 Okay. So hello everyone. So this is the last lecture in the Paris-Bugine-Turkish seminar. So the seminar will stop but our collaboration will continue in different forms and I would like to take this opportunity to thank all speakers during the past 10 years and my co-organizers from Tokyo, Takeshi Saito, Atsushi Shihou, Takeshi Tsuji, from Beijing, Yunshun Hu, Ye Tian, and Wijiu Jiang, and from Paris, Fabrice Ogogozu. I would like also to thank four more organizers, Christophe Bray, Ariane Lizar, and Ichao Tian. And it's my great pleasure to introduce the last speaker, Christophe Bray, who will speak on modular representations of JLQN for ununified air. Thank you very much. Thank you very much to all of you for this nice invitation. So I'm going to talk on joint work with Florian Erzich, Yunshun Hu, Stéphane Mora, and Benjamin Schrin. Okay, so the contents of the talk. We have three parts. In the first part, I will recall past results. In the second part, I will state a new theorem. And the last part of the talk, which will actually be the longest part, will be some ideas on the proof, fairly precise ideas on the proof. Okay, so let me start with an explanation of the of the setting and of past results. So throughout the talk, P will be a prime number, and F will be a finite field of characteristic P, which will be my coefficient field for all representations, either on the GL2 side or on the Galois side. And I will assume it is big enough in the sense that it will contain all hexagon values and so on, so that I don't have to worry about that. F will be a totally real number field, where P is unramified, and I will fix V, a place dividing P, a place of F dividing P, which will be my fixed place till the very end of the talk. I will only work at this place V. I will fix a quaternion algebra D over F, which is split at all places above P, and at exactly one infinite place. And finally, I will fix a continuous absolutely irreducible Galois representation of Galois F bar over F to GL2F, which is totally odd and which is modular. So the precise sense of modular will be clear in the next slide. And the general aim of this talk, and not only of this talk but of lots of work, is to understand better certain smooth admissible representation of GL2F V over F, which are associated to R bar, where FV is the completion of F at V. Okay. Okay, so I want precisely to define the representation of GL2F V I'm interested in, and it is called maybe improperly the local factor at V associated to R bar, which I recall the definition, at least the idea of the definition, because it's a big technical. So first recall that for any compact open subgroup of the finite adders of the group D cross, I have a Shimura curve xk over F, which is a smooth projective algebraic variety over F. Okay. And the first representation one can consider is the following smooth representation of these finite adders over F. First to take the inductive limit of the H1 et al of these Shimura curves with coefficient in F, this inductive limit being taken over the compact open subgroup k. So k is getting smaller and smaller in the inductive limit. And then I take the R bar isotopic part of this galois representation. Of course, there's a galois action because it's a tachomology and I assume it is nonzero. This is what I mean by being modular. Okay. I'm not interested in modularity questions here. Although at some point, there are hidden somewhere, but I want to study something related to this representation, which of course I assumed nonzero. Okay. So as I told you, I'm not instructed directly to this representation. I want to study representation of gl to Fv. But the problem, you see, is that we do not know so far if this representation pi of R bar as a restricted product decomposition, as a decomposition as a restricted product of smooth dw cross representations over all the finite places w. It's called, in the classical case, it's a node result due to flat. It is conjectured here, I guess. It's a node conjecture now by, I think, Buzzard, Diamond, and Jarvis. But it is not known. So you cannot define a local factor at V just by using this. You cannot use this. So you have to proceed in another way, which will be sort of ad hoc way, and which will require some weak technical assumption on R bar. And let me just mention that if one day one is able to prove that there is a flat decomposition like this, then it has already been checked that in that case, the local ad hoc factor that I'm going to define coincides in that case with the factor at V of such a decomposition if it exists. But one can define it directly. Okay. So I need to assume some weak genericity assumptions on R bar from now on. So let me give them to you right away. I mean, this is not so much important for the talk. Oops. So P is bigger than five. R bar is absolutely reducible, restricted to this open subgroup of the Galois group. I need some weak genericity assumptions on R bar W, which are the restriction of R bar to the corresponding decomposition Galois group for places W different from dividing P that I do not give here. Not very important. I also need a condition at some places where which are prime to P. If D ramifies at W, I want R bar W to be non-scale. This is not very much important. So here's how one can define the local factor we are interested in. I do not give all the technical details here. This is not so much important and this is not new anyway. So first, one can prove that under these conditions, one can define an optimal open compact subgroup Kv of the finite address of the outside of V and then a certain smooth finite dimensional representation MV of Kv outside of V over F, which has to be thought of as a time or a reduction of P of a type somehow. And then this local factor can be defined as follows. First, you take the Kv invariant homomorphism from this finite dimensional MV to pi of R bar and this is not enough. You need to take some subspace for some a few hacker operators at finally many places different from V. So anything that is going on here is at places different from V. We do not touch V and the purpose of this representation is to get rid of multiplicities that are coming from places different from V because you will see in the rest of the talk, I'm going to use multiplicity one theorems. If I do not take this representation, I do not have multiplicity one. I have an artificial multiplicity different from one, which maybe can be dealt with later on. For the moment, we don't want to be bothered with such problems. So we can define such a we can get rid of these these problems like this. So this local factor was defined. So we don't know it is local. I mean, it's still only only depend. It's a gel to every representation, but it clearly fully depends on R bar. So it was defined in the paper myself with Fred Diamond, I guess 10 years ago, and then what generalized in the paper by Emerton, G and Savit that we mentioned again in this paper in this talk. Okay, so by V of R bar is a smooth admissible representation of gel to F V over F and it has a central character. Okay, which is this way. Okay, so I am going to recall some known results about this by V of R bar. The first one, of course, is the case of gel to QP. More precisely, the case where F equals Q and D is gel to. And in that case, by V of R bar is fully known. So this is a work of Emerton building on work of Colmes, of myself, of Kizin, of Laurent Bergeri, and of other people. It was 10 years ago. So in particular, we know the following three things on by view of R bar. We know that the Gelfand Kirillov dimension is one. I will recall just afterwards what the Gelfand Kirillov dimension is. We know that by view of R bar is a finite length as a gel to QP representation over F. And we also know that it is local in the sense that it only depends on the restriction of R bar to the decomposition group at V, R bar V. And I should mention before defining the Gelfand Kirillov dimension that this theorem, I guess, should in fact hold as soon as F V equals QP, because then we have DV is gel to QP. But as far as I am aware, this is not known. This is known in some cases in the literature where F is not Q and D is not gel to, but F V is QP, but not in the generality. But it should, I think it should be true. Okay, but in the stock, we are not going to be interested in gel to QP anyway. Let me recall now the Gelfand Kirillov dimension. There are several definitions. I give to you the most, maybe the most direct one. So first, I recall to you the definition of the congruent subgroups KV of N, which is 1 plus P to the N M2 OFV. So M2 is the 2 by 2 matrices, which is an open compact subgroup. And KV is a maximal compact open subgroup, gel to OFV. Okay, so we have all these congruent subgroups. And then here's the definition of the Gelfand Kirillov dimension. So I guess it is due to Gelfand Kirillov, but this precise definition can be found in a recent paper by Emerson and Pascunas. So let pi V be any smooth admissible representation of KV of 1 over F. KV of 1 is the first congruent subgroup. Well, I could, it's an asymptotic definition. So I could even take KV of N for arbitrary N, but so there exists a unique integer, GK of KV, which is between zero and the dimension of the PADK analytic group KV as a ZPADK analytic group. So in particular here, it is four times the degree of FV, such that you, the following ratio here, the dimension of the invariant of KV by KV of N, which is a finite dimensional vector space because it is admissible, divided out by the P to the N times this integer is bounded by two strictly positive real numbers. So they have to be strictly positive because you could here take a bigger integer and then it will tend to zero. Okay, so you don't want this, of course. So very roughly, you can think about the Gelfand-Kilov dimension as an integer that measures the dimension of these finite dimensional vector spaces when N is getting bigger and bigger asymptotically. Roughly, okay? Okay, so let me now recall some known results when we are not with GL2QP. So of course, much less is known. So I need a few notation. Well, F will be the degree of my field FV, which I recall is un-ramified. Q is the cardinality of the residue field. And I will denote by K and K of one, respectively, with the maximal compact subgroup at V and the first congruent subgroup. So I get rid of the index V. So I think these notation are quite a standard. So you think you can remember them. This one maybe is less standard. K mod K of one will be denoted by gamma. This is just the finite group GL2 of FQ, where FQ is the residue field. And Z of one will be the center of K of one. And finally, I need to call MK, which is the maximal ideal of the Iwazawa algebra of K of one, module of the center Z of one over F. So maybe I should have called it MK of one, but we don't use really the Iwazawa algebra of K mod Z of one, only of K of one mod Z of one. So okay, so maybe I should recall that Z of one are trivillion by V of gamma. This comes from the condition of the central character. That's why in the stock, everything will be module of Z of one. And then here is the one state, one nice statement, which is known in that situation for arbitrary FD and Roba, I mean, as they are before. So when has the foreign theorem, which, well, let me state it and then I'll say, I'll say something about the names. So we are concerned with the invariant of by V of Arba under the first congruent subgroup K of one. So this is, of course, a finite dimensional representation. And this is, and it has an action of K mod K of one, which is gamma. So this is finite dimensional gamma representation. So it's a tiny, tiny piece of by V of Arba. So it is also the kernel of by V of Arba for the action of the maximal ideal of MK. Okay. And this finite dimensional gamma representation, even though you may think it is a small part of by V of Arba was not so easy to determine. And it is expected explicitly known, in particular, it is local, it only depends on Arba V. And most importantly, for, for me, it is multiplicity free, meaning as a representation of gamma. So all the irreducible constituents are distinct. Yes. And that's will be the thing that I'm going to use in the sequence. So this theorem was the first proven in the case of the UOLV of the property will be by emerton G and savvy, this paper that I already mentioned that I will mention again in this talk, they made the main breakthrough to prove this result. And they, they used patching filters. That was the main tool they used. And then it was generalized by three kind of words. First, the paper in, I don't know, I think it's chronological. Yes, the paper by Daniel Leigh, Stefano Moro and Benjamin Schrein, then some work of Yon Chan-Hu and Aaron Wang, and then another paper by Daniel Leigh. And all this was built on my paper with Pasquunas of many years ago, which itself built on the seminal paper by Buzzard, Diamond, and Jones. Okay, so we have this multiplicity free result. I will come back to this theorem too later in the talk. So it is important for this talk. So I should now make clear that if DV is not TL2P, apart from the theorem, which is not exactly what we had for TL2P anyway, none of the statements in theorem one are known. So let me recall to you that this statement where the Gelfon carried off the finite lengths and the fact that the representation is local. Okay. Okay. So now I want to state our main theorem. Okay. So first I need some hypothesis on R bar v. I need some precise generosity hypothesis on R bar v that is stronger than the weak generosity hypothesis I had as a running hypothesis in the beginning. So for this I need the search fundamental character of level f and 2f. Okay. So if I want to define them as I'm going to do, I need to fix embeddings because it's not very important. I mean, I guess you all know what are these fundamental characters. So first I will assume that R bivari is semi-simple and I will not read by Roba. So till the end of the talk, R bivari now is semi-simple. So I should mention that in all these questions about cell weight and so on and these representations, this is always the first case that is usually considered. And then once we understand this case, usually we go to the non-summysimple case right afterwards, but afterwards. So I assume it is semi-simple and I want some generosity hypothesis. So let me give it to you. Okay. So don't maybe this is a bit technical. You can of course write the restriction to inertia of Roba in terms of cells fundamental characters up to twist. So you have certain powers. Of course that occurs. And I want the digits in the p expansion of these powers to be sort of very much in the middle. So between 8 and p minus 11. That's the bounds we need. So in particular this implies that p is bigger than 19 till the very end of this talk. Okay. So I should mention that we have not tried to optimize this generosity assumption. But it could be that working harder we could get 19 and then even working harder we could get sorry we could get 17 and then working harder we could get 13 and so on. But for the moment we find this. So p is large bigger than 19. Okay. Now I want to state our main result. It is the following. Under this assumption we have the Galifant-Kirilov dimension which is f. Okay. So on f, d and r bar this is the assumptions as in the previous theorems and on r bar it is r v bar it is semi-simple and sufficiently generic as in the previous one. So I should mention now three remarks on this theorem. First that of course these assumptions on the r bar being semi-simple and sufficiently generic should be unnecessary. One should always have that Galifant-Kirilov dimension is f. The second statement is that in the paper and by g and Newton they prove that the Galifant-Kirilov is always bigger than f. They prove this using the patching techniques. So they know what is going on at this infinite level at infinity and then they mod out to get down to five-year bar and when you do this you don't exactly know what you lose or not when you mod out. So that's why they only have an upper bound a lower bound by f. So our main result is that f is also an upper bound and finally let me make clear right now that even under these assumptions on the r bar and even knowing the Galifant-Kirilov dimension so far we do not know if pi v of r bar is a finite length g l to f v representation over f and even less if it is local meaning only depends on the restriction of r bar to the local decomposition group at that we have the Galifant-Kirilov dimension. So the the rest of the talk which is you see will be longer than if you look at the timing will be devoted to give you a fairly precise idea of the proof of this theorem. Yeah some ideas on the proof. So we are going to use two intermediate theorems the one which I call the first one and a second one which will come in in two minutes and I will explain the proofs of these two theorems and when you put them together you get the Galifant-Kirilov dimension. So the first one is the following extension of theorem two. So theorem two let me recall to you right away it was this way oh sorry it's after this way it was that when you take the kernel of pi v of r bar for the maximum ideal for the Wasawal-Dibrov the first congruence subgroup you have something which is multi-stiff free as a gamma representation and equivalently as a k representation. So what we do in the theorem the first intermediate theorem is that we take mk to the square so of course it's not anymore a gamma representation but it's a k representation which is finite dimensional and we still prove it is multi-stiff free. So you see we need general city assumptions for this because in general it is not it's not going to be multi-stiff free but for the moment we assume this we need this multi multi-stiff free things. So that's the first intermediate theorem and the proof of it is following the same techniques as for the proof of theorem two in particular by Merton G and Savit and the followers in particular we need patching fontus but it's technically much harder as you will see but this is not this theorem that we are going to use directly we're going to use a corollary which is not very hard to derive from the theorem but with concerns the Iwari subgroup not the maximal compact k. So let me recall first that the Iwari is the matrices in k that are upper triangular modulo p. So p here is my uniformizer because everything is in rami fact fv is in rami fact and i of one is a propi Iwari so it is the group of matrices that are upper unipotent multiply and I really not as I did for k m i which will be the maximal ideal of the Iwari Iwari algebra of the propi Iwari modulo zero one okay and the corollary we are going to use is that if you consider now pi v of arba and then you take the kernel by this maximal ideal to the cube so it is a representation of the Iwari and it is multiplicity free so if you take an irreducible representation smooth irreducible representation of the Iwari in characteristic p then the propi Iwari acts trivially on it because it is irreducible hence it is a representation of I mod I of one but I mod I of one I of one is a finite torus which is a nabillion group and of cardinality prime to p okay so the irreducible representation of the Iwari over f are just characters so this statement means that all the characters that occur as sub quotients of this representation are all distinct so you see that you of course need generosity assumption for that and we are going to use this corollary okay now the second intermediate theorem is the following which is entirely on the Iwari side so the first I mean apart from this corollary the first intermediate theorem will be entirely on the uh somehow k and k of one side and the second intermediate theorem is entirely on the Iwari side it is the following take pi v which is any smooth admissible representation of the Iwari mod z of one over f such that the kernel of pi v by the this ideal mi to the cube is multiplicity free as we had as we know in the case of pi v of r bar but here this is any pi v then in that case the gulfon kilo of dimension of pi v is smaller than x okay so we recall that the gulfon kilo of dimension is something asymptotic for the compact open subgroup so I can perfectly it is perfectly defined for a representation of the Iwari okay and then it then directly follows from this previous corollary in this theorem that the gulfon kilo of dimension of pi v of r bar is smaller than f okay and by g newton for for the reverse inequality we get the main result okay so now I will explain the proofs of these two intermediate theorems and I will start with a second one not the first one because the second one is in fact shorter although it was for us the hardest one so I need some further notation so first let me know the by pi v with the stranger symbol the algebraic dual of of pi v okay and then of course if so it is a module of the uaz the uaz of algebra of the propiwari so when I mod out by the maximal ideal of this uaz of algebra I get the dual of the invariant and the i of o which is a finite dimensional representation a finite dimensional um representation of i mod i of one so it is just a bunch of characters the direct sum because i of i of one is from two people so we have certain characters chi alpha and finitely many which are what they are which are all distinct by assumption let me denote now by projective proj chi alpha the projective envelope of this character chi alpha in the category of compact so here it is truly the uawari the uaz of algebra for the uawari group not the propiwari but in fact it is just the tensor you take the uaz of algebra of i of one mod z of one and you transfer by chi alpha and then the uawari acts on this and this is the projective envelope of chi alpha so we know that chi alpha does not appear in mi by visual mod mi cube because we use our assumption that by visual mod mi cube by visual is multiplicity free with the finite dimensional representation of i which is multiplicity free and since chi alpha already appears in the quotient by mi it doesn't appear in the kernel and then using this it is not difficulties for more using these these these definitions together with the universal property of projective envelopes to prove the following so if you I mean there will be some three things coming they might look a bit technical but they are not hard to prove so first there yeah so one can prove there exists for each alpha uh i equivalent maps ash alpha from uh two f copies of project the projective envelope of chi alpha to itself to just one copy such that we have the following property first the image of h alpha is in inside mi to the square of chi alpha the induced map posh a current i chi alpha mod mi to f copies to mi square mod mi cube is injective and finally and most importantly for us by v dual will be a quotient of the direct sum of the cookernol direct sum of alpha of the cookernols of all these alpha so in all of this we only use this not to be see three things and universal properties of projective envelopes i mean and and and easy stuff on it was our algebra so you see that theorem five the one bounding the gelfon cannot dimension it i mean first we obviously have that the gelfon killer of pi v will be smaller than the maximum of alpha because of the last segment here of the gelfon killer of this cookernol except you have to dualize back okay so here there's a hidden duality between discrete and compact modules so you here you are on the compact side you do a lot you dualize back to get back on the side of smooth admissible representation of the worry and you can compute the gelfon killer of dimension of this cookernol assuming we have these one and two here and you take the maximum of alpha and this is bigger than gk because of three gk of pi pi v but in fact it is not very difficult to prove that the gelfon killer of dimension of such a cookernol is smaller than f and this ultimately boils down to calculation in the graded ring for the powers of the maximum ideal of this he was our algebra and it turns out this credit the credit ring was actually computed in a nice paper by Laurent Closet of course it can also be derived from results of lazar and so on i mean this is not so hard but it was nice to have this paper of Laurent Closet at hand and using a not so hard calculation we get the gelfon killer of bar okay so i should mention before i switch so that's the end of the second into my theorem so you know it's not so hard except that it took us a long long long time to find this slide okay the rest somehow is and the reason is that the rest there's already an existing strategy but not for this one but for the first one there is an existing strategy which is the one of a martin g subit which we are going to and and and the followers which we are going to push one step further okay so now we leave the world of you already and we enter the world of the maximal compact so this is the world of cell weights and all these things so let me recall that the cell weights is a nearly it's a nearly simple representation of gamma over f finite dimension of course and i really not as i did for characters proj k sigma the projective envelope of sigma in the category of compact modules of a z was a algebra of k i need k here so this is an infinite dimensional representation which if you take the if you do a nice back in the world of smooth representation of k is admissible and the reason we introduced this projective envelope is that it is enough to prove this by just using the universal property of proj k sigma let me recall that the first intermediate theorem i recall it is here this is this one the kernel of pi of arba for for the mk to the square is multiplicity free so the irreducible constituents are cell weights and we want them to be all distinct so in particular we certainly want uh well we want this to be to be true okay and in fact it's even enough to prove the statement for some specific cell weights which are called cell weights of robot which are those cell weights which we know already embeds into pi v of arba so such that the home k sigma to pi v of arba is non-zero so of course in that case we already know that the dimension here is bigger than one bigger or equal than one so we need to prove that this is exactly what uh so now sigma from now on sigma will be a cell weight of robot and the main tool for that will be the patching functor m infinity of m atom g sabit which itself builds on the patching technique of taylor wiles and of kizin so i'm not going to recall exactly what it is because uh wouldn't i mean this would uh be a bit too too too technical and would require too much time but let me just say this is an exact covariant functor from a continuous representation of k of a finite type wf module so wf is the fit vectors uh well there's an assumption with the central character that you can forget here to finite type r infinity modules which of course satisfies several properties in terms of support uh when you apply it to some types and so on which i uh uh if you want to know them you can check the paper of m atom g sabit so here our infinity is a usual patch deformation ring which in our situation because of our denericity assumption will be a full power series ring over the v vectors okay so of course the this functor is uh yeah depends on uh many many choices it depends on the global setting but also on many choices but so it's highly highly non canonical but we just use it and end of its many properties that i will recall when i use them on the sequel of the talk and they are extremely useful of course okay so um i will now restate the thing we have to prove in terms of the patching functor so somehow we are going to lift everything to infinity because it seems impossible to prove this directly so let me denote by i mean m infinity gothic m infinity the maximum ideal of this local ring this power series ring and let me take v which is any finite dimensional representation of k so the gl2 o fv over f then from one one thing we get from the concept from the properties of m infinity is the following equality you can compute the k invariant homomorphism from v to pi v of our bar this local factor at v in terms of uh the dual of m infinity of v applied to this v here mod the maximum ideal of r infinity so recall that m infinity of v is a finite type r infinity module so when you mod out by the maximum ideal it is a finite dimensional aspect of space and i just take the dual okay here also this is finite dimensional because the representation is admissible so we want to prove the theorem for the multi-piece the theorem for the multi-piece three part which is uh most important follows from the fact that this dimension is one which equivalently is the the fact that the r infinity module m infinity of um uh this uh posh k sigma about mk square is cyclic cyclic meaning that you only did one generator or in other terms that is a quotient isomorphic to a quotient of r infinity so if you know this of course then when you mod out here you get one dimensional vector space so the dual is also one dimensional and you are done i mean you are done for v equals posh k sigma but mk square so this is now what uh i am going to do in the rest of the talk is to give you an idea how one can prove the cyclicity so uh so far i mean this reduction to the cyclicity is not due to us this is something that is due to m r and g sabit and the the followers so there's no new idea now this is now we we we really start to be um uh analyzing this representation so uh first there's something you can consider is this the kernel that you can mod out by mk instead of mk square but if you mod out by mk then you are back in the world of gamma representations and so it is actually the predictive envelope of the cell weight sigma in the category of gamma representation over f okay uh but here we do not mod out by mk we mod out by mk square so it's not anymore representation of gamma so we have to understand this guy and we can uh let me this is not so hard let me recall what it looks like first i need there's an algebraic part so let me denote by v2 tau the following algebraic representation of gamma i recall that gamma is g2fq as a residue field so g2fq fq acts on seam 2 f to the square if you fix an embedding fq into f which i do take an arbitrary embedding then i i twist by uh minus one debt to the minus one and everything uh uh is for the i mean is using the embedding fq inside f which is tau so i i put tau here and i have as many such algebraic representation as i have such embeddings which which is f i have f such embeddings f liter f okay then you can prove that proj k sigma but mk square as a k representation is an extension of two gamma representation you have proj gamma sigma as a quotient and as a as a subrepresentation you have a direct sum of the all embedding stow of this proj gamma sigma tensor by v2 tau okay and this is a non-split extension for all the all the push void all the direct summons here that you can consider let me just also mention that we know what this tensor product is i mean when you transfer something which is projective you always get something which is projective so we know that this thing is a direct sum of projective envelopes of some cell weights and we know which which are the cell weights so you need three cell weights well you recover proj gamma sigma but you have two other cell weights which are a small modification of sigma in the direction of the emitting tau which i do not recall explicitly but which are everything can be made completely explicit okay so this is the k representation proj k sigma module mk square and for the rest of the talk i will need to introduce the following quotient of proj k sigma but mk square which i will call q tau for each embedding tau so this is the unique quotient of proj k sigma but mk square which is non-split extension here so this is a push forward i cancel anything that is not at the embedding fixed embedding tau and for the fixed embedding tau i have this tensor product which is direct sum of two and i i i cancelled this proj gamma sigma in the middle so i get a non-split extension like this okay and i will use q tau in the next slide uh okay okay so to to to proceed to prove this so i recall that we want to prove this theorem m infinity proj k sigma mk square is c-click so we are going to apply m infinity to all these protective things but uh we also need to leave the k representation proj k sigma of mk square as a lattice as a free wf module with a continuous section of k because then we will be able to relate it to gala representations and fontan theory that's why we lift it but yes sorry i don't have a question yes so we mentioned tensor products but do you can you tensor product those things only when non-factor is finite dimensional or you have also some completed tensor products when no everything is finite dimensional here i mean i'm going to ah because of this okay because you are working with uh module mk square then you need mod mk square in the in the otherwise indeed this is fine this is infinite dimensional but mod mk square because this is an the dual of an admissible representation okay this is finite and and this tensor product is uh okay this is also finite dimensional yeah everything is finite dimensional okay okay basically till the end of the talk uh except the very last the two last two slides everything will be either finite dimensional over f or a finite rank over the bit vectors free or finite rank over the bit vectors so this is what i need to know here and i'm going to do here i'm going to lift the scale representation as a free wf module with a continuous action of k which reduces mod p to proj k sigma but mk square so it is easy to lift proj gamma sigma because actually there's a unique representation of gamma lifting proj gamma sigma as a free module of a wf so this is you know this is an old result due to brawler i guess which you can find in the sears book uh so what's the representation linear they go for instance um okay it's also easy to lift sorry oops ah no i don't want okay it's also easy to lift the algebraic part uh as uh as a so here this is a representation of gamma here this is a representation of k not of gamma it is not even smooth it is algebraic so i lift v2 tau as v2 tau tilde which is seem to have two copies of the vectors and there's a twist by the determinant and to make k to make k act on this i need to fix also an embedding say okay k is g2 of o f v and uh o f v uh embeds into which is the ramified embeds into w f via the embedding uh of fq into f uh okay so this is and uh here's the first thing one can prove so i need a comment if you take this tons of product here just as it is forget about the one over p one second and if you reduce it mod p then you get the tensor product uh this one v2 tensor proj gamma sigma which is a direct sum of proj of this projective here okay we are not we do not want this we want to find q tau so q tau he's is you take the same projective envelopes except that you put an extension in that in that order okay so it turns out that there is a lattice when you invert p in this finite dimensional vector space there is a lattice which is not the tensor product of these obvious lattices which are not a lattice but which exists such that when you reduce it might be you exactly find this non-speed extension in the right order okay this is the first result we prove uh here and the second result is that now we we from this we can get the lattice lifting proj k sigma but mk square we take the following kernel so we use this lifting here this sort of power lifting we map it uh we reduce it mod p to proj gamma sigma and we embed it diagonally diagonally into f copies of proj gamma sigma that's for the definition of this map on this direct sum now the definition of this map on this direct sum is just that l2 tau reduces mod p to l2 to mod p which subject onto proj gamma sigma okay because here the subjection is here and so for each embedding you map it to one copy of proj gamma sigma you have f embedding so you have f copies and you take the direct sum of these morphisms and you take the kernel of this then so here this is free over wf here this is in characteristic zero okay so this is somehow a lattice inside this uh this thing when you invert p uh and this lattice mod p is exactly the projective envelope of sigma but mk square so we are going to apply the patching filter to all these these guys and and indeed we want to prove that m infinity of l is a click if we do this we are done so we know that already by previous work of uh Daniel Le, Stéphane Morabin, Germain Schran, and Yon Chanou, our one that m infinity of this proj gamma sigma is simply proj gamma sigma till the lift of proj gamma sigma uh remember i mean we are in the there are other work on this but here i really remember that we are in the semi-simple case for our our v bar i mean sigma is a cell weight of a semi-simple a representation of Galois v bar mod mv and uh the first thing one can prove is a foreign proposition is that the r infinity module m infinity of l2 tau mod p and hence by an application of Nakayama m infinity of l2 tau both are cyclic meaning our quotient here it is a quotient of r infinity mod p and here it is a quotient of r infinity uh and well so i don't know how how time i have left uh yeah i have 14 minutes thank you so um let me say that the techniques to prove this are are standard with respect to what is already in the papers by uh Daniele uh so maybe i'm not going to insist on on on this the techniques are not uh are not new maybe i will uh this is a standard devisage uh and uh okay let me skip this so uh to to proceed to the next step so the next step is a foreign so remember we want we want to we are interested in l which is the kernel of this direct sum to f copies of posh gamma sigma but before going to l we are going to proceed step by step adding one embedding after the other and in particular we start with l tau which is the kernel the same kind of kernel except we only take one embedding l2 tau so there's according to one copy of posh gamma sigma here and we take the kernel of this so this is just a fiber product a fiber product here you've got three w f representation of k here and here which we know have the mod p have the common posh gamma sigma quotient so we take the fiber product and now i will explain why m infinity of l tau is cyclic so by exactness of m infinity m infinity of l tau is also a fiber product but m infinity of posh gamma sigma which we know is cyclic m infinity of l tau 2 uh which we know is cyclic over m infinity of posh gamma sigma which we also know is cyclic because this one is cyclic however it could be that the fiber product is not cyclic of course so we have to prove it is cyclic and the proof for l for for why you you add all the embeddings direct sum of all all embeddings here can be reduced to this case by just an induction to a once we know this one is cyclic we're going to add another embedding we're going to have another fiber product and so on okay so now uh i will explain why this uh fiber product is cyclic and here we enter the world of gallery representation so uh let me denote by r v which is the r square rho bar rho bar is r bar v this is our local gallery representation semi-simple uh this is the local a notarian ring parameterizing frame deformations of rho bar and sense of kizin laser so there are no conditions except there's a condition of the determinant that i will uh i will forget here that's important so here's what follows from previous cyclicities that i just mentioned so first uh we have our infinity which i told you was a full power series ring of a w of f but in fact before being a full power series ring of a w of f it is a full power series ring of r v this here which in the particular case here because of our generated assumption turns out to be also a full power series ring but let me forget it here okay so we know that m infinity of projama sigma is a quotient of r infinity and in fact because of this variable this patching variables play no role at v we know it is a quotient only on r v so there's an ideal j such that it is isomorphic to this like was for the other one because we know these two things are cyclic r infinity modules and of course same thing for for the reduction of p here by exactness of m infinity it is just r v much pj okay but in fact we know what r j because now we can we are in specific situation we know sigma is is a cell weights of a semi-simple robot and we can compute things everything is quite explicit and in fact we can prove that r v much j exactly parametrizes potentially crystalline lift of a robot of any tame type type here is bushler kutsco type whose reduction what p contained the cell the cell weight sigma and with parallel hot shape weight one zero so it's not i should mention that it's not the kind of usual deformation range that one usually considers because we it is a multi-type deformation ring i mean we take all we take several types and not just one usually you fix one type you fix such a weight and you consider potentially crystalline lifts of robot with this type and this budget weight here we consider all types all tame types meaning by the way tame means they are representational to fq in characteristic zero so level level zero if you like whose reduction what p contains sigma take all these types and we know actually this is exactly the quotient we have so this is the place where i think moderates statements are some somehow hidden because moderates events are hidden in the support of these infinity modules and the reason we know this is we derive it from the fact that if we just fix one of these same types and for these hot shape weights then it's also the usual deformation range for one of these types we actually know it is domain we prove it is a domain so since we know the support is a very visible component for fixed time type it must be everything and now when we put them all together it's we can derive that we have must have the full check like was for this guy except we have cell weights uh hot state weights two minus one at the embedding type which of course are coming from the algebraic parts uh that uh the previous story so we again compute everything explicitly so it's a little bit more complicated because now we have to deal with hot state weights two minus one that is up to twist our shape weight three comma zero uh so the computations are more difficult but but it can be it can be done it can be done even by hand uh and likewise the the single type deformation range with these hot shape weights are also domain so we we can prove it and oops and uh uh yes and so now if you forget about these extra variables the thing you need to prove is that this fiber product here which so you only now consider these guys forget about these matching variables you need to prove it is a quotient of rv if you notice it will be cyclic so meaning one generator of rv and you will be done and to prove this it's easy to see that you need to prove that j plus j tau is exactly p comma j and for this it is enough to prove to prove that p belongs to j plus j tau so so what we know here is that a priori j plus j tau contains a power of p but we have to prove that it contains p really and not and not only p p cube and so on so in other terms this is something like we have to prove that the uh the potentially crystalline representation here with hot shape weights zero and everywhere and the potentially crystalline liftings here with hot shape weights outside tau and two minus one at tau are as little congruent as possible and this can be done by hand and explicitly we because we can of course if we want to prove this we can check it mod p square if you prove this mod p square you have done and this this is something you can do uh by hand okay and this finishes the proof of the main result we have cyclicity for m infinity of m so i want to derive one application of this uh Gelfand Kirillov business which which well was sort of nice for me uh it is an application to the Piedic Langlands program so it is based on the following theorem which is a theorem uh of doto and daniele which itself builds on work of uh cariani, emerton, jeep, gertie, baskunas and shin and it has to do with big patch modules so so far i was considering i was patching uh things like hum k hum k invariant homomorphism from some finite dimensional v to pi v of r and maybe the dual of that okay which was finite dimensional and i was patching this this m infinity everything was a finite rank of r infinity and so on but it turns out you can also patch the full dual of pi v of r v which is of course infinite dimensional now okay and of course this is not anymore this is something which is a finite generated over r infinity double bracket gl2 of o fv which is k uh but this is not anymore finite generated over r infinity and it has a compatible action of gl2 fv okay so that this one can do this and this is done in the paper a recent paper by doto and uh building on on previous work but here they exactly do the thing we need for the for this local factor and so on and so the corollary of our main result is the following which was known i mean we is not new that if we had the gulf on hill of dimension then this would follow back it's nice to recall it so take any map from r infinity to o any specialization somehow of dublev algebra where e is a finite extension of qp so containing wbf um then the corresponding specialization and m infinity m infinity tensor or infinity oe except you have to dualize back back so here this says i think a sheik of dual something like that i mean you have to be careful about duality and you invert p well this is non-zero and i mean then if it's non-zero it is a invisible unitary continuous representation of gl2 fv over e lifting by v of r bar it is a banner space which has a unit ball which is preserved by the gl2 action and which lifts by v of r bar but the thing it is non-zero and to prove the idea of the proof is that you need flatness you need to prove that m infinity is flat over infinity and then you don't if you know this then you know that specialization or non-zero but but this follows from the gelfand scale of our main result together with a result that m infinity is a quen macule over this non-commutative ring so here quen macule is in the sense of our slender books bomb and so on there's only one x i which is non-zero which is a result of g newton and this implication is a so-called miracle flatness so in non-commutative set so we have this result so i think now it is a yeah almost almost time so it is good but i'll just have one slide so i should mention that so far robar was semi-simple but we think the case robar non-simple will work as well and this is actually ongoing work of yun shanhu and aohan mong so of course we need some genetic assumptions but robar will be non-simple and finally one other thing we hope to get so maybe before i so we prove that the gelfand scale of dimension of this sort of minimal representation of gf to fv where we forget about what we have forgotten as much as we could about multiplicities coming from other places different from v so we prove that the gelfand scale of dimension is f but of course if you add multiplicities coming outside of v infinitely many for instance if you don't take the right exactly the right compact open subgroup and so on you will get something like several copies of by v of arba but this way this won't change the gelfand scale of dimension so we shouldn't need these multiplicity one assumptions and so on to prove in fact in the end the gelfand that the gelfand scale of dimension is f and maybe we can prove it in fact we hope to prove that at least for suitable level k of v outside of v compact open subgroup of the finite adels of t cross outside of v so we i do the same thing as i did at the very beginning to define pi of arba but i only take the inductive limits over open compact subgroup of gl to fv with a fixed prime to v level and then i take the arba is a typical part so this is a pretty bigger than pi v of arba many many copies of pi v of arba queries but we hope to prove that the gelfand scale of dimension is still f so of course then we we have to deal with things which are not anymore over 51 okay so i guess uh i guess this i guess i'm done okay thank you thank you christoph for this very nice lecture so we have time to take few questions are there any question or comment so when you take the home from arba to something yes some chronology what do you know about this chronology like can you tell sub questions which are arba but not sub okay yeah i don't know we only take arba as a sub you are you are right i don't know about other kind of things you can do of course this is infinite dimensional whether an arba is two dimensional so indeed it could be that there are things of which are not as a sub but then then i don't usually i mean in that setting when usually uh consider such things and and we're very happy to be able to prove something about this are there any questions so he okay yeah he wrote it okay so how about higher powers of m g so uh we think we can okay so let me go back to the from this if you assume generosity enough let me see where is the yeah i think but we didn't we didn't write it i think that from five of our mk by some induction uh business we can probably get higher powers we can go probably go a little bit further by some kind of induction if you assume sufficient generosity mk cube and k4 and so on but uh so far it was not clear to us that we would gain so much from uh proving these things so i mean mk square for what we have in mind seems to be enough so maybe in the future it would be interesting to have higher powers but of course in general you cannot expect of course to be multiplicity free even if you very generate because less i mean you have finitely many cell weights and this is an infinite dimensional representation so okay so i don't see other questions do you see something so so in the definition of this beautiful k of dimension so you have you take ratio and it's bounded by ah okay okay so yeah in the very beginning so now you know this existence so so you can you can take consider the limit so uh you can do you know if it's converges or the meaning of the value so sorry i'm not quite sure i understand the question so you you can you can take the limit with this to end oh yeah and can you say something about no i mean in fact this this is not exactly the the definition we use we use the definition in terms of oslo under books one theory things like that yeah so we i don't know also you are asking whether this thing is maybe just has a limit instead of just being bounded right this is your question yeah yeah so i'm not going to go if we don't know anything about that i see thank you okay so if there are no other questions another another thing of dimension i am not so so you have some compact module but of course you can dualize and does it get from Kirillov dimension comes from the fact that somehow after you dualize you get finitely generated module over something i don't know this is a smooth admissible okay so this no no when you dualize you get something finitely generated over yes over the the the you are sourcing yes yes and so you can look at the dimension exactly the sense of non-commutative analog of dimension in the terran rings and this is the help on Kirillov dimension yes so in analogy with the theory of the Hilbert function and so on the commutative case you expect that and in fact in this case Lazar and so on it means there is some theory for those kind of non-commutative rings so so the question before was whether there is some kind of a Hilbert polynomial or something similar in this non-commutative setup for non-commutative certain non-commutative rings which are close to be commutative in the sense that you you have enough here some filtration and i mean like this kind of people sour so i think yeah well i think so i i forgot a little bit but i think this uh yeah you know if you know that the dimension if you know the dimension Kirillov dimension indeed you know that this must be something like a polynomial of degree in n of degree uh so this dimension of degree the Gelfon Kirillov dimension plus one maybe or let me see if the Gelfon if the Gelfon Kirillov dimension is zero which means that this thing is bounded for any n which means that then pi v is finite dimensional so this is a constant polynomial yes this must be this so i think you can prove that uh this is one of the uh aspect of Gelfon Kirillov dimension if i'm not mistaken maybe Yonshan Lu can correct me that this dimension is actually a polynomial for n big enough is actually a polynomial in n of degree Gelfon Kirillov dimension plus one no no but it cannot be because you put p to the n in the denominator ah no no no i mean the upper i mean the numerator no no no but if you write in the denominator okay okay so it's okay the variable is not n it's maybe p to the n yeah Daniel Li confirmed polynomial in p to the n yeah sorry you're right yeah not n p to the n yeah of course thank you for the clarification okay so then we thank Christophe for this nice lecture and thank you for the invitation and have a safe and nice summer vacation goodbye