2/4 Analytic number theory around torsion homology


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2/4 Analytic number theory around torsion homology
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Venkatesh, Akshay
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Institut des Hautes Études Scientifiques (IHÉS)
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were there so if a and and so
I just remind you quickly at 1 point we were yesterday so last time the thing we discussed was the focus of all is and ordering them imagine recorded fields something like this where he is less than 0 and so element of this and we defined the congruence subgroup in the city where the insurance these number 1 2 0 0 the chairman this 1 and in divide scene and then the the phenomenon which we would discuss is that when we take this group and abelian Isaac so this that and the size of this or the size of its torsion part of it if if it's finance grows exponentially so this was with In squares pockets of this with this is the right side I gave yesterday more precise conjecture and will revisit it later today that was kind of phenomenon at discussing and is of plight lectures on this is that of and I just like that so I'm still in this so after his present this up I described the heuristic old but which at least 6 were very crudely explains why this should be large and some ominous spend more time talking about that and so I discussed will finished discussing the heuristic from yesterday and we'll talk about another heuristic and probably by the end of the day will start on this analytic court so so this 1st heuristic from yesterday so the Oracle heuristic from yesterday suggested that when you take this group making 2 billion good should we should model that says so can be model that as modeled as interviewers where so for some of the 1 PGD there let's say random vectors "quotation mark which some consists of RIC Apple would entries but say minus 1 0 won a case of this came from looking at how presentation finished this group coming from geometry might look like when I and II this here G should be very roughly on the order of and squared audiences what we discussed yesterday now I want make a 2nd but I would make it explain 1 way in which this heuristic is indefinitely inadequate so so this heuristic as I said at the crude years figured very I correctly predicts the size of this but if you ask where anything finer it's it's really wrong so let's say it looked examine what this says about so far I just discussed in size less lucrative actual group structure what this suggests that the construction of pocket so if evident over is a cyclic group or proper cyclic groups and so on and so explained that this heuristic really gives you a wrong picture at just such a fine quest is this something essentially missing witness and sort of important to be aware of that OK so over the top of the group structure you might as well look at Prime by Prime OK so if you believe this heuristic let's just look at a single prime p of the loop and a look at the key part of his words I throw away everything that's troll way all that portion that's primed to peak OK so they just keep whatever factors of the monarchy you might be squared and so on so if you believe this you would think all right so I'll write this mean this can be modeled as you take now p-adic integers synergy quotient by so if you believe this is reasonable value that it would that legislative Peapod everywhere where these want to be the on UK the chose to you choose random vectors the community up here there's some 0 minus 1 1 everybody don't expect that should have been with hoop structure of its economy are comedians constraint on OK so if you win you think maybe this is so this is a reasonable model for other key partners behaves you take some fairly large number GE you take BP Virginia "quotation mark hindered by the now this part of this is some way
to produce a P group and that it's it's a this way has already shown up yesterday in in a different contexts so so in fact that you can very precisely understand what about what this process produces a case so if you take so quotient thing Zepeda indeed bye random elements to be achieved at least in the limit as G goes to infinity it produces exactly the same distribution that you see in the corner structure of sticks so I also dismissed as Slick uses the call and when stress distribution official by this I mean that if you saw it GE because of finite abelian key groups then populace look at the probability the probability that is toxic take something to Jiji pick so to speak in gene by gene matrix of p-adic numbers at random and do this quotient thing by the rows that may affect the probability that you get G it will be exactly the probability that we see in Cornland stress which is the product from 1 to infinity 1 minus 1 over PDI divided by the number of automorphisms of yes or a look at the knees could say it's not nearly so Roman story but in a 2nd all say why you expect to see this exactly the same thing in the corners for context of candidates separate but interesting story but what I would do what I say is that in particular just as food we just discussed in his lectures the probability of getting a group like Zeman PDK is very small the case of over discusses the monthly consumer 3 disease wanted the case probability will be on the scale of Peter the minus case squared and if you look at data if you go look at it's on the table of data of this year's a bunch of different and you look at the bill as they should you see that if this is not realistic of PUC many factors of the Monty much more often than you would expect focus of this doesn't match the data but they thought say this is so this up I think this is a clear guess as to what it is missing from this heuristic but limited having a smaller side about why the same thing shows up in the corners for contacts of any court questions appear here it is true that the problem with the reality of the Jerusalem as to have a cases for the ease of analysis slip they we choose them at random with respect to the HA measure OK however in practice that I know they're there Theron's the Sauber not I kind of know what the state of the art has been practicing if you simulate this you find pretty much anywhere you picked and you'll get the same distribution organizers might find it any reasonable distribution it seems it's very insensitive how you picked experimentally but when I make a statement like this let's say that I mean the DIR chosen uniformly with respect you know questions OK so a slight aside the photos worth making because we're so were also talking about the heuristics and another costs so why should this show up y those tests show up when you model when the study class groups 0 1 1 thing I want to say is disequality here it all looks so much as start limits has due to infinity the the convergence here is extremely fast OK so long as G is slightly more than number generators for genes were already close and the proof of this is very it's very is an elementary exercise algebra OK it's not much harder than knowing the number of convertible matrices over a why this is so why should the same this vision which I've said is not correct in this case but what why does the job inciting class proves the point is that you can also so so that if you take Khazanah Majorie quadratic field OK so the class of K is by definition the group of all ideals quotient in by the Group of principal ideals OK now this is the quotient of 2 infinite groups and so but you can replace this by from a kind of finite and easier to analyze follows so so let let's say Let S B a large said of prime ideals so for example if you take all the ideals with more preachy all the Norman Western some very large number the indigenous should depend on caseloads on big number and so it is this is large enough you can replace despite finite version where we just look at those ideals generated that the ideal generated by Prime's Esq copies you look only those ideals without all the factors and ask and you quoted only by the principal ideal intersect set of basic principle intersect the same set of now this top thing here it so it's what you get for all crimes in essence clearly of free abelian group of ranked equal to the size of St OK the bottom think it comes from all elements of K principal idealistic Geremi K but they should have all the factors OK so this in other words comes from OK I this disk sorry about that this comes from the S units spaghetti elements of K old whose prime factors lie in it's like only in S & directly their shows you this is also a free group of rank but the that's where that the fact that imaginary quadratic is used for real quadratic UBS plus 1 that's what you get to the point is the class group of of magical killers naturally presented as a cautioned in other words of some 3 billion group modulo the same number of relations OK so in some sense so that pattern explains heuristic me why
you should expect this distribution of it would need to be some conspiracy among these relations for it not to happen so it that this model of group shows up and understated which also showed up in other contexts very natural model that you take some number of a large number generators and pushing buttons in number of relations OK now for the 2nd says going to say yes war what has gone wrong here had any questions so there yeah the Shiite majority which is gone wrong is that this so let me say this where I'm I'm missing something with all the answers suppose we looked none of this but we looked at again that were similar a group that was not cosmetic focus you take all 1 will boast to take something like a hyperbolic 3 manifold it is not arithmetically you take its fundamental group and you did that and you go through the same story was in fact Nathan downfield Thurston show that under leased under a certain model of random such groups you really do get scorn stress distribution OK so the fact is that doesn't seem to hold the summit is related to the Ahmadis city also it somehow related to so this is this is related to that the failure of the model to account for for arithmetically cemeteries which could have and by that I mean that the operators this model knows nothing about the hockey operators undercover is greatly changed the rights of limited space out there isn't this a very similar phenomena activists failure on this comment on that and then will be done with this heuristic OK but so in other words at that the end of this year is consistent with the neighbor this heuristic doesn't account for the head the operators in this this is 1 way in which it becomes visible gets the group structure right so you know what we did here as I took a random Dubai GP attic matrix and formed a group out of it and that's as distribution and the story it has some parallels exist this has some parallels with With the credit classical right matrix visitors to give you an example of such a parallel I'm here I wrote that so this probability of that that you get to see Monty decay OK I said it's very small why they're so small it's because this may baiji matrix is most unlikely doesn't want to have many eigenvalues near 0 OK so this is actually exactly the same phenomena that if you take that I think about it said could be right in the Genuity matrix ensembles a random Hermitian matrices the probability that you have KID IN values In a window in windows scaled let Epsilon has exactly the same behavior it's excellent the case square so there is some sort of corn master story has some reflection of what you see in random matrix dated much easier than random matrix the is but it you see some parallels and in particular this failure is a parallel failure to to do with random matrix
theory so the failure part of this naive model 2 "quotation mark OK above so it's kind of similar to those of an analytic failure which is that too this the failure of Sangeeta of random matrix also random matrix theory to model the ideal values of of its if you don't know where I'm going about here and it's often a matter of status to model the eigenvalues all of gamma proper have playing if Gamez Arafat so something physicist notice about 20 years ago is that typically if you take a course in the upper half plane and you take it for plusher 19 values that the gap statistics look like random matrices but they don't have the tariff medic became that what sort of presumably because the presence of hockey operators forces it to behave in a generic way as alluded to this failure is kind of parallel to that 1 it is so aware there are some similarities OK so that I think that's the end of the 1st year and now I wanted so that they do not know I want to acquire the 2nd heuristic which is much more specialized but also gives me a chance to talk about old some of the reasons 1 might be interested in this In this story in the 1st place and any questions odds of sofa here that me I have to describe a little bit the relations of destined to the line 1 story so so let me start by by saying 1 a land that is about briefly saying something which happens in the classical case over to you and then described the parallel phenomena in this setting and why it's it's no way more interesting so Over Q price a look if we have F which is a way to do so after the weight to whole Moffat formal level again and so it would rational coefficients and had the idea the principal sites on Sunday and so I coalition were amid this wonderful discovery that they could associate an elliptic curves solicitude more III 27 of which an elliptic curves and it has this amazing property that this AP here is related to the number of points on this over FP appears a number of points of this over a P solicited by likely the insurer but people 1 minus peace and in fact it would be you don't on there's a way you can disguise future is Yuri never remember
the fall of the curve ball 1 can I just remember the gallery representation the Tate Modern of cases so if you don't know what it is don't worry about it all so Galois representation signed saying this would if you've seen it but it'll make contact with about to next but it doesn't matter if you I have not seen it and this has the props some property trace of road that it's traces on for being its elements Our related to safety but even so that's the story of acute now let let me now say the course for the corresponding story over our
quadratic so over years in amounts reported failed now the applicant has just remind you that will act as I said at the start of yesterday if we didn't know about the space in which to forms we could nonetheless extract all the information in it from the distant the unionizing the OK so as far as the tanker operators could have exactly the same information so over all right right now that I for example here is there's also speak medical patients not rational and just stopped indispensable so again he eliminated them the simple statement I can and I'll say something to try to make it a little more palatable and then I'll talk about the size of it's involves the work of many people so limits so symbols gun and restricting the simple case make it simple to make so let's suppose that we are in a setting where the teapot of this but a gallon on in here that we in this I'm not now Nova is imagining critique field let's suppose that the people of this just consists of a single factors in 1 the foreign portability just makes that statement as clean as I can make it right and then in this setting Rossellini taken element and non-zero elements here which call health so here you can make a gal representation so it goes in the Galois group of it's OK it's going to be the corresponding quadratic field so goes the Gallo group of care are over OK to jail to after linear deal doesn't want peace at the same K St. here here the right and I'll also that's the route and his specialty full of trying to make a little more friendly so and has the following properties and has many properties not listing all them that 2 important 1 . it is unratified outside in times and for any prime Q the way that they could seem tacky Maximus Alpha is by multiplication by trees onto for being used to if you've never seen this kind of thing this looks like like sort of incomprehensible so let me just make a special case and tried it and try to make this will mark friendly and sold by the way this is a lot harder at his home and talk about the history of this and a 2nd man is a lot about the deck is much harder so let's just to illustrate the the kind of what this means that left take a case when a piece of the action be road hazards :colon Road and is score rose up and setting I just had this 1 thank you for this P is primed to and from there I don't think it matters but maybe this is is only 1 and but I don't think it matters of .period only 1 his his friend and only at a given many key should be printed OK so that is think example when he is to focus on the sides spend some time saying that this you know how how should 1 think about this case because it so when appears to let me in practice so let's appears to say you interviewing actors and you see that the two-part just as you want to in practice the exodus lot that you'll have the noises won't be subjective them status objected so say that role this subject so what I'm introduced translators right-hand side into something more down there to road goes from the Gallo group of key barbara k and goes to jail too z 1 Tuesday and this is a group with 6 elements at isomorphic symmetric group on 3 letters OK so to give this map to give them after the governed S 3 is exactly the same as giving a CU field OK so Rhode determines so to give row is equivalent to giving a CU field which is von Gal OK so that CU extension and yes it's fun gal annually get Street extension from and now translate these other 2 properties In the so a says that this the discriminative is divisible only by only by Prime skew dividing intensity and conditions the says to be translated to discuss translation what it says is that if you OK so I have this here's is off of 1 thing sort of said here without explaining is that there really is a way the Hakki operators act on this because I saw that I said this at the very start of when we were discussing weight informs us that the Hickey operators act here this kind they compatibly act here OK you sort of think through this and figure out how to translate the heck action here to hear the same exactly the same thing works over here again never I never wrote adding that I never actually boarded rebounded definition but when To do so that they can have many Group Indian has it that the operators and the other was a devoted to celebrity it automatic in this case because I I assumed that the people to 1 piece so right and this other assertion says that what is being feted Alpha is non-zero if and only if Q so Q is a prime of K it is not in doubt OK so we knew
so something that at least some pointed to try to make myself feel better about this for small piece you can translate his assertions that things might expose number of fields and the way the price splitting and you can even going tested numerically and it's very satisfying to see so I was using it for small pieces really the statements the conference there is something very explicit OK so old and they talk a bit about the history is any questions alright this I was so on the 1 hand you in some way you expected perhaps my analogy with this but there's something it was also the 1st in this has recently been proven very recently by Peter Schulz but I would always say some of whites it's so much more subtle than this and some of that history when we look at it in the case of a Q as many as I said yesterday this thing is essentially it's more or less a free group of side said yesterday it sums up many copies of the Apostle small finite group of now the problem you have here is some a situation like this this is the marquee the is not detected by anything in that classical theory of automorphic forms in classical 31 Norfolk forms it has things objects that exist only in characteristic 0 little plot functions home forms and so it has no hope of seeing this and act isn't there and the reason I got interested in problem the 1st places it is somehow all the standard techniques of automorphic so this thing this Alford behaves just like a modular form has every right to be called a modular forms but not a lot of the existing story of uh the classical techniques applied to it and the fact that so the idea that you should you should be looking at these on so that 1 should think about the possibilities has portion and that you might have a sort of story goes back so far so I think it was predicted and maybe close only knows more about the history I think about have ash Close Owen Taylor already knew about this from the 19 eighties this the decision was of the new law appear so that there something to be said here it's not obvious this is something you should be looking at and it's not obvious that there is this portion phenomena here all so it is that I think it's thing that is insight to realize this is something worth worthy of study anyway added to this goes back to the wall on 1 of the main motivations for studying this would be done with conjecture made with Niccolo Bergeron was to show that this kind of phenomenon this portion is actually complete it's not some sort of strange exception but it's In it for half once you go away from the group groups like a cell to ensure more varieties actually the norm but this is what you see is primarily portion and you see very little and characteristics OK so all this was this is not was just to give you some context and to this story fits and some would have that must be a part of the program again and still so now shelter has proved a sample of the I pollution Morris story but the amount we know about this still vastly vastly OK so very few of other tools that we know about characteristics 0 modular forms can be applied there's also a Converse this assertion that it was is not known so that the problem anything on this underscored in "quotation mark Sears conjecture Sears conjecture is usually under understood as a statement about you the rational province but so do this is that it should be so it's a conjectural conference and I said this is the kind of very a simple case was as the 1 P but even if that is this is the 1 piece global whatever decision there is some precise statement you can make whereby you can exactly predict this In terms of by something very short description of how this relates language program now I will come back to the question of its size and any questions you have to follow through there is going to have a problem the more we always used to go say so now OK so site when and what I mean is that notation that fairly clean energy might be but suppose that as you said supporters lifted characteristics 0 then correspondingly this would lift to a p-adic presumably geometric represents any other questions so prices and back to the question of size society quite so if you have some description of this In terms of Galois theory you I tried to figure out how enough how large is by seeing how likely it is that such through exist so I'm gonna say this very this deserves more explanation I have time for about sold by anon Dili inversion all the Kong-listed heuristic so ii explaining why citizen 1 moment the probability that that such a really exists should be roughly 1 over people OK so let me just compare here at let me say once mentioning corners wrestled to the usual connoisseur heuristics say that for the at mentally word K they say it's OK for kids demand scenery quadratic the probability that P divides the size of class group is this number which His one-liners that number up there sold 1 minus the product from 1 to infinity
1 minus 1 of EDI and sir now things is a so so that it's president about the G equals 1 at so this is right now this is roughly 1 of repeated first-order OK but we can we interpret this in the following way OK the same as the probability that there is a lot nontrivial homomorphism from the class group dizzying 1 PC and my class field theory this is the same as they're being on ratified homomorphism from the Gallo group OK so this way you can think about the :colon here 6 as telling you so measuring something about how likely is it that you have abelian extension of a number field In this context the answer isn't is supposedly 1 of few so when thin on billion version the heuristics what I mean is the some something that describes for you the chance of you having the Home Office and the GAO represents 1 the earlier OK and this exist side I would say so I go largely Due to to work embargo Barcelo so lips a bottom of formulated this for the symmetric group and then if you if you take what he did and then you applied very recklessly you come to this this claim OK so so suppose you believe that for a moment the is most of the rest of the world we are you know yes and some I think yes and some ideal situation when there is no are Canadian story .period are very problematic some elements of the force to be awarded to so but if there were no ah comedian places yeah that that of excitement maybe I should say that's also another way how would you guess what these heuristics are is related to what Emanuel said he didn't know them 1 way you can do is you can think about a function field case and there's could have a clear guess of what the answers so to tell you might discover this kid OK are where are we so if you accept this or that and I with somehow deep in the realm of heuristics here but it if 1 accepts this that says that the size of a Seoul how big is this group going to be what it's going to be at least for every P every prime occurs to the probability of 1 over P. and that this is infinity so you might think it's problematic this heuristic predicts Infinity for integer but I but I think that is so if you work you can apply the corners heuristics the same way the predict try to guess if he just went prime I prime apply the caller refuses to guess the size of a class group it would predict that imaginary feels of infant class group and real fields have 5 OK so so they should take this with a pinch of salt at least this suggested it should be large for 1 advantage of this tired of heuristic is it works very well when you replace when you general setting as you play Cecil to buy a bigger group we replaced the dramatic ridiculed by another field is also always gives an answer and the answer seems as far as people being able to tested due to really be experimentally seems so far to be right chemical and talk about that but that's the main virtue this year's stickers that it it is very it's much more generally applicable in the previous OK so that's I think the end of heuristics and now I'm going to talk and we will talk about the next about analytic portion and to start that I'm just going to talk about it all introduced as hyperbolic free space at the start of this be making having definitions for today so that I can we can look at the formula next time of any questions yet so I gave him the head of the leading reason has been given no leeway it's unclear you know it's somewhat open to interpretation but that I II interpreted in the way that I want to which is that the torch in part is getting very large yacht yeah I like for example if you literally applied this year stick it predicts that the probability that a modular forms exist is 0 so so that have to be a little bit careful about what was the realms of the world where you applied but am anywhere the things I found this to be quite reliable as from this year be cautious about how you use it right so now let so of cases and I wouldn't go back analytic portion of what may I mean right the formula now and then the .period job will be to define the quantities therein OK so this formula will again let me attributed after I write it so what I'm about to say is true so the true is "quotation mark idea if it did it would be true this caution or Compaq and if not there's it's much more delicate story all right follows is literally true in this case the size of gamma Norton torsion part abelian eyes times are this regulator which will be the year I think we will which I now feels that the most interesting part of the story this is equal power now divided by the volume hyperbolic volumes I and so I'm going to talk about the hyperbolic space and will define the quantities 1 by 1 is equal to determinant of Seoul over here it will be too look at the side Delta Norton Delta 1 will be looked lost operators and I will while called the findings reflections on gamma Norton was extreme this is the wildlife in on functions at his usual month forms this the function 1 forms muscles somewhere and we need to discuss how you take determinant of such a thing this is that really it's a remarkable formula because it says that this objects some thing of discrete group theory you can compute in analytical way
and so so was conjectured by rain singer and I think so in particular is that when received injections in particular have been defined as in the 1st place this as far as I understand it there the definition of this I think indeterminate something dimensional was not was was students so it's tremendously insightful the biggest such a formula could exist all say something extra about how you might guess that but it's it's an incredible thing just a guess it's there and it was proved to be much eager annual independently and also calls a couple of words but this formula has nothing to do with number theory it's also version of this is true for any of versions is true with gamma Norton what it's 3 replaced by any contact Riemannian manifold so what I said yesterday another way and think of his right hand side as a special value of a Solberg executive function so of also it's Seymour a little bit about that next time to will but you were for now while I haven't I wanted the final the quantities here this 1 would seem to litany that defined Hi Vermont 3 states in the volume all and 1 thing to say that these are the SAR has to be defined too but for now I just say are as 1 Lawrence 1 if there's a billion ization rectified are as to make it some measure the size of the 3 parts of the regular alright so maybe just a few minutes on the hyperbolic space at all under any questions right so just like so I've already written written it up here so did just as S O 2 are so the hyperbolic space right so that when you study has sold the basic factors it and even the bigger group vessel to act on the hyperbolic plane there which our writers each 2 for the moment so in other words that so that's all that as a space just positive half of the complex planned but what's useful about 1 of the things which is important about this is that it has a metric that's invariant reminding metric where and that's invariant by Beisel to are and there will not be just the thing so to some extent we try to study a group like this to the geometry of so analog in this case up here is that the cells for example and to all Oh is my used square day the sits inside a cell to see and acts on this summer but and now the story 1 dimension higher it's on the sale of Baros some of their outline of it acts as a three-dimensional analog of the space and tackle everything is very similar just one-dimensional so this space is now you can think of three-quarters X 1 X to Y for wise positive the metric is exactly the same formula but the action here the action by fractional linear transformations here you you can see this being my fractionally transformations when you think of these and some returning on a quick but instead of riding the this show you how some a few basic classes of things here act so so for example the element 1 not so sometimes I I think of it's we can also think of the space we will think of it as being yeah so it sometimes nice to think of this as being inside the complex plane times a positive rails where the complex coordinates are X 1 Class 5 A X 2 and why fear and if you think of it this way so 1 Alford 0 1 pacts on since X 1 post this this thing here translates into complex plane X 1 post-fight to profit of wine and this element here this violates both that the interest-rate sensitive 1 for flights to why 50 squared 1 likes to and the absolute value of the squared wife again finally that element an element Exira 1 1 0 acts by version so in a way that you have this geometry now too and all 1 1 thing which you did you get even before you do anything fancier Is that so just the fact that these acts on this geometry gives you something already so the volume of you conform the quotient OK so that it is in the simplest cases you just like the fundamental Maine frazzled to the looks like this In the small discriminatory the final demand for just looks like a sort of three-dimensional version where you have like something like that typical picture final them Infiniti is a kind of buildup of things that so right so you you can make you can now measure this because you have this metric and just that fact is useful this gives a kind of an intrinsic measure follow the complexity or of both look this gambit Oregon complexity or if you want size a benefactor gives gives you some number which means in some way measures how large this group so in
terms of this number and I were conjecture last and now let me let me rewrite it it is slightly more generalities so my conjecture Persia on itself the conjecture of mind Bergeron states that and it's naturally phrase this way is that if you take the logarithm Of the size of this portion it's a scare elastomer skill by in squared but the natural thing to skill-biased this volume then you get this should go to 1 Over 6 .period look at that is not the way I stated last time so to recover the form from last time so version from last time you can use the Humbert the volume formula which was proven by Humber in the 19th century again what that says is that the volume of hyperbolic 3 space modulo SL to all people where always in that order of discrimination is given by I have the value the 3 halves over 4 pi squared times the zeta function of Hughes squared minus D 2 OK so in the case I had last time this view was minus 1 the was minus will be was pulse 1 understated function cure alive at 2 is piste where there were 6 times as the Quayle function and is so this constant which appeared last time really it comes from the value of zeta function of this quadratic field because visits it's really a measure of hyperbolic volume nothing more OK so maybe that's a good point to stop the next time I will will discuss the other day what what the other things in this formula mean and a little bit about what we can do it it was ridiculous this is the use of the word we had a formula of a handful of the Hillary asked what the problem isn't even dimension this tends to use the severe crisis in some way it's 1 for trivial reasons individuality so it has interest only and the one-dimensional this was not twisted somewhere all of the we the police


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