar flat is, is, is, is, is, in this A and B then,  in C, yeah, in C then X bar, X bar flat is, is a quotient of X bar and X bar is, is W  here, here, here, W is a subspace, a subspace and, and, and, and, and, and, and, and, and,  and, and, and, and, and this is, and B in the case of B, B then X, B X bar flat is, is,  again, W and, mu, yeah, and then, this is, W is in V and, and then, such that W, W, the  manipulator of W is contained in W and, and then, mu is in, in X, W, W, this and, and,  and, and this, yeah, and then the map from X bar to X, B is, is a, just a, a, P, P mu  goes to V zero, mu restricted by, just mu zero and, and, and, and, yes, this is the  definition, I know the B, this, this is for, for A and C and for B then, it is P mu goes  to, to, to, to V zero, to, to, to V zero, B minus, mu, mu, mu zero and, yes, no, no,  no, no, no, that is, fine, yeah, yeah, yeah, yeah, that's same, same thing, yeah, yeah,  yes also, yeah, this is same, yeah, for, for, for all A, B, C, yeah, yeah. And so, that  if a subjection and, and so the X bar flat has a coefficient topology of X bar, or one  This is another.  Also, we can define the sataketopology,  which may not be the image of the sataketopology of X.  The search complicates things.  But I don't discuss such too much details.  And again, Seorem is that again,  this is how it's looked compact.  And for A and B, then this is a result of sataketopology.  This is proved by him in 1964, I think,  in his paper in the analysis of mathematics.  And so C may be new.  And we can have, again, X bar, flat.  This is also a compact space.  This is compact, first rule.  And this has a sataketopology, which  is different from this space.  And this is so-called compactification by Uena of the  pre-war-techs building.  In the case of C. And A and B, in the case of AB,  it is already studied by sataketopology.  In another paper, his paper in 1964, something like that.  This paper also appeared in analysis of mathematics.  So when he was young, sataketopology, two papers  in analysis of mathematics.  One is this one, and the other is this one.  Yeah, yeah, so then these stories are actually,  and we have nothing to return down,  but maybe the results are maybe generalized to all.  All reductive groups are not only GLD or PTSD.  And the problem should be the compactification of the  understanding of the building of the space  of norms may be not so good for generalization.  And then we have to directly consider  the building, I think.  But similar arguments may be OK.  It can be down there.  And also the version of SU.  Of course, in the case of number fields, then those are  already known, essentially.  And for X bar, X2 bar, then people already studied all  reductive groups.  Yeah, and so the version for S, S are finite set of  primes, S are arithmetic subgroups.  S for S are finite set of primes, set of places,  including all arachnidian places.  Then X is now the XV for all V. And then we can define X bar  and for X bar, S are, and then the S arithmetic subgroup  group is has no compact.  This is in written.  This one is written in that paper, which will be submitted  in archive soon.  So there is some such S version.  So these are remarks.  But I did not introduce this generalized version because  the story becomes too much complicated.  So I just considered the case of just one place.  And then the final thing which I will talk about is the  story of colloidal compact fiction.  But as I said, already the paper is not yet written.  Just we are starting to write.  And so here can happen as usual that some terrible  problems appear in the writing and then searching.  At present, I hope that the things can be solved.  But I'm not so perfect sure.  But I write the picture section three.  The first is we only consider B and C.  In the case of GLD of the Arachnidian number field,  then there is no analytic structure for the search space  feature appears here.  And so we consider B and C. And then the MKB, in the case of  B, this is a modular space of, so K, G is GLV.  And this is a B case, GSP.  And this is GLV.  And K is an open compact subgroup of G, AF, finite part.  And the finite part means just this is without infinity.  And then we can define such modular space.  And modular space, MK is the modular space of polarised  Arbillian varieties with K-level structure.  This is for B. And this is the modular space of  the dream-field modules of dimension G. And this is  modular space of dream-field modules of rank E with K-level  structure.  This is B and this is C. And then, as is well known,  you have a presentation of MKC.  So I talk about only analytic theories.  We hope also the algebraic theories can be done, but  today, and only I present a very rough story.  MKC is, it is well known that MKC is written as  K. So this is for B and C. The omega was, in the case of C, then  omega was dream-field upper half-plane, upper space.  And then, this is then let MKB be flat, MK flat B. This is  modular space, the compactification of MK by Bayley-Borrell,  in the case of B. And the capra pink in the case of C. And  then, and also MK bar, bar-troll, the toroidal  compactification. In the case of B, then it is constructed by  Manfred and other collaborators.  And in the case of C, then I believe that we can construct  this, but the paper is not yet well.  So, but I hope to present the picture.  The picture is that the, so this is not unique, not unique.  And there's no, and here this is a standard one, unique, unique  one exists, standard one exists.  And then MK bar, bar be the inductive limit of all  toroidal compactifications, all MK bar, MK bar, there are many.  And so then, then we have inverse limit.  And then we cover diagram, cumulative diagram.  MK bar, bar, bar-troll, MK bar, bar flat.  And here you have, you have X bar GF, Gaf, FF, divided by K.  And you have GF, X bar B times Gaf, F, flat K.  So, that is, then, not that we have a map from this to GF, X times.  So we have a map from monogat to X, so you have a map already here.  And then what I am saying is that this map may be phi.  Phi can be, phi is extended to continuous maps.  Phi is extended to continuous maps.  And this one, this one is in the case B, case B, that is a national number,  this is case, this is our Avalian variety case, this is a homeomorphism.  That is, this is the, this is a Sadake compactification.  So this is just a political space, but this is an analytic space.  And so that is Bailey and Borrell put a complex analytic structure.  So maybe C should be here.  Complex analytic structure on the topological space feature, Sadake constructed.  And so this is the picture.  And then what I have five minutes left, but I, in any case, the paper is not written,  so I can give only a love story.  So then how, how the, so concerning this then, I have one, so just small thing.  So that, that is, so the love story of log dream belt modules.  But I am hoping to describe this Barl version.  Without Barl then the, the, without Barl, the Trojan compactuation should be defined  as the quotient of this Barl version.  So then I describe this in short, so, well, but if it's module then.  So let S be a log smooth, rigid analytic space.  I am talking about the analytic situation.  So log smooth, rigid analytic space over, so we are considering C, C, C, yeah, yeah,  analytic space over C.  Then, and the log smooth then we have S Barl that is a limit of, of, of blowing up of,  of S along log structure.  If the log is given this way then we, we blow up, then you have this and then we blow up  this, this is intersection then, then you have this is S and this is limit of, limit of  this is S Barl is the limit of, of, of, of such blowing up.  And then the, the log dream belt module is a, is a, log dream belt module, module over  S infinity over S, S, S Barl is, so that is, that with, with cable level structure.  So that K, K is assumed to have, we are, assume, assume the, for simplicity the, the G, GL,  the OV, other, that is, yeah, then, then is a, is a such thing is, is a, is a pair of,  pair of V and L and V and lambda, here L is a, L is a line bundle on S Barl and, and  so the S, inside S, the, you have open set U, the, this is a part where log is trivial,  the part log is trivial.  And then V, V is a, V is a, and then you have, you have then here you, the blowing up doesn't  change U, so then you have J here and then, then V is a, V is a inside here and, and then  this is a, this is a, this is a D dimensional F space, C for V, C for V, D dimensional F  space and lambda is a, is a level structure, K level structure, the, the, the A, F, F, T,  but, cosidate module, K, the, the isomalism is, the only, J, J, oh yeah, thank you very much,  thank you very much, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah,  and then, this should satisfy the, the, the, the, the, the conditions are, conditions are,  there is some, for example, there is some condition is the following, yeah, so for all such that,  so which topology is it, is it?  Ah, yeah, yeah, yeah, yeah, that is, I think there's a rigid, rigid topology, oh such thing, yeah,  this is,  The limit of the rigid topos,  Oh yeah, yeah, I think so, yeah, yeah, such that, such that the, all T in this, but then,  and all S in this, we have the following, so maybe just T is enough, yeah, all T then,  then we have, we have, we have, this is, suppose, stock at T, V1,  in the stock of V then, we have this, and, and, and, such that, satisfying having the following property,  that property is the first, V0 is the, the inverse limit of, you know, inverse image of, image of,  of, of OS, VAR inside j star OU, ah, ah, ah, ah, ah, intersection, intersection, sorry, sorry,  V is V, V, sorry, V, V, and intersection OS, OS VAR inside, inside j star OU, yeah, yeah,  and, and furthermore, the condition is that, then, from OS VAR to, to, to, to see at T,  the, you have a, is, is, then, then induces, induces, ah, injection, this, this is injection,  so that is the, the, the, we have a norm of, for this, so then, so from this then, ah, norm 1,  on, on, V0 appears, ah, V0 infinity, ah, this is V0 infinity, appears, and the second condition,  so that we are, we are, how, how I am actually showing how we get such V, V, V, V, V, V,  the, the point of the, of the, ah, X bar, yeah, yeah, and then, then, ah, the, the, then,  another condition is that, second one is that, ah, sorry, I need to, to, to, to, to,  thank you very much, I am already, I need to, ah, sorry, sorry, ah, so then, then, ah,  another is that, so, there is a, you have a, you have, you have, let VIO, FBR, VR, the,  the, then, you have here, by, in this here, you have, or V, and then, by taking the inverse  image here, ah, you can, you can have a, this defines the V of the inverse image, the integral  structure is defined, and then, ah, then you have VIO, then, then for all I, ah, ah, ah,  yeah, the, you have a, VIOF divided by VIO minus 1, VIOF, ah, from here, then, you have  a, JOU, divided by VIO minus 1, VIOF, and then, you have a exponential of VIO minus 1, VIOF,  ah, this is a Dlingel exponential map, ah, and that is the, this is the additive map, ah,  so, ah, then you have J star of U, and, and the condition is that, by, by, the, by, by,  by, by, ah, then, the image of, image of, VIOF minus, minus, minus 0 is in, in m minus 1  inside, inside J star of U, this is the, m is the log structure, ah, that is the, ah,  CF feature, ah, yeah, that is the log structure, and then, that is the inverse of the log structure,  and then, then, ah, the last, last thing is that, the, another condition is that, the,  the further condition is that, condition is that, there is some homomorphism from, from  m of, of, of, of, of, plus, this is the log structure, S, S, S bar, T, 2, 2, 2, 2, R, additive,  such that, the, the, the composition, lambda, OF minus, lambda of minus 1, OF minus 0, ah,  M inverse, and then, ah, this is by, exponential, ah, yeah, then, then, M, F, close to F inverse,  H, all, all are D and T, yeah, yeah, the, this composition extends to, to a, to a norm on,  on, on, on V infinity, V infinity is the, I don't know, V i, V i minus 1 infinity, it  is, it is a, F, F infinity times lambda i, OF divided by lambda i minus 1, OF, so, ah,  so, ah, so the norm, also, norm is defined on, on, on, on, on this, so, ah, then, by this  then, ah, the, the, that, X bar and, and the, the, X bar and, ah, the, the, this, toroidal  compactification is related, and so, the, the, the, this, Mk bar, bar, tol, bar is, is a,  is a modular space of such, such object, so then, if we have such thing, then, you have  a morphism, extending S, Soda, yeah, yeah, Soda, that is a fine modular space of, of,  of, of, in, in, in this sense, yeah, so, so then, ah, yeah, sorry, sorry, the last part  is, the, the, the, two-love story, ah, finish here, yeah, yeah, yeah.  Thank you.  Question.  What is the relation between the curly end and the rest of the data?  Ah, ah, ah, sorry, sorry, sorry, sorry, very, very sorry, that was, this, this, this should  be eres addicted to you, sorry, the, that is, I was just misunderstand the L's, oh, you,  the, the, the, the, the locale, locale exists oh yeah so, I was oh, oh ah, oh ah, yes,  I was sorry, sorry, sorry, so this, that is, so, ah, here, this is locale,  locale is, localeおおowany, I was, I was thinking of locale sorry, yeah, yeah, yeah.  KAVARTY  Is capital lambda the capital V over there?  Capital lambda I, O, F?  Capital lambda I, O, F, sorry.  What is that?  Oh, V, V, V, V.  Sorry, sorry, V.  Yeah, yeah, I was confused.  Sorry, sorry. This is V, yeah, sorry, sorry.  V, yeah, that's V, yeah.  Yeah, yeah, sorry, sorry, sorry.  Oh, this is just, so I want to have some fun questions on the end.  So when you consider J-Law style, if you do it in rigid geometry,  then you could consider sections with possibly essential singularities or just morphic.  Yeah.  So what do you mean here? Is it...  Yeah, terrible singularity of the functions appear.  But the condition is that the M is reasonable functions,  so like this is just something like generator of the device,  generated by...  If you go...  We went to...  Yeah, yeah, we went to the inverse limit, but in the middle, then this one is reasonable thing.  Here, this is just reasonable thing.  And this is invertible element of this and generated by invertible elements  and some device, generators of the device.  So the condition is that the here, we have a reasonable object.  Having not so terrible singularity, just they have poles.  They have just, at worst, they have poles there.  Yeah, this is having poles.  But in general, it has essential singularity.  Yeah, yeah, so that is...  Every terrible singularity appears, but the condition is...  This is the condition.  Yeah, yeah, the assumption, the condition is...  This is the condition, the big condition.  The assumption, this is condition one is this.  And the condition two is this.  And the conditions that you put is relative to a C-valued point of the inverse limit space.  So does it imply the same condition for other points for the Berkowitz point?  Or do you just consider C-points?  C-points, so I only... C-points, yeah.  C-points only.  So this is also VI.  And then this is...  Here is what we have.  Yeah.  If there are no other questions, let's thank Mr. Casano again.  Thank you.
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