 Thank you very much for the invitation. It's a great pleasure for me to be here. Actually, it occurred to me that this is my first talk at this venue. I have been a guest of the IHS several times, but I never gave a talk at this venue. So for me it's a new thing. Okay, so this is, I will be talking about a, let me say this before I forget it, joint project, this being your shear. As you heard about rigid character varieties or character groups and relation to lupine-tate theory and phi-gamma modules. And so let me, although I'm sure that all of you know very well what classical phi-gamma modules are, what they are good for, nevertheless to warm up and to set up some notation. Let me remind you of other quickly about a few things. So somehow I used geometric notation. So let, let B be the open unit disk on the QP. And then let me write down the usual list of rings, or at least some of them. So there is O B, which are the holomorphic functions on B. Then there is O little B on B, which are the bounded holomorphic functions. And then there is R B, which are somehow the functions holomorphic, I mean let me be vague, holomorphic near 1. So the robber ring. And then there are, so inside there we have the E dagger of B, the bounded elements in the robber ring. And this carries a certain, carries a norm, a natural norm. And then you can complete and you get for tens ring E of B. So which is the norm completion. So all these rings, let me maybe use a generic letter A for, any of them are rings of power respectively long series in one variable set with coefficients in QP and of course certain depending on the ring certain conditions on this power or long series. So they carry natural topologies, carrying natural topologies. And then in addition the multiplicative monoid, so p of the integers minus 0, acts on any A by the variable gets sent by an element A in this monoid to 1 plus C to the A minus 1. So we have this monoid action and then the usual, I mean the translation is that gamma is the group of units and phi is the action of p. Now in the bounded holomorphic functions it's usually some weak topology, so this is a, but I mean this is, I don't go into the details of this, this is not important for the following. Now just vaguely, let me remind you vaguely speaking, a phi gamma module is a module of some kind, some additional conditions over A with a semi-linear action of this monoid. I mean there are additional conditions, but just as a, to remind you of the basic idea. So now we let the Gallois group, the absolute Gallois group of QP, so we have this cyclotomic character, kind of sick. And maybe let me denote, I think I need this notation. Do you know the kernel here by HQP sick? And then the theory of fields of norms, right, they allows us to construct some kind of unremifed extension, a tilde of A with action of the Gallois group such that the, somehow the A is the fixed ring under this kernel, and then we have a residual action and the residual action is via this character, kind of sick, the above monoid action, the above, I mean of the unit group, the above CP star action. So and maybe I also just write as a typical, typical application, this contains theorem, this shows what this all is good for, that the CP representations of the Gallois group are the same as the etal gamma modules over the unit ball in this ring, for example, in this last ring. Just by sending such a CP representation, so it turns out with the tilde version over CP and take this fixed vectors under this kernel. Okay, so now there are, I mean as you also, I'm sure you know, I mean I restrict it here to QP as base field for specific reason, I mean it's certainly the case that all of this, this works for any finite extension L over QP with one exception, with the exception that the monoid action can no longer be made explicit. So this is one problem, now there's another philosophical problem, so there is somehow maybe a philosophical, philosophical problem, I mean if you think about periodic length, so let me maybe, let me write this from the point of view of the length, because I mean, so this variable, so the variable C stands, I mean if you look at this representation theoretically, then the variable C stands for the unipotent radical, I should say the one-dimensional unipotent radical, so let's say the upper one, so here we have any entry in CP in GL2 QP. And the monoid action corresponds to conjugation by certain diagonal matrices. So now, but now if you go to an extension GL2 of L, right, and you look at this unipotent radical, it's no longer one-dimensional as a periodic league group as dimension degree of the field, whereas the theory here, this theory of recombinance of these rings is still the same kind of power series rings, they are still one-dimensional power series rings, so this is kind of a philosophical difficulty, which we should keep in mind. So now let me, next let me describe, let me say a few words about the P-Zine-Ren theory, but maybe one should, I mean this builds upon previous work by Colmess, and there is also related work by a student of Colmess-Foucault. So now we take a finite, we take any finite extension, let's say of degree D, I may need this number later, and let's pick a prime element. So somehow I go from behind, yeah, this was the classical Frequemont-Module theory, now comes the Lubin-Tate theory. So we also pick corresponding to this prime element, pick a corresponding Lubin-Tate formal group law, which allow me please to just denote this by LT, it's kind of a careless notation, but nevertheless let me just denote it by LT, and then I mean it's a Lubin-Tate formal group law, so in particular we have a, we have an homomorphism right from the ring OL into the endomorphism ring of this formal group law, which usually is denoted by sending A to square brackets A. I do need this notation, this is among others, this is a power series over the ring of integers and has no constant term. So now you can use, you see I said, I mean there was this exception that in general this theory cannot be, the monoid action cannot be made explicit anymore, now having such a formal group law we can do the following. So first of all just as a notation I use, have the same rings as above, but with coefficients in L, so then I decorate this notation by an index L. And now we can do the following, I mean using this formal group law, so now we can let act the monoid OL minus 0 simply by A lower star of the variable is this power series, A apply to C. And now of course I mean now I write gamma L is the units in this ring and phi L is the action of the prime element. So we have of course now, correspondingly we have notions of phi L gamma L modules and now the Gallo group, I mean correspondingly, now the absolute Gallo group is GL of the field L. Now we have the Lubin-Tate character to OL star, it describes the action of the Gallo group on the Tate module of this formal group law and now I denote the kernel by HL upper LT, so it's the kernel of Tate. And then let me just say an analog, for example an analog of fontane theorem holds true, so we are now of course I mean the HQP-sync is replaced by this group. So in some sense this resolves, so this resolves in some sense the exception above the fact that the monoid action was not explicit any longer, so this now it is explicit using this device. Okay, but of course the rings have not, the nature of the rings has not changed, so the philosophical problem remains. Okay, so now let me go to the question, so what is the, what is the representation theoretic meaning of this open unit disc? And the answer is given by Fourier theory, so let me make a list here maybe. So we are back now, I'm back now to the first, I'm back to the original situation up there over QP. So on the one hand I have the open unit disc, now on the other side I consider all continuous characters from the additive group Cp to let's say to a varying field, but for example I mean large field Cp star. And let me mention right away, because this becomes important later, here continues, this is automatically the same as locally QP analytic, so expandable map is, the character is expandable locally in a power series with coefficients in QP. So there is the following rather simple bijection, so this is kind of, if you want, this is kind of the character group of the additive group, this means the additive group are Cp, so Cp hat. And there's the following bijection, so an element in the little set in the open unit disc goes to the character which maps an integer x to 1 plus c to the x. So this is a bijection, and so this means that b is the rigid, this naturally of course is a rigid variety, the rigid character variety of the additive group Cp, where more precisely here I should maybe say Qp league group, so we are over Qp. So this is one point. Now as far as the rings are concerned, and we have the, let's say for the first ring, we have the Fourier isomorphism, which in this context is a theorem by Amis, which says that the, that so representation is theoretically forming, somehow having such a league group, we can form the distribution algebra of the league group. This is the coefficient somehow, the values of the distribution, so this is the locally Qp. So now one really has to say this, because for arbitrary functions, of course, continuous is different from locally analytic. So locally Qp analytic distribution algebra, a distribution algebra of this league group Cp. So this just a continuous dual of the, of the space of locally analytic functions on the league group, naturally an algebra, and on the other hand we have this Ob, and they are isomorphic by the usual Fourier recipe. So if I have a distribution Mu, I send it to the function which turns out to be a power series, if Mu in a point little c is just the distribution evaluated at this character chi c, right? I mean the character is in particular locally analytic function, that's why I mentioned this explicitly, and so I can evaluate any distribution on it. Okay, so that gives a, gives a follow up representation theoretic interpretation for the ring, and moreover the monoid action, so the monoid or monoid Cp minus zero, right? It does not only act on the ring, it actually acts on the character variety, acts on Cp on the character group Cp hat by this one element a, now it's kind of an upper star, but this is not important of a character is just the same character, but shifted multiplicatively by a, and this induces why are the Fourier isomorphism the previous action on the ring? So also the monoid action has a geometric meaning. Okay, so now let's go back to to an L, so again now L over Qp finite with the previous notations, and then then the additive group OL, now in the ring of integers in this field, is an L leagope, leagope over the field L, I think I should clean now, I suppose. Okay, and then we have the following theorem by Tidalbaum and myself. First of all it says that the, now we can form again the character group OL hat, which is all locally, now continuous doesn't work anymore, locally L analytic characters is a rigid variety, I'm used to use the letter X for this, so X is naturally a, so let me collect some of the properties, a connected smooth one-dimensional rigid group variety over L, and I mean technically this plays a role in some sense, let me add and is a steinspace like the open unit disk is, so this is one thing, and the second statement 0.2 over there, this was 0.1, so 0.2 is okay as well, the Fourier isomorphism, right, same recipe, don't have to repeat this, the Fourier isomorphism between the locally L analytic distribution algebra, and now the ring of holomorphic functions on X is okay, holds true as well, so this is now a new ring here, so now let me just for fun tell you how we can make a corresponding list out of this ring, so first of all, right, the monoid, first let's look at the monoid action, the monoid acts, I should say which one, the monoid again, OL minus 0, and acts by multiplication on this rigid curve, right, exactly the same recipe as here, you just multiply, you shift the character multiplicatively, and hence on the ring, okay, so this is still an explicit description, there is no, I mean okay I will maybe say a little bit more about this later, so now just more or less for fun, let me show you that we can build up an analogous list of rings, so we do have OL of X already, so now of course we have OL bounded, so these are the holomorphic functions, then we have the log of the oba ring, which is just the union, so we consider any, in the curve X, we consider any open affinoid Y and remove it, this is still an admissible open subset in X, take the functions and take the union over all these, I mean removing some whole larger and larger affinoids, so that's the analog of the oba ring, and then the E dagger LX is the same, but you do this with the bounded functions, and these complements, oh L, so this carries this carries a certain limit norm, and so one can take the norm completion getting EL of X, okay, so that's just to, yeah okay, and I mean as before the monoid action, the OL minus zero action extends through these rings, and so have notions of phi L gamma modules over these rings, no I don't care, any actually, no no it is a union because I mean, no no, I mean actually you see this, I mean I wrote it down abstractly in order not to have to introduce notation, but you actually can somehow introduce an increasing sequence of very specific affinoids which contain any other one, so yeah, so now let me make one observation which also maybe gives you at least an initial idea how the X is, how the X arises, so if you the OL, not as an L lead group, but as a QP lead group, as a QP lead group, of course the OL is just isomorphic to D copies of the lead group ZP, right, and so now if we take the character group of this here, then we get of course just D times the character group of ZP which was B, so we have D times the open unit disk, and then the X is a specific analytic sub-variety given by specific equations of this D-dimensional open polydisk if you want, so then of course the functions here OLX, I mean maybe I should say here over L, so to be precise, so then the functions here, they are quotient of the functions on this open polydisk, so with a big kernel, but if you restrict to the bounded functions on both sides, the map of course restricts, this is here, right, and now the restricted map is injective. This comes just from the fact that the characters of finite order, right, they are on the one hand they are locally analytic, L, whatever you want, on the other hand they are dense in the continuous functions, so that's I mean the bounded function has something to do with the dual of the characters of finite order, and that's the, this density is the reason that this restriction suddenly becomes injective, and this of course is a power series ring in D variables, okay, so this at least indicates that maybe this ring allows some transport, although this is a one-dimensional curve here, allows some transport between different dimensions. I do not know anything in so far about the image, what the image of this map is, unfortunately. Okay, so now I maybe first should say at this point we have no analog of the tilde construction, I mean there is a, there is a natural embedding of this OLX into BDRAM, but it doesn't seem to be the, although it's kind of natural, it doesn't seem to have the good properties, which one would like to have it, so I actually believe that maybe the, this is another thing for the theory of perfect rate space is to do these tilde construction in this context. But one can do something else, so for this I have to tell you first a kind of a third part, there's a third part to this theorem of tile moment myself, ah, but I guess I first need, maybe I first need some, some notation, I think I first need some, some notation, so fix a, fix a generator t prime zero of t prime of the dual tape module, the dual tape module t prime of the lubin tape formula. And I mean this t prime you see can be viewed as formal homomorphism, so have formal homomorphism, but over Cp only, for simplicity I also denote it by t zero prime, going from the lubin tape formula to the multiplicative formula, so I think that's the notation I need, right? I mean this is as a, as an OL module, this is free of rank one, so that's, that is meant by a generator. And then we have the following third part of the theorem, so the map kappa from the open unit disk over Cp to x over Cp, which sends a point z to the following character, kappa z of x is right, I apply the power series, x is in OL after all, I apply the power series x to the point z, and then I send this to the formal multiplicative group to buy this formal homomorphism. So this is an isomorphism of rigid group varieties with the monoid action, with OL minus zero action, but I mean let me stress again over Cp, so the x actually, the x is a, is an l form of the, of the open unit disk, and it is non, I mean that's maybe, I don't write this down informally, if you want the fourth part of the, of the theorem, it's non-trivial, it's a non-trivial if l is different from Qp. So this means, yeah, so it's really, this has various consequences that this ring is not, I mean it's not a power series ring, there's no variable, it's not even, you see we know by Lazar for example that the OB is a Bezou ring, right? Every, every invertible ideal, every finally generated ideal is principal, every closed ideal, all these notions coincide, it's no longer the case for the OX, the OX is only a prüfer ring, but which is for most purposes still nice enough, so there's still a divisor theory on this ring, but it's no longer the case that all closed ideals are principal. So we can now use this isomorphism, so let me, let me draw a picture, so we have here OCPX, right, it's isomorphic to OCP of B, then in here we have O l of B, and here we have O l of X. Now this here of course is right, these are just the convergent powers here, yes, F of Z with coefficients in Cp. So let me treat this kappa, so treating kappa as an identity, I mean let me view it on this side because I can write the elements down explicitly. This ring must be the fixed ring of this ring under a certain twisted Galois action, so we have that O l X is OCPB fixed by Galois under twisted, I denote this by an additional star for the, what do you call it, the twisted Galois action given explicitly, but if you work out the following explicit formula, so if you have such a power series then sigma acts twistedly, Galois automorphism sigma acts twistedly on it just by its usual action on the coefficients, Galois action on the coefficients, but then you apply tau, the character tau, I tell you in a moment what it is, to sigma inverse giving you an element in O l star, and you take the corresponding power series, apply it to the variable, so you replace the variable by this power series, where tau is the character of the dual-tate module, t prime, so it's actually simply the inverse of the lumintate character times the cyclotomic character. Okay, so now let me tell you the many sides we have so far, so first a little bit, I mean I will come back to this, there are basically two results, and the second one I will come back to this twisted action, but let me first mention another thing without giving too, I mean without even notationally giving too much details, so let me just say first of all k-scenes construction works well, so when there is the following theorem, I mean there is this construction by Keasing, you start with a five-dimensional filtered L vector space with a linear automorphism phi, and so I just use an analog of his notation mod x phi L gamma L n, so this is the category of finally generated projective O l x modules with, and now I have, because this is kind of technical and this cost me too much time, so let me just vaguely say with phi L gamma L action in quotation mark in the sense of Wach modules, so it's not a literal, I mean the gamma L acts literally, but the phi L of course is not a literal automorphism of the usual Wach type conditions on this object, so this is, and there is an equivalence of categories also in this context between the two, and I mean let me just say that this any object here gives rise to a by further base extension, it becomes a true phi L gamma L module over this robot type ring R L x, so this is one without going into any details, this is one very clean result, the reason being that it's kind of all you need for this is in some sense geometry, and the geometry although it's a little bit more complicated here, works quite well, so now let's come back to this twisted action, so back to twisted action of G L, so let's consider let M be a, let M be a phi L gamma L module over the usual robot ring in the sense of Kiesin-Ren, so then you see what you can do is you can extend then we may define a twisted G L action on M, sorry not on M, so we base extend the coefficients to R C P B by sigma star of let's say F tensor M is the original twisted action on this power series tensor, and then we use our torque sigma inverse giving us an element in O L star, which acts of course, because this is a phi gamma module acts on M, so times M, to easily check that they both have the same same linearity properties, so this is a well defined, well defined action, so then there is the following proposition, so now we pass to the M index X, I just take the fixed elements under this twisted action, so this is a phi L gamma L module over R L X, that sounds good, but possibly zero, and actually it is sometimes zero, so it's always finely generated projective with the usual this action, and the action is as it should be, I mean it's yeah, but it can be zero, and it is zero, let me so now it works backwards, I mean replace torque by torque inverse and then the same holds backwards starting from from X to B, so now let me make the following conjecture, okay I still have a few minutes, so M to M X gives an gl, yeah gl, there is a there is an addition, and you see on the you still have the I mean the monodaction commutes with this Galois action, so the monodaction you still have it, but it comes from the monodaction M, this commutes with each other, gives an equivalence of categories between the L analytic phi L gamma L modules over R L B and the L analytic phi L gamma modules over R L X, and L analytic means you see you can derive this, you can I mean these actions these the gamma L actions on these modules they are not only continuous they are even locally QP analytic, so you can derive them to an action of the Lie algebra of gamma L which is just the field L acting on these L vector space, they are all L vector spaces, but because the action is only the original action is only QP analytic, this L action on an L vector space nevertheless is in general only QP bilinear, and if it happens to be L bilinear then the the object is called, so if somehow the original action is even locally L analytic then the object is called L analytic, so using, so let me say a few words to finish let me say a few words about the evidence for this, so using the Tate-SEN method in families, so this is this paper by Berger and Kolmes, I mean one can show that one can descend, one can do in Tate-SEN theory, one can descend to the cyclotomic extension, so if I only, if I define Mx-sick I take only the fixed elements with respect to HL-sick twisted, and this is always fairly generated projective, of course now over bigger RL-sick completed X of the same rank as the original M, and one even can descend it, I mean now it's getting into technicality, so I don't write it down, so even descends in some sense to some finite extension in the cyclotomic tower, and that means that one has the SEN operator on this descended thing, and then the conjecture actually comes down, so conjecture is equivalent, I think it's equivalent, at least it's a consequence to if M is L analytic then the SEN operator is zero, and therefore the thing fully descends in the sense of fixed vectors. Now for these days we have two kinds of evidence, so this is actually a statement now in abstract, if you want an abstract Tate-SEN theory you can make this into a statement removing all the geometric objects here, and then for example it is true, you can prove it if you start with a finite dimensional OL star representation, then and you do this twisted thing then you have that, if it's L analytic then the SEN operator is zero, and then last week I was visiting Laurent Berger and he suggested a strategy to prove this under the assumption that, under one assumption I believe namely that the kernel of the cyclotomic character contains the kernel of the Lubin Tate character, then somehow he can use his theory of locally analytic vectors in the Tate-SEN theory to approach this conjecture. Okay, thank you very much. Are there any comments or questions? So if this conjecture would be true, would this have would this have some consequences for the Yannick Langland's correspondence for GO2L? Probably, I don't know. I mean that's the motivation of course, but I don't can say anything specific there. Or in this particular, no I mean this was this observation, I mean one has to go to the bounded, one has to do something bounded. Can you say something for this conjecture in the one dimensional case or one dimensional model? I mean I have not seriously thought about it, but I would think that one can show it in this if one dimensional model itself. Let's say free one dimensional. You see on this there is another difference, I mean that this projective was not, this is not my, these are not my antiques, I mean over OLB every finally generated module, projective module is free, but over OLX is no longer true, so you really have even rank one projective module. Is there something about the properties of those rings? The one ring in OLB, the closed, I see the closed ideals are principles? And in OLK what is the statement exactly? OLX is a proof ring, so there the closed ideals coincide with the finally generated ideals coincide with the invertible ideals. In OLX, OLB has the better property that every finally generated ideal is even principle, the besuring.