Okay. So, hello everyone. So, this is the last lecture in the Paris-Bijin 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, Atsushi Shiho, Takeshi Tsujin, from Bijin, Yunxuan Hu, Ye Tian, and Wei Zhejiang, and from Paris, Fabrice Ogozou. I would like also to thank former organizers, Christophe Breuil, Ariane Bizar, and Yichao Tian.

And it's my great pleasure to introduce the last speaker, Christophe Breuil, who will speak on modular representations of GLQN for un-monified N. 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 Aertisch, Yunxuan Hu, Stéphane Morin, and Benjamin Schramm. Okay, so the contents of the talk will 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 will be my coefficient field for all representations, either on the GL2 side or on the Galois side.

And I will assume it is big enough in the sense that it will contain all Hecke-Hagen values and so on. So that I don't have to worry about that. F will be a totally real number field where P is unramified. And I will fix V, a place dividing P, a place of F dividing P, which will be my fixed place till the very end of the talk.

I will only work at this place V. I will fix a quaternary 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 F is the completion of F at V. 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. But it's a bit technical. So first, recall that for any compact open subgroup of the finite adeles of the group D cross,

I have a Shimora curve, XK over F, which is a smooth, projective algebraic variety over F. And the first representation one can consider is the following smooth representation of these finite adeles over F.

First, you take the inductive limits of the H1 et al. of these Shimora curves with coefficient in F. This inductive limit being taken over the compact open subgroup K. So K is getting smaller and smaller in the inductive limit. And then I take the R bar isotypic part of this Galois representation.

Of course, there's a Galois action because it's et al. cohomology. And I assume it is non-zero. This is what I mean by being modular. I'm not interested in modularity questions here, although at some point they are hidden somewhere.

But I want to study something related to this representation, which of course I assume non-zero. So as I told you, I'm not instructed directly to this representation. I want to study representation of GL to FV. But the problem you see is that we do not know so far

if this representation pi of R bar as a restricted tensor product decomposition, as a decomposition, as a restricted tensor product of smooth GW cross representations over all the finite places W. It's called, in the classical case, it's a node result due to Flach.

It is conjectured here, I guess. It's a node conjecture now by, I think, Buzzard, Diamond, and Jarvis. But it is not known. So you cannot define a local factor at V just by using this. You cannot use this. So you have to proceed in another way,

which will be a sort of ad hoc way, and which will require some weak technical assumption on R bar. And let me just mention that if one day one is able to prove that there is a flat decomposition like this, then it has already been checked that in that case, the local ad hoc factor that I'm going to define

coincides in that case with the factor at V of such a decomposition if it exists. But one can define it directly. OK. 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. So p is bigger than five. R bar is absolutely irreducible, 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 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 with 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 a few Hecke operators at finitely many places different from V. So anything that is going on here

is at places different from V. We do not touch V. And the purpose of this representation is to get rid of multiplicities that are coming from places different from V. Because you will see in the rest of the talk, I'm going to use multiplicity one theorems. If I do not take this representation, I do not have multiplicity one. I have an artificial multiplicity different from one,

which maybe can be dealt with later on. But 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's still only depends...

It's a GL2FV representation, but it clearly fully depends on R bar. So it was defined in the paper, myself with Fred Diamond, I guess, 10 years ago. And then was generalized in the paper by Emerton G.N. Savitz that we mentioned again in this paper, in this talk. Okay, so pi V of R bar is a smooth admissible representation

of GL2FV 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 pi V of R bar. The first one, of course, is the case of GL2QP.

And then more precisely the case where F equals Q and D is GL2. And in that case, pi V of R bar is fully known. So this is a work of Emerton building on work of Colmes, of myself, of Guizin, and of 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 GL2QP dimension is one. I will recall just afterwards what the GL2QP dimension is. We know that pi V of R bar is a finite length as a GL2QP representation over F. And we also know that it is local

in the sense that it only depends on the restriction of R bar to the decomposition group at V, R bar V. And I should mention, before defining the Galle-Fonkerov dimension, that this theorem, I guess, should in fact hold as soon as FV equals QP, because then we have DV is GL2QP.

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 GL2, but FV is QP, but not in this generality. But I think it should be true. Okay, but in this talk, we are not going to be interested in GL2QP anyway.

Let me recall now the Galle-Fonkerov dimension. There are several definitions. I give to you 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, M2O FV, so M2 is the two by two matrices,

which is an open compact subgroup, and KV is a maximal compact open subgroup, GL2O FV. Okay, so we have all these congruent subgroups. And then here is the definition of the Galle-Fonkerov dimension. So I guess it is

due to Galle-Fonkerov, but this precise definition can be found in a recent paper by Emerson and Pascunas. So let pi V be any smooth admissible representation of KV of one over F. KV of one is the first congruent subgroup. Well, I could, it's an asymptotic definition, so I could even take KV of N

for arbitrary N. So there exists a unique integer, GK of pi V, which is between zero and the dimension of the PID kinetic group KV as a ZPID 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 would tend to zero. Okay, so you don't want this, of course. So very roughly, you can think about the Gelfand-Hilov dimension as an integer that measures the dimension of these finite dimensional vector spaces

when N is getting bigger and bigger asymptotically, roughly. So let me now recall some known results when we are not with GL2QP. So of course, much less is known. So I need a few notation.

Well, F will be the degree of my field AV, which I recall is unramified. Q is the cardinality of the residue field. And I will do that by K and K of 1 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 notations are quite standard, so I think you can remember them. This one maybe is less standard. K mod K of 1 will be denoted by gamma. This is just the finite group GL2 of FQ, where FQ is the residue field. And Z of 1 will be the center of K of 1. And finally, I need to call

MK, which is the maximal ideal of the Iwasawa algebra of K of 1 modulo the center Z of 1 over F. So maybe I should have called it MK of 1, but we don't use really the Iwasawa algebra of K mod Z of 1, only of K of 1 mod Z of 1.

Okay, so maybe I should recall that Z of 1 act trivially on pi V of Arba. This comes from the condition of the central character. That's why in this talk everything will be modulo Z of 1. And then here is the one

nice statement, which is known in that situation for arbitrary FD and Robert, I mean, as they are before. So when has the following theorem, which, well, let me state it, and then I'll say something about the names. So we are concerned with the invariant of pi V of Arba

under the first congruent subgroup, K of 1. So this is, of course, a finite dimensional representation, and it has an action of K mod K of 1, which is gamma. So this is a finite dimensional gamma representation. So it's a tiny, tiny piece of pi V of Arba. So it is also the kernel of pi V of Arba for the action of the maximal ideal of K.

And this finite dimensional gamma representation, even though you may think it is a small part of pi V of Arba, was not so easy to determine, and it is explicitly known. In particular, it is local. It only depends on Arba V. And most importantly,

for me, it is multiplicity-free, meaning as a representation of gamma. So all the irreducible constituents are distinct. Yes, and that 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 proper UOV, by Emmerton, G, and Sabic, this paper that I already mentioned, that I will mention again in this talk. They made the main breakthrough to prove this result, and they used patching functors. That was the main tool they used.

And then it was generalized by three kind of works. First, a paper in, I don't know, I think it's chronological, yes, a paper by Daniele, Stéphane de Moroy, and Jean-Marc Chran. Then some work of Yonshan Hu and Aoran Wang. And then another paper by Daniel Lee. And all these were built on my paper with Pascunas

of many years ago, which itself builds on the seminal paper by Bezard, Diamond, and Jarmus. Okay, so we have this multi-PC3 result. I will come back to this theorem too later in the talk. So it is important for this talk. So I should now make clear

that if DV is not here to PUP, apart from the theorem, which is not exactly what we had for DV to PUP anyway, none of the statements in theorem one are known. So let me recall to you that these statements were Gelfand-Kerilov, the finite lengths, and the fact that the representation is local. Okay, so now I want to state

our main theorem. So first I need some hypotheses on R bar V. I need some precise generosity hypotheses on R bar V that is stronger than the weak generosity hypotheses

I had as a running hypothesis in the beginning. So for this, I need the self-fundamental character of level F and 2F. Okay, so if I want to define them as I'm going to do, I need to fix embeddings because it's not very important. I mean, I guess you all know what are these fundamental characters. So first, I will assume

that R be very semi-simple. And I will be not read by a whole bar. So till the end of the talk, R V bar now is semi-simple. So I should mention that in all these questions about Serwitt 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-semi-simple case right afterwards, but not afterwards. So I assume it is semi-simple and I want some 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 rho bar in terms of

Serw's fundamental characters up to twist. So you have certain powers. Of course, that occurs. And I want the digits in the P expansion of these powers to be sort of very much in the middle. So between 8 and P minus 11, that's the bounds we need. So in particular, this implies

that P is bigger than 19 at the very end of this talk. Okay, so I should mention that we have not tried to optimize this genericity 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're fine with this. So P is large, bigger than 19. Okay, now I want to state our main result. It is the following. Under these assumptions, we have the Galleform-Kirilloch dimension, which is F.

Okay, so on F, D, and R bar, this is the assumptions as in the previous theorems, and on the row bar, row bar is R of E 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 row bar being semi-simple and sufficiently generic should be unnecessary. One should always have that Galleform-Kirilloch dimension is F. The second statement is that in the paper by G and Newton, they prove that the Galleform-Kirilloch is always bigger than F. They prove this using the patching techniques.

So they know what is going on at this infinite level, at infinity, and then they mod out to get down to pi V of R 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 row bar and even knowing the Galleform-Kirilloch dimension, so far we do not know if pi V of R bar is a finite length GL2 FV representation over F,

and even less if it is local, meaning only depends on the restriction of R bar to the local decomposition group F. But we have the Galleform-Kirilloch dimension. So the rest of the talk, which is, you see, will be longer than, if you look at the timing,

will be devoted to give you a fairly precise idea of the proof of this theorem. Yeah, some ideas on the proof. So we are going to use two intermediate theorems, one which I call the first one and the 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 Galleform-Kirilloch dimension. So the first one is the following extension of theorem two. So theorem two, let me record to you right away. It was this one. Oh, sorry. It's after this one. It was this, when you take the kernel of

of R bar for the maximum ideal for the Uazawa algebra of the first congruent subgroup, you have something which is MALC piece T3 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 M K 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 multiplicity free. So you see, we need generosity assumptions for this, because in general, it is not. It's not going to be multiplicity free. But for the moment, we assume this market,

we need this multiplicity free things. So that's the first intermediate theorem. And the proof of it is following the same techniques as for the proof of theorem two, in particular by Emerton G and Savit and the followers. In particular, we need patching functions, but it's technically much harder as you will see.

But this is not this theorem that we're going to use directly. We're going to use a corollary, which is not very hard to derive from the theorem, but it concerns the Iwaris subgroup, not the maximal compact K. So let me recall first that the Iwaris is the matrices in K

that are upper triangular modulo P. So P here is my uniformizer, because everything is in Ramey fact, if V is in Ramey fact. And I of one is a propi-UoE. So it is the group of matrices that are upper unipotent matrices. And I really note, as I did for K, M of I,

which would be the maximal ideal of the Iwaris algebra of the propi-UoE modulo Z of one. Okay. And the corollary we are going to use is that if you consider now pi V of r bar, and then you take the kernel by this maximal ideal to the cube,

so it is a representation of the Iwaris and it is multiplicity free. So if you take an irreducible representation, smoothly reducible representation of the Iwaris in characteristic P, then the propi-UoE acts trivially on it because it is irreducible. Hence, it is a representation of I mod I of one. But I mod I of one,

I of one is a finite torus, which is an abelian group and of cardinality prime to P. Okay. So the irreducible representation of the Iwaris over F are just characters. So this statement means that all the characters that occur as sub-coations of this representation are all distinct. So you see that you of course need generosity assumption for that.

And we are going to use this corollary. Okay. Now the second intermediate theorem is the following, which is entirely on the Iwaris 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 Iwaris side. It is the following. Take pi V, which is any smooth admissible representation of the Iwaris mod Z of one over F such that the kernel of pi V by this ideal M I 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 Gelfand-Kirchhoff dimension of pi V is smaller than X. Okay. So recall that the Gelfand-Kirchhoff dimension is something asymptotic for the compact open subgroups. So I can perfectly, it is perfectly defined

for a representation of the Iwaris. Okay. And it then directly follows from this, the previous corollary in this theorem that the Gelfand-Kirchhoff 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 with the stranger symbol,

the algebraic dual of pi V. And then of course, so it is a module of the Iwasawa algebra of the propi Iwaris. So when I mod out by the maximal ideal of this Iwasawa algebra, I get the dual of the invariant under I of O, which is a finite dimensional representation,

a finite dimensional representation of I mod I of 1, so it is just a bunch of characters. It's a direct sum because I of I of 1 is prime to p. So we have certain characters chi alpha, phi et al, which are what they are, which are all distinct, phi assumption. Let me denote now by

projective Proj chi alpha, the projective envelope of this character chi alpha in the category of compact. So here it is truly the Iwaris, the Iwasawa algebra for the Iwaris group, not the propi Iwaris, but in fact, it is just the tensor. You take the Iwasawa algebra of I of 1, mod I of 1,

and you cancel by chi alpha, and then the Iwaris acts on this. And this is the projective envelope of chi alpha. So we know that chi alpha does not appear in mi by the dual mod mi cubed because we use our assumption that pi v dual mod mi cubed pi v dual is multiplicity free.

It's a finite dimensional representation of I which is multiplicity free. And since chi alpha already appears in the quotient by mi, it doesn't appear in the kernel. And then using this, it is not difficult, it is 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, yeah, so when there exists for each alpha,

I equivalent maps hash alpha from two F copies of projective of chi alpha to itself, just one copy, such that we have the foreign property. First, the image of H alpha is inside mi to the square of chi alpha.

The induced map, roge a for each i chi alpha mod mi to F copies to mi squared mod mi cubed is injective. And finally, and most importantly for us, pi v dual will be a quotient of the direct sum of the coke anole,

direct sum of alpha of the coke anoles of all these alpha. So in all of this, we only use this non-GPC free things and universal properties of projective envelopes. I mean, and easy stuff and it was our algebra. So you see that theorem five, the one bounding the Gelfand Kirillov dimension. I mean, first we obviously have that

the Gelfand Kirillov of pi v will be smaller than the maximum of alpha because of the last statement here of the Gelfand Kirillov of this coke anole, except you have to dualize back. So here there's a hidden duality between discrete and compact modules. So you, here you are on the compact side, you dualize back to get back

on the side of smooth admissible representation of zero. And you can compute the Gelfand Kirillov dimension of this coke anole, 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 Gelfand Kirillov dimension

of such a coke anole is smaller than f. And this ultimately boils down to calculation in the graded ring for the powers of the maximum ideal of the Siwazawa algebra. And it turns out this, the coded ring was actually computed in a nice paper by Laurent Clozell. 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 Clozell at hand. And using a not so hard calculation, we get the Gelfand Kirillov bar. Okay, so I should mention before I switch, so that's the end of the second intermediate theorem. So, you know, it's not so hard, except that it took us a long, long, long time

to find this slide. Okay, the rest somehow is, and the reason is that the rest, there's already an existing strategy, but not for this one. But for the first one, there is an existing strategy, which is the one of Emmerton G. Savit, which we are going to, and the followers, which we are going

to push one step further. Okay, so now we leave the world of UoOe and we enter the world of the maximal compact. So this is a world of cell weights and all these things. So let me recall that cell weights is an irreducible representation of gamma over f, finite dimensional, of course.

And I will denote, as I did for characters, Proj k sigma, the projective envelope of sigma in the category of compact modulus over z was the 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 unmissable. And the reason we introduced this projective envelope is that it is enough to prove this by just using the universal property of Proj k sigma. Let me recall that the first intermediate theorem, I recall it, is here. This is this one.

The kernel of pi u of r bar for the mnk to the square is relativistic free. 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.

And in fact, it's even enough to prove this statement for some specific cell weights, which are called cell weights of rho bar, which are those cell weights, which we know already embeds into pi v of r bar, 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 one. So now sigma, from now on sigma will be a cell weight of rho bar. And the main tool for that will be the patching functor M infinity of M atom G sub it,

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 I wouldn't, I mean, this would be a bit too technical and would require too much time. But let me just say this is an exact covariant functor

from a continuous representation of K of a finite type WF module. So WF is the fit vectors. 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 sub it. So here R infinity is a usual patch deformation ring, which in our situation, because of our generosity assumption, will be a full power series ring over the V vectors. So of course,

this functor is, 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 in the sequel 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 functor. 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 maximal idea of this local ring. Its power series ring. And let me take V, which is any finite dimensional representation of K. So the GL2 of V over F. Then from one, one thing we get from the properties

of M infinity is the following equality. You can compute the K invariant homomorphism from V to pi V of r bar, this local factor of V, in terms of the dual of M infinity of V applied to this V here, mod the maximal ideal of r infinity. So recall that M infinity of V is a finite type

r infinity module. So when you mod out by the maximal ideal, it is a finite dimensional F vector space. And I just take the dual. Here also this is finite dimensional because the representation is admissible. So we want to prove the theorem four, the multi piece,

the theorem four, the multi piece three part, which is the 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 need one generator, or in other terms, that is isomorphic to a quotient of r infinity. So if you know this, of course, then when you mod out here, you get one dimensional vector space. So the dual is also one dimensional and you are done. I mean, you are done for V equals Posh k sigma,

but 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, so far, I mean, this prediction to the cyclicity is not due to us. This is something that is due to M. F. 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. This is the kernel you can model by M k instead of M k square. But if you model by M k,

then you are back in the world of gamma representations. And so it is actually the predictive envelope of the same weight sigma in the category of gamma representation over F. But here we do not model by M k. We model by M k square. So it's not anymore 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 V two tau, the following algebraic representation of gamma. I recall that gamma is GL two F Q as a residue field.

So GL two F Q F Q acts on SIM two F to the square. If you fix an embedding F Q into F, which I do, I take an arbitrary embedding. Then I twist by minus one det to the minus one. And everything is for the, I mean, is using

the embedding F Q 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 little F. Okay, then you can prove that Proj K sigma but MK square as a K representation

is an extension of two gamma representation. You have Proj gamma sigma as a quotient. And as a sub-representation, you have a direct sum of all embeddings tau of this Proj gamma sigma times by V two tau. Okay, and this is a non-split extension

for all the all the push void, all the direct summands here that you can consider. Let me just also mention that we know what this tensor product is. I mean, when you tensor 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 embedding 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 over MK square, which is a non-split extension here. So this is a push forward. I cancel anything that is not at the fixed embedding tau. And for the fixed embedding tau, I have this tensor product, which is direct sum of two

and I cancel 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 theorem.

M infinity Proj K sigma MK square is cyclic. So we are going to apply M infinity to all these projective 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.

Because then we will be able to relate it to Galois representations and Fontaine's theory. So that's why we lift it. Yes? Sorry, Gemma has a question. Yes. Sorry? You mentioned tensor products, but can you tensor product those things only when one factor is

finite dimensional? Or you have also some completed tensor products when... No, everything is finite dimensional or here? I mean, I'm not... Ah, because of this. Okay, because you are working with a modulo MK square then you need mod MK square in the

otherwise, indeed, this is infinite dimensional, but mod MK square because this is the jewel of an admissible representation. Okay, this is finite dimensional. And this tensor product is... Okay, this is also finite dimensional? Yeah. Everything is finite dimensional. Okay.

Basically, till the end of the talk, except the very last, the 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. I'm going to do here. I'm going to lift the scale representation

as a free WF value with a continuous action of k, which reduces mod p to proj k sigma with MK square. So it is easy to lift proj gamma sigma because actually there's a unique representation of gamma lifting proj gamma sigma as a free module of a WF. So this is, you know, this is an old result

due to Brauer, I guess, which you can find in the sales book. So what's the representation linear of the group sigma 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, 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 of 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 gN2 of OFV and OFV embeds into, which is the ramified embeds into WF via the embedding of FQ into F. Okay, so this is,

and here is the first thing one can prove. So I need a comment. If you take this tonsor 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 tonsor product, this one, v2 tonsor project gamma sigma, which is

a direct sum of project, of these projective here. Okay. We are not, we do not want this. We want to find Q tau. So Q tau is, you take the same projective envelopes, the same, except that you put an extension 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 tonsor product of these obvious lattices, which is not a lattice, but which exists such that when you reduce it much p, you exactly find this non-split extension in the right order. Okay. This is the first

result we proved. And the second result is that now we, from this, we can get a lattice lifting project k sigma with m k 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 project gamma sigma

and we embed it diagonally into f copies of project gamma sigma. That's for the definition of this map on this direct summand. Now the definition of this map on this direct summand is just that l2 tau reduces mod p to l2 tau mod p which subject onto project gamma sigma.

Because here the suggestion is here. And so for each embedding, you map it to one copy of project 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 3 over w f. Here, this is in characteristic 0.

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 m k square. So we are going to apply the patching from dirt to all these guys. And indeed we want to prove that M infinity of L

is cyclic. If we do this, we are done. So we know that already by previous work of Daniel Lee, Stéphane Desmois, Benjamin Chan, and Yung-Chan Wu, and Wang, that M infinity of this project gamma sigma is cyclic, project gamma sigma tilde, the lift of project gamma sigma.

Remember, I mean, we are in the, there are other work on this, but here we 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 m V. And the first thing one can prove is the following proposition

is that the R infinity module M infinity of L 2 tau mod p and hence by an application of Nakayama M infinity over L 2 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 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

Emmert and Gisavito, Daniel, Stephano Mora, Benjamin, and so on. So maybe I'm not going to insist on this. The techniques are not, are not new. Maybe I will, this is a standard devisage. And, okay, let me skip this.

So to proceed to the next step. So the next step is the following. So remember, we want, we want to, we are interested in L, which is the kernel of this direct sum to F copies of project gamma sigma. But before going to L, we are going to proceed step by step,

adding one embedding after the other, the other. And in particular, we start with L tau, which is the kernel, the same kind of kernel, except we only take one embedding L2 tau. So there's according in one copy of project 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 mod P have the common project gamma sigma quotient. So we take the fiber product. And now I will explain why M infinity over L tau is cyclic.

So by exactness of M infinity, M infinity over L tau is also a fiber product. M infinity of project gamma sigma, which we know is cyclic, M infinity over L2, which we know is cyclic over M infinity of project 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 where you add all the embeddings, direct sum of all embeddings here, can be reduced to this case by just an induction. Once we know this one is cyclic, we're going to add another embedding. We are going to have another fiber product and so on. So now I

will explain why this fiber product is cyclic. And here we enter the world of Galois representation. So let me denote by R v, which is R square rho bar. Rho bar is r bar v. This is our local Galois representation, semi-simple.

This is the local noetherian ring parameterizing frame deformations of rho bar in the sense of Kiesin, Maser. So there are no conditions, except there's a condition of the determinant that I will forget here. So here's what follows from previous cyclicities that are just

mentioned. So first we have R infinity, which I told you was a full power series ring over W over F. But in fact, before being a full power series ring over W over F, it is a full power series ring over R v this here, which in the particular case here because of our generated assumption turns out to

be also a full power series ring. But let me forget it here. So we know that M infinity approach gamma sigma is a quotient of R infinity. And in fact, because of these patching variables play no role at v, we know it is a quotient only on R v. So there's an ideal J such that it is isomorphic to this. Likewise 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 R v mod P J. But in fact, we know what R J, because now we can we are in specific

situation. We know sigma is a cell weight of a semi simple robot, and we can compute things. Everything is R v mod J exactly parametrizes potentially crystalline lift of a robot of any time type. Type here is Boushneh-Kutzko type, whose

reduction mod P contained the cell weight sigma. And we spot a hot shake 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 it is a multi-type deformation ring. I mean, we take all, we take several types, not just one. Usually you fix one type, you fix hot shake

weights, and you consider potentially crystalline lift of robot with this type and this hot shake weight. Here we consider all types, all tame types, meaning, by the way, tame means they are representation of FQ in characteristics zero. So level zero, if you like, whose reduction mod P contains sigma. Take all these types, and we know actually

this is exactly the quotient we have. So this is the place where I think modularity statements are somehow hidden because modularity statements are hidden in the support of these M-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 for these hot shake weights,

so the usual deformation ring for one of these types, we actually know it is a domain. We prove it is a domain. So since we know the support is a union of irreducible components for a fixed M-type, it must be everything. And now when we put them all together, we can derive that we must have the full check. Likewise for this guy, except we

have hot shake weights, two minus one at the embedding type, which of course are coming from the algebraic parts of the previous tool. So we again compute everything explicitly. So it's a little bit more complicated because now we have to deal with hot shake weights two minus one, that is up to twist hot shake

weight three comma zero. So the computations are more difficult, but it can be done, it can be done even by hand. And likewise, the single type deformation rings with these hot shake weights are also domains, 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, you need to prove it is a quotient of RV. If you know this, it will be cyclic, so meaning one generator of RV, and you will be done.

And to prove this, it's easy to see that you need to prove that J plus J tau is exactly P comma J. And for this, it is enough to prove that P belongs to J plus J tau. So what we know here is that a J plus J tau contains a power of P. But we have to prove that it contains P,

really, and not only P cubed and so on. So in other terms, this is something like we have to prove that the potentially crystalline representation here with hot shake weights zero and everywhere, and the potentially crystalline liftings here with hot shake weights outside tau and two minus one

at tau are as little congruent as possible. And this can be done by hand. explicitly, because we can, of course, if we want to prove this, we can check it mod P squared. If we prove this mod P squared, you are done. And this is something you can do by hand.

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 Geldfang field of business, which, well, was sort of nice for me. It is an

application to the pediatric language program. So it is based on the following theorem, which is a theorem of the Dotto and Daniele, which itself builds on work of Karayani, Emerton, Gerti, Pascunas and Shin. And it has to do with big patch modules.

So far, I was considering, I was patching things like hom K invariant homomorphism from some finite dimensional V to pi V of R, and maybe the dual of that, which was finite dimensional. And I was patching this as this M infinity. Everything was a finite rank of R

infinity and so on. But it turns out you can also patch the full dual of pi V of R bar, which is, of course, infinite dimensional now. And of course, this is not anymore, this is something which is finitely generated over R infinity double bracket

GL2 of dimension. So one can do this, and this is done in the paper, a recent paper by 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, it's not new that if we had the GL2 dimension, this would follow that. It's nice to recall it. So take any map from R infinity to OE, 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 times the R infinity OE, except you have to dualize back. So here this is, I think, a Sheikov dual, something like that. I mean, you have to be careful about duality. And you invert P. Well, this is non-zero.

And, I mean, then if it's non-zero, it is an invisible unitary continuous representation of GL2 FV over E, lifting by V of R bar, which is a banner space, which has a unit ball, which is preserved by the GL2 action, and which lifts by V of R bar. But the thing is, it is non-zero.

And to prove, the idea of the proof is that you need flatness. You need to prove that M infinity is flat over infinity, and if you know this, then you know that specializations are non-zero. But this follows from the Gelfand K-loof, our main result, together with a result that M infinity is a quen

Macaulay over this non-commutative ring. So here quen Macaulay over the miracle flatness in non-commutative settings. So we have this result. So I think now it is

almost time, so it's good that I just have one slide. So I should mention that so far, Rho bar was semi simple, but if we think the case, Rho bar non-semi simple will work as well. And this is actually ongoing work of Yongshan Hu and Aohan Mong. So of course we need some genericity assumptions, but Rho bar will

be non-semi simple. And finally, one other thing we hope to get. So maybe before I, so we proved that the Gelfand K-loof 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 proved that the Gelfand K-loof dimension is F. But of course, if you add multiplicities coming outside of V, finitely many, for instance, if you don't take exactly the right compact open subgroup and so on, you will get something like several copies of by V of R bar, but this won't change the

Gelfand K-loof dimension. So we shouldn't need these multiplicity one assumptions and so on to prove, in fact, in the end, that the Gelfand K-loof dimension is F. And maybe we can prove it. In fact, we hope to prove that at least for a suitable level K of V outside of V, compact open subgroup

of the finite ideals of D cross outside of V. So 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 GL2 F V, with a fixed prime to V level. And then I take the R bar isotopic 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 K-loof 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'm done. Okay.

Thank you. Thank you, Christophe, for this very nice lecture. So we have time to take a few questions. Are there any questions or comments? So when you take the home from R bar to something, homology, what do you know about this homology? Like, can you tell some questions?

which are r-bar but not sub? Okay, yeah, I don't know. We only take r-bar as a sub. 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 which are

not as a sub, but then I don't know. Usually, I mean, in that setting when usually I consider such things and we're very happy to be able to prove something about this. Daniel has a question. Okay, yeah, he wrote it. Okay, so how about higher powers of m g?

So, we think we can, okay, so let me go back to the, from this, if you assumed generosity enough, let me see, where is the yeah, I think, but we didn't write it. I think that from by view of m k, by some induction

business, we can probably get higher powers. We can probably go a little bit further by some kind of induction if you assume sufficient generosity, m k cubed, n k four, 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, m k 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 better generate, because

I mean, you have finitely many weights, and this is an infinite dimensional representation, so. Okay, so I don't see other questions. Do you see something? So, in the definition of this, you define k dimension, so you have, you take a ratio

and it's bounded by p. So, yeah, in the very beginning, so now you know this existence, so you can take the limit. So, do you know if it's converges or the meaning of the value?

Sorry, I'm not quite sure I understand the question. So, you can take the limit with respect to n. Oh. Yeah, and can you say something about the limit? In fact, this is not exactly the definition we use. We use the definition in terms of

Oslander-Buxbaum theory, things like that. So, 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 going to go, yeah. We don't know anything about that. I see, thank you.

Okay, so if there are no other questions. Another thing, I mentioned, I'm not so, so you have some compact module, but of course you can dualize. And does it, Gelfand Kirilov dimension comes from the fact that somehow after you dualize, you get finitely generated module over something. I don't know, this is smooth

submissive, but okay, so this, no, no, when you dualize, you get something finitely generated over the, over the, the, the, the, yes, yes. And so you can look at the dimension in the sense of non-commutative analog of dimension in the Therian rings,

and this is the Gelfand Kirilov dimension. So in analogy with the theory of the Hilbert function and so on, in the communicative case, you expect that, and in fact, in this case, Lazar and so on, it means there is some theory for those kind of non-commutative rings. So, so the question before was whether there is some kind of a Hilbert

polynomial or something similar in this non-commutative setup for non-commutative, certain non-commutative rings, which are close to be commutative in the sense that you, you have enough, you have some filtration. I mean, like this kind of people, Sawa.

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 of dimension, indeed, you know, that this must be something like a polynomial of degree in N of degree. So this dimension of degree, the Gelfand Kirilov dimension plus one maybe, or let me see if the Gelfand,

if the Gelfand Kirilov dimension is zero, which means that this thing is bounded for any N, which means that then pi V is finite dimensional. So this is a constant polynomial. Yes, this must be this. So I think you can prove that this is one of the aspect of Gelfand Kirilov dimension. If I'm not mistaken,

maybe Yongshan Wu can correct me, that this dimension is actually a polynomial for N big enough, is actually a polynomial in N of degree Gelfand Kirilov 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 upper, I mean the numerator. No, no, no, but if you write in the denominator. Okay, so it's, okay, the variable is not N, it's maybe P to the N. Yeah, then you only confirm polynomial in P to the N. Yeah, sorry, you're right, yeah, not N, P to the N, yeah, of course. Thank you for the clarification.

Okay. So then we thank Christophe for this nice lecture. Thank you for the invitation. And have a safe and nice summer vacation. Goodbye. Same to you.