So first of all, my apologies to Takeshi Saito and all the people in Japan because the original plan was to give this talk in June in Japan. But, well, at least I can give it now in Paris. And also my apologies to the people on the product board because I will give the same talk as I gave there.

So, I want to talk about Timura varieties with internet level and join in the cohomology of locus metric spaces. And I want to first start with these Timura varieties. In fact, I probably restrict myself to discussing this in the case of the modular curve, which is already interesting in itself. So, let me recall what the classical theory of the complex numbers.

So, what we see is we get the modular curve by looking at the upper half plane. So, a set of complex numbers of positive and imaginary part. And then you fix the congruent subgroup gamma in SL2 of the integers.

And so there's X on the upper half plane. And then you get the modular curve at level gamma quotient of H minus omega subgroup. And so this parameterizes some elliptic curves for the structure. And so the picture I want to, there's a picture there I want to get also over the periodic numbers.

And so the picture is the following.

So, you have H, which is roughly, well not quite exactly, but roughly the inverse limit over the modular curves at all levels. And so, just by definition, this Poincare upper half plane embeds into the projective space over the complex numbers.

And, in fact, you can give a modular interpretation to this. So, you can consider H also, in terms of this modular description, as parameterizing pair E alpha, where E is the elliptic curve over C. And alpha is an isomorph, it's a trivialization of the first singular homology of E.

And, in terms of this modular interpretation, what is the map? So, you take a pair E alpha. So, orientation preserving the realization, if you want. Yes, sorry, orientation preserving.

Well, in fact, I mean, later on I was also confused. I mean, this is the modular curve, but usually you take several connected components and that will confuse the two points of view later on. So, anyway.

And what you do is you send this to the Hodge filtration. So, what's the Hodge filtration? So, via the map alpha, you can identify C2 with the complex homology of E,

via some of the exponential map. Anyway, this ejects onto the algebra of E, which is the Hodge filtration. And so, this gives the one-imaginable quotient of 6, 2, and n vector space, and thus an unlimited U1.

And so, we want a similar picture over Cp. And so, clearly we need some Hodge filtrations talking about this.

So, thus I want to recall a little bit about Hodge filtrations over Cp.

Or, in fact, I mean, we can replace Cp by any algebraically closed extension of Qp. So, that's C over Qp, the algebraically closed in Qp. And let's take any proper smooth variety over C.

And, well, then there's this classical theorem of Fodge. Probably, if I put only proper here, then maybe I also need some additional reduction arguments, maybe two by three.

So, there is a Fodge-Durham Spectral Sequence that I need to generalize. So, the Durham Spectral Sequence pushes up too high.

So, it starts from the one page, and at the one page you have Fodge cohomology. And this converges to the Durham cohomology.

And so, the theorem is that this degenerates at the first page already. And so, this means that, so of course this is proved by the left chest principle, choosing an isomorphism maybe between C and the complex numbers,

and using complex analysis there. So, it's degenerates that you wanted, so thus we get a Fodge-Durham situation on the Durham cohomology,

whose successive quotients are Fodge cohomology. But, if you look back at what we wanted to do here, so we wanted to have a filtration on a singular cohomology. And of course, over C that's not a problem,

because the singular cohomology, if you change it over C, is going to be isomorphic to the Durham cohomology. So, the Durham filtration is good, but over C pieces this is no longer true. So, if you take the entire cohomology of X,

to keep the coefficients there, and it's up to C, it's not canonically at least isomorphic to the Durham cohomology. And so we can't use this to produce a filtration on a cohomology.

And then, of course, if you don't get a cohomology, then you wouldn't even expect such a statement to be true, but you would rather expect that such a statement is true after your tensor is either wrong, but not even then it's true,

as my base field C doesn't embed into the contents field either wrong. So, if this guy was defined over some discrete value of the extension, maybe this perfect residue field, then such a statement would be true, but if my writing starts live only over an algebraically closed field,

that's no longer true. So, we need some other large filtration to make sense of this, and fortunately there is such a filtration. So, there is another spectral sequence which looks very similar,

but has some interesting twists in it. So, let's just start to take the spectral sequence. Which again starts from Hodge cohomology, but this time on the YouTube page. And I'm not making any typos here, I'm really being careful.

So, it interchanges i and j, and you have to put a tape twist. And converges again, well not again, but this time it converges to a cohomology.

So, as I'm only working over C, it probably doesn't make much sense to put a tape twist here, but what is meant to say in this way is that it's canonical if my variety is defined over some smaller fields. It converges to H i plus j.

Oh, sorry, H i plus j, sure. It's defined over smaller fields that with these tape twists, everything will be Galois-covariant. And this also degenerates already, it's the first page, which is just 22.

And thus, you also get a filtration on a talk called homology tensor with C whose successive functions are again Hodge cohomology. This filtration I call the Hodge state filtration.

It is the one which gives the Hodge state decomposition. So, assuming your variety is already defined over a discretely-valued extension of QP with perfect residue fields, then it follows from this Galois covariance that it has to degenerate because there can be no differentials between the Galois-covariant differentials in here.

And also there are no extensions and no maps between those, so this implies that actually this filtration is canonically split by using the Galois action. But that's only true if it's defined over discretely-valued extension. And so, in general, this is really just a filtration,

it's not canonically split.

So let me give an example for this, what this looks like.

So, in the case of interest, so that we can take an elliptic curve over a periodic field now, then, of course, we have the, we can look at the first-round homology, say, of E, which you can define, say, as the algebra of the universal vector extension or also as the dual of the ground homology.

And so this is a filtration with a quotient, it's not the duality one. And for elliptic curves, maybe I wouldn't have to talk about the dual elliptic curve first. Well, in general, if you have put in a real variety here, then you would have to use a dual real variety here.

So you take the dual elliptic curve, take the three algebra, and then the dual vector space. And then there is another filtration where you take the tape module of E, and this is up to C. And so now the two terms are interchanged.

This term is a co-kernel, and the new algebra appears as a sub. And in fact, to make it canonical, you have to put a tape list by one here.

And so again, this is defined with some small fields, and this implies that this is not a tape representation, and it's in fact canonical to the sum of the two terms, but not in general. And over C, those two terms would be isomorphic, and also over C, this is also canonically split.

Let's get back to the modular curve now. So we want to define a map which,

So a parallel truck which is just given by associating an elliptic curve to this filtration here, and to do so, we of course have to trivialize the K-Tape module. But note that this time you only have to trivialize the K-Tape module, and not the whole integral homology. So you don't have to do anything that's primes away from P.

And so, more taking. Let this be the elliptic space, that's just a fancy version of a rigid analytic variety,

over C, associated with my modular curve. Well, up here I can find this as something over C, but say,

well I could argue that it's defined over a cube, if I put some additional components, or I could also simply choose an isomorphism between C and QP bar, if that's what you see.

And, so we want to trivialize the K-Tape module at P, so we have to take an inverse limit of all the levels at P. So, let me choose the following notation, so that gamma P to the n inside gamma be the principal congruent subgroups.

So it would probably be more standard to write gamma cap gamma P to the n, to keep notation simple. Let's do it this way. No, again, I have to pay attention to the numbers.

Not too high. OK. OK, and then the theorem is that this picture, which we had over complex numbers, doesn't make sense over the periodic numbers. So, for this of course, we first have to make sense,

give some sense to this inverse limit of all levels at P. And, we have to do some rigid analysis geometry, and so this means we have to take a lot of completions, and of course completions are not generally well behaved, if you're leaving the mysterious setup. But, if you want to take this inverse limit, we really have to leave the mysterious setup.

So, we might run into a lot of technical problems. Well, actually we don't, so... Maybe there's some special kind of non-hysteria spaces, which are well behaved, and these also have these perfectoid spaces.

We'll see, such that... Well, I would like to say it is the inverse limit of these spaces, but as it turns out, the category I'm working in doesn't admit inverse limits,

so one has to use a slightly stronger notion, which I call being similar to the inverse limit.

So, what does it mean being similar to? So, these added spaces, they're actually locally arranged spaces, equipped with some variations, so you don't have to talk... So, they really have an analytic topological space, and not just some growth-neak topos. And, this condition means, first of all, that you're getting homeomorphism on analytic topological spaces.

That's a condition which says, roughly, that if you take the direct limit of the structure sheets at finite level, match this to the structure sheet at infinite level, then this has dense image.

The problem is somehow that there is no canonical topology you can put on this direct limit. So, there would be the direct limit topology, but it's not of the good kind.

So, you take the topology, the bound topology, which makes the open unit topology set of far-wanted elements in here. Well, anyway, it's a technical issue. But, the second part now says that on this infinite level guy, this log-state period map is defined.

So, let's just take the period map on my log-state period at infinite level,

going to a space associated with this. And, it has a bunch of good properties. So, it's defined in some sense,

meaning that there is a covering by this p1 by a finite subspaces whose pre-images are finite perfectoid subspaces of this infinite level guy. And, it also commutes with heck operators. And so, what does this mean? So, this is defined over a periodic field, this p1.

So, gogqp certainly acts on this guy. And, of course, if you write this here, as we know at infinite level, we have an honest group action of this group on this space, not just some heck of correspondences. And so, it's gogqp-equivariant.

Also, it's equivariant for the heck operators away from p. What does this mean? So, we have some non-trivial heck operators here. But, it's not clear how they should act here. But, in fact, they just don't act.

Okay, what else? So, there's a natural ample line bundle given by the dual of the real algebra of the universal elliptic curve

on this modular space here. Then, there is, of course, a natural ample line bundle on p1. Or, of one. In fact, they're compatible. So, the natural line bundle omega on this modular space is just a pullback.

And, in general, there's such kind of a statement for automorphic vector bundles. So, in general, for some sumo variety, you will get some flag variety. And, on this flag variety, you will have naturally defined some automorphic vector bundles.

And, if you pull them back, you get some automorphic vector bundles on your sumo variety. Alright, that's, after all, how you define automorphic vector bundles in the complex setup. The same is looking here. And, also, it's true that this map extends to the minimal compactification.

The power of minimal compactification is also good. It keeps it perfect. Yeah, so, if I could put, and if I denote by star the minimal compactification,

then, uh, I mean, this theorem is also true. I should put star here.

And, actually, the moment I say this, I realize that to make this f-file, I actually have to talk about the minimal compactification. Because, otherwise, I remove some constants. I should put some minimal compactification here. Of course, I should say what's in that.

So, if you take some guide-infinite level, which is essentially an elliptic curve plus a trivialization of the periodic tape module and maybe some additional structure away from it that I don't care about.

Then, this maps this to well, this goes back in there. Sitting inside 2p over 3 times e to the c which is why I have this isomorphism alpha the same as c squared.

But, note that there are some interesting twists happening here namely over the complex numbers the real algebra appeared as a quotient whereas now it appears as a sub-module from 0 to 1.

I want to make a cover of remarks if that's it. The first thing I want to say is the reinterpretation of this p1 that appears on the left on the right-hand side and also what that actually does.

So, for this I have to recall a theorem that appears in joint work with period Weinstein and says the following. Again, I have my algebraic closed and complete extended in c

with o, c, and c in the ring of integers. And then I can look at p equals over the groups over the string of integers and it turns out that

the sort of state filtration that we find depends only on the periodic and so in some sense you would expect that it only depends on the associated periods of a group and so this turns out to be true so there is also a state filtration defined for p equals over the groups. And that actually classifies those. So, the category is equivalent

to the category of pairs lambda w say where lambda is a tape module so it's a finite tree

and so the snap sends the p to group g to the periodic tape module and so on. So this is some kind of periodic analog of

Riemann's specification of a b in varieties and the answer was a complex numbers and of course it's also used in a similar way in here. And I want to make some remarks that if I have a b in variety over o, c

then the large state filtration of a

is the same as the large state filtration of c, p equals over the group which has an independent definition which is in fact much simpler so there is a very simple definition of large state filtration of a p equals over the group for the first appeared in the work of Faltings

I'm exactly sure and that was a used lot by far and so that's where I learned this. So, what does this mean? This means that we can regards, so after trivializing this periodic tape module

giving the large state filtration is the same thing as giving the p equals over the group and so this means that in geometric points at least on the locus of good production so by the way that's just a b in variety not over c

but really over the being of integers the large state period map is a map taking the curve to its associated p equals over the group instead of taking some p equals over c

remembering the trivialization of the tape module unfortunately there is no generalization of this material at Weinstein which works for other rings so not doesn't work in a relative set it doesn't even work for smaller fields

so I mean only on geometric points on the locus of good production there is this very direct description of what this map does and in general somehow slightly different but

in fact from this description of geometric points you can actually use the covariance properties and you can also do it differently

because it depends only on the p equals over group somehow it doesn't matter if you use some hack operator away from p so the next thing I want to explain

is how this map looks like explicitly

you have your infinite level modular curve and

I mean on the special fiber you have a stratification into the ordinary locus and the super singular locus and say if I would compactify I would put all the cusps also into the ordinary locus and just by putting this back in the specialization map you get a similar stratification on the generic fiber

so this is actually stratified into an ordinary part and super singular part and so this maps to the p1

and there is also a classical stratification studied on the p1 as an edX space namely you have the rational points inside there and you have the complement which is usually called 3-infless upper half space and these compositions

turn out to correspond so I can put these arrows here meaning that the ordinary locus

is exactly the preimage of p1 of QP and the super singular locus preimage of omega2 and so what I want to do is define and explain what these maps are on this strata and they turn out to be very different so already from this diagram you see that the ordinary locus is essentially contracted to points

and something non-trivial geometrically only happens on the super singular locus so in particular this map is not at all injective as was the case of the complex numbers it really contracts a lot of stuff so what does

k2 and map do on the ordinary locus? on there it just measures the position of the canonical subgroup so because in the canonical case

you have the canonical subgroup and in the ordinary case you have the canonical subgroup which gives you a canonical one-dimensional sub of your irreducible group and if you pass the tape modules to get a canonical one-dimensional sub of your gated tape module and that's after you turn over C this gives exactly the linear algebra there

and that's what happens there but on the super singular locus it's more involved so what is the super singular locus? so that's somehow at level 0 you just have a union of

open disks for all super singular points and then as you pass up all the levels then what happens at all these disks is that you get these Leibniz-Tate towers and so at infinite level what you get is just a finite disjoint union of Leibniz-Tate spaces at infinite level

so in joint work with Jared Weinstein in the same work we also proved that this is actually a perfectoid space

by purely local arguments and Jared Weinstein can even write down explicit descriptions for what the space is which is pretty amazing because at finite levels you don't have explicit descriptions but some of this infinite level guy is in some way simpler than all the finite level guys individually

I mean Jared Weinstein is also studying these explicit aspeniate subsets and they're trying to show that they realize local language and for this it's very useful for him to work at infinite level because at finite level you don't have this explicit description

okay now there's this strange isomorphism between the two towers between the Leibniz-Tate and the Wienfeld tower which is due to faultings and then developed in more detail and it's a precise setup that we need I mean as an isomorphism of perfectoid spaces

in this paper with Jared Weinstein that this is the same as the Wienfeld space at infinite level but Wienfeld space just by its definition so there's a tower

of finite detail covers of Wienfeld space so by its very definition it will map down to Wienfeld's upper half plane and so that's actually what the logistic period map does

so the strange isomorphism between the two towers is somewhere built into this logistic period map

and it feels a little strange that there is actually some map of edX spaces which realizes this because if you think that you start somewhere in the ordinary locus and then you run into the supersingular locus then for a long time you will stay constant and then at some point you start to walk only but

so that's not kind of behavior you know from finite type geometry but in this infinite I mean in this highly non-inferior situation that's actually possible and maybe

I'm only at three everything I said here for the modular curve also works for Shimura writes a fodge type I'm very lucky that I can just write

Shimura writes a fodge type without having to think much because as it turns out he first proves this for the Ziegler modular space of principally polarized linear varieties

where it's quite a bit of work but then any Shimura variety of fodge type is a closed sub-variety there and then you can by formal arguments deduce everything you want so that's very nice okay and now I want to talk about some applications of this

and so the crucial theorem is the following

it says that you can always find congruences between Eisenstein series and custom forms in some sense and so the setup is fine so let me take some Q bar Shimura variety which is a fodge type

and then the theorem is that any system of Hecke-Eigenvalues which you can find in the compact to support

cohomology of Shimura variety with a torsion coefficient let's say I work with 4xFP can be lifted to a system of Hecke-Eigenvalues of a classical constant so that's doing two things

so it's saying that any torsion class can be lifted to characteristic 0 which is a very interesting result and it also says that anything which comes from the boundary can also any Hecke-Eigenvalues which come from something from the boundary can also be realized as

coming from something which is a cast form so even if some of this classically this associated Eisenstein series does not vanish more p as a cast you can still find these congruences and maybe I should say that in order to get this classically cast form

I may have to increase the level of p yeah I write something like that I only consider

at this point I still consider the whole Hecke algebra away from p to talk about Hecke-Eigenvalues maybe I want to be commuted ok

and because by the work of many many people there are now very strong theorems asserting the existence of Galois representations for cast forms one can use one can then reduce these Galois representations again and get Galois representations for something which appears in this

torsion call module here and so from this you get from the first corollary is the existence of Galois representations for torsion classes so let me fix some totally real LCM field and let

xk is a locally symmetric space for the general linear group of any dimension over this field f and so except in very very few cases this will not be a shimua variety

so if f is q and n is 2 then this is a modular curve but essentially outside this case it's never a shimua variety so if f is totally real it's very closely related to someone so already if f is q and n is 3 this is just some real manifold or if f is totally real, totally imaginary

and n is 2 then this is some 3-dimensional hyperbolic manifold which has no algebraic structure at all but still the Battico homology of this real manifold knows about Galois representations so

homology of the existing

Galois representation rho rho and continues

to ln fq bar associated with that meaning that in particular the traits of Rubinius elements will be the eigenvalues of some Hecker operators so it's an old idea of

Plouzelles that you should try to realize these locally symmetric spaces here as boundary components in the Borel-Sar compactification of some unitary or symplectic shimua varieties so you have this unitary or symplectic

shimua variety which has a compactification as a real manifold with corners this Borel-Sar compactification and in there these locally symmetric spaces will appear and just like homology will contribute to the homology of this space as a boundary

contribution and then you lift this which is some upper Eisenstein contribution to something which is actually a cusp form and then for this cusp form you know how to attach power representations to it and then well here we go

well there's a long exact sequence where two terms are the usual homology and the compact support homology of the space and this other theorem is essentially just gone let's ignore the other strata you can do this by induction and so this means that the eigenvalues will

appear either the usual homology or the compact support homology and but by homoradiality it's somewhat enough to make and shift them from one to the other I mean you have to do a lot of representation in the end

so you know that the eigenvalues will the system of eigenvalues will appear either in the homology or in the compact support homology or in the shimua variety but if it appears in the usual homology only then you can use homoradiality to get something the compact support homology again but for this exactly we're kind of sure that the boundary

is like a hyper-surface no no well the there's also a version of the theorem where

I take ZMOP to the M-coefficient and then in the inverse limit you can get some results about characteristic zero and so this way you get the second corollary which says that

for any so-called regular algebraic caspo caspo-automorphic representation pi of gln over f

f as above and any isomorphism between C and QL bar or QP bar there exists a continuous well actually it's almost everywhere on a Remi file

associated with this and so I should say that this was proved

before I did by Harris-Lamptey and so on

and so the point is that you can realize these regular algebraic representations in the cohomology of these locally symmetric spaces with some coefficient systems

it's conjecture that you can do without the irregular but then you have no idea where to find these representations in which cohomology groups are the simplest case would be the case of mass forms of eigenvalue the quarter for the Laplace Laussian and the case where the ground field is Q at n is equal to 2 so one has no idea how to attach

representations to them and so in the last few minutes let me try to explain the proof of this theorem here

so let me use the same notation for the associated eddic space or I'll probably also choose in some such isomorphism here and let's listen to C

the first step is to use some Pierre Descartes theory to rewrite this etacromology group there but obviously we're interested in some torsion groups so we need some kind of integral Pierre Descartes theory isomorphism and usually those only work if your variety is good or same as several reduction or something like that but as I want to

increase the level at P indefinitely I can't expect that I can't get through with such statements because it's very hard to find same as several models for the Shimura varieties and so one needs a different kind of comparison result whereas the thing you compare to is a priori is still something mysterious

it's a following statement, it's a comparison of torsion coefficients which appears in my work on Pierre Descartes theory but is mirrored on a result of faulting so it says that

if I look at the etacromology group torsion coefficients

so it doesn't matter whether I compute it on the etac space or on the variety, it's the same a general comparison result and if it tenses us up to oc mod P so you get a map to the cohomology

well actually I have to go to the minimal compactification here and that takes the cohomology of the following string sheave sheave i plus mod P I'll explain it in a second and in fact this is not an isomorphism but it's almost an isomorphism in the sense that faulting is almost mathematics

so in all these proofs there will be a lot of statements which are only almost true but it won't matter in the end so what is i plus? i plus is the intersection of i and o plus inside the structure sheave

where this is the functions which vanish at the boundary of cost forms and this is the sheave of functions bounded by one

and so this kind of cohomology group which appears here has several strange properties so it's computed on the characteristic zero space but it's the sheave is a characteristic P-sheave still and it's still extremely important that you compute it on the etaal side of the space

so this makes it upper-order extremely hard to understand well

that's the first step and now I pass to

infinite level at P so the direct limit of all levels at P of this group coefficients is an almost isomorphism

too etaal cohomology now computed at my infinite levels for more variety with the same level kP same sheave

and so now the task is to understand this group here and so remember that in the end we wanted to go to classical cost forms

and we're already one step there because we already have the sheave of cost forms which appears here so the second step is to get rid of the etaal side here so it says that you can actually also compute this now now that you're at infinite level

it doesn't matter anymore so this is almost the same as and so this left-hand side can actually be computed

by a shear complex with respect to some affinity cover and so

what this relies on is some version of the almost purity theorem and in fact some of the almost purity theorem our priority only does something for the finite etaal covers and we really want to consider the whole etaal side here so one needs some slight

refinement of it and prove it with the same methods and it says that if I have some perfectoid-affinite algebra G-algebra say and if I look at the associated perfectoid space then you can compute

the etaal cohomology on X of this O plus sheaf and then there's a similar version for this I plus sheaf and it's actually what you think it should be so at least almost

so it's R plus in degree zero and zero in positive degrees so that's some kind of version of Tate's aseclicity theorem for the etaal side but Tate's theorem would only apply if I don't put the plus here so if I invert P but I really want the version without inverting P

and in classical rigid analytical geometry this statement with the plus here is absolutely not true so there's a whole lot of torsion in these groups in general in fact unbounded torsion if you don't put some at least normality hypothesis but if you pass

to this infinite level then all this torsion will go away okay and so this means that so what are the terms in this complex here so there are the

sections on U I plus in fact I can model P afterwards but this results there I use some finite P's and so these are cost forms of infinite level

all defined on a finite subset and remember that in the end we wanted to get some Hecke eigenvalues in a classical cost form so what's left to do is approximate

these cost forms of infinite level which are only defined on some of the finite subsets by globally defined cost forms which are of finite level without messing up the Hecke eigenvalues

classical meaning in particular of finite level one extra minute

So the step 2 I use that the space is perfect, while in the step 3 I will use the logitype period map. So what are these usual techniques, going back to Katz's paper on periodic modular forms,

how to make this procedure. So usually U would be the ordinary locus, and then the solution is to multiply by the Hasse invariant.

So what properties of the Hasse invariant do you need for this? So you need that the vanishing locus is exactly the supersingular locus, so you can remove all poles by multiplying by a high enough power of the Hasse invariant. And the other property you need is that this commutes with all Hecke operators away from P,

so that it doesn't mess up the Hecke invariant values. So we need some analog of the Hasse invariant, and where do we get it from?

So the solution is to pull back 4 of 1, say.

One wire is a logitype period map to get some kind of fake Hasse invariant. So because this logitype period map commutes with all the Hecke operators away from P, any function that you pull back from there will automatically commute with this Hecke operator,

so this gives you this property. And then you need to see that there are enough of these fake Hasse invariants in some sense. And this exactly comes down to the fact that this map is affine, so you can actually choose an affinity cover of your,

well, it's not P1 in general, but in the P1 sense, it's a modular curve case, pull it back, so you get an affinity cover there, and that's the affinity cover with respect to which you compute the stretch complex there. So it's a weight one? It's a section of a vehicle?

It is a section, yeah, probably. So for which affinity cover, you know that the pullback is out of the... Well, for example, for the standard cover of P1 by 2 closed bolts there. Well, the point is that this property, that the pullback is affine, it's stable under a quasi-generational subset, and it's stable under the action of the gl2qp.

Yes. And so whenever you have some open subset, so there is an open subset which is true, which contains a rational point, and then you can make the subset very large under the gl2qp action,

and then basically any subset will be rational of all of those, and then you get a form of all open subsets. So whenever you give me some explicit guy, I can somehow verify that for this subset it's okay. But you don't know it for all rational domains in P1? I guess for all rational domains it should be okay. For all rational domains it should be okay, let's say.

Ah, because you can put it inside... I can put it inside something where it's rational and then... I mean, because I have very big subsets for which it is true using the gl2qp action, and then it passes to rational subsets, I can get it for all rational domains, yeah. Ah, in all affinity domains are rational, okay.

Ah, are they? In P1? Well, in P1 maybe, yeah. Yeah, I think that it was classified wrong, but anyway. Then if P1 is the paper one. But I mean, I only need to produce one affinity cover, if that's okay. Okay, let me stop now.

Start with questions from Tokyo, then Beijing, and then Paris.

Thank you very much. Are there some questions for you? So, then you can tell us briefly how you define this project, will you not? First I show that this whole shape spectra sequence exists in families in some sense,

or at least in filtration exists in families. So, let's say I can define it on the open Chimo variety. And then I have to prove that it extends to the memory compactification. And for this I use some version of Riemann's, I don't know how to say it in English, in Riemann-Siebakheitsatz, which says that bounded function,

so that you have some normal complex and logic space, and you have some things that are close inside there, and you have a function which is bounded on the complement of the closed subset, then it will extend through all the variety. And I proved some analog of this in the setting of project-wise spaces to show that this whole shape period map automatically has to extend.

Because in the period neighborhoods of the normal complex configuration, I can show that the image will be bounded, and then it makes no difference. So, you told us in the first step of the distribution,

that you think it's related to the fighting style of the Muslim. So, can you come up with your construction, just the fighting style of the Muslim, from your construction? I think I didn't understand the question.

Ah, sorry. So, you mentioned this fighting style of the Muslim. Yes. So, my question is that you recover from your construction, his isomorphism? It's a slightly different isomorphism. So, fighting takes us to form a model,

and then computes us from one to the other to form a model. Whereas I computed on the original generic fiber, which is something that is actually related to you. My theory of applies is because, as long as there is a specialization there, from the total of the generic fiber to the total of the formal scheme,

and I can compute what happens when I push forward on this map, so essentially taking mentioning cycles. And if you do this, then you recover from this isomorphism, I think. You need that the model is nice. Yes, you need that the model is nice, but... Which isomorphism is it also?

It's just... This isomorphism between the total of all utensils of C mod P and C mod G of this O plus mod P C. Ah, yes, but in fact, we found things, it doesn't consider the rigid data, it just works with... Yes, it just works on the form scheme, that's what I said. Okay, but one can show using your perfect activity, it's equivalent to...

If the form model is sufficiently nice, so... The computation of vanishing cycles is easy? If the form model is sufficiently nice, yes. Okay, so, other questions from Tokyo? Okay, can I ask a question? Yeah.

So, can you prove that some toroidal compactions of higher dimensional similar varieties is perfect? I saw a little bit about this. In some sense. Well, I certainly expect that it's true, but for the moment I can't prove it, I think. So the problem is the following.

So, how do I prove it for the minimal compactification? For the minimal compactification, I first prove it on some specific subset called the Chibula neighborhood, a strict neighborhood of the anti-canonical tower. So, that part goes through. But then, for the minimal compactification, I have all of G02QP acting on them.

And so I can move the subset to cover all of the minimal compactification. But if I wanted to do this argument for the toroidal compactification, then it can, because on the toroidal compactification you don't have all the Hecke operators acting.

So it depends on the choice of this cone decomposition and so on. So the Hecke operators are not acting on it, so this argument of spreading it around doesn't work a priori. Yeah, you have to make some other argument. So you use Hecke operators or you use G02QP?

Sorry, I use G02QP. So I show that the subset of which I can control everything explicitly, it's G02QP translates for covers of all space. And similarly for zero spaces? Yeah, similarly for zero spaces. I mean, yeah, the Hecke space I use is synthetic.

So maybe there is some other questions from Tokyo or then from Beijing? Is that part of the question? Maybe not for Tokyo. Thank you. So from Beijing? Okay, so do you have any questions?

No. Okay, then I have one question. So in your camaraderie for regular algebraic class representations, you constructed a Galois representation. And do you know anything on the local behavior of this Galois representation?

For instance, do you know the local compatibility away from p and the ground is at p, for instance? Well, I mean, there is a student, Richard Taylor, who was working on these questions, and I guess I won't, I mean, and the student answers these questions to a large extent.

So there is a potential. Okay, there is a potential. Yeah, yeah. Okay. Well, then, okay, there are other questions. Okay, thank you. So are there any questions?

Yeah. About the corollary too, I mean, there is something mysterious because you don't have any Shimura variety. Well, as I said, you write this as a- There's no such type, I mean. No, no, but I mean, for the fp, I understand, but now you get something of a-

Well, I mean, and you can write this as the cohomology of some local system of this xk, and by passing to large-level p, you can also forget about its local system. And so you essentially just need a version of corollary one, which is not only with fp bar coefficients, but with z mod p to the m coefficients. Because in the inverse limit, you will recover what you want.

And you get the good ek eigenvalue. I mean, the Frobenius eigenvalues are the same as the ek. Yes. So at all good times, yeah. So you go over z, over p and z, and you do this? Yes. Okay. So I use the corollary one for all z mod p to the mz's, and then go to the inverse limit.

Other questions? If not, let's thank the speaker again.