 Thank you very much for all of the organisers for giving me the opportunity to talk in a material web seminar. I'm very excited to give this talk. So indeed I'll be talking about Chalmers, an unbanishing conjecture, and this is all joint work with Peter Coimans and Mark Schustermann. And essentially the result is the following that if you fix a prime power, three or four, we prove that if you range over the quadratic character, the function field, the rational function field, FQT, and you order them by the size of the discriminant, this is a physical polynomial, where FQT, then for 100% of them, you have that value of the L function, the half point is nonzero. So this is a probabilistic version of Chalmers unbanishing conjecture, and essentially this is a very high level overview, but I will see more details of this. We achieved this by controlling a certain sequence of higher Chalmers groups attached to this problem that are able to eventually determine this 100% result. So the key aspect of this, the key technical aspect of this segment is that within the technologies developed so far, they suffer from a certain very specific pathology, and I will explain what we do to overcome the problems created by this pathology. All right, so the broad overview of the talk is that I will first provide some general context for Chalmers unbanishing conjecture, both over the rational, which is the case Chalmers consider first over the function field FQT, and I will introduce these mysterious sequence of spaces that I mentioned via CHI, and I will explain how these spaces are able to essentially govern the unbanishing of the L function at the half point, and I will overview the methods that are currently known to control such the distribution of two infinity Chalmers groups in quadratic twist families, and what are the limitations, what is the reason for the limitations created by the pathology. So we'll see very soon what the pathology is, and I will explain what is the key additional new input that we introduced in this very specific case to overcome the pathologies. So let's get started with the statement of Chalmers conjecture. So Chalmers conjecture that if you take actually any English character and you take the L function, then a half which will never, literally never be zero of the function, but there are broader conjectures about linear dependency of the zero over the rational, so we're not going to them, but that's a special case of a broader set of conjectures, and cuts and sarnac and they have a more general framework that sort of give theoretical evidence for these conjectures, and under generation we're able to prove that if you order your characters by the absolute value of D, basically chi of D is the quadratic character of square root of D, then a positive proportion of that needs to be satisfied in Chalmers conjecture 15 or 16. And it was a big breakthrough in 2000 when Sander Adler proved that within the family of old square free, okay, here are some technical extra conditions, but essentially a positive proportion of quadratic character, 87.5% had non-vanishing at the off point. And this was the first such unconditional result, giving actual positive proportion for the non-vanishing in support of Chalmers conjecture and Geutilla previously at lower bounded, so was growing as X over X. All right, what happens actually over function? So Chalmers conjecture actually does not hold literally the bottom of the function field, one in D in 2018 was able to construct the power of X, X to the one-fifth quadratic characters of FQT for every single Q that going up to X, such that you are vanishing at the off point. So the exact formulation that for every single quadratic character you should have non-vanishing is false, and it's false even if you ask. Finally, many and there is even this lower bound growing as some power of X, X to the fifth in the worst case scenario. However, the sort of cut Sarnac type of heuristics leads to the belief that a probabilistic version of Chalmers conjecture should still be correct also over function fields. So the failures of the non-vanishing, so the number of vanishing should asymptotically be 0% of the totality of quadratic character. So if you order your characters up to X, you should see a smaller and smaller proportion of the action. So that's what I would call the probabilistic version of Chalmers conjecture. And there is quite a lot of support for such a statement. So work of Bowie and Florea from 2016 proves that at least 94.27% of the quadratic characters you have non-vanishing at the off point and as I mentioned before, this is not just for quadratic characters, conjecture is for any Dirichlet character, so you can pick other families and for instance David, Shandong the V, Alexander Floreva, Tidla Lenin, 2020, they were able to provide a positive proportion non-vanishing in the family of cubic characters. And finally, I also would like to mention to those 21, Ellenberg, Lee and Schusterman proved non-vanishing for a positive proportional function field, but the additional feature is that the proportion tends to 1 SQ goes to 10. So if you move Q to infinity, if you take it bigger and bigger, you get better and better, lower bounds for the non-vanishing. And I would already mention that in all these results, the half point is not playing a special role for this probabilistic, both in what is expected and what is these methods so far have been able to prove it should be the same for other points of the half line. All right, so that's an overview of what is known so far. So if I fix Q so far either, you know, straight away uniformly for Q94 to 20 percent, and if instead you move Q to infinity, you are able to get better and better lower bounds. So the result I want to present today as I mentioned at the beginning is a joint work with Peter Cohen and Schusterman, where he said we fix Q, but asking there has to be congruent to three model of four. And then we prove that within the family of imaginary quadratic character that's chi of FQT, we have non-vanishing of chi for 100 percent. So here Q is fixed and we have 100 percent of non-vanishing and imagine that here this is a function field so you might be a bit confused what do I mean by imaginary. It means that it is given by the square root of a polynomial that has odd degrees. So if you like in a more geometric formulation, it corresponds to an hyper elliptic cover of P1 that ramifies at the infinite point at infinity. So there are some assumptions here. First of all, we have the family of imaginary quadratics. We have the half point and I just mentioned a minute ago that both in terms of heuristics and in terms of results, people have considered other points on the offline. And these the methods we are using this so far really tailored to the half point. So the method is sensitive to the choice of the base point. And furthermore, okay, there is assumption that the character is imaginary as far as I can say I believe that there might be a moment. But at the moment we are working with this family and I would bet that that the result can be extended also to the case where it's either a split or inert at infinity so polynomials of even the degree. But what is instead fairly important is the assumption that is the Q is a prime power that is three model of four. So the case where Q is one model of four, which is the same assumption that basically your function fit fqt as a i as a fourth root of u. So this case is generally different than the same techniques that we're using are very sensitive of this change. And I believe this requires generally new inputs. And so this is really an important assumption is not just out of place. And that's why the imaginary quadratics I believe we might be able to. And the other base point is also something extremely sensitive. So for now we'll just stick to that. Okay, so this is the main theorem and hopefully there are functions, especially Q congruent treatment for will sort of come back during will become clear during the talk and now sort of want to gradually build up and tell you the ideas of the program. And so in the first place, what's the connection between vanishing of the at the half point. So I told you that we, I see this result by controlling the distribution of certain thermal groups in quadratic with family. It seems pretty far objects function at the half object like last group and so the connection is built as follows. You can view your character as the quadratic character attached to a square free non console say polynomial over fqt and the assumption that is imaginary precisely means that it is all degree polynomial. So now you fix an auxiliary prime and co prime to Q and the key thing is that there is more structure to that function. And to explain this additional structure, I'll introduce the hyper elliptic curve attached to the function field square root f so C of chi y square is f of t is an affine equation defined by fq and this corresponds essentially uniquely to the character. So f here is the same as chi equal to chi of f. So you have this nice curve and now the bail conjectures come and tell you that the vanishing at the half can be reconstructed in sort of linear algebra terms by essentially asking that the action of the Frobenius operator on the elliptic model so L here is the additional auxiliary prime that I'm using on the Jacobian of C of chi. So this is rabbinian variety attached to C of chi. This should have a square root of q as eigenvalue. So you're vanishing if and only if square root of q is an eigenvalue of the Frobenius operator with a elliptic takedown. So there is some sort of linear algebra if you want or matrix theoretic interpretation for every L you can choose your favorite auxiliary prime L and you can measure the vanishing by checking whether the eigenvalues of Frobenius are or are square root of q. If one of them is then you have vanishing. Okay excellent and by the way that's exactly how van der Lee has constructed the examples. She starts with a single example of vanishing which you can always achieve either with an elliptic curve or with the restriction of an elliptic curve according to whether q is a square or not. Anyways there is a single example of vanishing and then you consider the family of polynomials f that are obtained by essentially base changing this example by a map of p1. So you compose f0 composed with g and then this is constructed in such a way that the resulting elliptic curve maps on your starting example and then there is a theory of momentum that this will basically tell you that the Jacobian has in his isogenic class one as a factor one of either this elliptic curve or this a bigger surface which detects the vanishing. So if if g if f0 as vanishing automatically is not composed with g as also vanishing and then it's a method of checking that in this way you produce many different f in principle a modular square. So many different square free f and you can sort of if g is degree five it's reasonable you get that power that I wrote x to the one so that's the family of counter examples to the literal Chaoula conjecture so the asking literally the del function never vanishes and then you might think that essentially an approach could be okay I just check modulo l in this Frobenius matrix or this analytic matrix just check whether modulo l for some auxiliary part it so happens that square root of q is not an eigenvalue so you take the L torsion of the Jacobian you take this eigen space and you might try to prove that for instance it's the 3d eigen space so then it certainly means that you have non vanishing at the half point and a way of looking at this is in some sense we are trying to control the analytic absolute value instead of the infinity absolute value of the Archimedean absolute value of del function instead of saying that is a big in the Archimedean sense to prove that it doesn't vanish you prove that it's a big in some with other place of q some other auxiliary prime. This is the rough idea you might decide to follow and then the question you have to face is what do we know about this sequence of eigen spaces on the L torsion of the Jacobian as you bear the quadratic chi so such questions are as I'm going to convince you hopefully in the next slide very much. Sorry such interpretation that you said L functions is zero if and only if school do we have such interpretation in the case of q a rational number of q? No not for the honest assumptions as was suggested to me by Jordan L and where you have it for t adding functions there is something very similar and actually that was really what motivated Eva Savo developed Eva Savo theory was to somehow what are the type of L functions over q that have a similar sort of Galois theoretic interpretation about the zero so the function so yes unfortunately no this is really something a story either over fqt or a story over q for not the usual L functions but for things like piadical functions. Yes so for example L function at one then is non-zero we have directly theorem infinitely many primes but for one half maybe at one half we have to think about some primes. The half point is not aware of any interpretation like this I would be really surprised if I had any ideas on that. Maybe we have to think about Gaussian primes that are ordinary primes for example I see. Yeah maybe I'm happy to talk more about this if you like. Thank you. Okay so yes but thanks a lot for the question this is really the place where you see the function fielding of this story coming out that the L functions over function field is much sort of more algebraic structure behind there where you can really take an ellavic matrix and read off the the vanishing of the function at the point in terms of the behavior of the eigenvalues of this of this matrix. All right so the question if you take such an approach you have to ask the question what do we know about this eigen space of the L torsion and I want to convince you in the next slide that this is very much a question in the style of the Cohen-Lenster heuristics so let me make some sort of analogy instead of the more complicated polynomial from q square minus q I just take the polynomial of x minus one so the one eigen space from q minus one and I look instead what do we know about the behavior of the L torsion of the from q minus one. So since Kai is imaginary it's not hard to convince yourself that this is nothing else than the L torsion of the class group of sqt root x so if you're useful the right number here is verbatim exactly the the class group constructed exactly in the same case as for or imaginary quadratic and maximum order you you do it over sqt and you ask the behavior of the L torsion of the class group as you bear your polynomial f among all the degree polynomial so what do we know about this the behavior of the L torsion of the class group conjecture there is a very nice conjecture of Cohen-Lenster which I guess at least when q is not one of well it's essentially already in the paper of Cohen-Lenster over q but it has a straightforward version of the function field adaptation of the function field which gives you a conjecture about the distribution of this space essentially what is the probability that this space is zero dimension one dimension two dimensions so for every possible rank of this vector space it tends to the expected probability for achieving rank equal to i r is your favorite rank so we expect conjecture at this eigen space the identity eigen space follows a very nice distribution unfortunately this is out of for l op and i'm fixing here l an outline this will come back soon why i made this choice but so far let's just fix l op and for l op this is completely open at the moment however over function fields something fairly close to the Cohen-Lenster heuristic is no and this was a breakthrough in 2009 of Ellenberg Venkatesh and Vesterland that introduced the method coming from topology the study of the stable homology of orbit spaces that give them upper and lower bounds that are approaching for the behavior of this torsion that are getting closer and closer to the Cohen-Lenster heuristic as you sent you and very very roughly there is way more to it you can look at the paper if you want the non-vanishing result of Ellenberg Venkatesh and Schusterland adopt the techniques that you have from Ellenberg Venkatesh and Vesterland to this different more complicated eigen space from two square minus two and the you get in in this way lower bounds the proportional convention that are approaching one as two infinity as i mentioned before so this this is this this work was sort of the first place where over function fields one takes this new approach of not estimating the Archimedia and absolute value of the L function in the house that you take another auxiliary prime and you try to estimate the logic absolute value to obtain progress towards probabilistic Charles conjecture however the limitation the current limitations for L op that we have for the Cohen-Lenster heuristics are so that at the moment we only get approximate results that become sharp and sharp when you send the parameter q of the function field to infinity and then what else can you try to do for a fixed you so now you fix you and you are not allowed to allow these out of question at the moment at least for proving Cohen-Lenster type of results for such an eigen space but then what else can you do well I said that L is beyond and then there is one case left this idea and that's the case of n equal to and that's exactly the prime that you exploit here and maybe you start getting a feeling of why the common that's related to the fact that we take a congruence model for you all right so now that I told you this I should explain why in the first place in this Cohen-Lenster heuristics and why I excluded so far the case of n equal to so why actually I sorted out l equal to in the Cohen-Lenster heuristics and why don't we directly try to prove that the two torsion maybe of the eigen space from q square minus q the charles eigen space is actually three why why don't we do that so to understand that properly let us go back to the analogy to the simpler eigen space from q minus one this is just the classroom so if you're familiar with with class group of imaginary quadratic fields you will be familiar with what I'm gonna say next and indeed let me remind you that this is the two torsion of the Jacobian there from human is one eigen space is nothing as the two torsion of the class group of the quadratic order fqt root f and it is a very classical fact or genius theory that the dimension of this space is entirely predictable and actually the space itself is entirely predictable the two torsion space itself as dimension equal to the number of prime factors omega f minus one of f minus one so typically for a random polynomial f or if this was over the integer for a random integer the you get a lot of prime factors so typically this this two torsion space is gigantic so there is for a it seems to go in the opposite direction for 100 percent you get you get the mass of two torsion so that's not good and it's also not good for the coincidences if you want to add a distribution there is no distribution whatsoever you can hope for because the dimension of the first space is already exploding so with probability one you don't get any given crack as a two as a dimension of the total torsion. However it was well known and expected that it was very much expected that if you get rid of this massive exploding piece inside of the two parts of the class the rest should be exactly like in the case of the coin that should grow in the same with the same flows and this was very much open until 2017 and where Alex Smith introduced a number of methods and then his work was further generalized he further generalized his work for more general settings and so you can find the two final preplains the date is 2022 and despite the work is over number fields if you sort of use the method it's not hard to literally translate the method and prove and here you already see this assumption coming that when hue is congruent to three model four then once you get rid of the two torsion so you double your group you will kill the two torsion then the rest of the group should follow exactly the coordinates. All right so that's regarding the one against space and then let's go back to the nambanage thing so if you follow the original naive idea and look at the two torsion you might hope already to make it with the two torsion unfortunately that's really not the case typically you get something similar to omega f even more than that actually for the size of the two torsion with the eigen space from q square minus q so you have a really massive amount of classes which are essentially going in the opposite direction but then following this progression that you see in the previous two points you can then make the following plan at least perhaps after the two torsion whatever being on top should typically be fine so in other words you can plan to do to extend the means resulted from the one against space to the square root q against space and prove that determining the distribution of the two infinity torsion of the from q square minus q against space and so you prove a stronger result which has a very special case the consequence that 400 percent of the time space is fine in this space being fine and remember is exactly the same as uh saying thanks to the veil conjecture saying that you have known vanishing at the half point so 400 percent of the time so the plan that you might decide to follow is to determine the distribution of the double of the eigen space the two infinity all right and this I will now guide you through this now I arrive finally at my promise sequence of spaces f p and kites sort of Selma spaces that are f two vector spaces and are detecting the known vanishing so it would be convenient to switch from the Jacobian that was in the previous slide to sort of the dual object and you can do this essentially thanks to plastic theory you can detect these eigen space in the Jacobian also in terms of un-ramified extensions of your function field fqt root f and for that I have to introduce just a little bit of notation there is a there is a character there is a modulo modulo t kai that as an abelian group is always the same but the gaula model structure changes as you change the character basically you twist it by acting there is minus one here to the kai sigma so you essentially change the gaula model structure of the of z two root q q is three model of four and kai is imaginary so z two root q is just nice pid it's a ring inside the ramified extension q two root q of q and inside you have this kind of dual object q two root q modulo z two these are two modules so far it's so good but the point is that when q is three model of four and kai is imaginary it is a true statement that you are punishing at a function if and only if you can construct a gaula extension on top of fqt root f that is basically this semi-direct product t kai semi-direct with z two times f two where the action is given by the formula above essentially which is ramified above fqt kai so if you are lost here and you're followed until the previous slide so until here we have this plan of determining the two infinity of the uh Jacobian and for technical reason is much more convenient to talk about ramified extension and class field theory gives you basically a dictionary between these two things and this formulation for technical reason is essentially the same as the same plan as before but instead now we do it we approach it with ramified extension and essentially to find such a homomorphism is the same as finding a one cosine valid in t kai if you take the coordinate this is a very general thing of semi-direct product that the coordinate function in a semi-direct product is the same as a one cosine valid in t kai and that justifies why one introduces the space that finally introduced the pronunciate vector spaces over f2 b and kai which are basically the ramified quadratic extensions of your fqt test that come by multiplication by one minus two to a suitable power times the next power so this this condition n plus two and plus three makes makes it sure that you are in s2 so you have something in vector space so this looks a bit technical it's what it is the upshot you can take for now in case you're confused is that this is just a sequence of f2 vector spaces and that is detecting the non-vanishing through class field more precisely the non-vanishing is the technique as follows that you have that the health function does not vanish at the out point if and only if this sequence of vector space which by the way are one inside the next is eventually one dimension so the decreasing sequence of vector space and it's not hard to check that there is always one element inside of this sequence of vector space which is the constant field extension if this sequence collapses to this element then you have no match so now i deliver the promise of providing a decreasing sequence of vector space that the text the non-vanishing of the health function and so the goal is to determine their distribution and to do that i will for now just say that you can attach a dual sequence of vector spaces and that comes together with the pairing and the property of the pairing is f2 value it is exactly the text the next space vn plus one for when you take the left hand so the things that completely kill wm and the text the next space also from wm by taking the right hand things that completely kill vn inside wm so there is more structure here this sequence of vector spaces come with a sequence of dual vector spaces and together with a sequence of pairings and the ideas of smith i will explain in a second why the case of the class group the use of the one against space can be reformulated in such a in such terms and become something better I'm entering that case what is bm what is wm and what is r10 i will tell you in a second what that is but let me just start to mention what the pathology is that i said at the beginning of the topic that we had to deal with is that the ideas of smith work very well as long as the first spaces are disjoint a bit something a bit stronger than that but certainly you are in triangle in case the two spaces coincide and I hope to tell you something about that and for us it is the case that the two spaces are two starting spaces are always exactly the same space so before going next in the explanation of the strategy how you accomplish such a goal I want to explain to demystify a little bit the sequence of vector spaces the sequence of pairings into something that should be more familiar to you using this analogy with the one against space so this is the usual class group and the dm chi here is not the analog of the dm chi is nothing else than the two torsion classes that are divisive the dual two torsion classes that are divisible by 2 to the n that in the dual class group they you can divide by 2 to the n that class you can find another side that such that 2 to the n times psi is equal to chi prime and w n is the same in the dual world for so for the usual class group and you is the same space space outside from the classes two torsion classes that are divisible by 2 to the n and you see again these two exponents that differ by one for the same reason so that you learn the two torsion okay and the arting pairing i is defined is defined by choosing a random lift here to the n psi is chi prime and then you just evaluate you evaluate with reciprocity law you send b the arting symbol of b in the lift and then you can use that b is already divisible by 2 to the n to check that this is where the fun it doesn't depend on the choice of the links and it has the property exactly of detecting what is divisible by 2 to the n plus one in both the dual class group and the usual class it's a completely elementary fun and okay so whatever this space is of course i gots that i introduced in the case of chaula where they are the analog of this very elementary abelian groups story and let me now tell you how to include the sequence of space so both for the one against space and for the square root qi in space so for the chaula in space it turns out it is essentially my generalization of something that dates back to the work of the in the 1930s of last little day he was the first to give a description of the fourth portion of the class group of the quadratic field and that generalizes and the spaces b1 and w1 in this story are decided by a matrix of the chaula symbol so you take chi you take all the prime divisors you compute all you put them as rows and r rows of a matrix sorry that's column rows of matrix and you take all the mutual general symbols and the last kernel of this matrix is b1 and the right kernel and stuff and once if you're able to do that if you're able to decide what is the probability that this matrix of the genre symbol is a given dimension then what i'm going to overview in the remaining nine minutes is how you control the sequence of successes so you now know what v1 and w1 are and to know v2 and w2 you need to control the pairing art1 from v1 and xw1 and essentially at the very high level the matrix goes back over that this pairing art1 is every discipline it's all possible matrices with that given r of v1 and v1 and the way you prove such a distribution statement are to what are called reflection principles and i will explain what they are they are essentially relations among art and pairings and when you change the field the character chi and these reflection principles is exactly this place of the proof these relations between that the pairings as you change a character is exactly a list that breaks down when this pair is b1 so this is a very very high level over what goes into the method about the method is subdivided and let me explain these reflection principles so these are useful relations among the entries of the pairing when you take nicely arranged sets of characters and what nice here means exactly is a little bit lengthy but okay these are essentially the cubes of things so where the length of the cube is exactly is exactly n plus two where n is the n of the art and pair sorry this s here should be so you arrange you take a two to the n plus two different fields so and these are primes they are all co-prime with each other they are all think they are all co-prime to have not and this is a way to write two to the n plus two different fields and you sort of take there are here two further cubes where you stop your index n plus one and then so these are c2 c1 and c0 three cubes and the reflection principle goes as follows you assume that you have for each point on this cube you have elements a chi and b chi that are in the do in the respective spaces bn and wn and they are changing in two different bits so a is there and here and b is there and here and for the rest they are the same character literally the same for all the fields at the same time and what you prove is that under some inductive assumptions there is a basically an equation that expresses the variation of art total variation of art and pairings model two so you sum these art and pairings along this cube the total cube that's governed by the difference of the art and symbol in a field that depends only on the cubes you want and depends in a kind of asymmetric way where the first n indices play a symmetric role and the last index the one of a is especially so this is slightly intricate combinatorial setup but it's a combinatorial setup where one finds useful relations among art and pairing of art and pairings of moving fields so you find such equations and the the the key input is that you have to ask here that a and b are moving different bits so as i mentioned a is moving here but here there is a and b is moving here so for example a cannot be equal to b so what to do when instead for instance a is equal to b in that case for instance when you have b1 equal to w1 you you simply don't have such such reflection bits and yes so such a reflection piece but once you have it then the only way is still of very huge combinatorics and maybe invented by Alex Smith to use these relations to prove every distribution of the art and pairing and this this issue of the a equal to b of is pathological phases which for typical families of chi don't happen but sometimes happen it was phase first in this joint work with Peter Coimans where in 2022 we prove seven angles conjecture and the possibility of the negative perturbation and that also came down to the distribution of settlement groups in in in a pathological case where b1 is equal to w1 so what did we do there we found an exotic reflection principle so a reflection principle for a against a and you have to lower to a cube of one dimension less and essentially the output is that you under the new combinatorial arrangement where a is there on all the points of the cube we found under similar inductive assumptions as in the Smith case that the sum of a versus a is equal still to the splitting behavior of the prime pn plus one pn plus two in a field depending only on the two to the n dimensional cube and yet there is minus one and that's exactly the character minus one that is characterizing this family and is the responsible for the symmetry at the first stage the the reason why your b1 is equal to w1 is exactly that you go in a family where all the primes are splitting complete are one module four and that forces the quadratic reciprocity to be perfectly symmetric and forces these two spaces and somehow you exploit this character and you do this using a bunch of reciprocity laws that we discovered for this sort of maps find only on on such cubes and these reciprocity laws they are a generalization of the low reciprocity law per day and we call them higher radii reciprocity and we use them repeatedly to prove this exotic reflection principles and in this thing it was in this work was a crucial that the space w1 came with a special class and the special class you it was automatically in the space and it played a crucial role in the mechanics of the proof and in chaula there is no analog of such a special class and you should try to do the same thing you simply don't know where to start however you can get inspiration and the inspiration comes from the fact that these spaces always contain a special character which is the constant field extension character which plays exactly the same role as kaiman is one and it's a little bit technical but let me tell you that the unlike the class now there is also a sort of slightly bigger part of these gaula motions that are bearing with kai that is fixed and that is the two-dimensional space which is non-trivial as a gaula motion because kai epsilon acts non-trivial in this gaula motion and essentially you can get inspiration from what happened in the negative pair case to find the different combinatorial arrangement where some part of this character is varying along all the cube and there is an exotic reflection principle where minus one is replaced by this this special character here all right so summing things up with exotic reflection principle we can prove relations between arc and pair is also when you have symmetry and then you can prove that the sequence of spaces v and kai follows a variant of the coin lens but heuristics and this shows in particular that the spaces v and kai are almost surely collapsing to a one-dimensional vector space and thanks to the big conjectures uh sorry that that is thanks to plastic here that is a viewer to say that root q is not a variable for b for 100 percent of the kai and thanks to the big conjectures you know they almost totally uh their function doesn't match and that's a good point