 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. But, well, at least I can give it now in Paris. And also my apologies to the people in Kota Boru because I would give the same talk as I did there. So I want to talk about Shimura varieties with internet level and toy in the form of looking at these metric spaces. And I want to first start with these Shimura varieties. And in fact, I'll probably restrict myself to discussing this in the case of the modular curve. It's already interesting in itself. So I'm going to talk about the classical theory of the complex numbers. So we'll see if we can get the modular curve by looking at the upper half plane. So the set of complex numbers of positive and imaginary card. And then you fix the Hungary subgroup gamma in SL2 of the introduced. And so this acts on the upper half plane. And then you get the modular curve at level gamma, quotient of h times n of the 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 particular upper half plane embeds into the projected space over the complex numbers. In fact, you can give a modular interpretation to this. So you can consider h also in terms of modular description as parameterizing pair e alpha where e is the elliptic curve of a c. And alpha is the trivialization of the first singular homology of e. And in terms of some modular interpretation... So you take a pair e alpha... So orientation preserving the variation, if you want. Yes, sorry, orientation preserving. Well, in fact, I mean later on I will also confuse... I mean I said 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 sense this to the virtual iteration. So what's the virtual iteration? Why is it that alpha you can identify c2 with the complex homology of e? Why some of the exponential map... Anyways, it's subject to the algebra of e, which is a virtual iteration. And so this gives a one-dimensional equation of 6 to the right of vector space and doesn't have a limit of u1. And so we want a similar picture over cp. And so clearly we have to need some virtual iterations talking about this. So thus I want to recall a little bit about virtual iterations over cp. Or in fact, I mean we can replace cp by any algebraic equals extension of cp. So let's c over cp e over cp equals cp. And let's take any proper smooth variety over c. And well then there's this classical theorem of FORGE. Probably... If I put only proper here, maybe I also need some additional reduction arguments maybe future would be. That there is such a wrong spectral sequence I need to generalize. So the wrong spectral sequence pushes up to high. So it starts from the e1 page and it's e1 page to the FORGE coordinate. And it converges to the wrong coordinate. And so the theorem is that this degenerates at the first page already. And so this means that... So of course it's proved by the left chest principle choosing an isomorphism maybe between c and the complex numbers using complex analysis there. So this degenerates at e1 so thus we get a hodged-rump filtration on the wrong homology. With successive quotients of hodged-rump homology. But if you look back at what we wanted to do here so we wanted to have a filtration on the singular homology. So of course over c that's not a problem because the singular homology as you can see is connolly to the wrong homology. So the wrong filtration is good. But over c pieces is no longer true. So if you take the entire homology at x to keep the coefficient say. Then it's up to c. So it's not canonically a piece as a morphic to the wrong homology. And so we can't use this to produce a filtration on the wrong homology. And then of course if you look at equities and you wouldn't even expect such a statement to be true but you would rather expect that such a statement is true after you change it with 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 discreetly valued extension, maybe this perfect residue field then such a statement would be true. But if my writing starts live only over another very close field that's no longer true. So we need some other homology filtration to make sense of this. And fortunately it is such a filtration. So there is another spectral sequence which looks very similar but has some interesting twists in it. Let's just start to take spectral sequence. Which again starts from Bosch homology 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 take twist. And converges again, well not again, but this time it converges to a powerful homology. So as I'm only working over c it probably doesn't make much sense to put a take twist here but whereas Mantis said in this way it's canonical and my writing is defined over some smaller fields. It converges to h i plus j. Sorry h i plus j is correct. If it's defined over smaller fields that with these take twists everything will be galawike invariant. And this also degenerates already. That's the first take which is this 22. And thus you also get a filtration on a taco homology term that we see these successive functions are again hosh homology. This filtration I call the hosh take motion. It gives the hosh take the composition. It is the one which gives the hosh take the composition. So assuming your variety is already defined over an extremely weighted extension of qp with perfect residue field, then it follows from the galawike invariance that it has to degenerate because there can be no differentials between the galawike and 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 galawike action. But that's only true if it's defined over a discrete value of the extension. And so in general this is really just a filtration and it's not canonically split. So let me give you an example for this. What does this look like? So in the case of interest so that you can take an elliptic curve up here in the field now, then of course we have the you can look at the first around homology say of E which you can define say it's a duality of the universal vector extension or also it's a duality around homology. And so this is a filtration with a quotient. It's not a duality. But for elliptic curves maybe I wouldn't have to talk about the duality curve first. Well in general it's kind of if you have put in the real variety here then you would have to use the duality of the real variety here. And if you take the duality curve take it three times the dual vector space. And then there's another filtration where you take the take module of E and this is up to C. And so now the two terms I need to change. You know, this term is a core kernel and the real algebra appears as a sub and in fact to make it canonical you have to put a T for a square one here. And so again if this is defined with some small field then this implies that this is our state representation and this is the state canonicality of 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 canonicality split. Let's get back to the modular curve now. So we want to define a map which is just given by associating an elliptic curve to this filtration here. And to do so we of course have to trivialize the take module. But note that this time you only have to trivialize the take module and not the whole integral homology. So you don't have to do anything that's primes away from P. And so what do you do? Is this the edict space that just depends on a version of a rigid analytic variety? Over C associated with my modular curve well upper I would find it as something over C but say well I could argue that it's defined over a Q if I put some additional components or I could also simply choose an asomorphism between C and QP bar that is C. And so we want to trivialize the take module at P so we have to take an inverse limit over all the levels at P. So let me choose the following notation. So that gamma P to the N inside gamma be the principal comb of N sub-primes. So it would probably be more standard to write gamma cap, gamma P to the N but to keep notation simple. Let's do it this way. Again, I have to pay attention to the numbers. Too high? Okay. And then the theorem is that this picture which we had was a complex number so it doesn't make sense over the period of numbers. So for this of course we first have to make some sense this inverse limit over all the levels at P and we have to do some original 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 in a serious setup but if you want to take this inverse limit we really have to leave in a serious 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-serious spaces which are all behaved and these are also these perfect word spaces. As you can 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. What does it mean being similar to? So these edit spaces they're actually locally earned spaces equipped with some variations so you don't have to talk so they really have an unaligned topological space and not just some broken ectopos and this condition means first of all that you get an homomorphism on topological spaces that's a condition which says roughly that if you take the direct limit of the structure sheets at finite level that's just the structure sheet at infinite level that's a systems image. The problem with some of that there is no canonical topology you can put on the direct limit so there will be a direct limit topology of the good kind so you take the topology which is the bar topology which makes the open unit ball a set of 4.1 kilometers in here Well anyway, it's a technical issue but the second part now says that on this infinite level guy this hot state period limit is defined so it makes this state period limit on my little character infinite level going to a space and it has a bunch of good properties so it's affine in some sense meaning that there is a covering by this p1 by a fielded sub-spaces whose pre-images are fielded perfected sub-spaces of this infinite level guy and it also commutes with hack operators and so what does this mean? So this is defined over a purely fielded p1 so g of g2p certainly acts on the sky and of course if you write this here as we know at infinite level we have an honest group action of this group on the space not just some hacker correspondences and so it's g2pp-acquiriant and also it's acquiriant for hack operators away from p what does this mean? so we have some long-term hack 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's of course a natural ample line bundle on p1 of 1 in fact they're compatible so the natural line bundle of omega on this modular space is just a pullback and in general there's such kind of a statement for automatic vector bundles so in general for some dual variety you will get some map to some flag variety and on this flag variety you will have naturally defined some automatic vector bundles and if you pull them back you get some automatic vector bundles on your dual variety that's after all how you define automatic vector bundles in the complex setup and also it's true that this map extends to the minimum of compactification the cover of minimal compactification is also good gives a perfect yeah so if I could put if I denote by star the minimum of compactification then I mean this here is also sugar I should put star here and actually it's more when I say this I realize that to make this f fine I actually have to talk about the minimum of compactification because otherwise I remove some cast I should put some minimum of compactification here of course I should say what's meant is if you take some guide infinite level which is essentially an elliptic curve plus a trivialization of security take module and maybe some additional structure waveforms that I don't care about then this map is to all this coefficient here sitting inside which is which is why I have this item of an alpha the same as c squared I've noticed there's some interesting twists happening here namely over the complex numbers the algebra appear to the quotient where it now appears as a sub-module from p1 I want to make a cover from Mark's inventors the first thing I would say is the reinterpretation of this p1 that appears on the right hand side and also what the map actually does so for this I have to recall a theorem that appears in joint work with Jared Weinstein and says the following so again I have my algebraic ghost in complete extending c with oc and c the ring of integers and then I can look at people with other groups with the ring of integers and it turns out that the sort of state filtration that you find of course it depends only on the periodic component so in some sense you would expect that it only depends on the associated periods of the group and so this turns out to be true so there's also a state filtration defined for periods of the groups and that actually that actually classifies those the category is equivalent to the category of pairs lambda w is the same where lambda is the take model so it's a finite free and so this map sends a group G to its period take model so this is some kind of period gather log of P1 specification of the B of varieties over the complex numbers and of course it's also used in a similar way in here and I want to make a remark that if I have a new variety of O C then log state filtration of A is the same as log state filtration of course the period of the group which has an independent definition which is in fact much simpler so there's a very simple definition of log state filtration of the period of the group for the first period in the work of faultings I'm not exactly sure and that was a used log by far again so that's where I learned this so what does this mean this means that we can regard so after trivializing this period take model by giving this log state filtration the same thing as giving the period of the group and so this means that we have to find the perfect points at least on the locus of good reduction so by giving the B of variety not that we'll see but really over the period of integers the log state period map is a map taking the curve to its associated P of the group unfortunately there is no generalization of this mysterium mysterium at Weinstein which works for other rings so not doesn't work in a relative setup it doesn't even work for smaller fields only on geometric points on the locus of good reduction there's this very direct description of what this map does and in general it's almost slightly different but in fact from this description on geometric points you can actually use the covariance properties maybe you can also do a different view because it depends only on the P of the group somehow it doesn't matter if you use some hacker operator away from P so the next thing I want to explain is how this map looks like you have your infinite level modular curve and I mean on a 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 under the specialization map you get a similar stratification on the generic fiber so this actually stratified into an ordinary part and super singular part so this maps to the P1 and there's also a classical stratification studied on the P1 as an egg space namely you have the rational points inside there and you have the complement which is usually called reinforced upper half space and these compositions turn out to correspond so put these arrows here meaning that the ordinary locus is exactly the P image of P1 of QP and the super singular locus is P of omega 2 so what I want to do is define and explain what these maps are on the stratum 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 particularly this map is not at all injective as was the case over the complex numbers but it really contracts a lot of stuff for what does this much take period 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 P visible group and if you pass the take modules to get a canonical one-dimensional sub of your get a take module and that's after each other we'll see this gives exactly the new algebra there that's what happens there but on the super singular locus it's more involved so 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 is that all these disks you get these tributate towers and so at infinite level what you get is just a finite disjoint union of tributate spaces at the 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 as finite subsets and they're trying to show that they realize local length and this it's very useful for him to work at infinite level because at finite level you don't have this explicit description okay and now there's this strange isomorphism between the two towers between the tributate and the dreinfeld tower which is due to faultings and then developed in more detail where far again it's a precise setup that we need I mean as an isomorphism of perfectoid spaces it's in this paper of Jared Weinstein that this is the same as dreinfeld space at infinite level but dreinfeld space just by its definition so there's a tower of finite detail covers of dreinfeld space so by its very definition it will map down to dreinfeld's upper half plane and so that's actually what the autistic period map does so the strange isomorphism between the two towers somewhere built into this autistic period and it feels a little strange that there's actually some sort of map of attic spaces which realizes this because if you think that you start somewhere in the ordinary locus and then you run into the super singular 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 the kind of behavior you know from finite type geometry but in this infinite I mean this highly non-issuing situation that's actually possible and maybe I'm only at 3 everything I said here for the modular curve it also works for a shimura varieties of hoch type I'm very lucky that I can just write shimura varieties of hoch type without having to think much because as it turns out the first proof is for the Ziegler modular space of principally polarized opinion varieties where it's quite a bit of work but then any shimura variety of hoch type is a close-up variety there and then you can by formal arguments deduce everything you want so that's very nice and now I wanted to talk about some applications of this 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 the following so let me take some there you have a Q bar shimura variety which is a hoch type and then the theorem I said any system of Hecker-Argen values which you can find in the complex support homology so it's a shimura variety with a torsion coefficient let's say I work with over Fp can be lifted to a system of Hecker-Argen values of a classical custom form so that's doing two things so it's saying that any torsion class can be lifted to characteristic zero which is a very interesting result and it also says that anything which comes from the boundary can also any Hecker-Argen values which come from something from the boundary can also be realized as coming from something which is a custom form so even if some of this associated Eisenstein series does not vanish more P as a cusp you can still find these converances and maybe I should say that in order to get this classical cusp form I may have to increase the level of P yeah right from P maybe I only consider at this point I still consider the whole Hecker-Argen value away from P to talk about eigenvalues and also to be commutative ok and because by the work of many many people there are now very strong serums asserting the existence of color representations for cusp forms one can then reduce these color representations again and get color representations for something which appears in this torsion core module here and so from this you get in the first corollary the existence of color representations for torsion classes so let me fix some totally real RCM field and that xk is a locally symmetric space here a group of any dimension of 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 not a shimua variety let's do someone so already f is q and n is 3 this is just some real manifold or if f is totally imaginary and n is 2 then this is some three dimensional hyperbolic manifold so which has no algebraic structure at all but still the Batico module of this real manifold knows about color representation so there exists a color representation real from the absolute colors of the f and continues of course to deal with people it's associated with the meaning that in particular the traits of Frobenius elements will be the eigen-based of some heck of the values so it's an old idea that you should try to realize these locally symmetric spaces here as boundary components in the Borel's circle magnification of some unitary or synthetic shimua varieties so you have this unitary or synthetic shimua variety which has a compactification as a real manifold with corners this Borel's circle magnification and in there these locally symmetric spaces will appear and that's the the homology will contribute to the homology of this space as a boundary contribution and then you lift this which is an upper Euren-Eisenstein contribution to something which is actually a cast form and then for this cast form you know how to attach color representations to it and then how do you know it really contributes there is a boundary map co-boundary map well there's a long exact sequence where two terms are the usual commode and the compact support commode gives us space and essentially this is gone let's ignore the other strata you can do this by induction and so this means that the eigenvalues will appear either in usual commode or in compact support or homology but by comparability it's somehow enough to make and shift them from one to the other I mean you have to dualize things because our representation is here so you know that the the system of eigenvalues will appear either in the compact support commode or in the system of variety but if it appears in the usual commode only then you can use comparability to get something in the compact support commode again but for this exact sequence the boundary is like a hypersurface no no for the bio-sacrification as a property that's the inclusion of the open part into all of it exactly but okay and so there's also a version of this here a teximo p2zm coefficient and then in the inverse limit you can get some results about characteristic zero stuff and so this way you get the second corollary which says that for any regular so-called regular algebraic hospital automatic representation pi of gln over f fs above they exist to be and any isomorphism between c and ql bar there exists a continuous well actually it's almost everywhere on ramified associated with this system and so I should say this was proved before I did by Harris-Lam Tate and so on and so the point is that you can realize these regular algebraic representations in the corollary of these local isometric spaces with some coefficient systems it's conjectured that you can do results irregular but then you have no idea where to find these representations in which corollary groups the simplest case would be the case of mass forms of eigenvalue a quarter for the Laplace-Lassian and the case with a ground-fuse q and n is equal to 2 so one has no idea how to attach corollary representations to them and so in the last few minutes let me try to explain the proof of this theory here so let me use the same notation for the associated edict space where I probably also choose some such isomorphism here let's listen to C the first step is to use some p-adecoch theory to rewrite this etaco-module group there but obviously we're interested in some torsion groups so we need some kind of integral p-adecoch theory isomorphism and usually those only work if your variety is good or same as the reduction or something like that but as I want to increase the level at p indefinitely I can't expect that I can get through with such statements because it's very hard to find same-estatal models for these two more varieties and so one needs a different kind of comparison result whereas the thing you compare it to is a priori still something mysterious it's a foreign statement so it's a comparison result for torsion coefficients which appears in my work on p-adecoch theory but it's mirrored on the result of faultings so it says that if I look at the etaco-module group torsion coefficients so it doesn't matter whether I compute it on the etic space or on the variety it's the same a general comparison result and if you turn this up to or c-mod p so you get a map to the cormology well actually I have to go to the minimal compactification here and that takes the cormology of the following strange sheave it's sheave i plus 1p we'll explain in a second and in fact it's not an isomorphism but it's almost an isomorphism in the sense of faultings it's almost mathematics so in all this 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 is intersection of i and o plus inside the structure sheave where this is the functions which finish its boundary of course forms and this is the sheave of functions bounded by one and so this kind of cormology 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 entire side of the space so this makes it a priori extremely hard to understand well that's the first step and now I pass to an infinite level at p so the direct limit overall that was at p of this group of coefficients entered up to osu!p is an almost isomorphism too eta cormology now computed at my infinite level shimua 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 you want to go to classical cost forms and there are 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 this eta side here so it says that you can actually also compute this now now that you are 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 sheave complex with respect to some affinity cover and so what this relies on is some version of the almost periodic theorem and in fact in some of the almost periodic theorem our priority only does something for the finite eta covers and I really want to consider the whole eta side here so one needs some slight refinement of it but I can prove it with the same methods and it says that if I have some perfectoid of unit algebra and if I look at the associated perfunerate perfectoid space then you can compute the eta cormology on x of this O plus sheave and then there is a similar version for this I plus sheave and it's actually what you think it should be so at least almost so it's R plus in degrees 0 and 0 in positive degrees so there is some kind of version of Tate's a specificity theorem for the eta 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 originality geometry this statement with the plus here is absolutely not true so there is a whole lot of torsion in these groups in general in fact unbound the torsion if you don't put some at least normality hypotheses but if you pass to this infinite level then all this torsion will somehow go away okay and so this means that so what are the terms in this complex here so there are the sections on you for I plus in fact I can mod out P afterwards by this result there we use some infinite P's and so these are cast forms of infinite level defined on a finite subset and remember that in the end we want to get some Hecker-Argan values in in a classical cast form so what's left to do is approximate these cast forms of infinite level which are only defined on some infinite subsets like globally defined cast forms which are of finite level without messing up the Hecker-Argan values classical meaning in particular of finite level one extra minute so the step 2 I used as a space perfectoid and the step 3 I will use in a long time period so what I mean there are first these usual techniques like going back to cut this paper on panic modular forms um how to extend how to make this procedure so usually you would be the ordinary locus and then the solution is to modify by the Hussein variant so what properties of the Hussein variant do you need for this so you need that the vanishing locus is exactly the super singular locus so you can remove all poles by multiplying by high enough power of the Hussein variant and the other property you need is that this commutes with all Hecker operators away from P so that it doesn't mess up the Hecker-Argan values so we need some analog of the Hussein variant and where do we get it from so the solution pull back let's say one wire to get some kind of fake Hussein variant so because this Hecker period commutes with all the Hecker operators away from P any function that you pull back from there will automatically commute with these Hecker operators so this gives you this property and then you need to see that there are enough of these fake Hussein variants in some sense and this exactly comes down to the fact that this map is affine you should choose an affine red cover of your well it's not P1 in general but in the P1 say it's more in our curve case pull it back to get an affine red cover there and that's the affine red cover with respect to which it computes this Chech complex there and so it's a weight 1 Hussein it's a section of omega it is a section yeah probably so for which affine red cover you know that the pull back is well for example for the standard of P1 by 2 close ports the point is that this property that the pull back is affine red it's stable under passing through rational subsets and it's stable under the action of the GL2QP and so whatever you have some open subset so there is an open subset for which it is true which contains a rational point and then you can make the subset very large under the GL2QP action and then basically a subset will be rational of one of those and then you get almost all open subsets so whenever you give me some explicit guy I can 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 I mean for all rational domains it should be okay I would say because you 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 passes to rational subsets I can get it for all rational domains and all affine red domains are rational okay in P1 well in P1 maybe I think directly it was classified well anyway then in P1 it's okay for all of them but I mean if you produce one if you recover it okay start with questions from Tokyo then BG and then there's okay thank you thank you so can you just briefly how you define this first I show that this is a sequence of system families in some sense so let's how I can define it on the open sheen more variety and then I have to prove that it extends to the minimum of compactification and for this I use some version of Riemann's I don't know how to say it in English it says that bounded function so if you have some normal complex and logic space and you have something seriously close inside there and you have a function which is bounded on the complement of the seriously close subset then it will extend through all the variety and I prove some analog of this in the setting of the specialized spaces to show that the such set period automatically has to extend because in the period neighborhoods of the minimum of compactification I can show that the image will be bounded and then it makes more concrete so you told us a little description that you so you said it related to finding size of motion so can you come up with your process this finding size of motion for your construction I think I didn't understand the question so you mentioned finding size of motion yes so my question is that you can come up with your construction it's slightly different than some autism so far things fixes a formal model and then computing is only on the formal model whereas I computed on the original the generic farther which is something that is actually new to you my theory of life is because there is a specialization from the total purpose of the generic fiber to the total purpose of the formal scheme and I can compute what happens when I'm pushing forward on this map so essentially taking mentioned cycles and if you do this then you recover from this isomorphism I think you need that the model is nice yeah you need that the model is nice which is a morphism this isomorphism between you talk about all the trends of this this isomorphism but in fact it doesn't consider the real detail it just works with yeah it works on the formal scheme that's what I said but one can show using your perfect utility equivalent to if the formal model is sufficiently nice the competition of vanishing cycles is easy if the formal model is sufficiently nice yes I have a question from Pepe can I ask a question yeah can you prove that the total compactation of higher dimensional similar varieties is perfect I saw a little bit about this I certainly expected it's true but for the moment I can't prove it I think so the problems are following so how do I prove it for the minimum competition I first prove it on some specific subset called a street neighborhood of the anti-canonical tower so that part goes through but then for the minimal competition I have all of geo2qp acting on them and so I can move the subset to cover all of the minimal competition 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 hacker operators acting so it depends on the choice of this conical position and so on so the hacker operators are not acting on it so this argument of spreading it around doesn't work up priori yeah you have to make some other argument so you use a hacker operator so you use geo2qp sorry I use geo2qp okay so that the subset on which I can control everything explicitly it's geo2qp translate will cover the whole space and similarly for Zika's space yeah similarly for Zika's space I mean yeah for Zika's space I use this in vacuum okay so maybe there is some either questions from Tokyo or then from BG maybe not from Tokyo thank you so from BG okay so do you have any questions no okay so then I have one question so in your foray for regular edge bracket representations you construct that Galawara representation and do you know anything on the local behavior of this Galawara foundation for instance do you know a global compatibility away from P and the drama is at P for instance well I mean there's a student of Richard Taylor who was working on these questions and I guess I mean and the student answers these questions to a large extent so there is a potential okay there is a potential okay okay there are other questions okay thank you any questions about the corona like too yes I mean there is something mysterious because you don't have any Shimura but I feel well as I said you write this I think no no but I mean for the FP I understand but now you get something about well I mean you can write this as the commodity of some local system of this XK and by passing to the large level at P you can also forget about the scope of the system and so you essentially just need a version of this Galawara I1 which is not only with FP bar coefficients but with Z about P to ZM coefficients because in the inverse limit you will recover what you want and you get the good HECA eigenvalue I mean the Frobenius eigenvalues are the same as the HECA yes so at all good times so you go about Z about P and Z okay so I use the Galawara I1 for all the FP to ZM's and then go to ZM's other questions if not let's turn to the speaker again