 Hello and thanks to the organizers, to Halina, Michael and Philip for inviting me. The work I'm going to talk about is based actually on two papers, one of which is a joint with Martin Orr and Yuri Zahrin, and another with Martin Orr. So we consider a billion varieties and cave resurfaces of a number fields. And the aim is to explore various conjectures in the literature and logical links between them. So this conjecture state that certain variants take only finitely many values when the degree of the number field is bounded. And in case of a billion varieties, we also want the dimension to be bounded. And okay, the notation is mostly standard. K is going to be most of the time a number field with algebraic closure k bar. Gamma is the absolute Galov group. And x bar for a variety x of a k denotes the same variety of an algebraic closure of k bar. So just a word about the terminology, a lattice in this talk means a free finitely generated abelian group with a non degenerate integral symmetric bilinear form on it. And let me start with the conjecture, which is due to Robert Coleman. I think he never published this conjecture as far as I know, but this is mentioned in several places in the literature. So I believe this probably was done around 1990, but maybe earlier, maybe later. And it is stated as follows. So fix two positive integers, d and g, and consider all abelian varieties a of dimension g defined of a number fields of degree g. Or if you prefer, we consider a variety of dimension at most g defined of a number fields of degree at most d. It doesn't make any difference. Then there are only finitely many isomorphism classes among the rings and a bar. So the ring of anti bar means the endomorphism of the abelian variety a bar, which is the same as a, but considered of a k bar. You can state equivalent versions of the same conjecture by replacing the endomorphisms of the biline variety of the, of the right closure of k by simply by the endomorphism of a, which is of course the same as Gullin, Gullin variant part of the endomorphism of a bar. So I'll just take this opportunity to ask a question posed by Bjorn Pudin in the chat. And the question is, does, does a Coleman conjecture reduce to the case d equals one by restriction of scalars? So to the rational field. Oh, can we reduce the Coleman conjecture to the case of the ground fields by restriction of scalars? I don't know. It's a good question I need to think. At the moment, I can't say anything. Sorry. Okay. So let me just continue. So one motivation for the, for this work was the result of Gaël Rémond, who proved that Coleman's conjecture implies the uniform boundedness of torsion subgroups of, of groups of k point biline variety. Well, of course, of bounded dimension, which is a well known hard conjecture. So Coleman's conjecture is very hard. It also implies that the boundedness of the minimal degree of an isogen between isogen subgroups and varieties. Okay. So I'm sorry for this terrible, please move this one away, but it's it anyway. So sometimes it doesn't move away, but anyway. Okay. Now let me talk about K free surfaces. The situation is to a certain extent similar and a replacement for, for the endomorphism of the biline variety is an erroneous very group. I'll try to argue that there is some similarity between the two. So, so there are also very group of a, of a K free surface is a lattice. So it's the same as the Picard group of flex bar. And it has a natural bilinear form given by intersection pairing, which is not degenerate. So we get a lattice of rank at most 20. And Shafarevich in one of his last published papers stated a conjecture, which is sort of morally similar to Coleman's conjecture. So let's fix a positive integer D. Then there are only finitely many lattices L up to isomorphism for which there exists a K free surface defined over a number field of degree D with a Neuronsivari lattice isomorphic to this lattice L. And in his paper, he says that Sarah told him about Coleman's conjecture. So this is one of the sources that we know that Coleman did make his conjecture like this. So Shafarevich was aware of Coleman's conjecture when he was stating that. And okay, we can state equivalent versions. So one way of talking about this is replace lattice by its discriminant. So we know that there are only finitely many lattices of bounded discriminant and bounded rank. And therefore, in a cool way of stating this conjecture would be to say that if the degree of the number field K is bounded, then the discriminant of the Neuronsivari lattice is bounded. And we can replace, if you like, the Neuronsivari by its Galo-Iquivarian sublattice or by the Picard group of X, which is a subgroup of finite index in Neuronsivari invariant and the Galo-I. And it's possible to show that all these versions are equivalent. Now, let me explain that. So I mentioned that the two conjectures are kind of similar. Let me explain how to restate Coleman's conjecture in terms of lattices. So consider the tensor project, tensor product of the endomorphism of A with Q. This is a semi-simple algebra of a Q. So it can be written as direct sum of simple components, which are algebras of matrices with entries in some division algebra, Di. So if we call Ki the center of Di, then we can define the reduced trace as follows. So on each simple component, we have the usual reduced trace of a center of simple algebra from the matrix algebra with entries in Di to capital Ki to the Ki. And then we can compose this with the usual trace from the finite field extension Ki of Q to Q and take the sum of all simple components. So let's call this the reduced trace. And then it's clear that the ring endomorphism ring of A is an order in this semi-simple Q algebra. Endomorphism tensors with Q. The ranks are bounded. Yes. And the important thing is that the value of the trace on the endomorphism of A gives us integers. So and it turns out that we do get a lattice, so that we get a non-degenerate symmetric integral by linear form given by a trace, trace of x, y on the endomorphism of A. And thus we can restate the conjecture. So we can define the discriminant of A as the discriminant of this integral form on the endomorphisms. And an equivalent version of Coleman's conjecture says that if the degree of the field and the dimension of a binivariate is bounded, then the discriminant of this form is bounded. So both conjectures of Coleman and of Shafferovich can be stated as conjectures about certain integers, which should be bounded, provided that degree of number field and the dimension case of binivariate is both bounded. Okay. Now let me move on to the third group of conjectures. So there will be more conjectures now in a minute. And these remaining conjectures will concern the Brouwer group. So the Brouwer group is understood as a homological Brouwer group. So it's a second et al. homology group of Schemax with coefficients in the et al. multiplicative group. So the situation over k bar is quite clear. So I mean after growth and dig these things are well-known. So let's assume that k is a field of characteristic zero and let raw be the pecan number that is the rank of the Neuronsi vari group. So for the binivariate A of k, the Brouwer group of A bar is described as in the picture. So this is the product of a certain finite number of copies of Q mod z. And namely you need to take, I mean so using kumar exact sequence, you need to take basically the rank of the second homology group minus the rank of the Neuronsi vari. So this is how many copies you need to take. And when we're talking about the k-free surface x, then the Brouwer group of x bar looks similar. So it's a product of a number of copies of Q mod z. Namely you need to take 22 minus raw. So 22 is the rank of the second homology group of a k-free surface. And it is known now that if x is an abelian variety or a k-free surface defined over a finitely generated field of characteristic zero or of positive characteristic, then the Galois invariant subgroup of the Brouwer group of x bar is finite. And with the caveat that if characteristic is positive, then we stay away from p-torsion characteristic p. So it's like modulo, the p-parametric torsion subgroup. So this is due to myself and Zarin in and also to Cosohurator. So Cosohurator did the sign for characteristic p equal to 2 in the case of k-free surfaces. Okay, so let me now state a conjecture made by Tony Varela-Varado a few years ago. So he said that, so it's a conjecture. If we fix a positive integer d and we fix a lattice l, then if x is a k-free surface defined over a number field of degree d, such that it's neuron-severe lattice is isomorphic to l, then the size of the Galois invariant subgroup of the Brouwer group of x bar is bounded. Okay, so well, of course, it's not very hard to make conjectures. So if, I mean, we look at this and we might as well say, let's state a stronger conjecture, let's omit the reference to the latches. I mean, so here's what we get. If we need more conjectures, then it's quite easy to produce them. So let's state this conjecture as follows. Let d be positive integer, then for any k-free surface defined over a number field of degree d or at most d, the size of the Galois invariant subgroup of Brouwer x bar is bounded. And once we are talking about this sort of thing, let's make the same conjecture for a billion varieties. So okay, I'll label this conjecture Brouwer of k-free and Brouwer of av. Okay, so the conjecture is what you expect. We fix d and g and then there's a constant depending only on d and g, such that for any a billion variety of dimension g defined over a number field of degree at most d or equal to d, the size of the Galois invariant part of Brouwer y bar is bounded. And the main results, let me state it as this diagram of logical links. So we have implications as follows. So the genuine results here are three implications in the middle and in the left. And because the implication in the right-hand side is a triviality. So it's quite obvious that if you combine the conjecture of Shafarevich and conjecture of Varai Varado, then we automatically get the uniform boundness of Galois invariance of groups of Brouwer groups of k-free surfaces. So the right-hand side of this diagram is more like a joke. So let's concentrate on the middle part of the diagram and the left side. So the left side of this picture is the world of a billion varieties and there the result is that Coleman's conjecture implies the boundedness of Brouwer groups of a billion varieties. So I decided against going into the proof of this because it's not involved. And the middle implications go from statements about a billion varieties to statements about k-free surfaces and they are based on the uniform Cougar-Sataki construction. So Cougar-Sataki construction has been exploited quite a lot recently. So in the original paper of Cougar-Sataki this was purely transcendental construction which produces an a billion variety of our complex numbers from complex k-free surfaces but then there was the extremely important work of Deline about the weld conjectures for k-free surfaces and sort of modern incarnations of the same construction are stated in terms of Shimura varieties which are closely related to the modular space of k-free surfaces with a given polarization and modular spaces of a billion varieties with a certain polarization. So the important thing in the proof of this sort of horizontal implications was to make the Cougar-Sataki construction uniform so that it doesn't depend on the degree of polarization. So we don't need to treat like infinitely many modular spaces. So morally this is a some sort of replacement for Zarkin's quotient trick for billion varieties which is extremely useful because it allows to prove results for billion varieties with some polarization if similar results are known for principally polarized a billion varieties. So in the case of k-free surfaces this is based on lattices so this construction uses embeddings of lattices into bigger lattices. So basically what one does is one looks at all possible polarizations of k-free surfaces all possible degrees so basically all even numbers and corresponding lattices are embedded into one ambient lattice and then the Cougar-Sataki construction is applied to this just kind of one lattice one with this hot structure which takes care of all polarizations simultaneously and produces some sort of a billion varieties. So that's basically how it works. So this has been worked out in the joint paper with Martin Orr and independently in the paper by Xi. Okay so some other comments that comments conjecture is known in dimension one and it follows from the Brouwer-Zegel theorem. And another comment is that all the conjectures mentioned in this diagram are true in the CM case and I will talk about this in the later part of this talk. All right so that's the main result of the main part of the talk. Let me now say that of course if we find these conjectures difficult as we do then we might want to consider not all a billion varieties but maybe a billion varieties in specific families and maybe k-free surfaces specific families. So what is known in this direction is that for example if we go just all the way through just look at one parameter families then conjectures of Kaulman and Chufarevich hold so they hold for one parameter families and this is not so hard to deduce from results of Kadorin Tamagava and results of Huy. Another result of another result of Kadorin and François Charle is about one parameter families of k-free surfaces so if let's fix the prime number L and consider not the not what we want to consider but just the L primary torsion subgroup of Brower X bar invariant on the Gala then this number which we know is finite is bounded in in an arbitrary one parameter family of k-free surfaces that's what they prove. Alexey we have a question in the chat. Yeah we too you want to unmute and ask away. Yeah that's fine oh let's see. Oh hi sorry I was just wondering about your construction so I think your own type some response. I was just wondering if you apply your construction to k-3 and then you apply Kuga Satake do you recover Zaryn's trick on a billion varieties? Not at all no no it's completely different no no what we do is a construction with lattices you see so you look at k-free surfaces of fixed polarization degree so it's some even number and then the second complex homology group of the k-free surface is a particular lattice of rank 22 okay so we are even lattice in you know we know it's structure now the polarization is a class in this lattice so what we need to talk about when so when we we need to deal with the polarized hodge structure so this means that we need to talk about the orthogonal complement to the polarization in the k-free lattice so this gives us a sub lattice of rank 21 and absolute value of discriminant you know the degree of polarization and what we do is I mean using some results of Nicollin which are you know so Nicollin created this very nice theory of lattices and then you can embed this lattice says into one so for every degree of polarization you embed them into some lattice it doesn't mean it doesn't matter which one you could I think you can take a lattice of rank 25 I mean I'm not sure I mean in paper we I mean some rank some fixed rank and and then you you have a k-free sorry you have a hodge structure of k-free type on this given lattice and then you look and then you kind of imitate everything that you do you create I mean look at modular space of hodge structures on this so you just need to run the kugosaki construction kind of once for one modular space of k-free so it's and then you just need to consider a given shimura variety you know or for sovereign type yeah yeah exactly and then you and then you you apply kosaki and then you land in the world of a billion varieties so you get like a modular space of a billion varieties of some dimension with fixed polarization that's how it works okay so okay so it has nothing to do with zarkin streak stick to speaking what I meant what I wanted to say is that it plays morally the role of zarkin streak because it allows us to instead of considering like one polarization to time to consider them all simultaneously okay okay so oh so like actually so Bjorn commented on in the chat saying that every k-free is related via kugosaki to some abelian variety but not vice versa but I my question was I was wondering if it like somehow recovers zarkin streak when you only look at the what the abelian no no we're only going one direction now we go from k-free to abelian I mean the construction goes from k-free to abelian varieties it doesn't go back in a way okay okay so so so it allows you to to deduce results about k-free if you know them already for abelian varieties oh but not okay not this but not the other it's not not in the other direction okay all right thanks thanks for the clarification yeah no problem so any other questions if okay yes in fact Karlo Kaspari had a question okay Karlo can you hear me yeah the question is it's just written is in this in this results by Kaduriya and Tamagawa and Sharks the families are complete or or you you can allow bad reduction I think it doesn't matter you just look at open curves and no it doesn't matter no they don't have to be complete okay thank you sure all right so in the last result I wanted to mention uh sorry yeah yeah just speaking about one parameter family so I would like to mention another result uh due to Varela Varado and and Virae who produced explicit bounds for the size of the Galoa invariant part of Brau-Rex bar for specific families for specific one parameter families of k-free surfaces okay so so things are definitely easier if you restrict the specific families okay so as I told you I decided Gains going deeper into technicalities so what I plan to do now is to talk about the CM case so I mentioned earlier let me just go back that these conjectures are known to hold in the CM case and let me just maybe spend some time talking about the previous paper which is a joint work with Martin or so um so we proved like three main results there theorem A theorem B and theorem C theorem A says that um there are only finally many so if you if you can't see my screen because of this please move this one away which we can't move away it's q bar it's not q there um I hope this will move away for a second to make to make the bar appear yeah okay here we go so there are only five to many q bar isomorphism classes of a billion varieties of CM type of given dimension which can be defined of a number fields of given degree so this result is in fact not very hard to prove because as a so we did use it as a rather quick consequence of the result of of Zimmerman so using standard tools like classical bounds of massive wood holds for the minimal degree of isogeny between the billion varieties and and classical sarkin's quotient trick so we deduce it from result of Zimmerman who produced a level bound for the size of Gala orbits of c m point so this these things are well known to the to the experts just one word so this is a consequence of the average calmest conjecture proved by andrietta gore and howard moda pusipera and also by young john um so this theorem A as I say this is not so hard to deduce from from Zimmerman Zimmerman's result and of course as a immediate corollary you get that Coleman's conjecture holds for c m a billion varieties great um and next I would like to so this is not so well known maybe I would remind you the definition of a c m k free surface so a k free surface x is uh has complex multiplication if it's monfortate group is commutative so if you're if you're not sure what monfortate group is let me not define it but I'll give you the equivalent definition so consider the transcendental lattice of my k free surface x when I consider it our complex numbers so we need to fix some embedding of the field of efficient to complex numbers and then t of x c is the transcendental lattice so by definition this is the orthogonal complement to the neuron severity lattice inside second classical complex homology group a second basic homology group of x of x c so that's the transcendental lattice it carries a hutch structure of k free type and we can consider the endomorphisms of this lattice tensors with q so this is a q hutch structure um so we can consider its endomorphism which respect the hutch structure okay and the condition I mean the definition of a k free surface of c m type is that this hodge endomorphism ring is a c m field so it's a purely imaginary aquatic extension of a purely of totally real field and there's a second part in the definition we need also to require that when we look at this rational transcendental lattice as a vector space over over over this c m field e then it has dimension one so so if both of these conditions hold then we say that our k free surface has complex multiplication so these surfaces have been introduced by patesky-chapir and shaffer-erich soon after they proved the derelict theorem for k free surfaces in in a different paper I think it was 1973 and they proved in that paper that every k free surface with complex multiplication can be defined over a number field and so these standard examples include k free surfaces of maximal possible pica rank which is 20 in this case the transcendental lattice is of rank two so it's a uh an imaginary so so when you're tensored with q uh and look at the endomorphism because this is the imaginary quadratic field also we can look at we can we have examples given by kuma surfaces attached to c m abelian surfaces but these always have from pica rank uh at least um 18 but let me tell you and prove that in fact for every even value of the pica rank there exist um k free surfaces with complex multiplication so in fact there are many many others and we just don't know how to make them explicit if you like so okay so the results in in the k free case size follows uh so theorem b says that there are only finally many q bar again when this annoying window will disappear you'll see the bar yeah there are only finally many q bar isomorphism classes of k free surfaces with c m that can be defined over number fields of given degree um and again this follows from theorem a uh in a way that i just described so when i was talking about the uniform cook cycle construction it's in fact has been um created to to deduce theorem b from theorem a okay i think i've explained how this works to to large extent uh so in his paper uh where shafarevich stated his conjecture which i mentioned in the beginning of the talk this is 1996 paper uh shafarevich proved this in the case of k free surfaces of maximal pica rank 20 um so this this was a starting point of our far investigation let me also mention that dominica valoni uh has given an easier proof uh so so it's sort of annoying that to prove these results uh we rely on on this massive work that went to the proof of average conjecture so in fact there are like essentially easier maybe i should say even to some extent elementary proofs what not really elementary i mean based on on something but but on much less so dominica valoni has found an easier proof in the case um but it's unfortunately not completely general it it works in the case of uh cm by the ring of integers of of this field e of the reflex field it's it's not um it doesn't seem to work in the case where we have complication by some order um in that field but nevertheless it's it maybe maybe it's possible to find an easier proof um that would be nice okay and as a immediate consequence of theorem b we we obtained the corollary that shafarevich's conjecture holds for k free surfaces with cm okay i mean the next result is actually is not about complex multiplication so let me um so this result is about forms um of varieties and the brauer groups so let's consider a smooth projective geometrical variety x defined over sorry not the number field but okay we can allow it to be defined more generally our refined and generated uh field of a key so so increasing zero but finally generated i recall the well-known definition of a form so it's a standard definition so a variety y of refilled l which is an extension of k contained in in a given algebraic closure is called k bar over l form of x if the base change of the ground field from l to k bar makes it isomorphic to x bar so so these are varieties which become isomorphic to x over k bar all right the standard definition of a form and and using work of many many many people including kodore, moonen, kret, tankev, andre and others uh uh we we prove in this paper with martin order the following theorem so theorem c let me state it for a billion varieties with cm or arbitrary k free surfaces with or without cm so it says that in this case um for each positive integer n there is a constant depending on n and x such that for every form y of x defined over l which is an extension of k of degree at most n the size of the Galois invariant subgroup of braver y bar is bounded by this constant so this is a little hard to understand but i think it already makes sense to think about this in the case where l is equal to k so in this case n is one and i'm just talking about all k bar over k forms of um my given variety x so this is just the usual forms of x over the same field over which x is defined so and then we say that in the case of a billion varieties with complex multiplication or arbitrary k free surfaces if we look at it's all possible forms of the same field then the size of the Galois invariant part of the brau group is is bounded now why why is this um difference between arbitrary k free surfaces and the billion rights with cm so in fact the resultant to prove is slightly stronger it holds whenever the variety satisfies the so-called integral monfortate conjecture and uh let me not uh sort of maybe just just try to define it uh let me let me just say that um if an a billion varieties satisfies the usual monfortate conjecture then by result of Kaderi and Monin uh it satisfies its integral version so and then the monfortate conjecture is known for cm a billion varieties on the other hand the monfortate conjecture by work of Don Cave and Andre is known for k free surfaces and that's why um and that's why this theorem is um stated for cm a billion varieties and arbitrary k free surfaces okay so let me let me kind of illustrate what i'm so maybe it's a little obscure i mean since i'm talking about forms let me illustrate uh this result in one particular case so here's a here's a straightforward result i mean like like a like a the first thing one can think of when um talking about forms so if i consider the formac chaotic surface so the sum of four fourth powers equals zero um in b3 then i can also consider arbitrary diagonal chaotic surfaces by taking arbitrary arbitrary nonzero coefficients a b c d um and and these are of course forms because these two um i mean the surface with coefficients and the surface with coefficients all equal to one become isomorphic over some finite extension of k so these are forms so this theorem says that um and of course there are like if you feel this the field of rational numbers then there are infinitely many forms and and and this theorem says in this particular case that uh the size of the Galois invariant part of the bra group is bounded for all the forms and this can be used to deduce that um all the bra group um when i consider it modular the image of the bra group of the ground field um then then this then this is also finite and and the size of this is bounded so of course this is the fact that formac chaotic surface is a k3 surface so um in this particular case since i'm talking about the situation then um sharp explicit bounds are known in this case when k is q or q of square root of minus one so this this is the work of many people and uh concluding the recent paper that i wrote with damian views maybe it's a good time to answer questions of course see there are there are two questions in the chat yes yeah maybe you too maybe you uh mute and ask away oh um i was just asking about the proof of the previous theorem like do you use this tankiv lemma which identifies like the special monfortate group was i think restriction of scalar of some some collective group or something like in this proof because you said you used the work of tankiv and andre no no the word of tankiv and andre is just used uh because i mean we just used the fact that the monfortate conjecture is known for k3 surfaces our fine tangerine fields of a q so that's that's we just we just use this result okay so the proof works quite generally as i said the proof works quite generally for arbitrary varieties so let me just quote um okay so maybe i maybe i should state it in the more general okay so i think um we need to know that x suffice the integral monfortate conjecture that's all we need to know but unfortunately i don't know of very many cases when this is known so this is known for um a billion varieties satisfying the usual monfortate conjecture and and for k3 surfaces but i mean the maybe i'm not really an expert but maybe there are other cases known but uh those were the two cases which um seem to be quite well known somehow i see okay okay but but like in your proof do you need like a very explicit description of some monfortate group or the special monfortate group no no okay okay so maybe it's not yeah it's not the result i was okay thanks any other questions so um there are two more questions but maybe let's defer them to the end of the talk that's fine okay yeah yeah whatever yeah sure um okay so um i'd like also to mention that uh theorem c is also valid in characteristic p if you consider the result modulo um p torsion and this is due to a million ombrosi okay and um actually i'm i think i've i'm i'm gradually coming to the end of my talk so we will have plenty of time um for questions and discussion so if we if we just combine theorems a and c that what we get is um the results i stated about a billion varieties of cm types so so fix positive integers m and g then there is a constant say depending on only on m and g such that the size of the gulloy invariant part of the bra group of x bar where x where x is any form of an a billion variety so it could be an a billion variety or it could be a torsion um that's fine um this is allowed so so so when when it's the billion variety with cm of dimension g um and x is defined over um number field of degree n then then we have this bound they have this uniform bound and a similar result of course holds for k-free serfs so for any positive integer n there is a constant depending only on n such that the size of the gulloy um invariant part of the bra group of x bar for any um k-free surface x with cm defined over number field k of degree n is bounded by a constant that depends only on n so yeah i think this finishes my little survey and yeah maybe maybe we'll we'll have um yeah we'll have time for yeah for many questions so