On modular representations of GL_2(L) for unramified L
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Breuil, Christophe
Let p be a prime number and L a finite unramified extension of Q_p. We give a survey of past and new results on smooth admissible representations of GL_2(L) that appear in mod p cohomology.
 Okay, so hello everyone. So this is the last lecture in the Paris-Bijine Tokyo 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, and Sushi Shihou, Takeshi Tsujin, from Bijin, Yon Shun Hu, Ye Tian, and Wiju Jiang, and from Paris for this overview. I would like also to thank four more organizers, Christophe Bray, Ariane Lizar, and Yichal Tien. And it's my great pleasure to introduce the last speaker, Christophe Bray, who will speak on modular representations of G2L for Anonymous My Fair. 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, Yon Shun Hu, Stefan Morat, and Benjamin Schraen. 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 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 would 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 un-ramified, 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 GL2FV over F, which are associated to R bar, where FV is the completion of F at V. Okay, so I want precisely to define the representation of GL2FV 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 Schimuracker 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, you take the inductive limit of the H1 et al of these Schimurack curves, which 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 of this Galois representation. Of course, there's a Galois action because it's et alcomology. 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, well, 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 JL 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 tensor product decomposition, as a decomposition as a restricted tensor product of smooth DW cross representations over all the, the finite places double. It's called a, in the classical case, it's a node result due to flat, flat. It is conjectured here, I guess, it's no conjecture now by I think buzzer, diamond and javies, 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 always already been checked that in that case, the local ad hoc factor that I'm going to define coincides in that, 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 type 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 metabasitity 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 problems like this. So this local factor was defined. So we don't know it is local. I mean, it still only only depends. It's a gel to every representation, but it prior fully depends on R bar. So it was defined in the paper myself with Fred diamonds, he gets 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 pi 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 one. 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. And I'm more precisely the case where F equals Q and these 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 Colmez of myself, Kizin, of Laurent Berger and other people. It was 10 years ago. So in particular, we know the following three things on pi V of R bar. We know that the Gelfon Kirillov dimension is one. I will recall just afterwards what the Gelfon Kirillov dimension is. We know that pi V 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 our bar to the decomposition group at V, our bar V. And I should mention before defining the Gelfon Kirillov dimension that this theorem, I guess, should infile 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 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 Gelfon 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, just one plus P to the N M2 OFV, so M2 is the two by two matrices, which is an open compact subgroup. And KV is the maximal compact open subgroup, gel to OFV. Okay, so we have all these congruent subgroups. And then here's the definition of the Gelfon Kirillov dimension. So I guess it is due to Gelfon Kirillov, but this precise definition can be found in a recent paper by Emehrhorn 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 Pi V, which is between zero and the dimension of the PIDK analytic group KV as a ZPIDK 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-Hilloff 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 GL2 QP. 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 be dot 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 is, will be directed 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 Iwaza algebra of K of one, modular 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, it was our algebra of K mod Z of one, only of K of one mod Z of one. Okay. So maybe I should recall that Z of one are trivially owned by V of gamma. This comes from the condition of the central character. That's why in the stock everything will be modular Z of one. And then here is the one state, one nice statement, which is known in that situation for arbitrary FD and robot. 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 5 of our bar under the first congruent subgroup K of one. So this is of course, a finite dimension or representation. And this is an, and it has an action of K mod K of one, which is gamma. So this is a finite dimensional gamma representation. So it's a tiny, tiny piece of pi V of our bar. So it is also the kernel of pi V of our bar 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 pi V of our bar was not so easy to determine. And it is expected explicitly known in particular, it is local. It only depends on our bar V. And most importantly, for, for me, it is multiplicity free, meaning as a representation of gamma. So all the irreducible constituents are distinct. Of, yes, and that's will be the thing that I'm going to use in the sequence. So this theorem was first proven in the case of the uOV of the property will be by emerton G and savvy. This paper that already mentioned that I will mention again in the stock, they made the main breakthrough to prove this result. And they used patching filters. That was the main tool they used. And then it was generalized by three kind of works first the paper in a, I don't know, I think it's chronological. Yes. The paper by Daniel Le, Stefan Moran and Benjamin Schrin. Then some work of Yon Chan-Hu and Aaron Wong. And then another paper by Daniel Le. 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 James. Okay. So we have this multi PC free result. I will come back to this theorem two later in the talk. So it is important for this talk. So I should now make clear that if TV is not yielded to QP, apart from the serum, which is not exactly what we had for jail to QP anyway, none of the statements in serum one are known. So let me recall to you that these statements were the Gelfon carry love, 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 our Barbie. I need some precise generosity. I put hypothesis on our Barbie 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 two F. 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, guess you all know what are these fundamental characters. So first I will assume that our be very semisimple and I will not read by robot. So till the end of the talk, our Vivaal now is semisimple. So I should mention that in all these questions about serweight 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 semisimple case right afterwards, but afterwards. So I assume it is semisimple 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 robot in terms of sers 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 eight 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 technique. So they know what is going on at this infinite level, infinity, and then they mod out to get down to 5-year-old 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 b 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 F. But we have the Galifant-Kirilov dimension. So the rest of the talk, which 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, one which I call the first one and a second one, which will come 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 it to you right away. It was this way. Sorry, this way. It was this, when you take the kernel of pi b of R bar for the maximum ideal for the Wasawal-Dibrov's first congruent subgroup, you have something which is MULK-Pistifri 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 MULK-Pistifri. So you see, we need general city assumptions for this, because in general, it is not. It's not going to be MULK-Pistifri. But for the moment, we assume this, we need this MULK-Pistifri thing. So that's the first intermediate theorem. And the proof of it is following the same techniques as for the proof of theorem 2, in particular by Merton G and Savit and the followers. In particular, we need patching filters, 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 we concern the Iwo is a group, not the maximal compact K. So let me recall first that the Iwo is the matrices in K that are upper triangular modulo P. So P here is my uniformizer, because everything is in Ramifac, Fv is in Ramifac. And I of one is a prop Iwo, so it is the group of matrices that are upper unipotent matrices. And I really not, as I did for M i, which should be the maximal ideal of the Iwo algebra, of the prop Iwo Rii modulo zero one. And the corollary we are going to use is that if you consider now 5v of Abba, and then you take the kernel by this maximal ideal to the cube. So it is a representation of the Iwo Rii, and it is multiplicity free. So if you take an irreducible representation, a smooth irreducible representation of the Iwo Rii in characteristic P, then the prop Iwo Rii 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 is a finite torus, which is an abelian group and of cardinality prime to P. So the irreducible representation of the Iwo Rii over F are just characters. So this statement means that all the characters that occur as sub-quotions of this representation are all distinct. So you see that you of course need genericity assumption for that. And we are going to use this corollary. Now the second intermediate theorem is the following, which is entirely on the Iwo Rii side. So the first, I mean, apart from this corollary, the first intermediate theorem will be entirely on the, somehow, k and k of one side. And the second intermediate theorem is entirely on the Iwo Rii side. It is the following. Take pi v, which is any smooth admissible representation of the Iwo Rii mod Z of one over F, such that the kernel of pi v by this ideal mi to the cube is multiplicity free, as we had, as we know in the case of pi v of Arba. But here, this is any pi v. Then in that case, the get form kilo of dimension of pi v is smaller than F. Okay, so we recall that the get form kilo dimension is something asymptotic for the compact open subgroup. So I can perfectly, it is perfectly defined for a representation of Iwo Rii. Okay, and then directly follows from this previous corollary in this theorem that the get form kilo dimension of pi v of Arba is smaller than F. Okay, and by G Newton 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 the 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 by pi v was the strange symbol, the algebraic dual of pi v. Okay, and then of course, if so it is a module of the Iwo Rii, the Iwo Rii algebra of the POPI Rii. So when I mod out by the maximal ideal of this, it was a algebra, I get the dual of the invariant and the Iwo, which is a finite dimensional representation, a finite dimensional representation of I mod I of 1. So it is just a bunch of characters, the direct sum because I of I of 1 is prime to pi. So we have certain characters, sky and fire, finitely many, which are what they are, which are all distinct by assumption. Let me denote now by projective proj kai alpha, the projective envelope of this character kai alpha and the category of compact. So here it is truly the Iwo Rii, the Iwo algebra for the Iwo Rii group, not the POPI Rii, but in fact it is just the tensor, you take the Iwo algebra of I of 1, mod I of 1 and you tensor by kai alpha, and then the Iwo Rii acts on this. And this is the projective envelope of kai alpha. So we know that kai 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 kai alpha already appears in the quotient by mi, it doesn't appear in the kernel. And then using this, it is not difficulties formal using 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, yes, one can prove there exists for each alpha, i equivalent maps, h alpha from two f copies of projective envelope of kai alpha to itself, just one copy, such that we have the following property. First, the image of h alpha is inside mi to the square of kai alpha. The induced map h a kai alpha mod mi to f copies to mi square mod mi cube is injective. And finally, and most importantly for us by the dual will be a quotient of the direct sum of the kukenol direct sum of alpha of the kukenol 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 easy stuff on, it was our algebra. So you see that theorem five, the one bounding the Gelfon Kirillov dimension, I mean, first we obviously have that the Gelfon Kirillov of pi v will be smaller than the maximum of alpha because of the last statement here of the Gelfon Kirillov, this kukenol, 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 you dualize back to get back on the side of smooth admissible representation of the e-worry. And you can compute the Gelfon Kirillov dimension of this kukenol, 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 v. But in fact, it is not very difficult to prove that the Gelfon Kirillov dimension of such a kukenol is smaller than F. And this ultimately boils down to calculation in the gradi drink for the powers of the maximum ideal of this, it was our algebra. And it turns out this credit, 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 Kirillov back. Okay, so I should mention before I switch, so that's the end of the second interim. 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, not for this one. But for the first one, there is an existing strategy, which is the one of a Martin G. Sabit, which we are going to end and the followers, which we are going to push one step further. Okay, so now we leave the world of Iwo Ori 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 is a nearly simple representation of gamma over F, finite dimension of course. And I will denote as I did for characters, proj k sigma, the projective envelope of sigma and the category of compact modules over Z was a algebra of k. I need k here. So this is an infinite dimensional representation, which if you take the, if you dualize 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 that I recall it is here. This is this one, the kernel of pi of r bar for m and k to the square is multiplicity three. So the irreducible constituents are cell weights and we want them to be all distinct. So in particular, we certainly want, we want this 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 r bar. So such that the home k sigma to pi v of r bar 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. So now sigma from now on, sigma will be a cell weight of robot. And the main tool for that will be the patching function m infinity of m atom g subit, 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 wouldn't, I mean, this would be a bit too too technical and would require too much time. But let me just say this is an exact covariant phonter from a continuous representation of K over finite type Wf modules. So Wf is the fit vectors. Well, there's an assumption with central character that you can forget here. To finite type r infinity modules, which of course satisfies several properties in terms of support when you apply it to some types and so on, which I, if you want to know them, you can check the paper of m atom g subit. So here r infinity is a usual patch deformation ring, which in our situation, because of our denerity assumption, will be a full power series ring over the v vectors. So of course, the swanter is, yeah, depends on 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 of its many properties that I will recall when I use them on the SQL of the talk. And they are extremely useful, of course. Okay, so I will now restate the thing we have to prove in terms of the patching phonter. So somehow we are going to lift everything to infinity because it seems impossible to prove this directly. So let me denote by m infinity gothic m infinity, the maximum idea of this local ring. It's power series ring. And let me take v, which is the same as the final dimension 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 best local factor v in terms of the dual of m infinity of v applied to this v here, more than the maximum ideal of our infinity. So recall that m infinity of v is a finite type r infinity module. So when you mud out by maximum ideal, it is a finite dimensional aspect 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 is free path, which is most important follows from the fact that this dimension is one, which equivalently is the fact that the r infinity module m infinity of this posh k sigma, but m k 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 notice, of course, then when you mud 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 m k square. So this is now what I am going to do in the rest of the talk is to give you an idea how one can prove the cyclicity. So far, I mean, this prediction to the cyclicity is not due to us. This is something that is due to MF and G Savit and the followers. So there's no new idea. Now this is, now we really start to be analyzing this representation. So first, there's something you can consider is the kernel that you can mud out by mk instead of mk square. But if you mud out by mk, then you are back in the world of gamma representations. And so it is actually the predictive envelope of the Savit sigma in the category of gamma representation over F. Okay. But here we do not mud out by mk. We mud out by mk square. So it's not any more representation of gamma. So we have to understand this guy. And we can, I mean, 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 acts on sim2f to the square. If you fix an embedding, fq into f, which I do, take an arbitrary embedding, then I twist by minus one, that to the minus one. And everything is for the, I mean, is using the embedding fq inside f, which is tau. So I put tau here. And I have as many such algebraic representation as I have such embeddings, which is f. I have f such embeddings, f liter f. Okay. Then you can prove that proj k sigma with mk square as a k representation is an extension of two gamma representation. You have proj gamma sigma as a quotient. And as a subrepresentation, you have a direct sum of the all embedding tau of this proj gamma sigma, times by v2 tau. Okay. And this is a non-split extension for all the push forward, 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 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 with mk square, which I will call q tau for each embedding tau. So this is the unique quotient of proj k sigma by 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 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. Okay. Okay. So to proceed to prove this, so I recall that we want to prove this zero. m infinity proj k sigma mk square is a kick. So we are going to apply m infinity to all these productive things. But 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. Okay. Because then we will be able to relate it to Galois representations and Fontaine's theory. So that's why we lift it. But yes. Sorry, I have a question. Yes. So we mentioned tensor products, but 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 not out. Ah, because of this, okay, because you are working with modulo mk square. Oh, yeah. Then you need mod mk square in the, in the, otherwise, indeed, this is finite, this is infinite dimensional, but mod mk square, because this is the dual of an admissible representation. Okay. Okay. This is finite dimensional. And, and this tensor product is, ah, okay. This is also finite dimensional. Yeah. Everything is finite dimensional. Okay. Okay. Okay. Basically, till the end of the talk, except the very last, the two last two slides, everything will be either finite dimensional over F or a finite rank over the V vectors. Free of finite rank over the V vectors. So this is what I need to know here. Ah, 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 project k sigma, but mk square. So it is easy to lift project gamma sigma because actually there's a unique representation of gamma lifting project gamma sigma as a free module of a WF. So this is, you know, this is an old result due to Broward, I guess, which you can find in the search book. So what's the representation linear, the group, for instance. Okay. It's also easy to lift. Sorry. Oops. No, no, I don't want. Okay. It's also easy to lift the algebraic part. As a, 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 the two of all FV and all of the embeds into it, which is the ramified embeds into WF via the embedding of FQ into F. Okay. So this is, and 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 much P, then you get the tons of product. This one V2 tons of proj gamma sigma, which is a direct sum of proj of these projective here. Okay. We are not, we do not want this. We want to find Qtow. So Qtow, he's, is, you take the same projective envelopes, same cell, 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 tons of 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. Here. And the second result is that now we, we, from this, we can get a 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, we reduce it much P to proj gamma sigma. And we embed it diagonally into F copies of proj gamma sigma. That's for the definition of this map on this direct sum. Now the decision, the definition of this map on this direct sum is just that L2tow reduces mod P to L2tow mod P, which surject onto proj gamma sigma. Okay. Because here the suggestion 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 thing when you invert P. And this lattice mod P is exactly the projective envelope of sigma with mk square. So we are going to apply the patching pointer to all these guys. 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 Daniel Le, Stéphane Mourabien, Germain Schran, and Yon Chanoua, that m infinity of this proj gamma sigma is cyclic. Proj gamma sigma, the lift of proj gamma sigma. 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 V bar. I mean, sigma is a cell weight of a semi-simple representation of Galois F V bar mod Mv. And 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 here it is a quotient of r infinity mod P, and here it is a quotient of r infinity. And well, so I don't know how much time I have left. Yeah, I have 14 minutes. Thank you. So let me say that the techniques to prove this are standard with respect to what is already in the papers by Hamerton, Jisavi, Daniel, Stefan, and so on. So maybe I'm not going to insist on this. The techniques are not new. Maybe this is a standard devisage. And okay, let me skip this. So to proceed to the next step. So the next step is the foreign. So remember, we want to, 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 L to 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. Here, you've got three WF 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. M infinity of posh gamma sigma, which we know is cyclic. M infinity of L tau two, 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 where you add all the embeddings, direct sum of all embeddings here can be reduced to this case by just an induction. So 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 I will explain why this fiber product is cyclic. And here we enter the world of gallery representation. So let me denote by rv, which is the r square rho bar. Rho bar is r bar v. This is our local gallery representation, semi-simple. This is the local notary ring parameterizing frame deformations of rho bar in the sense of kizin, laser. So there are no conditions except there's a condition on the determinant that I will forget here. So here's what follows from previous cyclicities that I just mentioned. So first, 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 rv. 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 gamma sigma is a quotient of r infinity. And in fact, because of this variable, these patching variables play no role at v, we know it is a quotient only on rv. 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 the reduction of p here by exactness of m infinity. It is just rv mod pj. Okay. But in fact, we know what rj, because now we can, we are in specific situation. We know sigma is a is a ser weights of a semi simple robot. And we can compute things. Everything is quite explicit. And in fact, we can prove that rv mod j exactly parametrizes potentially crystalline lift of a robot of any time type type here is Bushnell Kutsco type whose reduction might be contained the ser, the ser weights 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 rings 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 lift of robot with this type and this much it way here we consider all types or tame types, meaning by the way, Tami means they are representational to fq in characteristic zero. So level level zero if you like, whose reduction would be content in sigma take all these types. And we know actually this is exactly the quotient we have. So this is the place where I think moderate these statements are some somehow hidden, because moderate statements 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. Likewise, for this guy, except we have cell weights, hot state weights two minus one at the embedding time, which of course are coming from the algebraic parts that the previous story. So we again compute everything explicitly. So it's a little bit more complicated because now we have to deal with such state weights two minus one that is up to twist our shapeway three comma zero. So the computations are more difficult, but it can be it can be done can be done even by hand. And likewise, the single type deformation rings with these high shaped weights are also domain, so we can prove it and oops and 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 into proof 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 to is exactly P comma J. And for this, it is enough to prove to prove that P belongs to J plus J to. So what we know here is that a period J plus J to contains a power of P. But we have to prove that it contains P really and not and not only P cube and so on. So in other terms, this is something like we have to prove that the the potentially crystalline representation here with H8 weights zero and everywhere. And the potentially crystalline liftings here with H8 weights outside two and two minus one at all 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 are done. And this, this is something you can do 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 Gelfand-Kirilov business, which, which, well, was sort of nice for me. It is an application to the Piedic Langlands program. So it is based on the following theorem, which is a theorem of Dotto and Daniele, which itself builds on work of Karajani, Merton, G, Gertie, Pascunas and Sheen. And it has to do with big patch modules. So, so far, I was considering, I was patching 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 with this M infinity. Everything was a finite rank over infinity and so on. But it turns out, you can also patch the full dual of pi V of R, which is, of course, infinite dimensional now. Okay. And of course, this is not anymore, this is something which is a finite generative of R infinity, double bracket, G2 of O, FV, which is k. But this is not anymore finite generative of R infinity. And it has a compacted production of G2 FV. Okay, so that one can do this. And this is done in the paper, a recent paper by Dottwin, building on previous work, but here they exactly do the thing we need for this local factor and so on. And so the corollary of our main result is the following, which was known. I mean, we, it's not new that if we had the GFK dimension, and this would follow, but it's nice to recall it. So take any map from R infinity to O, any specialization somehow of double F algebra, where E is a finite extension of QP, so containing WF. Then the corresponding specialization and M infinity, M infinity, tensile R infinity OE, except you have to dualize back, back. So here this says, I think, shek of dual, something like that. I mean, you have to be careful about duality. And you invert P. Well, this is nonzero. And I mean, then it is a missible unitary continuous presentation of the FV over E, lifting by V of R bar. It is a banner space, which has a unit ball, which is preserved by the G2 action and which lifts by V of R bar. But the thing is it is nonzero. 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 nonzero. But this follows from the Gelfand Kail of our main result, together with a result that M infinity is a quen makkoli over the stone commutative ring. So here quen makkoli is in the sense of our slender books one and so on. There's only one X i, which is nonzero, which is a result of G Newton. And this implication is a so-called miracle flatness. So in noncommutative setting. So we have this result. So I think now it is almost time. So it's good. I just have one slide. So I should mention that so far, robot was semi-simple. But we think the case robot non-simple will work as well. And this is actually ongoing work of Yon Shanhu and Ao Han Mong. So of course we need some genericity assumptions, but robot will be non-simple. And finally, one other thing we hope to get. So maybe before I, so we prove that the Gelfand Kail 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 Kail of dimension is F. But of course, if you add multiplicities coming outside of V, finitely 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 R bar. But this way this won't change the Gelfand Kail of dimension. So we shouldn't need these multiplicity one assumptions and so on to prove, in fact, in the end, the Gelfand Kail 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 add-als of the cross outside of V. So we, I do the same thing as I did at the very beginning to define pi of R bar, but I only take the inductive limits over open compact subgroup of GF to Fv with a fixed prime to V level. And then I take the R bar is a typical part. So this is a pretty bigger than pi V of R bar, many, many copies of pi V of R bar, but we hope to prove that the Gelfand Kail of dimension is still F. So of course, then we have to deal with things which are not anymore multiplicity one. Okay. So I guess, I guess this, I guess I'm done. Okay. Thank you. Thank you, Christophe, for this very nice lecture. So we have time to take few questions. Are there any questions or comments? So when you take the home from R bar to something, yes, a homology, what do you know about this homology? Like, can you tell sub questions which are R bar, but not sub? Okay. Yeah, I don't know. We only take our bars as a sub. You are, you are right. But I don't know about other kind of things you can do. Of course, this is infinite dimensional whether an R bar is two dimensional. So indeed, it could be that there are things of which are not as a sub, but then I don't usually, I mean, in that setting, when usually consider such things and, and we're very happy to be able to prove something about this. Are there other questions? Yeah. Daniel has a question. So he, Yes. Okay. Yeah. He wrote it. Okay. So how about higher powers of MG? So 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 write it. I think that from 5.0 about MK by some induction 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 so far, it was not clear to us that we would gain so much from 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're 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 little one K of the dimension, so you have, you take ratio and it's bounded by. Ah, okay. So, yeah, in the very beginning. So now you know this existence. So, so you can, you can take the limit. So you can use, 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 can, you can take the limit to, we just to end. Oh, yeah. And can you say something about it? No, I mean, in fact, this, this is not exactly the definition we use. We use the definition in terms of a Auslander books one theory, things like that. So we, I don't know. So you are asking whether this thing is maybe just has a limit instead of just being bounded, right? Yeah, yeah. So I'm not going to go. We don't know anything about that. Oh, thank you. Okay. So if there are no other questions. Just another, another thing of the Kirillov dimension. I am not so, so you have some compact module, but of course you can dualize and does it, does it Kirillov dimension comes from the fact that somehow after your 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 the, the, the, the, you are saw us this thing. Yes. Yes. And so you can look at the dimension, exactly the sense of non commutative analog of dimension in the terran rings. Yes. And this is the help on Kirillov dimension. Yes. So in analogy with the theory of the Hilbert function and so on the community of case, you expect that, and in fact, in this case, Lazar and so on, you mean 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, this kind of people saw us. So I think, well, I think, so I forgot a little bit, but I think this, yeah, you know, if you know that the dimension, if you know the dimension, you know, dimension indeed, you know that this must be something like a polynomial of degree in N of degree. So this dimension of degree, the Gelfon, you know, dimension plus one, maybe, or let me see if the Gelfon, if the Gelfon, your dimension is zero, which means that this thing is bounded for any N, which means that then by V is finite dimensional. So this is a constant polynomial. Yes, this must be this. So I think you can prove that this is one of the aspect of Gelfon, your dimension, if I'm not mistaken, maybe you can correct me that this dimension is actually a polynomial for N big enough. It's actually a polynomial in N of degree Gelfon, here of dimension plus one. No, no, but it cannot be because you put P to the N in the denominator. Oh, no, no, no, I mean the, 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, then you're a little bit more than 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. Goodbye. Thank you.
 Okay, so hello everyone. So this is the last lecture in the Paris-Bijine Tokyo 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, and Sushi Shihou, Takeshi Tsujin, from Bijin, Yon Shun Hu,  Ye Tian, and Wiju Jiang, and from Paris for this overview. I would like also to thank  four more organizers, Christophe Bray, Ariane Lizar, and Yichal Tien. And it's my great pleasure  to introduce the last speaker, Christophe Bray, who will speak on modular representations of G2L  for Anonymous My Fair. 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, Yon Shun Hu, Stefan Morat, and Benjamin  Schraen. 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 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 would 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 un-ramified, 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 GL2FV over F, which are associated to R bar,  where FV is the completion of F at V. Okay, so I want precisely to define the representation  of GL2FV 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 Schimuracker 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, you take the inductive limit of the H1 et al of these  Schimurack curves, which 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 of this Galois representation. Of course, there's a Galois action because it's  et alcomology. 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, well, 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 JL 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 tensor product decomposition, as a decomposition as a  restricted tensor product of smooth DW cross representations over all the, the finite places  double. It's called a, in the classical case, it's a node result due to flat, flat. It is  conjectured here, I guess, it's no conjecture now by I think buzzer, diamond and javies,  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 always already been checked  that in that case, the local ad hoc factor that I'm going to define coincides in that, 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 type 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  metabasitity 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  problems like this. So this local factor was defined. So we don't know it is local. I mean,  it still only only depends. It's a gel to every representation, but it prior fully depends on  R bar. So it was defined in the paper myself with Fred diamonds, he gets 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 pi 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 one. 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.  And I'm more precisely the case where F equals Q and these 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 Colmez of myself, Kizin,  of Laurent Berger and other people. It was 10 years ago. So in particular, we know the following  three things on pi V of R bar. We know that the Gelfon Kirillov dimension is one. I will recall  just afterwards what the Gelfon Kirillov dimension is. We know that pi V 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 our bar to the decomposition group at V, our bar V. And I should  mention before defining the Gelfon Kirillov dimension that this theorem, I guess, should  infile 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 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  Gelfon 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,  just one plus P to the N M2 OFV, so M2 is the two by two matrices, which is an open compact  subgroup. And KV is the maximal compact open subgroup, gel to OFV. Okay, so we have all these  congruent subgroups. And then here's the definition of the Gelfon Kirillov dimension.  So I guess it is due to Gelfon Kirillov, but this precise definition can be found in a recent paper  by Emehrhorn 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 Pi V, which is  between zero and the dimension of the PIDK analytic group KV as a ZPIDK 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-Hilloff 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 GL2 QP. 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 be dot 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 is, will be directed 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 Iwaza  algebra of K of one, modular 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, it was our algebra of K mod Z of one, only of K of one mod Z of one.  Okay. So maybe I should recall that Z of one are trivially owned by V of gamma. This comes from the  condition of the central character. That's why in the stock everything will be modular Z of one.  And then here is the one state, one nice statement, which is known in that situation for arbitrary FD  and robot. 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 5 of our bar under the first congruent subgroup K of one. So this is of course, a finite dimension  or representation. And this is an, and it has an action of K mod K of one, which is gamma. So this  is a finite dimensional gamma representation. So it's a tiny, tiny piece of pi V of our bar.  So it is also the kernel of pi V of our bar 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  pi V of our bar was not so easy to determine. And it is expected explicitly known in particular,  it is local. It only depends on our bar V. And most importantly, for, for me, it is multiplicity free,  meaning as a representation of gamma. So all the irreducible constituents are distinct.  Of, yes, and that's will be the thing that I'm going to use in the sequence. So this theorem was  first proven in the case of the uOV of the property will be by emerton G and savvy. This paper that  already mentioned that I will mention again in the stock, they made the main breakthrough to prove  this result. And they used patching filters. That was the main tool they used. And then it was  generalized by three kind of works first the paper in a, I don't know, I think it's chronological.  Yes. The paper by Daniel Le, Stefan Moran and Benjamin Schrin. Then some work of Yon Chan-Hu  and Aaron Wong. And then another paper by Daniel Le. 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 James.  Okay. So we have this multi PC free result. I will come back to this theorem two later in the talk.  So it is important for this talk. So I should now make clear that if TV is not yielded to QP,  apart from the serum, which is not exactly what we had for jail to QP anyway, none of the  statements in serum one are known. So let me recall to you that these statements were the  Gelfon carry love, 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 our Barbie. I need some precise generosity. I put hypothesis on  our Barbie 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 two F. 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, guess you all know what are these fundamental characters. So first I will assume  that our be very semisimple and I will not read by robot. So till the end of the talk,  our Vivaal now is semisimple. So I should mention that in all these questions about  serweight 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 semisimple case  right afterwards, but afterwards. So I assume it is semisimple 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 robot in terms of sers 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 eight 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 technique. So they know what is going on  at this infinite level, infinity, and then they mod out to get down to 5-year-old 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 b 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 F. But we have the Galifant-Kirilov dimension.  So the rest of the talk, which 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, one which I call the first one and a second one,  which will come 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 it to you right away. It was this way. Sorry,  this way. It was this, when you take the kernel of pi b of R bar for the maximum ideal for the  Wasawal-Dibrov's first congruent subgroup, you have something which is MULK-Pistifri 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 MULK-Pistifri. So you see, we need general city assumptions for this, because in general,  it is not. It's not going to be MULK-Pistifri. But for the moment, we assume this, we need this  MULK-Pistifri thing. So that's the first intermediate theorem. And the proof of it is  following the same techniques as for the proof of theorem 2, in particular by Merton G and Savit  and the followers. In particular, we need patching filters, 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 we concern the Iwo is a  group, not the maximal compact K. So let me recall first that the Iwo is the matrices in K that are  upper triangular modulo P. So P here is my uniformizer, because everything is in Ramifac,  Fv is in Ramifac. And I of one is a prop Iwo, so it is the group of matrices that are upper  unipotent matrices. And I really not, as I did for M i, which should be the maximal ideal of the  Iwo algebra, of the prop Iwo Rii modulo zero one. And the corollary we are going to use  is that if you consider now 5v of Abba, and then you take the kernel by this maximal ideal to the  cube. So it is a representation of the Iwo Rii, and it is multiplicity free. So if you take an  irreducible representation, a smooth irreducible representation of the Iwo Rii in characteristic  P, then the prop Iwo Rii 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 is a finite torus, which is an abelian group and of cardinality  prime to P. So the irreducible representation of the Iwo Rii over F are just characters.  So this statement means that all the characters that occur as sub-quotions of this representation  are all distinct. So you see that you of course need genericity assumption for that.  And we are going to use this corollary. Now the second intermediate theorem  is the following, which is entirely on the Iwo Rii side. So the first, I mean, apart from this  corollary, the first intermediate theorem will be entirely on the, somehow, k and k of one side.  And the second intermediate theorem is entirely on the Iwo Rii side. It is the following. Take  pi v, which is any smooth admissible representation of the Iwo Rii mod Z of one over F, such that  the kernel of pi v by this ideal mi to the cube is multiplicity free, as we had, as we know in the case  of pi v of Arba. But here, this is any pi v. Then in that case, the get form kilo of dimension of pi v  is smaller than F. Okay, so we recall that the get form kilo dimension is something asymptotic  for the compact open subgroup. So I can perfectly, it is perfectly defined for a representation of  Iwo Rii. Okay, and then directly follows from this previous corollary in this theorem that  the get form kilo dimension of pi v of Arba is smaller than F. Okay, and by G Newton 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 the 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 by pi v was the strange symbol, the algebraic dual of pi v. Okay,  and then of course, if so it is a module of the Iwo Rii, the Iwo Rii algebra of the POPI Rii.  So when I mod out by the maximal ideal of this, it was a algebra, I get the dual of the invariant  and the Iwo, which is a finite dimensional representation, a finite dimensional representation  of I mod I of 1. So it is just a bunch of characters, the direct sum because I of I of 1 is prime to  pi. So we have certain characters, sky and fire, finitely many, which are what they are, which  are all distinct by assumption. Let me denote now by projective proj kai alpha, the projective envelope  of this character kai alpha and the category of compact. So here it is truly the Iwo Rii,  the Iwo algebra for the Iwo Rii group, not the POPI Rii, but in fact it is just the tensor,  you take the Iwo algebra of I of 1, mod I of 1 and you tensor by kai alpha, and then the Iwo Rii acts  on this. And this is the projective envelope of kai alpha. So we know that kai 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 kai alpha already appears in the quotient by mi, it doesn't appear in the kernel.  And then using this, it is not difficulties formal using 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, yes, one can prove there exists for each alpha, i equivalent maps, h alpha from two f copies  of projective envelope of kai alpha to itself, just one copy, such that we have the following property.  First, the image of h alpha is inside mi to the square of kai alpha. The induced map  h a kai alpha mod mi to f copies to mi square mod mi cube is injective. And finally, and most  importantly for us by the dual will be a quotient of the direct sum of the kukenol direct sum of  alpha of the kukenol 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 easy stuff on, it was our  algebra. So you see that theorem five, the one bounding the Gelfon Kirillov dimension,  I mean, first we obviously have that the Gelfon Kirillov of pi v will be smaller than the maximum  of alpha because of the last statement here of the Gelfon Kirillov, this kukenol, 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 you dualize back to get back on the side of smooth  admissible representation of the e-worry. And you can compute the Gelfon Kirillov dimension of  this kukenol, 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 v. But in fact, it is not very difficult to prove that  the Gelfon Kirillov dimension of such a kukenol is smaller than F. And this ultimately boils down  to calculation in the gradi drink for the powers of the maximum ideal of this, it was our algebra.  And it turns out this credit, 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 Kirillov back. Okay, so I should mention before I switch, so that's  the end of the second interim. 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, not for this one. But for the first one, there is an existing  strategy, which is the one of a Martin G. Sabit, which we are going to end and the followers,  which we are going to push one step further. Okay, so now we leave the world of Iwo Ori 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 is a nearly simple representation of gamma  over F, finite dimension of course. And I will denote as I did for characters,  proj k sigma, the projective envelope of sigma and the category of compact modules over Z was  a algebra of k. I need k here. So this is an infinite dimensional representation, which  if you take the, if you dualize 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  that I recall it is here. This is this one, the kernel of pi of r bar for m and k to the square  is multiplicity three. So the irreducible constituents are cell weights and we want  them to be all distinct. So in particular, we certainly want, we want this 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 r bar. So such that the home k sigma to pi v of r bar 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. So now sigma from now on, sigma will be a cell  weight of robot. And the main tool for that will be the patching function m infinity of  m atom g subit, 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 wouldn't, I mean, this would be a bit too  too technical and would require too much time. But let me just say this is an exact covariant  phonter from a continuous representation of K over finite type Wf modules. So Wf is the fit  vectors. Well, there's an assumption with central character that you can forget here.  To finite type r infinity modules, which of course satisfies several properties in terms of support  when you apply it to some types and so on, which I, if you want to know them, you can check the  paper of m atom g subit. So here r infinity is a usual patch deformation ring, which in our  situation, because of our denerity assumption, will be a full power series ring over the v vectors.  So of course, the swanter is, yeah, depends on 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 of its many properties that I will recall when I use them on the SQL of the talk. And they are  extremely useful, of course. Okay, so I will now restate the thing we have to prove in terms of  the patching phonter. So somehow we are going to lift everything to infinity because it seems  impossible to prove this directly. So let me denote by m infinity gothic m infinity, the maximum  idea of this local ring. It's power series ring. And let me take v, which is the same as  the final dimension 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 best local factor v in terms of the dual of m infinity  of v applied to this v here, more than the maximum ideal of our infinity. So recall that m infinity  of v is a finite type r infinity module. So when you mud out by maximum ideal, it is a finite  dimensional aspect 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 is free path, which is most important follows from the fact that  this dimension is one, which equivalently is the fact that the r infinity module m infinity of  this posh k sigma, but m k 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 notice,  of course, then when you mud 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 m k square.  So this is now what I am going to do in the rest of the talk is to give you an idea how one can  prove the cyclicity. So far, I mean, this prediction to the cyclicity is not due to us. This is  something that is due to MF and G Savit and the followers. So there's no new idea. Now this is,  now we really start to be analyzing this representation. So first, there's something  you can consider is the kernel that you can mud out by mk instead of mk square. But if you mud  out by mk, then you are back in the world of gamma representations. And so it is actually the  predictive envelope of the Savit sigma in the category of gamma representation over F. Okay.  But here we do not mud out by mk. We mud out by mk square. So it's not any more representation  of gamma. So we have to understand this guy. And we can, I mean, 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 acts on sim2f to the square. If you fix an embedding, fq into f, which I do,  take an arbitrary embedding, then I twist by minus one, that to the minus one. And everything  is for the, I mean, is using the embedding fq inside f, which is tau. So I put tau here.  And I have as many such algebraic representation as I have such embeddings, which is f. I have  f such embeddings, f liter f. Okay. Then you can prove that proj k sigma with mk square as a k  representation is an extension of two gamma representation. You have proj gamma sigma as a  quotient. And as a subrepresentation, you have a direct sum of the all embedding tau  of this proj gamma sigma, times by v2 tau. Okay. And this is a non-split extension for all the  push forward, 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 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 with mk square, which I will call q tau for each  embedding tau. So this is the unique quotient of proj k sigma by 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  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. Okay.  Okay. So to proceed to prove this, so I recall that we want to prove this zero.  m infinity proj k sigma mk square is a kick. So we are going to apply m infinity to all these  productive things. But 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.  Okay. Because then we will be able to relate it to Galois representations and Fontaine's theory.  So that's why we lift it. But yes. Sorry, I have a question. Yes.  So we mentioned tensor products, but 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 not out.  Ah, because of this, okay, because you are working with  modulo mk square. Oh, yeah.  Then you need mod mk square in the, in the, otherwise, indeed, this is finite,  this is infinite dimensional, but mod mk square, because this is the dual of an admissible  representation. Okay. Okay. This is finite dimensional.  And, and this tensor product is, ah, okay. This is also finite dimensional.  Yeah. Everything is finite dimensional. Okay. Okay.  Okay. Basically, till the end of the talk, except the very last, the two last two slides,  everything will be either finite dimensional over F or a finite rank over the V vectors.  Free of finite rank over the V vectors. So this is what I need to know here.  Ah, 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 project k sigma, but mk square.  So it is easy to lift project gamma sigma because actually there's a unique representation of gamma  lifting project gamma sigma as a free module of a WF. So this is, you know, this is an old  result due to Broward, I guess, which you can find in the search book. So what's the  representation linear, the group, for instance. Okay. It's also easy to lift. Sorry. Oops.  No, no, I don't want. Okay. It's also easy to lift the algebraic part.  As a, 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 the two of all FV and all of the embeds into it, which is the ramified embeds into WF  via the embedding of FQ into F. Okay. So this is, and 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 much P, then you get the tons of product.  This one V2 tons of proj gamma sigma, which is a direct sum of proj of these projective here.  Okay. We are not, we do not want this. We want to find Qtow. So Qtow, he's,  is, you take the same projective envelopes, same cell, 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 tons of 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. Here. And the second result is that now we, we, from this, we can get a 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, we reduce it much P to proj gamma sigma. And we embed it  diagonally into F copies of proj gamma sigma. That's for the definition of this map on this  direct sum. Now the decision, the definition of this map on this direct sum is just that L2tow  reduces mod P to L2tow mod P, which surject onto proj gamma sigma. Okay. Because here the  suggestion 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 thing when you invert P. And this lattice mod P  is exactly the projective envelope of sigma with mk square. So we are going to apply the patching  pointer to all these guys. 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 Daniel Le, Stéphane  Mourabien, Germain Schran, and Yon Chanoua, that m infinity of this proj gamma sigma is  cyclic. Proj gamma sigma, the lift of proj gamma sigma. 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 V bar. I mean, sigma is a cell weight of a semi-simple representation of Galois F V bar mod  Mv. And 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 here it is a quotient of r infinity mod P, and here it is a quotient of r infinity.  And well, so I don't know how much time I have left.  Yeah, I have 14 minutes. Thank you. So let me say that the techniques to prove this are  standard with respect to what is already in the papers by Hamerton, Jisavi, Daniel,  Stefan, and so on. So maybe I'm not going to insist on this. The techniques are not new. Maybe  this is a standard devisage. And okay, let me skip this.  So to proceed to the next step. So the next step is the foreign. So remember, we want to,  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 L to 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. Here,  you've got three WF 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.  M infinity of posh gamma sigma, which we know is cyclic. M infinity of L tau two, 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 where you add all the embeddings, direct sum of all embeddings  here can be reduced to this case by just an induction. So 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 I will explain why this fiber product is cyclic. And here we enter the world of  gallery representation. So let me denote by rv, which is the r square rho bar. Rho bar is r bar v.  This is our local gallery representation, semi-simple. This is the local notary ring  parameterizing frame deformations of rho bar in the sense of kizin, laser. So there are no  conditions except there's a condition on the determinant that I will forget here. So here's  what follows from previous cyclicities that I just mentioned. So first, 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 rv. 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 gamma sigma is a quotient  of r infinity. And in fact, because of this variable, these patching variables play no role  at v, we know it is a quotient only on rv. 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 the reduction of p here by exactness of m infinity. It is just rv  mod pj. Okay. But in fact, we know what rj, because now we can, we are in specific situation. We know  sigma is a is a ser weights of a semi simple robot. And we can compute things. Everything is quite  explicit. And in fact, we can prove that rv mod j exactly parametrizes potentially crystalline  lift of a robot of any time type type here is Bushnell Kutsco type whose reduction might be  contained the ser, the ser weights 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 rings 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  lift of robot with this type and this much it way here we consider all types or tame types,  meaning by the way, Tami means they are representational to fq in characteristic zero. So level  level zero if you like, whose reduction would be content in sigma take all these types. And we  know actually this is exactly the quotient we have. So this is the place where I think  moderate these statements are some somehow hidden, because moderate statements 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.  Likewise, for this guy, except we have cell weights, hot state weights two minus one at the  embedding time, which of course are coming from the algebraic parts that the previous story.  So we again compute everything explicitly. So it's a little bit more complicated because now we  have to deal with such state weights two minus one that is up to twist our shapeway three comma zero.  So the computations are more difficult, but it can be it can be done can be done even by hand.  And likewise, the single type deformation rings with these high shaped weights are also domain,  so we can prove it and oops and 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 into proof 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 to is exactly P comma J. And for this,  it is enough to prove to prove that P belongs to J plus J to. So what we know here is that  a period J plus J to contains a power of P. But we have to prove that it contains P really and not  and not only P cube and so on. So in other terms, this is something like we have to prove that the  the potentially crystalline representation here with H8 weights zero and everywhere.  And the potentially crystalline liftings here with H8 weights outside two and two minus one  at all 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 are done. And this, this is something you can do 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 Gelfand-Kirilov business,  which, which, well, was sort of nice for me. It is an application to the Piedic Langlands program.  So it is based on the following theorem, which is a theorem of Dotto and Daniele,  which itself builds on work of Karajani, Merton, G, Gertie, Pascunas and Sheen.  And it has to do with big patch modules. So, so far, I was considering, I was patching  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  with this M infinity. Everything was a finite rank over infinity and so on. But it turns out,  you can also patch the full dual of pi V of R, which is, of course, infinite dimensional now.  Okay. And of course, this is not anymore, this is something which is a finite generative of R  infinity, double bracket, G2 of O, FV, which is k. But this is not anymore finite generative of R  infinity. And it has a compacted production of G2 FV. Okay, so that one can do this. And this is done  in the paper, a recent paper by Dottwin, building on previous work, but here they exactly do the  thing we need for this local factor and so on. And so the corollary of our main result is the  following, which was known. I mean, we, it's not new that if we had the GFK dimension,  and this would follow, but it's nice to recall it. So take any map from R infinity to O, any  specialization somehow of double F algebra, where E is a finite extension of QP, so containing WF.  Then the corresponding specialization and M infinity, M infinity, tensile R infinity OE,  except you have to dualize back, back. So here this says, I think,  shek of dual, something like that. I mean, you have to be careful about duality.  And you invert P. Well, this is nonzero. And I mean, then it is a missible unitary  continuous presentation of the FV over E, lifting by V of R bar. It is a banner space,  which has a unit ball, which is preserved by the G2 action and which lifts by V of R bar.  But the thing is it is nonzero. 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 nonzero. But this follows from the Gelfand Kail of our main  result, together with a result that M infinity is a quen makkoli over the stone commutative ring.  So here quen makkoli is in the sense of our slender books one and so on. There's only one X i,  which is nonzero, which is a result of G Newton. And this implication is a so-called miracle flatness.  So in noncommutative setting. So we have this result. So I think now it is almost time. So it's  good. I just have one slide. So I should mention that so far, robot was semi-simple. But we think  the case robot non-simple will work as well. And this is actually ongoing work of Yon Shanhu  and Ao Han Mong. So of course we need some genericity assumptions, but robot will be  non-simple. And finally, one other thing we hope to get. So maybe before I, so we prove that the  Gelfand Kail 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 Kail of dimension is F. But of course, if you add multiplicities  coming outside of V, finitely 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 R bar.  But this way this won't change the Gelfand Kail of dimension. So we shouldn't need these  multiplicity one assumptions and so on to prove, in fact, in the end, the Gelfand Kail 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 add-als of the cross outside of V.  So we, I do the same thing as I did at the very beginning to define pi of R bar, but I only take  the inductive limits over open compact subgroup of GF to Fv with a fixed prime to V level.  And then I take the R bar is a typical part. So this is a pretty bigger than pi V of R bar,  many, many copies of pi V of R bar, but we hope to prove that the Gelfand Kail of dimension is  still F. So of course, then we have to deal with things which are not anymore multiplicity one.  Okay. So I guess, I guess this, I guess I'm done. Okay. Thank you. Thank you, Christophe,  for this very nice lecture. So we have time to take few questions. Are there any questions or comments?  So when you take the home from R bar to something,  yes, a homology, what do you know about this homology? Like, can you tell sub questions which  are R bar, but not sub? Okay. Yeah, I don't know. We only take our bars as a sub. You are,  you are right. But I don't know about other kind of things you can do. Of course, this is infinite  dimensional whether an R bar is two dimensional. So indeed, it could be that there are things  of which are not as a sub, but then I don't usually, I mean, in that setting, when usually  consider such things and, and we're very happy to be able to prove something about this.  Are there other questions? Yeah.  Daniel has a question. So he,  Yes. Okay. Yeah. He wrote it. Okay. So how about higher powers of MG?  So 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 write it. I think that from  5.0 about MK by some induction 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 so far, it was not clear to us that we would gain so much from  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're 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  little one K of the dimension, so you have, you take ratio and it's bounded by.  Ah, okay. So, yeah, in the very beginning. So now you know this existence. So,  so you can, you can take the limit. So you can use, 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 can, you can take  the limit to, we just to end. Oh, yeah. And can you say something about it? No, I mean, in fact,  this, this is not exactly the definition we use. We use the definition in terms of  a Auslander books one theory, things like that. So we, I don't know. So you are asking whether this  thing is maybe just has a limit instead of just being bounded, right? Yeah, yeah. So I'm not going to go.  We don't know anything about that. Oh, thank you. Okay. So if there are no other questions.  Just another, another thing of the Kirillov dimension. I am not so, so you have some compact  module, but of course you can dualize and does it, does it Kirillov dimension comes from the fact  that somehow after your 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 the, the, the, the, you are saw us this thing. Yes. Yes. And so you can look at the  dimension, exactly the sense of non commutative analog of dimension in the terran rings. Yes.  And this is the help on Kirillov dimension. Yes. So in analogy with the theory of the Hilbert  function and so on the community of case, you expect that, and in fact, in this case, Lazar and  so on, you mean 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, this kind of  people saw us. So I think, well, I think, so I forgot a little bit, but I think this, yeah, you  know, if you know that the dimension, if you know the dimension, you know, dimension indeed, you  know that this must be something like a polynomial of degree in N of degree. So this dimension of  degree, the Gelfon, you know, dimension plus one, maybe, or let me see if the Gelfon, if the  Gelfon, your dimension is zero, which means that this thing is bounded for any N, which means that  then by V is finite dimensional. So this is a constant polynomial. Yes, this must be this. So  I think you can prove that this is one of the aspect of Gelfon, your dimension, if I'm not  mistaken, maybe you can correct me that this dimension is actually a polynomial for N big  enough. It's actually a polynomial in N of degree Gelfon, here of dimension plus one.  No, no, but it cannot be because you put P to the N in the denominator.  Oh, no, no, no, I mean the, 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, then you're a little bit more than 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. Goodbye. Thank you.
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