Height functions for motives, Hodge analogues, and Nevanlinna analogues
Kato, Kazuya
We compare height functions for (1) points of an algebraic variety over a number field, (2) motives over a number field, (3) variations of Hodge structure with log degeneration on a projective smooth curve over the complex number field, (4) horizontal maps from the complex plane C to a toroidal partial compactification of the period domain. Usual Nevanlinna theory uses height functions for (5) holomorphic maps f from C to a compactification of an algebraic variety V and considers how often the values of f lie outside V. Vojta compares (1) and (5). In (4), V is replaced by a period domain. The comparisons of (1)-(4) provide many new questions to study. The Equivariant Tamagawa Number Conjecture for modular motives with coefficients in Hecke algebras Séminaire de Géométrie Arithmétique Paris-Pékin-Tokyo / Mercredi 27 septembre 2017
 Thank you very much. So in mathematics there are talks about theorem and talks about conjectures with evidences. And then there are some talks about conjectures without evidences. I think you like one and two but you don't like three so much. But I am afraid that my talk is concerning this third subject. I got old and my brain already collapsed. I lost the ability to study those two. But because my brain is already collapsed, everything became the same in my brain. So I can connect anything, any different things as the same thing. So this is a really big advantage to become old. This subject, which video has more information about, compared toSPEAKER- tongue. This is the problem. So if you are to write up here, I will explain this in C. Now, in C, so here, in the denominator, and then here we have to, I know where it is. Here we, this is the number of here, and this is the algebraic variety of y. And I am going to say this is the disk talk. So if, and then we assume that we are using the y as the open object of the algebraic variety, then then the, and then m c, m c, we see the normal functions, functions of c, we are going to express the name to, to, to, to, we see. And then this is the, which are the, normal functions to, normal functions, if we now need to reverse such that, such that if c is not containing the boundary, this is minus c. And then this, this is about number theory, and this is about the, never in a theory. So the subject here, subject here is, is, is to study, is a question. How, how open, how open are f has similarity, or pole was 0 on, on, on c. And that is how open are, how open are f, fz belongs to, to, to the boundary. Yeah. And this is a, never in a theory. So I mean that, input is a, number theory, I mean, never in a theory, so I mean between the manyjectors, comparing with this two, I mean, when the water is not, L is the polarization. Oh, thank you very much. Now, but now we can, we can compare. What is R? T-f-r-i. T-f-r-i. Oh, yeah. Yeah, yeah. This is R. This R is, sorry. This R is non-negative R number. Yeah. So R moves. R moves. So, if R is, if F-C is not contained, then, how about that, it's not contained. Ah, ah, ah, not contained. Yeah, thank you. Thank you. So, now, but then we can compare R, R and R2. And then also, and the, the new functions are one, ah, it's a two, two, two, ah, each number. And two are in R. So, C, C, R, R, it stands R, F, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 3, 5 and 6 C's. Yes. Mines. Mines. C2 2, 2, 2, 2, 2, 4, 5, 6. There we go. And the XS is so-called P of the name defined first by Wies and generalized by other people. This is a structure with the same name as the XS. And this notation is called CXC's, a melodic CYL for wide for C and C. This is the projective smooth curve of the C. And the morphisms for M, M-hole, M-hole, A-hole, Y-hole, C-hole, this curve C, then morphisms for Y to XC's set of polysonsal, and then morphics for Y to XC's. And concerning the period domain, there is a notion of the horizontal map. And this is searching up with this condition. The horizontal map corresponds to a variation of structure with log generation, and the Y-hole is with the Y-hole, the Y-hole, the horizontal map. And also, this is also a morphism for wide to XC's, here the morphism is here the capia, the poro, the pancinjaiti here, the porus v Murubi but here they have a siebie on it, but this is the perium, a morphism. And then, a sati dhat, sati dhat, and you can draw it inside the dhat. And here, the exibartheism, exibartheism, parcell, parcell, compoptification, compoptification, or of exibartheism, I am not defined. But this is not to compoptification, but some parcell compoption, and then here, but here we don't talk so much about this one, so I don't explain so much. This is consistent, we report into all its, all its, in the theory of desegeration of all its structures and all its, with run-through, run-through, run-through dhat, run-through dhat, run-through. Yeah, yeah, this is, we argue some points, all the important all its to, to, to exceed them, you need to, exibartheism, I don't, we don't discuss so much. And then, so the degeneration in Satoshi, you have a, which is, variation of the H and the H and H correspond to the, in the, in the, in the degeneration of the DT-separ, this H, F-adge, F is a, period map, period map of, of H. It is at each, at each point of Y, then, then if you restrict this H to that point Y, then, a, a small Y, point of Y, then, then you get the, the point of, see the, the whole structure at each point. And so, then you, you get Satoshi, so that is for the, a period map, that is at each point of Y, you have a usual structure, this is a, so at each point you, you, you have a structure. And the, the, that is, but there are similarities also, because of the, you have a log designation, but, but if you use its, it's a thousand that becomes a, to, a, a, a, a, a, a, a, a, a, and then, so the, the, the, the, the, the TCC is a analog, but so the, the subject here, subject here, subject here, how often are, how often are, this H, H degenerates? Why, why, why, why, why, why, why, why is this, see what this is, yeah? Or, or, or, or, or, how often, how often, how often, how often are, are, is this in, in the moment, so that's, in mind, I'm not sure. Yeah, yeah, so, then, then this is a theory of designation of, which structure, so this is our, some, some, some nice subject, yeah, yeah. And then, now, I hope to, to define those, those high functions, those high functions, ah, yeah, that's right. And lambda, lambda is, lambda is a, yeah, I am defined lambda, and so then, and then those XC and R is not, this XC is not, not usually not an algebraic variety, and so we cannot, this subject is not contained in the usual, never-in-a-theory, we cannot directly apply the construction of, of high functions, neither, neither, neither, never-in-a-theory to, to, to this subject, yeah. And so, now, I, I, so, now I, I define a first, ah, first, ah, so, that's our first section, I define, ah, ah, XC, yeah, XC, yeah. So, we can, we can consider the next, the next, ah, construction, but today, I hope to make the, the story simpler, ah, I, I will consider the here, here, here, here, here, here, here, here, and then we will consider the here, here, that is, the, ah, we will, we will look at the here, here, here, here, here, here, so our, we fix the, ah, So now we have that filmmakers we rewife with this concept presentations for you to detail. So this presentation is a presentation for you to really see this is. this is a origin of. and and h i 0 almost all right and we also fix the h 0 h 0 q h 0 q is a is maybe h 0 j i i i i i i i h 0 q is a q depth of space of space of dimension h i the sum of h i and and fix the fix the h 0 q times h 0 q to q times 2 pi i to the minus w minus w this is a non-designate pairing a non-designate pairing a per linear home and and symmetric just an consig universeirect order with i l i l l l l X is written in the form of d plus minus times g g a u q divided by t. Here, this is a British domain, the period domain. But we take a little bigger one. This is the filtration, increasing the filtration zone on each c, the tensor product of c and h, q. The filtration is stored in a single sub-space. Such that the field divided by y is minus plus minus. This is decreasing filtration and the difference is that the dimension is h i. Then this is the polarization. That is, this is the filtration. Such that h i. Q is the filtration of this bearing. It is a polarized structure of weight. And h i. d plus minus is the same thing that here. Here we take this zero or minus of that. You replace this. This is replaced by. Then you have two plus minus. And g is the algebraic group of q. And g is the algebraic group of q. Then you have reaction and then k. And fq is the number of media component of the other thing. And then we have. And then we have. And then we have. M is the motion of the weight of weight w. And such that the dimension of the field divided by y is minus plus minus h i. And then other conditions are that the polarization are. And then. And lambda is the k plus of a f q tensor h 0 q to the total polarization. Here the k can act here. And then we take the mojirou k plus. And this should be also. And then there should be some conditions that there is some polarization. And also some on n. And some as morphism. This is basically. I forgot to say that f is a bit. Yeah. We have it. And this is a bit of realization. So. So that we have a morphism between this and h 0 q. And also this is the same also. Also some. Yeah. I'm perfectly precise. I wrote a preprint. It is. Actually that is all these things. But it is. Present. I have to collect many mistakes and the special version and also this particular version and part two. Will be. Maybe after two months or two weeks. So. We should have. Impressed that. I don't talk about this. Yeah. Yeah. And. Part three. This is a variation of the structure. And then. You have a variation. And then. And then. This is. So. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. 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 Thank you very much.  So in mathematics there are talks about theorem and talks about conjectures with evidences.  And then there are some talks about conjectures without evidences.  I think you like one and two but you don't like three so much.  But I am afraid that my talk is concerning this third subject.  I got old and my brain already collapsed.  I lost the ability to study those two.  But because my brain is already collapsed,  everything became the same in my brain.  So I can connect anything, any different things as the same thing.  So this is a really big advantage to become old.  This subject, which video has more information about,  compared toSPEAKER- tongue.  This is the problem.  So if you are to write up here, I will explain this in C.  Now, in C, so here, in the denominator,  and then here we have to, I know where it is.  Here we, this is the number of here,  and this is the algebraic variety of y.  And I am going to say this is the disk talk.  So if, and then we assume that we are using the y as the open object of the algebraic variety,  then then the, and then m c, m c,  we see the normal functions, functions of c,  we are going to express the name to, to, to, to, we see.  And then this is the, which are the,  normal functions to, normal functions,  if we now need to reverse such that, such that if c is not containing the boundary,  this is minus c.  And then this, this is about number theory,  and this is about the, never in a theory.  So the subject here, subject here is, is, is to study, is a question.  How, how open, how open are f has similarity,  or pole was 0 on, on, on c.  And that is how open are, how open are f, fz belongs to, to, to the boundary.  Yeah. And this is a, never in a theory.  So I mean that, input is a, number theory, I mean, never in a theory,  so I mean between the manyjectors,  comparing with this two, I mean, when the water is not,  L is the polarization.  Oh, thank you very much.  Now, but now we can, we can compare.  What is R?  T-f-r-i.  T-f-r-i.  Oh, yeah.  Yeah, yeah.  This is R. This R is, sorry.  This R is non-negative R number.  Yeah.  So R moves.  R moves.  So, if R is, if F-C is not contained, then, how about that, it's not contained.  Ah, ah, ah, not contained.  Yeah, thank you.  Thank you.  So, now, but then we can compare R, R and R2.  And then also, and the, the new functions are one, ah, it's a two, two, two, ah, each  number.  And two are in R. So, C, C, R, R, it stands R, F, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2,  3, 5 and 6 C's.  Yes.  Mines.  Mines.  C2 2, 2, 2, 2, 2, 4, 5, 6.  There we go.  And the XS is so-called P of the name defined first by Wies and generalized by other people.  This is a structure with the same name as the XS.  And this notation is called CXC's, a melodic CYL for wide for C and C.  This is the projective smooth curve of the C.  And the morphisms for M, M-hole, M-hole, A-hole, Y-hole, C-hole, this curve C, then morphisms for Y to XC's set of polysonsal,  and then morphics for Y to XC's.  And concerning the period domain, there is a notion of the horizontal map.  And this is searching up with this condition.  The horizontal map corresponds to a variation of structure with log generation,  and the Y-hole is with the Y-hole, the Y-hole, the horizontal map.  And also, this is also a morphism for wide to XC's, here the morphism is here the capia, the poro, the pancinjaiti here,  the porus v Murubi but here they have a siebie on it,  but this is the perium, a morphism.  And then, a sati dhat, sati dhat,  and you can draw it inside the dhat.  And here, the exibartheism,  exibartheism, parcell, parcell,  compoptification,  compoptification,  or of exibartheism,  I am not defined.  But this is not to compoptification,  but some parcell compoption,  and then here,  but here we don't talk so much about this one,  so I don't explain so much.  This is consistent,  we report into all its,  all its, in the theory of desegeration of  all its structures and all its,  with run-through, run-through,  run-through dhat, run-through dhat, run-through.  Yeah, yeah, this is,  we argue some points,  all the important all its to, to, to exceed them,  you need to, exibartheism,  I don't, we don't discuss so much.  And then, so the degeneration in Satoshi,  you have a, which is,  variation of the H and the H and H  correspond to the, in the, in the,  in the degeneration of the DT-separ,  this H, F-adge, F is a,  period map, period map of,  of H.  It is at each, at each point of Y,  then, then if you restrict this H to that point Y,  then, a, a small Y, point of Y,  then, then you get the, the point of,  see the, the whole structure at each point.  And so, then you, you get Satoshi,  so that is for the, a period map,  that is at each point of Y,  you have a usual structure, this is a,  so at each point you, you, you have a structure.  And the, the, that is,  but there are similarities also,  because of the, you have a log designation,  but, but if you use its,  it's a thousand that becomes a,  to, a, a, a, a, a, a, a, a, a, and then,  so the, the, the, the, the,  the TCC is a analog,  but so the, the subject here,  subject here,  subject here,  how often are,  how often are,  this H, H degenerates?  Why, why, why, why, why, why, why, why is this,  see what this is, yeah?  Or, or, or, or, or, how often, how often,  how often, how often are, are,  is this in, in the moment,  so that's, in mind, I'm not sure.  Yeah, yeah, so, then, then this is a  theory of designation of,  which structure, so this is our,  some, some, some nice subject, yeah, yeah.  And then, now, I hope to,  to define those, those high functions,  those high functions, ah, yeah, that's right.  And lambda, lambda is, lambda is a,  yeah, I am defined lambda,  and so then, and then those XC and R is not,  this XC is not, not usually not an  algebraic variety,  and so we cannot, this subject is not  contained in the usual, never-in-a-theory,  we cannot directly apply the construction  of, of high functions,  neither, neither, neither, never-in-a-theory  to, to, to this subject, yeah.  And so, now, I, I, so, now I,  I define a first, ah, first, ah,  so, that's our first section,  I define, ah, ah, XC, yeah, XC, yeah.  So, we can, we can consider the next,  the next, ah, construction,  but today, I hope to make the, the story  simpler, ah, I, I will consider the  here, here, here, here, here, here, here,  here, and then we will consider the  here, here, that is, the,  ah, we will, we will look at the  here, here, here, here, here, here,  so our, we fix the, ah,  So now we have that filmmakers we rewife with this concept presentations for you to  detail.  So this presentation is a presentation for you to really see this is.  this is a origin of.  and and h i 0 almost all right and we also fix the h 0 h 0 q h 0 q is a  is maybe h 0 j i i i i i i i h 0 q is a q depth of space of space of dimension h i the sum of h i and  and fix the fix the h 0 q times h 0 q to q times 2 pi i to the minus w minus w this is  a non-designate pairing a non-designate pairing a per linear home and and symmetric  just an consig universeirect order with i l i l l l l  X is written in the form of d plus minus times g g a u q divided by t.  Here, this is a British domain, the period domain.  But we take a little bigger one. This is the filtration, increasing the filtration zone on each c, the tensor product of c and h, q.  The filtration is stored in a single sub-space. Such that the field divided by y is minus plus minus.  This is decreasing filtration and the difference is that the dimension is h i.  Then this is the polarization.  That is, this is the filtration. Such that h i.  Q is the filtration of this bearing. It is a polarized structure of weight.  And h i.  d plus minus is the same thing that here.  Here we take this zero or minus of that.  You replace this. This is replaced by.  Then you have two plus minus.  And g is the algebraic group of q.  And g is the algebraic group of q.  Then you have reaction and then k.  And fq is the number of media component of the other thing.  And then we have.  And then we have.  And then we have.  M is the motion of the weight of weight w.  And such that the dimension of the field divided by y is minus plus minus h i.  And then other conditions are that the polarization are.  And then.  And lambda is the k plus of a f q tensor h 0 q to the total polarization.  Here the k can act here.  And then we take the mojirou k plus.  And this should be also.  And then there should be some conditions that there is some polarization.  And also some on n.  And some as morphism.  This is basically.  I forgot to say that f is a bit.  Yeah.  We have it.  And this is a bit of realization.  So.  So that we have a morphism between this and h 0 q.  And also this is the same also.  Also some.  Yeah.  I'm perfectly precise.  I wrote a preprint.  It is.  Actually that is all these things.  But it is.  Present.  I have to collect many mistakes and the special version and also this particular version and part two.  Will be.  Maybe after two months or two weeks.  So.  We should have.  Impressed that.  I don't talk about this.  Yeah.  Yeah.  And.  Part three.  This is a variation of the structure.  And then.  You have a variation.  And then.  And then.  This is.  So.  This is.  This is.  This is.  This is.  This is.  This is.  This is.  This is.  This is.  This is.  This is.  This is.  This is.  This is.  This is.  This is.  This is.  This is.  This is.  This is.  This is.  This is.  This is.  This is.  This is.  This is.  This is.  This is.  This is.  This is.  This is.  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Thank you very much. So in mathematics there are talks about theorem and talks about conjectures with evidences. And then there are some talks about conjectures without evidences. I think you like one and two but you don't like three so much. But I am afraid that my talk is concerning this third subject. I got old and my brain already collapsed. I lost the ability to study those two. But because my brain is already collapsed, everything became the same in my brain. So I can connect anything, any different things as the same thing. So this is a really big advantage to become old. This subject, which video has more information about, compared toSPEAKER- tongue. This is the problem. So if you are to write up here, I will explain this in C. Now, in C, so here, in the denominator, and then here we have to, I know where it is. Here we, this is the number of here, and this is the algebraic variety of y. And I am going to say this is the disk talk. So if, and then we assume that we are using the y as the open object of the algebraic variety, then then the, and then m c, m c, we see the normal functions, functions of c, we are going to express the name to, to, to, to, we see. And then this is the, which are the, normal functions to, normal functions, if we now need to reverse such that, such that if c is not containing the boundary, this is minus c. And then this, this is about number theory, and this is about the, never in a theory. So the subject here, subject here is, is, is to study, is a question. How, how open, how open are f has similarity, or pole was 0 on, on, on c. And that is how open are, how open are f, fz belongs to, to, to the boundary. Yeah. And this is a, never in a theory. So I mean that, input is a, number theory, I mean, never in a theory, so I mean between the manyjectors, comparing with this two, I mean, when the water is not, L is the polarization. Oh, thank you very much. Now, but now we can, we can compare. What is R? T-f-r-i. T-f-r-i. Oh, yeah. Yeah, yeah. This is R. This R is, sorry. This R is non-negative R number. Yeah. So R moves. R moves. So, if R is, if F-C is not contained, then, how about that, it's not contained. Ah, ah, ah, not contained. Yeah, thank you. Thank you. So, now, but then we can compare R, R and R2. And then also, and the, the new functions are one, ah, it's a two, two, two, ah, each number. And two are in R. So, C, C, R, R, it stands R, F, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 3, 5 and 6 C's. Yes. Mines. Mines. C2 2, 2, 2, 2, 2, 4, 5, 6. There we go. And the XS is so-called P of the name defined first by Wies and generalized by other people. This is a structure with the same name as the XS. And this notation is called CXC's, a melodic CYL for wide for C and C. This is the projective smooth curve of the C. And the morphisms for M, M-hole, M-hole, A-hole, Y-hole, C-hole, this curve C, then morphisms for Y to XC's set of polysonsal, and then morphics for Y to XC's. And concerning the period domain, there is a notion of the horizontal map. And this is searching up with this condition. The horizontal map corresponds to a variation of structure with log generation, and the Y-hole is with the Y-hole, the Y-hole, the horizontal map. And also, this is also a morphism for wide to XC's, here the morphism is here the capia, the poro, the pancinjaiti here, the porus v Murubi but here they have a siebie on it, but this is the perium, a morphism. And then, a sati dhat, sati dhat, and you can draw it inside the dhat. And here, the exibartheism, exibartheism, parcell, parcell, compoptification, compoptification, or of exibartheism, I am not defined. But this is not to compoptification, but some parcell compoption, and then here, but here we don't talk so much about this one, so I don't explain so much. This is consistent, we report into all its, all its, in the theory of desegeration of all its structures and all its, with run-through, run-through, run-through dhat, run-through dhat, run-through. Yeah, yeah, this is, we argue some points, all the important all its to, to, to exceed them, you need to, exibartheism, I don't, we don't discuss so much. And then, so the degeneration in Satoshi, you have a, which is, variation of the H and the H and H correspond to the, in the, in the, in the degeneration of the DT-separ, this H, F-adge, F is a, period map, period map of, of H. It is at each, at each point of Y, then, then if you restrict this H to that point Y, then, a, a small Y, point of Y, then, then you get the, the point of, see the, the whole structure at each point. And so, then you, you get Satoshi, so that is for the, a period map, that is at each point of Y, you have a usual structure, this is a, so at each point you, you, you have a structure. And the, the, that is, but there are similarities also, because of the, you have a log designation, but, but if you use its, it's a thousand that becomes a, to, a, a, a, a, a, a, a, a, a, and then, so the, the, the, the, the, the TCC is a analog, but so the, the subject here, subject here, subject here, how often are, how often are, this H, H degenerates? Why, why, why, why, why, why, why, why is this, see what this is, yeah? Or, or, or, or, or, how often, how often, how often, how often are, are, is this in, in the moment, so that's, in mind, I'm not sure. Yeah, yeah, so, then, then this is a theory of designation of, which structure, so this is our, some, some, some nice subject, yeah, yeah. And then, now, I hope to, to define those, those high functions, those high functions, ah, yeah, that's right. And lambda, lambda is, lambda is a, yeah, I am defined lambda, and so then, and then those XC and R is not, this XC is not, not usually not an algebraic variety, and so we cannot, this subject is not contained in the usual, never-in-a-theory, we cannot directly apply the construction of, of high functions, neither, neither, neither, never-in-a-theory to, to, to this subject, yeah. And so, now, I, I, so, now I, I define a first, ah, first, ah, so, that's our first section, I define, ah, ah, XC, yeah, XC, yeah. So, we can, we can consider the next, the next, ah, construction, but today, I hope to make the, the story simpler, ah, I, I will consider the here, here, here, here, here, here, here, here, and then we will consider the here, here, that is, the, ah, we will, we will look at the here, here, here, here, here, here, so our, we fix the, ah, So now we have that filmmakers we rewife with this concept presentations for you to detail. So this presentation is a presentation for you to really see this is. this is a origin of. and and h i 0 almost all right and we also fix the h 0 h 0 q h 0 q is a is maybe h 0 j i i i i i i i h 0 q is a q depth of space of space of dimension h i the sum of h i and and fix the fix the h 0 q times h 0 q to q times 2 pi i to the minus w minus w this is a non-designate pairing a non-designate pairing a per linear home and and symmetric just an consig universeirect order with i l i l l l l X is written in the form of d plus minus times g g a u q divided by t. Here, this is a British domain, the period domain. But we take a little bigger one. This is the filtration, increasing the filtration zone on each c, the tensor product of c and h, q. The filtration is stored in a single sub-space. Such that the field divided by y is minus plus minus. This is decreasing filtration and the difference is that the dimension is h i. Then this is the polarization. That is, this is the filtration. Such that h i. Q is the filtration of this bearing. It is a polarized structure of weight. And h i. d plus minus is the same thing that here. Here we take this zero or minus of that. You replace this. This is replaced by. Then you have two plus minus. And g is the algebraic group of q. And g is the algebraic group of q. Then you have reaction and then k. And fq is the number of media component of the other thing. And then we have. And then we have. And then we have. M is the motion of the weight of weight w. And such that the dimension of the field divided by y is minus plus minus h i. And then other conditions are that the polarization are. And then. And lambda is the k plus of a f q tensor h 0 q to the total polarization. Here the k can act here. And then we take the mojirou k plus. And this should be also. And then there should be some conditions that there is some polarization. And also some on n. And some as morphism. This is basically. I forgot to say that f is a bit. Yeah. We have it. And this is a bit of realization. So. So that we have a morphism between this and h 0 q. And also this is the same also. Also some. Yeah. I'm perfectly precise. I wrote a preprint. It is. Actually that is all these things. But it is. Present. I have to collect many mistakes and the special version and also this particular version and part two. Will be. Maybe after two months or two weeks. So. We should have. Impressed that. I don't talk about this. Yeah. Yeah. And. Part three. This is a variation of the structure. And then. You have a variation. And then. And then. This is. So. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. 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here, subject here is, is, is to study, is a in a theory. So I mean that, input is a, number 2, 2, 4, 5, 6. There we go. And the XS is C. This is the projective smooth curve of the C. for Y to XC's. And concerning the period domain, structure with log generation, and the Y-hole is with the Y-hole, the Y-hole, the horizontal map. And then, so the degeneration in Satoshi, you a, to, a, a, a, a, a, a, a, a, a, and then, so or, or, how often, how often, how often, how algebraic variety, and so we cannot, this subject high functions, neither, neither, neither, You replace this. This is replaced by. Then you