 The talk is about a generalization of a common functor. This is a joint work with Crystal Buhay, Flowing and Horsage, Stefan O'Mara, and Barbara Mishukhan. So first, I start with the introduction to the multi-local allowance program. So I fix prime number P and for N and positive integer and K, a finite extension of a QP. So the multi-local allowance program want to study the relation between on the one hand or on the one side, the multi representations of the Galois group of K. And on the other side, the admissible, smooth multi-orientations of a GLN K. Here we assume the center of this group GLN acts by a corrector. So we call this center corrector. So in this talk, we will only focus on the case N equals two and the case unverified extension. So we will replace FB bar by finite extension F of FD. So we assume it's very large. So it's hardly to make such an assumption. So I first recall the case GLQP. So so far, we know really very little about the multi-long-line correspondence. The only completely understood case is the case GLQP. So let me first review it. For the Galois representation, two-dimensional is very easy to classify. So you first have a reduce for case. So RUGAR can be written as an extension of two characters, subliter or not. And if you revise your useful, you can show that up to twist the bio corrector. It is an induction from GQP, QP square. So here QP square is a degree two unverified extension of QP from a corrector omega two to the power R. So here omega two is a series fundamental corrector of level two. And R is between zero and P minus one. So up to twist, you see they are exactly P irreducible representations of GQP, two-dimensional. So on the side of the GLQP, this is also studied, completed understood. So first of all, Pater and Leibniz, they classified all the principal series and also is sub-cautions. So they defined the other irreducible representations to be super singular. In classical notion, such representations are called a super-caspital. And then in 2001, Pater as classified all the super singular ones but only for the case of GQP. So the results of a Pater-Leibniz works for any local field, I mean, finite extension of QP or functional field. They work in general, but theorem only works for QP. So any super singular representation is isomorphic to, so again, up to twist by a corrector, is isomorphic to a compact induction. So here C induced means a compact induction from the Geo-to-ZP times QP cross. Of such, this is a, so SIMR is a standard orientation of Geo-to-ZP of dimension R plus one. And you should mod by some HEC operator. So the whole compact induction is not irreducible, but when mod by T becomes irreducible and is expressing. So when R goes over zero to P minus one, they give all these super singular rotations. So you see numerically, at least you get some biotection between super singular rotations of Geo-to-QP with two dimensional, you are useful rotation with a Galois group. So this suggests that you have some correspondence between them. So by defined as the so-called semi-simple multi-local-online correspondence. So for a robot, you attach a Geo-to-QP rendition pi robot. So when robot is irreducible, it's clear that you should match with some super singular, but a little bit more careful, there is so much shift of the degree, but it doesn't matter. Not so important. For the, sorry, reducible case, however, it's not so clear which principle, you should make put some principles here. It's not, I mean, clear which principles you should put because in the classical local-online correspondence, even for reducible Galois-de-Linear rendition, they also correspond to some irreducible rendition of Geo-to-QP. So here we have some reducible ones. Also, I should point out that unlike the corrector-receptive zero romanizations, this, there's no intervening operator between these two person series. They are not isomorphic, I should point out that. So when you have a direct sum of two correctors, you send this one to a correspondent to the semi-semitification of these two principle series because sometimes it can be reducible for each principle series in some special case. The comments found a factorial explanation for these correspondence and it can this correspondent to the non-semitical case, but it uses five gamma modules. Actually, it constructed a factor from the rotation of the Geo-to-QP to the Galois group, represent the Galois group. In fact, it constructed a factor from Geo-to-QP rendition to front-end five-man modules, which is equivalent to a Galois rendition by front-end. So I will recall this construction in the second part of the talk. So now let me review some main properties of this factor and you will see that this factor has a very important role in the whole getting allowance program for Geo-to-QP. So let's say that it's important to generalize it. So some known properties that comments first proved that this is in fact an exact factor. And similarly, this factor requires semi-simple multi-local allowance requirements, meaning that if pi over r is a Geo-to-QP rendition corresponding to low bar, then you apply this factor, you get back low bar. So this allows to attend as a correspondence to non-semitical case. So why this? Because you can construct, I mean, say low bar is of the form, a non-semitical extension of two characters. Then you can put pi over r to be some non-semitical extension between the two principal series. And because this is a factor, then you check, you prove it can give correspondence between the attention group on the one side of two characters, on the other side, attention group of the two principal series. So roughly, this is the edge for the multi-case. Now, this factor also realizes pi-article allowance correspondence project. So comments also constructed another factor in the other direction. So I mean, a study, you have a study from galvanization, so pi-article one, you can construct a binary of specification, pi-article. Then it's a theorem of a columnist, does finance school and pass school test says that if you apply a common factor to this big pile, you'll get back to the galvanization rule. So here, if you have a binary of specification, at least by the unitary, so I sketch this very quickly because I will not use that in the rest of the talk. You have a unit ball because this is a binary of specification, pi-article. So you can intent common factor to correct artistic zero limitation. In that way, you modify PN and you can apply the function V, then you take the inverse element. And finally, you invert P to get a bit of a big pie. Of course, when you do so, the exactness of V plays an important role, which allows you to do so. All this tells you that I mean, the common fact is very important if you want to generalize the pi-article elements to higher rent groups. So let me pass through the second part. First, I will review the construction of a common factor. So I recall the definition of a Fargo and Modules U2110. So a Fargo and Modules in the mod-P setting over F-T, the Lohan series field, is a finite Fourier F-T module M with a semi-linear commuting actions of a finite gamma. So here, on the coefficient field, phi sends T to T to the power P and gamma sends T to the, you can imagine that it sends one plus T to one plus the power gamma. So here gamma is in the group gamma isomorphic to, which is isomorphic to ZP crops by the cyclotomic corrector. And the Fargo and Modules is called et al. If the image of phi generous M when you linearize this mofida. So I mean, equivalently, if you have such an isomorphic. So this is a definition of a Fargo and Modules. And now let's look at how Cullman's constructed his factor. So the key observation of Cullman's is that inside the J2P, there is a monoid P plus defined in this way. So here, this is in this corner, you have a ZP minus zero. And this is just a monoid. And he observed that action of P plus can be translated to a structure of a Fargo and Modules. Because you can, if you look at the unimportant part, one VP01, the completed, I mean, group H-brown also called the Ibasava H-brown of this group is isomorphic to F double bracket and T, you sent the element 1101 to T plus one. Then the matrix P001, corresponding to the operator phi, the VP cross 001 is isomorphic to gamma. So you see the two metrics commute, obviously this correspond to the condition in the definition of a Fargo module that phi and the gamma commute. You can also check these two maps on the coefficients by doing the multiplication of matrix. So this is a key observation. But you see from the construction, this is very special to the QP case and also to the J2 case because the gamma in the infontance Fargo module is just isomorphic to the VP cross. So it's very special to J2 QP. This is why it makes generalization quite difficult to do so. There have been several generalization. So the first one is J2 Schneider and the Vinyahas. So I just copy the title here, but you can see it's in fact they are attached to a smooth in torsion orientation to a multivariable Fargo modules, I mean a generalization of Fargo modules. Because for example, when you replace J2 by Jn, you need to a big gamma, you need more variables. And that variety also continues their study on this multivariable Fargo modules. And the legend on Poheye also had a generalization. I will recall his construction very soon. And this will be the main object of my talk. A girl's clone also generalized this, but he started from a Propey Mahore head module and to get some Fargo modules. Not exactly from J2 QP, but just start from Hayk module. But anyway, all the difficult is that they cannot treat super singular pie because there's no classification except as a case of J2 QP. So we have very little understanding for such a super singular. So they cannot, you can formally generalize of common factor, but they cannot compute what you get for super singular pie. So this is, I think, one of the main difficulty. So let's continue. I will talk about Poheye's generalization. So our only focus on the case J2 and K is already fined expectation. So we consider the trace map from the integers of K, okay, to ZP. And I let it in zero to be the group, the unit for the group one, okay, zero, one. And in one to be the subgroup, one, the kernel, you replace okay by the kernel of trace. And you, of course, you have a n zero to be one, and one is isomorphic ZP. And now you take a pie, and this was most of my presentation of J2K, and you assume to have a central corrector. We consider the subspace of a pie, which are invariant under the action of n one, which are fixed by n one. So naturally it carries an action of the matrix one VP01 because n zero and one is isomorphic to a VP. And also we have an action of VP minus zero. The action is like this. You have, I mean, essentially this definition is not trivial, only when this X is like a P or P to the power N. You get something, a finite sum. Here is the finite sum for n one goes over this coset. Again, you can either check that, this is the action on pie n one. So here's the factor you use that the trace map is the beginning. So somehow you can see that you get something, similar to a common situation. Then we call a subspace V of a pie n one admissible. If V is stable on the action of a VP minus zero, and also we require that the invariant subspace of V under this unipotent group is finite dimensional. So this finite is very crucial because this correspondence that when you take the Conry-Yagin dual or the linear dual in this case, V dual, this becomes a finitely generated module over F double burgundy over this, I mean, this is the Iwasawa algebra corresponding to this group. So you put, you consider, I mean, you consider this dual, you put the control gradient action of a VP minus zero, you actually get a Psi gamma module over this ring. But let me recall it better what is Psi here. So you have phi and then Psi is roughly a left inverse of the operator phi. So you can similarly find what is Psi gamma module. You require in the semi-linear condition, you replace that by this one. So you get Psi gamma module, the action of Psi and the gamma also commute. So let me explain why you should consider the dual of V rather than V itself. This is because you start with a smooth orientation. So on V the action, I mean, because the action is smooth. So if you view V itself as a module over this F double burgundy, in fact, it's always a total one. Because finally to get a phagamodule, you will invert T. So if you work directly with V, you always get there because everything is total. That's why you need to take the Pagamodule to get something compound module, but you really total free module. You get something total free. But you lose the phi action, you only get a Psi action because you take the critical gradient action. Okay, but you have the following lemma. If V is a dimensional in this sense, the V dual and then you invert T, then this carries a structure of eta phagamodule. In fact, you prove that this guy is an eta phagamodule, and then the general fact that eta phagamodule is naturally eta phagamodule. So anyway, they are equivalent. So, okay. Now we pass to the definition of the way. So inside the pi, it takes all the admissible subspace of a pi and one. And for each V admissible, you first take the dual and the invert T, then you take the inverse limit for all admissible V. Because you have takes a dual, so here you need to take the inverse limit. For each admissible V, you get eta phagamodule, but now you have taken the inverse limit. So this is the so-called pro eta phagamodule. So our priori giving pi, it could be, you could get something in finite dimension. A priori, because you have no control about the V. In common situation, you can really prove that the dimension is finite because they can prove that inside all the admissible V, there is the final object. So Puhai can prove the following. He computed the D of pi. If pi is not super singular, that is if pi is a principle series or a sub quotient of a principle series, you can compute the explicit. And I also prove that D is a left exact function. And if you just restrict it to the non-super singular relations, it's even exact. But for super singular ones, it cannot say nothing, I think. So we don't know in general if the set of admissible V is M. So that means that you could get the pi equals zero. And we don't know if the pi is finite dimensional. I also already mentioned this. We don't know where D is exact. So we don't know that. The main result now is that a theorem of Puhai, Hussig, myself, and Laura and Sharon in this year. So on some suitable category C, which I will define below, D is an exact function and D of pi is finite dimensional over this field. That means you really get some finite dimensional color orientation and an exact function. But I need to tell you on which category C. So now let's define the category C. So I need to first introduce this subgroup, which is called Propi-Ivahori subgroup, defined in this way. I let Z1 to be the center of I1. Let lambda to be the complete group Etra, the Ivasava Etra of this group. And I let M denote the maximum ideal of this ring. This is a local ring because I1 is a propi group. Then is a classical result of the Nazar that lambda is a Nazar ring. And the Nazar already studied the structure of such Ivasava Etra. And recently called Kloder, he makes the structure in this, I mean, makes the structure exactly in the gel 2 case very explicitly. So we get that. If you look at the graded Etra, so the maximum ideal M will define a filtration on the ring. You can look at the graded Etra. This is isomorphic to the following ring. Here it is again non-commutative. You see, because I1 is non-commutative, it's non-Abelian, so lambda is non-commutative. Now when you pass a graded ring, it is again non-commutative. It is isomorphic to, in fact, isomorphic to the universal enveloping Etra of some Li-Etra. So here explicitly isomorphic to a transfer product for the index from I equals zero to F minus one. Generated by EI, FI, HI with the following relations. So EI and FI, here, this is a deep bracket E equals HI. And EI, HI, they commute and also FI, HI commute. For variables with different index, they all commute. So essentially the non-commutative relation is here, EI, FI equals to HI. In particular, you see that all the HI lies in the center of this ring, there the center. And when you multiply HI, you get actually a commutative ring because then EI, FI will commute to each other. Okay, now I define a decided ideal of this graded ring. Let J to be the ideal generated by EI, FI, and FI, EI. So in particular, HI belong to this idea. So note that if you mod by J, you really get some commutative ring. And its cool dimension is equal to F. The definition of the category C is that you take all the admissible smooth rotation pi, multiple rotation pi, such that the graded module, you take first the dual, they do will become because pi is admissible, this is a finite generated module over lambda, then you can pass through graded module. You did again, a finite generated module over the gradient ring. So we require that this gradient module is queued by some power of J, some finite power of J. So the first remarks that this is a opinion category and stable on the occasion. The reason is that when we define this property, in fact, on pi, you have other few equations, something equivalent to the M article. When you change it to other put the few equations, essentially this condition does not change. It does not depend on which filtration you take. So using lemma, RTE risk lemma, you can prove that this is a opinion category and that stable on the intentions. The second remarks that for any admissible pi, you have a notion of a given kilo of dimension. So this is the invariant for any such admissible rotation pi. It mirrors the growth of the dimension of the invariant subspace on the group for hn. So here hn is the i1 to the power pn. So you make n goes to infinity and this dimension function will growth because the group becomes smaller smaller so the dimension grows and this given kilo dimension mirrors the growth of the dimension. This is always finite bounded by the dimensions of this i1 effect. I remarked that if pi lies in this category c, then its tk dimension is bounded by f. So the reason is very, very simple because when you compute the given kilo dimension, in fact, it can pass to the gradient model and when pass to gradient model, say in the case, in the simply case, this gradient model is cubed by j, then you get a finite general model over commutative range which has cool dimension f and this tk dimension in the commutative case is just the cool dimension. So you really have an upper bound for this different dimension which says that in general, because giving a super singular dimension of pi, you have very little understanding. So in general, it's hard to control it's hard to control its gap on the kilo dimension. So which means that this category c, this condition is a very, very strong, I mean restriction of pi. So the natural question is that how large is this category c? Were it to be, I mean, just the smaller, just the contents, the principle series, we have a partial answer that this category contains some very interesting class of a pi, those coming from the common multi-core model. Let me now recall the setting. So I take f to be a totally real field. I assume it is already find at places about p. I also take a Cartesian algebra D with center f which I assume split about p and also split at places about infinity. So I say it's definite, but for simple, for some reason. So I fix a place about pv, that uv be a compact open subgroup of this one, d times, times a f infinity v, the finite adels outside of v. Then I can define as d uv f, this space. So first, if you fix open complex, open compact subgroup of d to fv, you can look at such functions. In this double coset to f, this is a finite dimension space. Then you take the, sorry, you take the inductive limit. Because you take all the open composite group, in fact, this space carries d to f, fv orientation, sorry, just fv orientation. Now, sorry, I forgot to, here I forgot to introduce what is arba. So sorry, so arba is a group representation of the galore group of the total relative to m, so arba. So giving arba, you assume that it's a continuous absurdity in the useful, you can attach a maximum ideal in some packet at random. And you can look at the subspace annihilated by this maximum ideal. It's also called the eigen space corresponding to the arba. So this is again, because action over d to fv commute with the action of a packet at random. So this is again, a representation of j to fv. This is even admissible, small subtraction. And we further assume that this representation is non-zero, that means that arba is modular. We will be interested in this j to fv orientation. In fact, the reason that this representation is expected to realize a modular balance correspondence to the restriction of arba to dfv, so I mean to be local. So, next we will study this representation. The first result is that on the some global conditions, for example, including the Taylor-Wall's condition. And we also need to assume a robot is generic. So here robot is a local component of arba at d. This generic condition, I will implant this in the next slide. So in particular, it implies that the prime p must be very large. So this is some condition not so, not good. So anyway, then the conclusion is that this rotation pi v dr lies in the category c. More precisely, we prove that the gradient module of this, the view of this rotation is actually queued by j. Not just a finite power, it's exactly queued by j. So as I already remarked that for any rotation in the category c, we have arba bound for the tk dimension by f. So in particular, for this rotation coming from what we call modality, we have arba bound. And actually, this was the main motivation to prove this theorem. So we prove this theorem before last theorem. So this is proven in last year. And the definition of c was largely motivated by this theorem because we found that some, the rotation coming from what we call modality certified this condition. So this suggested to look at such a case. Here, let me remark that this theorem is on the, it's due to our favorite people, Boye-Hosig himself, Maura and Shen, and also due to, I mean, Hu-Wang. So we find prove the case for Ruba semisim. We treat the case Ruba semisim. And in the work with Horan Wang, we treat the case non-semisim case, which means when Ruba is a non-semisim extension of two characters. Okay, this partially answers that the category c contains very interesting renditions. Now let me explain the generacity condition. I take an example in the reducible case. So I write this Ruba because it's reducible, I write it as this form after twist this way. I can assume this part is one and I write this character to be a power of omega f. So here omega f is a fundamental corrective theorem of level f and treat a time system already find it correct. Here R i goes over minus one to P minus two. So here Ruba is a generic, if R i certifies this for R i and strongly generic if certified there. So I do want to explain too much on this generic condition. So certainly if you allow P very large, you such Ruba exists. So, but for technical reason, we need to impose such technical conditions. Okay, we will measure the strongly genericity in the next theorem. This is even stronger because this depends on f here. So, which means that P must be bigger than four times that. Depends on f, not, yeah, it's even stronger, okay? So next question that you can ask. Now you know that because this rotation pi b r bar lies in the category C, you can ask when you apply a generalized factor b, what do you get? Can you compute them explicitly? Because when Ruba is irreducible, you can show that, I mean, this rotation in fact is a super singular. It contains something super singular. So it has been difficult, I mean, difficult problem to compute the figure module, what you get. So I let v of pi be the associated orientation to this factor module via the equivalent server contain. The partial answer is that if I assume Ruba is irreducible and I'm sorry, I should say strongly generic. So here, very generic means drunk generic, right? When Ruba is very generic, then the galore rotation is isomorphic to some copy. So here the direct sum d means some copy of the tensor induction rule, okay? I will now define what is tensor induction. So here d, if you ask what is the d here, this is because when I define here, when I define this rotation, I choose some uv. So when you change uv, the d will change. So this is why d appears, because it depends on uv. This is always a finite number of integers. So sometimes if you choose uv very carefully, you can make d equals one, the so-called mean markets. So now let me define this tensor induction. So for subgroup H of d of finite index n, you can write d as a left coset decomposition. So now if rho and v is a finite dimensional rotation of H, you can define the tensor induction. Unlike the uv induction, here we take a tensor product, not the direct sum of a GI tensor v. This is on the line space. Now you can give x of g. I mean, this is a very simple, very easy definition. It's a normal definition. So you get a rotation of g. So the dimension of a tensor induction is a dimension of v to the power n. If you look at the uv induction, the dimension is n times the dimension. So here we get tensor induction. Okay. So now let me finally pass through the proof of the theorem. Okay. Let me recall that. So n, 0, you know, this is a unimportant group. And we look at the complete Ibar-Sava-Edgar. Ibar-Sava-Edgar, which is isomorphic to a power series ring with f variables. So it is equipped with some m and 0 articulation. It takes the maximum ideal. You have a natural m articulation. You look at this muticative subset inside this ring, y0 times yf minus 1 to the power n. And you look at first you are localized at s and you take completion. Because when you localize, you again get a filter. When they can take complete completion of the filter. You can define the action of a phi and OK cross on the f and 0 by these formulas. So just by the mutification in matrix in a natural way. This action by continuity it tends to this completed filter with a. So now for, if you have a pi, a smooth admissible rotation with 0 to k, you first look at the dual and it takes a tensor product and completion. The A tensor pi dual, they take completion. This is more of you first localize at pi dual, then you take some completion with vectors on the equation. It's easy to see that this function is exact because this A is a flat module over f and 0 and when take completion it's also knowing that this is exact, when take completion is a exact function. Now we can define a continuous upgrade of psi on pi s dual, in fact, pi dual s. So you see, dI pi can be described as some completion of this localization of pi dual. When you have a continuous psi, you can attach psi on this dI of a pi and also OK cross action attached to this one. In this way you get a psi, OK cross module over. This is analog to the diagram of module in the south of the name. So first level is that if pi lies in the category c, then dI of a pi is a finite general module over n. So why is that? So the rough reason is that if pi lies in c then when you take a gradient module the grade dI of a pi is a finite general module over grade a. Then the standard fact that you have completed future module or over some sovereign, if the gradient module is finite generated then the module itself is finite generated. So of course it's key to check that why the gradient module is finite generated here it uses a definition of c. It uses the condition that when you take a gradient module over pi dual it is queued by some power of t. But I don't have time to plan this in detail. Now in general we cannot prove or we don't know how to prove if dI of pi is eta or psi of k, OK cross module. But however there always exists a largest eta quotient let's say denoted by dI of pi eta. So your quotient of pi is a mupin part you get a largest quotient which itself is eta of psi of k cross module. And then the theorem says that the eta part of dI of pi now is an eta of psi of k module and hence it is also an eta of pi of k cross module. And the factor if you send pi to dI of pi eta this is again exact. And finally dI of pi eta is a finite projective m module. So the key probably now here is that it's even a projective m module. This use is that it has an action of OK cross. OK, now I'll pass you the final part of the proof. So we need to know the relation to Borel's function. So you have the trace map from OK to ZT. So it induce a morphism between the evasive algebra which is of course objective. So hence when you pass through so here when you localize at S on the side of this F double branded T it means that you localize at T you send all the YI to T. So when you take completion but this part already complete so you get a suggestion from the ring A to FT. Now the key property is that I mean the relation to Borel's function D you look at you take the test product or rather you take the batch change of DI pi A dash you batch change to F of T you exactly get to the Borel's function D of pi. So now remember that the mainstream says D of pi you pi lies in C then D of pi is the final dimensional by gamma module. Now you know that DI pi A dash is the self-finance generated over A so it follows then you need to show why D of pi is the exact function but by the last theorem we know pi so this one is the exact and also this guy is the projective M module so when you pass to batch change you again get something exact function okay so roughly this is the proof of that so I finished that talk thank you very much for your attention okay thank you for the talk so are there any questions in the room? I have a small technical question that on one of your slides I saw some a little wise condition it appears it is here so what is it? it means that if you are a bar when you restrict so it is a representation of kalabubom F so when you restrict to you add F some piece power of minus 1 and then you get a subgroup which is a telewise condition when you restrict to the subgroup it remains to your loose points so this condition I think is already there starting from the first part of telewise for the proof of the last theorem so this is very hard to move are there some questions? no no questions so you studied one type of representations from the cohology is it the only type of representation from cohology so we study geo two representations are there other kind of representation from cohology that you would expect to exist and how they are also nice in the past see yes I mean in fact you can consider other global setup for example you can take a schmarr curve you can define similar space pi vr bar inside the h1 et al commode of schmarr curve you can also take other say unitary group some other global setup but it is expected that this representation because we expect that this is something local this is something local just related to global this is expected that it does not depend on which global setup are your truth but this is really very hard to prove I think no except jq there is no result on this part so in the case of jq they use because we have all the periodic local coordinates so everything is existing so that's yeah so we expect that it depends on the R which one you can choose anything you prefer and in fact all the argument in the proof of the argument is either in the periodic heart theory say we need to compute some local Galois differential memory this is purely local that does not depend on any global setup and all some representation theory of jq so again purely local so essentially you can change the setup and the proof works through so you mean in the other setup for example you can show that the representations are also in C yes in other one yeah in the same case again yeah but we cannot show it is independent of the choice we can show some common properties yeah okay all the questions no okay well let's send the speaker again