OK. Well, thank you very much for inviting me. So my title is indeed Figima Modules and Locally Analytic

Vectors. And what I'd like to tell you about is a joint work with Pierre Colmes. And it's really a short report about the attempt to define Figima modules but in the lubin-tate setting.

And so there will be some overlap with what Peter Schneider talked about. So let me first remind you of what I mean by in the lubin-tate setting. So the basic idea, as you know, the basic idea of Pielik-Hoch theory is that we start from a finite extension k of QP. And we are interested in the representations

of the Galois group of QP bar over k, so gk. And the way that we do that is that we defined an intermediate extension k infinity, which is between k and QP bar. And we try to choose this extension k infinity so that most of the ramification of QP bar over k

is contained in k infinity over k. So this translates into the fact that QP bar over k infinity is almost et al. And so let me denote this Galois group by hk and this one by gamma sub k. And we also try to choose k infinity

so that the Galois group gamma sub k is as simple as possible. And of course, the canonical choice for k infinity is to take the cyclotomic extension.

OK, and in this case, the cyclotomic character realizes gamma sub k as an open subgroup of ZP cross. And once you have made that choice, you have all the theories that we have in the study of Pielik representation. So for example, Sen theory, Pielik-Hoch theory,

the theory of Figma modules, and so on and so forth. And what we'd like to do with a view, of course, towards the Piedic local Langlands correspondence for GL2F is to replace this choice,

so this cyclotomic extension, with the extension generated by the torsion points of a Lubin-Tate formal module. So what happens if k infinity over k is a Lubin-Tate extension?

So what are the new phenomena that arise? And in order to tell you about this, let me set up some notation. So I'm going to work in the simplest possible setting. So let f be the unrammified extension of QP of degree h, of degree h.

Let me choose P as the uniformizer of OF. So this will simplify some of the things that I'll talk about. And let me denote, like Peter did, by LT, the formal OF module attached to P.

And k sub n is the extension of k generated by the P to the n-th torsion points of LT.

No, f is f. And it's probably helpful if f is contained in k. Pardon? I mean, the f LT is defined by P in the recipe.

But I mean, for the time being, k is not f. I mean, k is, you can think of it as something bigger. It's probably helpful if it contains f. Later in the talk, indeed, k will be equal to f, because that will make my life simpler. So you are anticipating on the sequel. OK, k infinity is what you imagine.

And then the lubin-tate character, k sub f, realizes gamma sub k as an open subgroup of OF cross. So what I'd like to tell you first is what happens if we do Senn theory not using the cyclotomic extension, but the lubin-tate extension.

So let me start with Senn theory. And first of all, let me very quickly remind you of what I mean by Senn theory in the classical cyclotomic case. So classic case. So for now, k infinity over k is the cyclotomic extension.

And the part of Senn theory that I'm really interested about is the following. So you start from V, a finite dimensional QP representation of G sub k. You let W be CP tensor V invariance under hk.

So of course, now this is a k vector space with an action of gamma sub k. And what Senn did was the following. So we define D Senn of V to be the set of finite vectors of W. So this is the set of W's of vectors

whose orbit under gamma sub k lies inside a finite dimensional k vector space. Orbit of W is inside a finite dimensional k vector space. So you said CP and not QP bar there.

There? Yes, it's really important that this is the completion of QP bar. Otherwise, probably all the vectors would be finite. If you don't, yeah. OK. So this is Senn's definition, a set of finite vectors. And of course, this is a very restrictive condition. There's no reason why a random vector would be finite.

So there's no reason why this space would be different from 0. The only thing that you can see up here is that this is a k infinity vector space. And then Senn's theorem is the following, that this k infinity vector space is larger, but not too large. So CP tensor d sine of v over k infinity

is the same as CP tensor v. So you recover CP tensor v from this construction. So you have enough finite vectors in W. And of course, you know that the point is that because the vectors inside d sine of v

are finite vectors, and there is an action of gamma sub k on this space, you can differentiate the action of gamma sub k. You get an action of the Lie algebra of gamma k. And because gamma k is one-dimensional, the action of the Lie algebra is really given by one

operator, which is Senn's operator. And that's really the point of Senn's theory, is to be able to define this Senn operator, which you then use to define the generalized Hodge state

weights of your representation v. But of course, this is an old definition of Senn that dates probably from 30 years ago. And if you are interested in having gamma sub k act, and if you are doing this now, you probably would not look at the set of finite vectors, but you would look at the more modern definition, the more

modern notion due to Schneider and Teitelbaum, of locally analytic vectors. Because after all, being locally analytic is a more natural condition if you want to look at the action of the Lie algebra of gamma sub k. So w is the same w as here. So it is a k vector space with a continuous action

of gamma sub k. And then we set w n to be the set of locally analytic vectors, so the set of vectors of w such that the orbit map from gamma sub k to w. So this orbit map is locally analytic on this group.

This group has a locally analytic structure because it's an open subgroup of z p cross. And so this definition makes sense. And it's a more natural definition if you're interested in doing differential algebra.

And the first theorem is that in this setting, so in the cyclotomic setting, this definition is the same as the previous one. So the set of locally analytic vectors is precisely equal to this n of v. So you don't get anything new.

And why is this the case? Let me tell you very briefly the principle behind the proof. So sense theorem and a little bit of Galois cohomology tells you that w is precisely k infinity hat. So the periodic completion of k infinity then sort this n of v.

And then it's not difficult to see that the set of locally analytic vectors is precisely equal to this. So the set of locally analytic vectors of k infinity hat then sort this n of v. So this is the first step. And the second step is to see that this set of locally analytic vectors of k infinity hat

is precisely equal to k infinity. And you do this using the Tate's normalised traces. OK. And it seems that if you want to replace

the cyclotomic extension by the Lubin-Tate extension, this definition is the right definition. Or at least it gives something interesting. So let me assume now that we are in the Lubin-Tate setting. OK, Lubin-Tate. OK, so k infinity over k now is the Lubin-Tate extension that is above Lubin-Tate extension.

So once again, w is Cp tensor v invariance under hk. And this is still, by an easy argument of Galois descent, a k infinity hat vector space of dimension d.

And it is, of course, still equipped with an action of gamma k. And before I state. d is the definition of the representation. d was the dimension. Yes, d is the dimension of v. Thank you. I didn't say that.

So the same dimension as v. And before I state the second theorem, let me tell you the following. So we have k infinity hat. And I can still look at the set of locally analytic vectors for the action of gamma k. And it's not hard to see that this is always a field, regardless of the gamma k that

is acting on k infinity hat. So here is the second theorem. So the set of locally analytic vectors in this setting is also of the right size. So we have the precise analog of sense theorem. So wn is a k infinity hat n vector space.

And when you extend scalars to Cp, you recover Cp tensor v. So we have a sort of analog of sense construction. But now, in the Loubinted setting. And no, no, that's just for simplicity.

But this is true regardless of whatever Loubinted group you choose. Thank you. OK, so you have this wn, which is a k infinity hat n vector space of the right dimension. Of course, when you see this, I guess the first question that you ask is, what does this field, k infinity hat n, look like?

What does it look like? So let me give you a brief answer to this question. So recall that we have this extension f over Qp. And let me take an embedding tau of f inside Qp bar. So from f to f, because f is Galois.

And you know that if tau is different from the identity, then there is a period in Cp for tau composed with chi sub f. So there exists an element x tau in Cp. But actually then, in f infinity hat non-zero,

such that tau, so g of x tau is tau composed with chi sub f times x tau. So this goes back at least to Serre, I think, about 50 years ago.

And of course, you know that if tau is the identity embedding, there is no such element. So this really reflects the fact this character chi sub f has h whole state weights, h minus 1 of them being equal to 0. So this gives you those elements, and one of them being equal to 1, which tells you there is no corresponding element when tau is the identity embedding.

And then the theorem, and I apologize for its formulation. So the field k infinity hat n is generated. And I'd rather not say precisely what I mean by generated, because it would take me a long time, by k infinity and those h minus 1 elements.

h, yes, it's the degree of f. So the logs of these elements for tau different from the identity.

Yes, locally QP analytic. I'll come back to that in a second. No, no, no, no, that's what I was asking. It's not necessary at all.

It makes my life easier, and it will make my life much easier in the sequel. You don't want to explain the generated? Yes, yes, yes, I'll say something in a,

well, I don't want to say too much, because if I give the precise definition, this will take me too much time. But it's basically power series in this log of x tau. And of course, if you think of a field of power series, this is not going to be a field. But you know that there are such objects, like if you take the local point of a Berkovich

point of type 4, it really looks like a set of power series. And it's also a field. So this field, k infinity hat analytic, is some kind of h minus 1 dimensional extension of k infinity. And the upshot of this, of this same theory for the lubin tet extensions, so that you start from v,

and you attach to it a vector space over this k infinity hat. You mean the composition for a suitable norm? For a suitable norm, yes, it depends what you mean by suitable, but yeah, yeah, yeah.

OK. So which is that this field is some kind of h minus 1 dimensional extension

of k infinity. So in particular, you see, if you have the p-adic Lagrange correspondence in mind, and you want to construct phi gamma modules over rings of power series in h variables, where h now is the degree of f, which is something that you might want to do in order to relate those phi gamma modules to GL2 of f, this looks like a good start.

Using the lubin tet setting, you can construct vector spaces or fields that, in some sense, have h minus 1 variables in them. And let me finish briefly by saying that, as Vita said, so this Wn is really the set of qp, locally qp analytic

vectors. So you look at the analytic structure on OF cross. That comes from the fact that locally, it's isomorphic to h copies of zp. This is your local analytic structure. But of course, you can look at Wf analytic using

the fact that, in this case, OF cross is a one-dimensional f-analytic group, analytic vectors. And it turns out that this is also of the right size.

So let me tell you what happens. If I look at the set of f-analytic vectors of k infinity hat, then I recover k infinity. And if I take a general representation, v, and I do this construction, I also recover cp tensor v.

So you have a choice that you can either get the set of locally qp analytic vectors, and this is a vector space over a field generated by h minus 1 variables. Or you can say that you are only

interested in the set of f-analytic vectors, and this still gives you something meaningful. And what I'd like to do is to have us bear this in mind as a template for what I'm going to explain now about phi gamma modules, which is a sort of generalization of sense theory, but a little bit more technical.

So let me tell you about some objects that Peter Schneider talked about in his lecture. Although I'm going to give a slightly different presentation, because I have a sort of different point of view about them. So quiescine-ren phi gamma modules.

And from now on, I'm going to assume that k is equal to f, because this now really makes my life simpler. So let me start by recalling the definition of the rings of power series over which a fountain's phi gamma modules are going to be defined.

So af is the set of power series in a variable y. So usually one uses x, but then I use y to differentiate the fact that x is a cyclotomic variable and y is the lubin-tate variable. So ai is in f, in of, sorry.

And ai goes to 0 as i goes to minus infinity. And bf is af invert p. So you have this local ring and this two-dimensional local field.

And these two rings are endowed with an action of gamma f. And the gamma f acts, you use the formulas that come from the lubin-tate formal module lt. So g of y is the power series coming from the lubin-tate

module in kf of g applied to y. And then you extend this action to make it into a ring homomorphism of, endomorphism of af. And it doesn't act on the coefficient. You're right, yes, g is f linear. And likewise, so if q is p to the h,

and I look at the Frobenius phi sub q, so the h-th power of the absolute Frobenius, this will also act linearly on f. And phi q of y is bracket p of y. So I have those rings of power series that are familiar from Fontaine's constructions with the lubin-tate action instead

of the cyclotomic action. And of course, if f is equal to qp, then I recover the cyclotomic rings, except with a different choice of the variable because of the way I've set things up. And so here is a fact that I'm really going to use as a black box.

So there exists an element u in this ring A tilde plus, so one of Fontaine's rings of periods. So A tilde plus is the width vectors over this limit of o, c, p over p. So I'm sorry I haven't adopted the more modern terminology. I'm sure I could put a tilt somewhere here

and everything will be done, but plug, plug, plug. Such that, so A tilde plus is one of these rings of periods of Fontaine. And in particular, it comes equipped with Frobenius and the action of Galois. And the point is that there exists an element here such that the action of Galois and the Frobenius

are precisely given by these formulas. The tensor is o f, no? Sorry? The tensor is o f, no? Well, no, because, aha, see now, f is on Ramey's side, so it's already included inside. I mean, yes, if I wanted to be very precise, I would say o f tensor o f A tilde plus.

But such that g of u, yes, yes, yes. So if g is in gf and phi q of u is p of u, and actually there are many such elements.

And you can choose one such u with a prescribed reduction modulo p. So plus prescribed reduction mod p. So is it unique, no? Yes, if you prescribe the reduction modulo p,

then it's unique. So you can find this in papers of Colmets. I'm not sure. I haven't done my homework. I'm not sure if Colmets was the first to notice this or not. I mean, it's an easy generalization of the principle of construction of Coleman-Bauer series.

And the point of the existence of this element is that you can lift the field of norms.

So this allows you lift of the field of norms. And this allows you to apply Fontaine's machinery of phi gamma modules. So you can apply Fontaine's machine of phi gamma modules.

And you get the following precise theorem, that there exists an equivalence of categories,

so equivalence of categories. So between, on the left-hand side, the f linear representations of g sub f. So a category of f linear Galois representations.

On the right-hand side, a category of eta phi gamma modules over b sub f. So eta. And now to be precise, it's phi q, the same phi q here, and gamma f modules over b sub f.

So this is really a straightforward consequence of Fontaine's constructions and the existence of this element, q. So I'm not sure who to attribute this to. It's certainly a theorem of Fontaine. But the first people to really write this down precisely

in this setting were Kissin and Wren, which is why I call my chapter Kissin and Wren phi gamma modules. Yes, gf is the notation. It's not there anymore. But gf is the Galois group of qp bar over f.

In fact, there, gf acts through gamma f, right? Yeah, absolutely. So chi sub f of g really means that gf acts through gamma sub f, yes. OK, so so far, I mean, maybe you are a bit disappointed because the Lubin test situation is precisely the same as in the cyclotomic situation, except that,

of course, now we have phi q gamma sub f module. And gamma sub f is a bigger group than zp cross. But apart from that, the story is really the same. But as you know, the next chapter in the story of cyclotomic phi gamma modules was to ask, which of those phi gamma modules are overconvergent?

Overconvergent. OK, so these phi gamma modules, if you choose a basis, then you can look at the matrices of phi under the elements of gamma. And their entries are elements of b sub f. And the elements of b sub f are power series

that, after you reconverge, nowhere. There is no reason why there would be any domain of convergence. And the overconvergent means you can choose a basis such that the entries of these matrices are power series that do have a known empty domain of convergence. OK, the point being that then you can evaluate those power series. And this allows you to relate phi gamma modules

and periodic H-theory. And the good news in the cyclotomic case was the following, is that if f is equal to Qp, so if we are in the cyclotomic setting, then all phi gamma modules are overconvergent. So if f is equal to Qp, all of them are overconvergent.

So this is a theorem of Charbonneau and Colmets. And the answer to this question when f is different from Qp is that not all of them

are overconvergent. So if f is different from Qp, so not all of them. And I think this was also suspected at first by Colmets. And the fact that they are not overconvergent

was proved by Foucault. And it's in a paper, Foucault and Chie. So let me write Colmets, Foucault, and Chie. And let me tell you a bit more precisely about those phi gamma modules that are not overconvergent.

So what Foucault and Chie proved precisely is that there exists a character delta, say, from gamma sub f to f cross, and an extension,

so v of f by f delta. So a two-dimensional f linear representation of Qf, that is an extension of f by f delta, such that this extension is not overconvergent. So the corresponding phi gamma module is not overconvergent.

That is not overconvergent. But is it OK that the overconvergent means that when you choose all the embedding of f into f, which are different? That's what I'm going to say right now.

Yes, thank you. OK, so now I'm going to do what Fontaine was saying. So I'm going to tell you about the reasons that make representations overconvergent or not. So let me start from v and f representation of Gf.

And now if I choose an embedding, once again, an embedding of f into Qp bar, so tau from f to f, then I can look at the same theory, so the classical same theory of v, but at this embedding. So what I do is I extend scalars from f to Cp,

but using this embedding tau. So f tau, and this gives you the classical same theory, but at tau.

And let me say that v is f analytic. So if this Cp tensor v is always trivial, so v is always Cp admissible at all the embeddings tau that are different from the identity. So if at every tau different from the identity,

so v is Hodge state with weight 0, or Cp admissible if you want, it's the same condition. So at each embedding, v has to be Hodge state, and the Hodge state weights have to be 0. So it's a rather strong condition. So for example, if you take f twisted by the lubin-tate

character, this is an example of an f analytic representation. At the identity embedding, there is one weight, which is 1, but at all the others, the weights are indeed equal to 0. So this is OK. But say if you twist f by the cyclotomic character,

then of course at all the embeddings, there is a weight equal to 1. So this is not going to work if f is different from Qp. So this is a rather restrictive condition.

And here is a theorem, which I think really explains the relevance of this definition to the theory of overconvergent lubin-tate phi gamma modules. So theorem. So if v is absolutely irreducible representation of Gf,

which is overconvergent, so on overconvergent,

then there exists a character delta from gamma f to of cos such that v twisted by delta is f analytic.

If you have an overconvergent phi gamma module, you can always twist it by a character of gamma and still get an overconvergent phi gamma module. So you really have to have some condition like this. But then the theorem says that up to this condition, there are no other irreducible overconvergent phi gamma modules. So the overconvergent here is in which sense? In the lubin-tate sense.

Otherwise, it would be a vacuous condition. I'm sorry. What's the exact relation between f analytic and overconvergence? It goes both ways? No, if you are overconvergent, then some twist is f analytic. But that's the only thing that can happen, is twisting. OK, so this gives you a lot of examples, of course,

of non-overconvergence representations. So this is somehow orthogonal to Foucault and Shea because here I assume that my representation is irreducible. So I'm sure that with a bit of extra work, you can drop the irreducibility condition and get a complete characterization of overconvergence representations. But I think I'm a bit too lazy, so I haven't gotten up to that yet.

Very good question. That's what I'm going to write next. So unfortunately, the converse is not a theorem. It's a conjecture. And what I'd like to do, one of the points of the rest of my lecture, is to explain some evidence for this conjecture.

So here's the conjecture. So if v is f analytic, then it is overconvergent. Yes, so that's what we conjecture. And let me mention some particular cases, so theorem.

So Kissin and Wren proved this conjecture under the additional assumption that v is crystalline. So if v is a crystalline f analytic representation, then it is overconvergent by a theorem of Kissin and Wren. When you say overconvergent, it means that the gamma module is an overconvergent.

Absolutely, yeah. It means that the, that's what I put here, yeah. But is the overconvergent extension unique when you say, or? Yes, yes, yes, yes. Yes, then there is a unique, I mean, if you have two bases in which the matrices are overconvergent, you can go from one basis to the other by an overconvergent matrix, so the extension is unique.

So this is true if v is crystalline. And Foucault and Shea also proved some cases of this conjecture for representations that are two-dimensional reducible representations.

So two-dimensional reducible representations.

Not reducible, triangulating, OK, thank you. Yes, the phi gamma module is reducible. OK, so there is a lot of evidence for this conjecture. And what I would like to do now is to tell you about phi gamma modules and their relationship with locally analytic vectors, which I haven't done yet, and explain

why I think that this is evidence for this conjecture. So third part, phi gamma modules and locally analytic vectors.

Well, at least I'm going to explain what I think is the right way of proving this conjecture, because if you are in a hurry, there's probably a quicker way. But I won't talk about this. So let me say a few things about classical overconvergent phi gamma modules. So let me go back to overconvergent phi gamma modules, but in the cyclotomic case.

So overconvergent phi gamma modules, so in the cyclotomic setting. So let me give you a few reminders about the objects that I'm going to talk about. So let i be a subinterval of 0 plus infinity.

And I can look at this set of power series Bf, but now look at those that satisfy some convergence condition related to i. So this is the set of power series,

except now I'm back with the variable x, because I'm in the cyclotomic setting, so with ai in f. And I want the power series to converge for, and I think this is the right condition, the valuation.

Sorry, the norm of x is p to the minus 1 over r. Whenever r is in the interval. The interval may be finite, or interval i. Extend to plus infinity, or it can be? Yeah, you can if you want. It's just that you cannot have plus infinity inside it. But yeah, it can go all the way to, it's not bounded.

Yeah, it can also be a finite interval? Yeah, yeah, yeah. Any sum interval that you want to use? It corresponds to an analysis. Yeah, absolutely. It corresponds to an analysis. When r goes to 0, this really corresponds to the radius to the norm of x going to 0. And when r goes to infinity, this corresponds to the norm of x going to 1.

So if you take for i the whole of this interval, then you get a power series that converge on the open unit disk. Because you got the other one under this. Otherwise, it's an analyst, absolutely. And this is a Banach algebra for the norm that you imagine.

If, of course, i is closed. And let me recall briefly that the phi gamma modules, the overconversion phi gamma modules that we're interested in, are modules over b.

Sorry? Yeah, but infinity is excluded in any case. So closed and bounded, ah, no, yes, yes, yes, thank you.

OK, so the usual phi gamma modules that we work with, for example, the ones that Gabriel used in his lecture, are defined over an annulus that is as inner radius r and is open with outer radius 1. And because this is not a closed and bounded interval, this leads to a little bit of trouble.

And it was an observation of Schneider and Teitelbaum and also of Kedleya that you can see modules over this ring as vector bundles on the corresponding annulus. And in particular, as a compatible set of modules on the closed subannually of the open annulus.

OK, modules, so they correspond to vector bundles. And in practical terms, this means that a module m over this ring will correspond to a collection. So m of, sorry, phi of rank D. So a phi B something

module of rank D will correspond to a collection of m sub s, where m s is a phi B, F, R, S module of rank D,

with, of course, the gluing conditions that you imagine. OK. It's a project. Yeah, it's a project. It's a vector bundle. I mean, there's no content.

And the reason I remind you of this is because these rings are now Banach algebras. And this will make what I say a little bit easier to understand.

So in particular, if you combine Fontaine's construction and Charboni and Colmett's theorem, so the construction of overconvergent phi gamma modules. So you start from V, a representation of G sub F. And to this, you can attach a vector bundle like this,

so a collection of D, R, S, V. So for S at least equal to R. And so if I write I for I, R, S for brevity, then DIV is a BIFV module free of rank D. So once again,

D is the dimension of V. And of course, because it's a phi gamma module, there is also an action of gamma on this module. And that Frobenius phi, that will send Di of V to Di prime of V, where I prime is the right interval

because Frobenius changes also diagonally with an action of gamma chi, and so on and so forth. And so how does one construct these modules? So you start from another of Fontaine's rings of period.

So B tilde I, which is the completion of A tilde plus, we already had. So we invert P. You invert the Teichmuller of any uniformizer. So let me take P tilde. So the same as in Weinstein's lecture.

So it's the completion of this ring for some norm that depends on I and is the one that you imagine on the ring of power series. And so you have this big ring of periods of Fontaine.

Inside this, you have B F tilde I, which is B tilde I invariant under H F. And if you know a little bit of Piedeker's theory, you can see that these rings B, F, I without the tilde

are contained inside this one. So B, F, I. And not only B, F, I, but also phi to the minus n of B, F, P to the n times I, which I denote by B, F, and I. So I think at this point, unless you already

know what I'm talking about, this looks very confusing. And it's an observation of Coleman, really, that this very confusing situation looks very much like sense theory. You have this big ring of periods B tilde I, which is a sort of analog of CP. Inside it, you have this B, this invariance under H F.

That's F infinity. And each of these rings of power series in phi to the minus n of x looks like the finite layers of the cyclotomic extension, so F and the F n's.

And in particular, you can apply the same kind of methods as in sense theory to construct this overconvergent phi gamma module. That's really a proof of the Charbonnet-Colmes theory. So you start from V. You extend scalar, so not to CP,

but to this bigger ring. And you get this D tilde I of V. And inside this D tilde I of V, you construct the analog of this n, which is going to be the union of this D I n of V,

where this notation reflects this notation. So here, by some decompression process. So these are the objects that appear

when you apply the Colmes-Centate method to phi gamma modules. OK, you take invariants, and then you descend to a finite level. But what I told you within the framework of cent theory, that you could replace these normalized traces,

this descent, with the construction of locally analytic vectors is still true for phi gamma modules. So theorem. So if I start from these invariants, so this is a Banach space with a continuous action

of gamma sub f. So I can look at the locally analytic vectors inside this. And this is the union of this D I n of V, or if you prefer, the union of the phi to the minus n of D p and I of V.

So the upshot is that once again, you can construct the classical overconvergent phi gamma modules, the ones that we use all the time in the periodic long lines for GL2QP, using locally analytic vectors.

And so if you want to construct phi gamma modules in the lubin-tate setting, well, let's try to see what happens if we do this construction. So let me now go back to the lubin-tate extension.

Yeah? So is this all OK for arbitrary animals, i, or do you have to take r sufficiently close to infinity? Yeah, yeah, r sufficiently close to infinity. It's the same as in, no, you don't gain anything. I mean, you have to be sufficiently close to infinity.

You don't gain any radius of convergence by doing this. I mean, in order to prove this in any way, I use the Charbonnier-Colmes theorem. Yes. And what happens in the lubin-tate setting is the following. So I do the same construction, so the set

of locally analytic vectors, but now for the action of the lubin-tate gamma f. And what I get is a free b f tilde i analytic module of rank d, so the dimension of v. And now, of course,

you are in the same situation as we were when we are talking about Cen theory. This looks nice, except you have no idea what this ring is made of. So in the same setting, in the setting of Cen theory, this was this h minus 1 dimensional extension of k infinity.

Here, on my front, things are a little bit more complicated. No, no, no. Even in the cyclotomic case, it's not a field. So I'm going to give you a lot of examples of elements of this ring.

So let me once again take an embedding tau of f from f

to f, and then there exists an element y tau in this ring. That has the following property. So if I look at g of y tau, so it's going to be bracket of k sub f of g,

but now twisted by the embedding tau. And phi q of y tau is still p of y tau. So I told you about this element y in the chapter on key sine and phi gamma modules that satisfy this property for the identity embedding.

So in this ring now, there are analogous elements that satisfy this formula for all the embeddings of f into f. So you get h elements y tau. And these elements are algebraically independent. They are even power series equally independent.

And so inside this big ring, you get a lot of multivariable power series in these h variables. So b, f, i, n contains multivariable, no, no.

In this case, it's perfectly symmetric in the y tau, yes, so tau. So what you get is a ring that looks like a ring of multivariable power series in h variable. And so we are getting closer to our objective

of constructing phi gamma modules over rings in a power series in h variables, so in as many variables as the degree of the extension. So unfortunately, there seem to be more elements in this ring. And it's still an open question to figure out exactly what is this ring, what is it made of.

So I still don't have a complete description of the elements of this ring. Yes, in some sense, yeah. So I mean, it's a question that I find very interesting. And once it's solved, I think we'll

have made some progress towards the construction of these phi gamma modules in several variables. So I'd like to take the last few minutes to tell you how this seems to relate to the Quisson-Rehn phi gamma modules and the conjecture that I had at the end of the chapter on Quisson-Rehn phi gamma modules. So I'd like to call this a periodic monodromy conjecture.

I think I'll call it a periodic monodromy question, because I have no other evidence for it other than the fact that if it was true, it would be very nice. And I'm glad that Kiran is in the audience, because he's shown that he's good at solving monodromy conjecture.

So I hope you can solve this one. OK, so let me go back. So now I'm back in the Lubin-Tate setting. I'm still in the Lubin-Tate setting. OK, so I have this set of locally analytic vectors. In particular, I have this action of gamma sub f. And gamma sub f is isomorphic to OF cross.

So in particular, for every embedding tau of f into f, I have a derivative in the direction of tau. So this is the element of the Lie algebra

that corresponds to taking derivative in the direction of tau. And how does it act, for example, on the power series in those h variables?

So that's a little exercise that's easy to do. So it is log lt, so the Lubin-Tate logarithm in the variable y sub tau, times some unit that we don't really care about, times the derivative of f with respect to the variable y tau.

And so this log, just like the cyclotomic log, so this element t, if you want, in Perikar's theory, this log of 1 plus x is an unbounded power series. It has a lot of zeros, so it's a little bit annoying.

But the part on the right-hand side is a genuine differential operator. And so for each embedding tau, you have a differential operator, partial tau. And if you have a differential operator, of course, you hope to be able to use the theory of periodic differential equations.

And so as I said, I can't really describe this whole ring, but what I can do is the following. I can describe not all of the qp analytic vectors, but the sub-ring of f analytic vectors. So this is the same as the set of analytic vectors

that are killed by all the derivatives in all the directions tau different from the identity. So nabla tau equals 0 for all tau different from the identity. And this is the following, is the union of the b, f, n, i.

So rings of power series in one variable with coefficients in f in the variable y identity. So y.

So what I get is a nice ring of power series in this one variable y identity. That was the y from the chapter on key sine n, phi gamma modules. And let me now finish. Once you do this, yes.

Once you kill all the other directions. So let me finish by writing down my question and telling you why I think it implies the conjecture in the key sine n chapter.

So question. And it's a sort of multi-dimensional analog of the classical conjecture of Kru. So if m is a b, f tilde i analytic module of rank,

so free of rank d, that is endowed with differential operators partial tau like this under Frobenius.

So with a set of partial taus and a compatible Frobenius in some certain sense. So be careful that this changes at interval i. Then the conjecture is that m has a full set of solutions

once you add the log of the white house. Then is m generated by horizontal sections?

So after adding the log of the variables. So if you know Kru's conjecture in one dimension, the classical monodromy theorem for peeling differential equations, this really is the analog in higher dimension.

A finite extension. So now in this case, you don't have to make a finite extension because you're left with a set of solutions that is defined over a ring that is big enough that Hilbert's theorem 90 will allow you to descend the solutions as you wish.

So this is a question that I want to ask. In the final two minutes, let me tell you why I think the theorem is a little star. So star means it will really be a theorem once I've sat down and written the proof. So if this is true, if the answer is yes,

so every f-analytic representation is over convergent. Mi chui, mi chui, merci. Representation is lubin-tate over convergent.

Yes, so now let me finish by explaining the steps of the proof. And then one just has to fill the gaps. So you start by constructing this d tilde i of v.

So if you want the analog of W instantially, you take the locally analytic vectors. And this, I told you, is free of rank d over b tilde f i n. And now you have to use the condition that v is f-analytic.

So as I said, this d tilde i v n is equipped with those derivatives, a nabla tau in the direction of tau. And if v is f-analytic, if and only if you can divide

these nabla taus by the lubin-tate log of Y tau. So if and only if this partial tau operator preserves this set of analytic vectors, the nabla tau respects it.

But then if you divide by log of Y tau, you divide by a power series that has an infinite number of zeros, and this is not going to respect it. But then it's an easy thing to see, once you know how to do it in the cyclotomic case, that this condition that v associates with zero is precisely the one that says that when you divide nabla tau by this log of Y tau, what you get is still indeed tilde i analytic.

And then once you know that, you apply the positive answer to this question. And this tells you that d tilde i v n, nabla tau equals zero for two different identity, is

three b, f, and i in Y module of rank t. And now you see that once you get a module over

this key sine ring of power series in one variable, you're basically done. So I hope I've convinced you that looking at local analytic vectors is a good idea if you want to try to generalize sine theory and figure my modules from the one-dimensional

setting to the multi-dimensional setting. And I also hope that somebody will be interested in this question. And therefore, I'll have to make some progress towards this conjecture. So thank you for your attention.

So think about these extra elements. If you take something which is killed by all the derivative outside of the identity,

you don't get extra elements this time. In some sense, yeah.

But you see, I can tell you exactly what the problem is. So this contains this power series in the white house. But it also contains, of course, the cyclotomic X. The cyclotomic X is a local analytic vector. And X is indeed a power series in the white house. But it never has the required radius of convergence. So there is some strange-

Is this solved? Or is there even more? I think that's all there is. So these elements that look, that want to be in there, that want to be a multivariable power series, but are not for some reason. Yeah. So to follow up on this, I guess, so there's this space sort of defined by this power

so this ring of power series maps to the usual big ring in p-adic, the b tilde range. And so you can, so this sort of defines a Berkovich point, and you can kind of localize around that. And does that, do those give you more of these, this kind of elements?

You're basically sort of asking for less, yeah, you're sort of asking for some kind of restricted convergence. So is that the kind of elements you're seeing? It's definitely a problem to do with the convergence. So I'm not sure- The series that don't converge on the full disk, but they converge on some.

You seem to insist to take the QP analytic vector instead of F analytic vector, the multidimensional setting, but in the beginning you said that F analytic vectors are sufficient to recover the original. Yes. So why don't you work it all the way through in one dimension?

So no, you're absolutely right to ask this. So the reason I mentioned this at the end of the chapter of Senn theory was also some evidence why this should work, because this is precisely also the set of F analytic vectors. No, you're right, I should have said this. So this is also d tilde i v f n.

And of course now the question is, is this of the right dimension? So in Senn theory I know it is, but as you're right to point out, in this setting I don't know, but I hope the analog of the construction will work. That's precisely my motivation.