 Merci beaucoup aux organisateurs pour l'invitation. Je vais essayer de faire un petit switch en anglais. Je pense que c'est assez utile de dire que ce travail est joint avec Pierre et Vitas Pascunas, mais je vais le faire. Juste parce qu'il y a un peu de différence dans le nom, donc joint avec, je ne vais pas dire avec Pierre Colmais et moi-même, mais avec Vitas Pascunas. Et donc le but est d'expliquer comment la première parole implique la bijectivité de la correspondance entre le GL2 de QP et les représentations de l'espace et les représentations de deux dimensions absolument réduisables. Donc, je vais commencer par rappeler que P est de la prime et je vais aussi l'éloigner, même deux fois. L sera une extension finie de QP et G sera le groupe GL2 de QP. Et juste pour un petit issue technique, je vais considérer une certaine catégorie de représentations, mais pour les absolument réduisables objets que nous sommes intéressés à, ils seraient les mêmes. Donc le rap de G sera la catégorie de l'admissable. C'est une conséquence d'une autre hypothèse que je fais sur les représentations. Donc l'admissable unitaire, mais maintenant je vais demander qu'ils aient un caractère central et qu'ils sont résiduels de la longue finie. Donc cela signifie que la reduction de l'admissable est de la longue finie. Et en fait, il se semble que cette condition, que c'est résiduel de la longue finie, est en fait équivalent de la longue finie topologiquement et c'est un conséquence de ce que l'on explique dans le premier cours. Et ça aussi s'il vous plait de cette théorie et de l'admissable, que le rap de G et cette catégorie de l'admissable de l'admissable de l'admissable ont les mêmes absolument réduisables objets. Donc, je me souviens que la théorie principale est que le functeur V, dont je vais revenir dans quelques moments, conduit une bijection absolument réduisable de l'admissable d'objets de cette catégorie de rap G de l'admissable et de l'admissable de l'admissable de l'admissable galouat de l'admissable. De l'admissable, de l'admissable. Donc, c'est que la pi va au V de pi. Et je vais ajouter quelque chose qui sera crucial pour l'admissable et c'est compatible avec la théorie classique dans le sens que si je regarde la détermination de V de pi ce sera equal à l'inverse de l'admissable de l'admissable de pi. Donc, le CC sera pour l'admissable de l'admissable. Et en fait, l'admissable de l'admissable sera la main partie de mon parler. Donc, je dois stresser que cette équalité est faite en général, si vous ne considérez pas absolument non-admissables représentations. Donc, cette équalité n'est pas un conséquence du fait que c'est une bijection mais un conséquence de la construction de l'admissable. Pourquoi j'insiste sur cette compatibilité? Parce que vous pouvez donc utiliser beaucoup de constructions qui apparaissent dans la correspondance de l'admissable. Une question qui est naturelle après voir Vita stock. Donc, que sont les représentations semi-simple? Simple représentations mod p de la forme Theta tensored with k semi-simplified for some Theta contained in pi which is absolutely irreducible. In Vita stock, in abstraction, they have to be either super singular or something which is contained in some direct sum of two principal series. Well, we are going to see that most of the time this inclusion is actually une equality. So the theorem is that if I take pi in band, oh sorry in this rep G absolutely irreducible then and I guess I should say non-ordinary or maybe I can say then we have an equivalence between the statements. The first statement is that pi is non-ordinary and the second statement is that pi bar semi-simplified is either super singular or this direct sum of two principal series. And maybe I should say that all this happens possibly after extending this field of scalars replacing it with quadratic un ramified extension. So actually even in the first theorem that Vita explained you might have a slight possibility that you have to extend the field of scalars. Ok, so it's not immediately clear that theorem 1 implies theorem 2 but I will try to explain that there is a certain yoga which allows you to get this result from the compatibility with class field theory and the compatibility between pi-addic and mod-p-languages correspondence. So let's see what has already been proved about theorem 1 Well essentially what we proved so far is that this map is well defined that the image of any absolutely irreducible non-ordinary object is to dimensional and absolutely irreducible. And so what we need now is that this is an injection and then a surjection where the argument works really differently for discussing surjectivity and then injectivity. So I will start with surjectivity. This is essentially known in all cases but one when P is 2 so this was known and actually I should say that the theorem itself was known by the work of Pascunas when P was at least 5. So known unless P is 2 the mod-p-galois representation row-bar simplify this trivial up to a twist. So this is known by the work of Colmez, Kissen Böckler and Chenevié By which I mean Colmez and Kissen proved a certain density result about crystalline representations in deformation spaces and then one has to check that you can construct crystalline representations on each irreducible component of the corresponding deformation space so you have to understand the irreducible components which means you have to understand the corresponding deformation rings and those were considered in the abstracted cases so when H2 is non-zero those were treated by Böckler and Chenevié and the only remaining case was this one and we have the following theorem if row-bar from GQP to GL2 of KL is the trivial representation then if I look at the framed deformation ring and I look at the generic fiber of that this one has two irreducible components so P is 2 here GQ2 and these are in bijection with those of spec are framed trivial character 1 over 2 that means you have two irreducible components which are completely described in terms of the determinant so it's rather easy to construct crystalline representations in each of those irreducible components and that gives you the density of crystalline points in the remaining case so the key point of the talk now is to understand the injectivity of the map now the injectivity will turn out to to be equivalent to a very innocent statement about the central character of some Banach representations except that it doesn't seem to be that easy to recover that central character from the Banach representations themselves so for that I will need to introduce a certain number of constructions but let me state an equivalent version and actually a stronger form of injectivity which can be expressed in very elementary ways so the statement is as follows if I take pi1 and pi2 in rap G absolutely irreducible and non ordinary then any P equivariant homomorphism continuous of course and linear between pi1 and pi2 is actually G equivariant so I will come back to this group P in a moment so here P is the mirabolic subgroup QP star QP01 alright so this is a purely representation theoretic result and I should remark straight away that if P is replaced with the Borel subgroup then this was known for quite a few years by work of it started with P and it was then refined by Pascunas so you might try to deduce the statement from the similar statement with the Borel subgroup because the two are very similar anyway the only problem is dealing with the central character in this issue so you have to prove that if you have let's say a continuous even an isomorphism a P equivariant isomorphism between pi1 and pi2 then necessarily pi1 and pi2 have the same factor, that's the whole issue because once you know that you can apply this result and you are done well unfortunately it doesn't seem to be possible to reduce easily to the case of the Borel subgroup so the proof will be a little bit round about by using a lot of techniques from the theory of Figma modules actually yeah I know it's always the same problem with this kind of issue you have a statement which is very nice looking but that's it ah so speaking about Figma modules I will need a certain number of preliminaries but I will not try to to go into the full details of the theory just to recall a few general things about this so first of all let me recall a few things about this Montreal functor so we have this category Rep G which is mapped by this Montreal functor or magical functor if you want to the category of finite dimensional continuous linear representations of this absolute Galois group of QP now here we have Cartier duality ah so it maps to this category itself and by Cartier duality I mean simply you take the representation you take the dual and then you twist by the cyclotomy character and ah then you have Fontaine's equivalence of categories with the category of etal Figma modules over this ring E which was denoted E of B yesterday in Peter Schneider's talk so we are not working with Figma modules over the roba ring here but really at integral level and when you compose all this you get a functor from Rep G to Figma etal of E which I will call P goes to D of Pi sorry now just a side remark V was covariant by construction ah so this one turns out to be contravariant because I composed with some duality here so this is contravariant and exact and now a natural thing to do is to understand if you start with a Banach space representation here you send it to a Figma module and you would like to reconstruct it from the Figma module itself so that you can translate everything in terms of Figma modules and that this can go in both sides so the way to do this is to generalize a certain number of constructions that appear in Colmez's paper in asterisk about the Piadik Langland's correspondence and there he considered the case when the central character is the good one predicted by the compatibility and actually it turns out that it's crucial to develop the whole theory with an arbitrary characters and then to compare the two approaches so here's how it works you start with any unitary character delta from QP star to OL star so of course this is continuous character and I have there is a a functor from the category Gama et Talof E into that of G-equivariant sheaves on the QP points of the projective line and I can compose with the global sections functor to get topological L vector spaces plus continuous G action and I will denote the corresponding functor as D goes to D cross delta P1 so this is an object which is rather familiar in the theory and now as topological L vector spaces D cross delta P1 is an extension of a Banach space representation sorry of a Banach space and the dual of a Banach space so there is an exact sequence of topological L vector spaces which works for any delta and for any et alpha gamma module D now the key point is to understand when you can have such a decomposition which is moreover G-equivariant so add the structure of G topological modules so if this is the case then you say that the pair D delta is G compatible so say D delta is G compatible if one can find a G-equivariant a decomposition star with pie and pie check in this category and now the whole point is the following in order to understand all the Banach space representations of G essentially you have to understand what are all the G compatible pairs well this is not completely straightforward and actually one has to redo a lot of constructions and results that are found in a pierce paper but we did all this in a joint work and then everything works well so we have the following theorem well I guess I should introduce some notation before that so let me call Mf as the Montreal functor of delta to be the image so I take D of pie when pie is in rep G with central character delta that means I restrict to those Banach representations whose central character is delta so this is naturally a subcategory of P gamma et tal of E and now the theorem is as follows first of all you can completely describe this in terms of P gamma modules but unfortunately there is a certain twist because we use Cartier duality at some moments there are some funny delta inverse appearing here so this is the same as those D such that D delta is G compatible and secondly so that completely describes the image of Colmezes Montreal functor purely in terms of the gamma modules secondly there exists a functor from this category Mf of delta inverse to rep G which I will denote pie goes to pie delta of D sorry it's D goes to pie delta of D such that several things happen but the most important one is the following D applied to pie delta of D is the Cartier dual of D by which I mean the Figma module which corresponds by Fontaine's equivalence of categories to the Cartier dual the Galois representation that's one point and probably the most important for us you can recover pie well not in all cases but in cases that we are interested in so if pie is super oh sorry not super singular I'm used with is absolutely irreducible and non ordinary yeah from time to time I will use the word super singular that will always mean absolutely irreducible non ordinary it's somehow easier to say one word instead of four but anyway so if pie is absolutely irreducible non ordinary then you can recover pie from the Figma module and the character and I should say with central character delta then pie is isomorphic to pie delta of D check where D is the Figma module which corresponds to pie so the philosophy as follows if you know delta which is the central character of pie and if you know its image then you get pie so this theorem combined with what Vitas explained already shows quite a lot of things for instance you have the following consequence that if D is of dimension at least 3 is absolutely irreducible then D delta is never G compatible you see the point being that if that was compatible then it would give rise to some Banach representation pie delta of D for which you know that you can control the reduction mod P and so you know that the image via colmesis functor has to be of dimension at most 2 but that's exactly D check so that one has to be of dimension at most 2 so you have to play with this all the time combining that theorem with information given by by the study of the reduction mod P but unfortunately that will never give you the injectivity of the Montreal functor so that's what I'm trying to explain now the key theorem that we prove is the following which actually combines the surjectivity and the injectivity is that if D is of dimension 2 and D absolutely irreducible then there is a unique character delta such that D delta is G compatible and you can see what this character is and actually delta is what we denote delta D taking the cyclotomy character inverse and multiplied by the determinant of D well of course you have to think of determinant of D as being a Figma modulo of rank 1 pass it through a intense equivalence of categories and seeing it as a Galois character and then local class field theory allows you to see this as a character of QP star which is unitary so that's really the key theorem which implies at the same time surjectivity because of that you simply apply it to Pi delta D of D and that gives you the surjectivity it also implies injectivity because it tells you that if you know Pi if you know D of Pi then you know that the central character has to be that one and that's it then you can recover Pi via this recipe or by using Pasquona's result and it also gives you compatibility with local class field theory by playing the game twice actually using at the same time surjectivity and then injectivity so this gives everything now the fact that this specific character works that it gives you a G compatible pair is exactly Kessin's argument with Zariski density of crystalline points you check that this is a G compatible pair in the crystalline case and then you have some Zariski density result but I would like to point out that there is some issue with that the experts are used with the following diagram which compares the deformation theory of GL2 of QP representations and Galois representations by saying that if you look at the image so you look at the deformation functor for something mod P and you map it via Colmais's functor to the deformation space of V of Pi and you would like to prove that the image is big enough so that you have surjectivity this uses the fact that we have a certain injectivity of this form that was one of the key arguments to be able to say that the image of this map is actually closed you check it on the tangent spaces and this has to be a closed immersion on tangent spaces which is equivalent to this statement here the only problem is that if you want to play this game when P is 2 we don't know how to compute those X1 and it's not so clear that this is injective so there is another argument which is already written down in Colmais's paper in Asterisk which allows you to do the interpolation argument starting from the crystalline case to the general case without using this injectivity result ok because you see in the remaining case this is the worst one when row bar is really trivial so if you apply V to that one you get 0 so this is not this is not very nice anyway so this deals with surjectivity of course there is a certain number of papers that you have to combine but at the end you have the statement that D and delta D is always D is of dimension 2 and now the interesting issue is to prove that this is the unique character for which this works so the argument goes in two steps and each of them will have two other steps so two steps for uniqueness of delta let's write delta as eta times delta D of course eta is going to be a unitary character and the statement is that eta is trivial so we need that eta is 1 and this will be proved in two steps the first step will be that eta is locally constant and then that eta is 1 now the problem is that each of these steps actually breaks up in two different steps in each case one distinguishes between trianguline and non-trianguline cases the proof is really very different in the trianguline case and in the non-trianguline case essentially what happens in the trianguline case is that so in the trianguline case this will be a very brief sketch because unfortunately dealing with the details is rather technical a trianguline case the key point is to go to the roba ring so one has a part of local analytic representations associated to dirig and the character delta so of course by taking global sections that will give you a certain topological g-module dirig cross delta p1 and you might hope that such a thing exists if and only if delta is equal to delta d which is not the case unfortunately even though the original statement is a uniqueness statement about delta there are many cases in which this module exists without delta being equal to delta d actually if you twist delta by any finite order character you still get such a shift however the important point is that in the trianguline case you can actually write down explicitly what this thing is so that was a very useful observation which started with some conjectures of Berger-Broy and Emmerton who conjectured what the local analytic vectors should be in the trianguline case and the key point was to understand this module here and the argument goes by imitating up to a certain point this proof because it doesn't apply really word by word however you end up understanding what this deri cross p1 is modulo the following statement that eta is locally constant otherwise the argument wouldn't be that easy so first you prove that eta is locally constant and here this is an argument which doesn't work only in the trianguline case it actually works in almost all cases however there is a very nasty case when the FIGAMA module is associated to a cp admissible representation so the RAM with equal odd state weights 0 for instance that case is rather complicated because you might play the same game with the over convergent module here so in some sense an intermediate between the integral and the roba version and that one exist for any character delta if the representation is cp admissible so that's kind of an intriguing case which has to be delta part so how does it work the fact that eta is locally constant at least in the trianguline case well the key point is that you can look at the action of the Lie algebra on this deri cross p1 and it turns out that this acts through a character well at least the center so the center acts by scalers here there is a distinguished element the casimir say this one will act via some constant c on this module and it turns out that you can read this constant from sent theory namely you have the following identity say in a moment what these objects are so this is the odd state weight of eta or equivalently you take the derivative of eta and you evaluate it at 1 so this is a number which I can see as a scalar operator this is the send operator of d and again this is going to be a scalar operator so this is an equality of operators on the send space a representation associated to d and now there is something funny happening you know that d delta is g compatible but you also know that d and delta d is g compatible that's the subjectivity issue so you write down the two equalities here and you compare them if theta send is not a scalar operator that will automatically tell you that this guy here is the same as the corresponding quantity for delta d and not for eta so if theta send is not a scalar that gives you that w of eta is 0 which means that eta is locally constant oh I'm sorry there is a mistake here this should be delta and not eta you write this equality for delta and delta d you subtract them and then you will have the difference I mean the quotient between delta and delta d which is exactly eta so that's why these two equalities give you that w of eta is 0 so that's the argument which as you can see does not use the fact that d is triangular but only this fact that the send operator is not scalar which is exactly saying that up to a twist the representation is not associated to some cp admissible representation so you see that this completely fails in that nasty case however it turns out that if you are triangular and irreducible then this can never happen so you're fine that tells you that eta has to be locally constant then you redo the work of understanding this module and you end up by proving that there is a certain map so eta locally constant plus work implies that there exists a map which of course is non-zero from some analytic induction locally analytic induction which is exactly we never know it should be delta 2 tensed with epsilon inverse delta delta 1 into pi locally analytic where I choose a triangulation of my FIGAMA module so the point here is that you start choosing a triangulation that will give rise to some characters delta 1 and delta 2 which are not unitary you prove that this will extend to a larger exact sequence of this form so you have a commutative diagram of this form which is G equivariant now you analyze each of these guys here each of them is an extension of a locally analytic parabolic induction by the dual of something else which is not the same locally analytic parabolic induction so you end up with four different parabolic inductions here and then you have to analyze a little bit the structure of these representations and this gives you this map here and now comes the interesting point you might try to inject this one into pi of course by definition and you would like to pass to universal unitary completions and to get a map from a certain banach space which is huge into that one the only problem is that this will work most of the time except when this universal unitary completions is too big so there are some cases for instance if D corresponds to some semi-stable representation there are some issues with the L invariant and things like this but once you understand sufficiently well all the universal unitary completions of these representations you are done this includes the exceptional case in the case p equals 2 when it was not known what the universal unitary completions is because there were some slight problems with the representation theory mod 2 but everything works well and in the end what you do is you construct a new FIGAMA module which has a different triangulation so I call this D prime rig so there is such an extension which is absolutely irreducible and everything you like and a map from pi of D prime to pi of D or to pi actually here by definition this is the same as pi delta D prime of D prime so this is the object that we know it exists for anything of dimension 2 so in the end you end up with a morphism from something very similar to this pi which is just pi of D by the way and now you are in good shape because this is irreducible this is irreducible, the map is non zero so this has to be an isomorphism and then you apply conmesis functor again and this will give you an isomorphism between D prime and D and now you are done because if you look at the determinant it has to be the same but here it's delta 1 delta 2 times eta while here it's delta 1 times delta 2 so eta is 1 so in some sense the triangulation case is okay because everything can be explicitly written down and that we know the universal unitary completions of anything which looks like a principal series representation okay the more interesting case is the non-trianguline one in which the module D-Recross P1 is completely mysterious you cannot write it down explicitly so the non-trianguline case actually really the key point is that D is not a finite height so really in the trianguline case the most difficult part of the argument is when D is a finite height which corresponds to the crystalline case essentially or crystalline case up to a twist so in the non-trianguline case what happens is the following you take D delta to be a G compatible and you look at this exact sequence which exists by definition of G compatibility and now you study the some operators which are rather simple you take alpha in OL star and you look at the map induced by restricting to certain subspaces so I take the kernel of the operator P001 minus alpha inverse of course this injects into D cross delta P1 the same kernel but now this can explicitly be described in terms of the FIGAMA module itself and in particular it does not depend on the choice of delta that's a crucial thing because this is the the usual psi operator on D and this is something which appears all the time in the theory of FIGAMA modules because this is essentially obtained by taking D twisting it by an unrhymified character corresponding to alpha and taking psi equals 1 which is exactly by Fontaine's theorem of V of D tensored with alpha to Vp so what you are doing here is that you look at the USL homology of all the representations obtained by twisting V by this unrhymified characters and you know that those guys are contain this module and now there is something even fun here that this injection is actually an isomorphism is an isomorphism of finite free or actually free of rank 2 modules over I take the USL algebra of Zp star 001 and I invert p but of course once you know that this is an isomorphism it's a quite classical thing that the USL homology of the representation itself is a finite free module over this USL algebra but now you see what is happening if I have a vector V which satisfies this then I can look at the operator W which is 0,1,1,0 and which is really the nightmare in the whole theory but I can use it and I can write this relation in this form right and now W this one times W this is just 1p on a central character I can still multiply by p1 and I get that delta of p Wv is the same as alpha inverse p1 Wv so the upshot is that Wv is also going to be an eigenvector of this operator but with a different eigenvalue which is given by taking into account the value of p of delta so Wv is well is in pie check dual on which p1 acts by I never know how it works yeah it's alpha delta of p inverse so the point is that if I apply this no no inverse then there should be a small mistake somewhere well ok probably it's no inverse yeah because I have another duality it's always a nightmare to compute when you have 2 or 3 dualities involved so we end up with this result saying that essentially this W operator permutes these modules here now the interesting part is that you can actually compute that W operator purely in terms of the Figma module itself and delta without any reference to this representation here and that everything is compatible so this W W acting on well not really deep so equals alpha I take its restriction to Zp star which is just 1 minus alpha Frobenius dpsi equals alpha so and this should be with quotation mark because it doesn't really act on that but it permutes them according to a similar recipe so this is explicitly given in terms of D and delta now so far so good but the problem is that when I say explicitly it's actually explicitly given by some extremely complicated convergent series which you cannot do anything with so now the point is that you're going to avoid dealing with W by taking into account that what you did here also works for delta D because D delta D is also G compatible and I started with any character for which this was G compatible so now you the two Ws play one against each other and I will write something which is nonsense now but which really gives the fundamental identity here W composed with W is supposed to be the identity except that since I have a lot of isomorphisms hidden there and some twists this is not really the identity when you look at them as endomorphisms so this is why I'm saying this doesn't really make any sense but this is the key so W composed with W is a certain operator M delta or actually M eta which goes from DPSI equals 0 to DPSI equals 0 this contains C alpha which is this module 1-alpha Frobenius DPSI equals alpha and it will map it isomorphically on to C alpha eta p so this is a Cartesian diagram this is an isomorphism here and this will be an isomorphism except that I did not tell you what is this operator M eta but I will tell you in a second this is also given by some complicated convergence series but it's much easier to deal with so M eta from DPSI equals 0 to DPSI equals 0 sends Z to the limit as N goes to infinity of the sum well I will just write this down so you have certain restriction maps from global sections to sections over I plus P to D and ZP all those guys are actually in D because this chief is an interesting one you the space of sections over any open compact subset is actually a subspace of the space of global sections so everything injects into the global sections and this series converges there if oh sorry but this is eta yeah so the key point now is that if eta is locally constant you don't need to take a limit that's quite reasonable because these guys satisfy the usual distribution relations I mean if you take restriction to I plus P to D and ZP you can decompose this into classes modular P to D and plus 1 and they satisfy the usual distribution relation so if eta itself is locally constant you don't need to take any limit if N is large enough on a un finite sum that's why we are trying to prove that eta is locally constant because we will be in a good shape we will have an explicit operator which is given by a finite sum which permutes those modules here that's the first key point and the second key point at some moment we have to use the fact that D is not triangle which we haven't used so far and now this is where it comes if D is not triangle then this C alpha have trivial intersection and actually this is an if and only if if D is not well almost I should say if D is not a finite height then this happens and the converse also holds so this modules have trivial intersection precisely when you are in the interesting case let's say and one can show using the fact that these modules have trivial intersection that if I have a locally constant character which permutes the modules in this way then necessarily it is trivial so really the key point in the non-triangle in case and this follows from the fact that if I look at pi check dual so this was a subspace of D cross delta P1 I can compose with restriction to any compact open subgroup and the composite map is going to be injective ok so in the end we started with a character delta for which D delta was G compatible we had some games and we arrived at this crucial equality of operators and now all we have to do is to prove that eta is locally constant but as I said if D itself is not a twist of something Cp admissible you are done by what I said in the triangle in case the only interesting case now is when D is a twist of such a represent of such a vegan module and here there will also be some Zariski closure argument but now argument but not in the same sense as in Vitas talk so let's introduce H the set of characters I don't know how to call them because I used all possible letters let's say chi chi from Qp star into 1 plus the maximal ideal of L such that chi of negative 1 is 1 and I have this equality M chi of C alpha is C alpha chi of pi of P for any alpha in OL star right so what I know is that eta is in age well that's certainly wrong a priori because I have something stronger here I ask that the image should be in 1 plus the maximal ideal should vanish at negative 1 however it's easy to see that this is a subgroup of the group of all characters satisfying these two relations because when you compose these operators if you compose M chi with M chi prime you simply get M chi times chi prime so that's a subgroup and so this statement which was wrong at first will certainly be true if I take a positive integer which is suitable suitable simply meaning that the values of eta to the end should be here and of course I can do that because my operator eta is unitary so I have some character in age and now the key technical ingredient in the proof is the fact that age is a risky close in T which is the set of the group of characters of this form well I certainly have to tell you what this means well let's think what does it mean to give a character from QP star which has these properties it simply means you have to give the image of P and the image of a generator topological generator of ZP star or 1 plus 4 times the corresponding Z2 so this is the same as 1 plus ML times 1 plus ML and of course there is a notion of being Zariski closed in there it simply means you are defined by equations with bounded coefficients formal power series with bounded coefficients right so the key point in the proof of this theorem is actually the fact that this modules C alpha as I said can be interpreted in terms of USR homology and now we had all the USR homology of the representations gained by twisting by these characters alpha to VP now when alpha varies in the unit bowl here these guys vary analytically in alpha and so one can prove that in a suitable sense these modules themselves themselves behave locally analytically and so that gives you the fact that this condition is of locally analytic type you can check that the coefficients are actually bounded and so this gives you the theorem the only problem in doing this is to check that is to control the torsion in the USR homology at the integral level when you go down in the cyclotomic tower but when the Galois representation is absolutely irreducible this is quite easy to do which is our case anyway ok, so once we have this we are done because we have the following result proposition if age is Zariski closed in this T and L is large enough well I guess it would be enough for it to contain a primitive pthroute of unity then either age is trivial which is very nice for us because it means that eta to the n is 1 and so eta is locally constant or age contains a non trivial torsion point and so again we are done because we can start with that torsion point and run again the whole argument that means that get eta locally constant and as I said before this implies that eta has to be trivial this is just this looks a little bit complicated but this is a very innocent statement about formal power series what it says is that if you have a and b in 1 plus the maximal ideal you can understand all the formal power series f such that f of a to the n minus 1 and b to the n minus 1 is 0 for all n so this is not something very frightening except that it turns out that you need some work in the case when log p of a and log p of b are linearly independent as we expect in this case over qp in this case you get that f is 0 and otherwise if one of those I mean if this one depends linearly on this one it's rather easy to deal with that case and so on and so ok well I guess I can stop here questions I don't know how to do it for gl2 of f just to make it clear no no no it's highly singular unfortunately but you can control the singularities it's regular in co-dimension 2 and the corresponding ring is normal it's complete intersection flat over zp I mean you have an explicit system of regular system of parameters and so on but we don't know how to compute the explicit desingularization thing so let's thank the speaker again