 I want to approach that it's not completely number theory but it uses strongly methods from number theory and it's a work which I did with Johan de Jong recently in November and it led us to find so maybe I start with this I just write it's going to give a new obstruction for a finitely presented group so this means an abstract finitely presented group to be so by one the topological by one of a smooth projective variety defined overseas that means the underlying topological manifold of the c-points of a smooth projective variety defined overseas so let me write x of c based at some point here where x here is a smooth quasi projective variety defined overseas so I have written this at the very beginning because it is a surprising output of what we had been doing and you are not really most of you doing really geometry but we have already obstructions for finitely presented group to be the topological fundamental group of a complex manifold when this one is projective because there is some opinions some hot theory in the background which gives some obstruction but here it's just an abstract quasi projective variety and a quasi projective variety when you look at its fundamental groups and it has the contribution of the loops around the divisor at infinity so here it is very strange and also present to see that we don't have to specify what are those loops here to find the obstructions that means whatever set of finite set you find in this group which you suppose to be the loops coming from infinity then it's going to be an obstruction and let me explain and just to to make it a little bit like a provocation here and this obstruction here is coming from two heavy ingredients one is a consequence of the Langland's program and which Langland's program here that's the conjecture which had been formulated by the linear in vail 2 on the existence of companions existence of LED companions and those LED companions are on x smooth quasi projective so as the assumption we had over the field of complex numbers here smooth quasi projective but over a finite field fq and l is a prime which is defined from the characteristic of q q is the power of p and those LED companions had been predicted by the linear as I just said and were on a curve so for x of dimension one this is a consequence of the Langland's program as proved by la forge laura la forge and even though there is no notion of Langland's program in higher dimension then dream field was able to prove it in in any dimension and then of course there is no Langland's program behind so the way to go from dimension one to higher dimension is in fact a fascinating proof so so that's one part which is behind here and the second part which is behind the COM which I'm going to present here is a so-called de Jong conjecture and de Jong's conjecture right now I don't mention precisely what what is the content of those conjectures and the content of the LED companions is going to appear in the course of the lecture here but de Jong's conjecture is technical to formulate but it has the consequence which was already seen by de Jong even though it was at the level of conjecture he could already see that his conjecture if true would implies the existence of arithmetic eladic shifts on certain x smooth quasi-projective again defined over finite field and when I say arithmetic eladic local systems that means I consider eladic local system on x over fq bar so ie so you have an eladic local system here let's say el which is on x over fq bar and to say that it is arithmetic it is said that it is equivalent to saying that this eladic local system in fact comes from an eladic local system on x over fq and maybe not fq itself but some finite extension of fq so let me write here fqs for some finite power s so el is defined here okay so that's the mathematics which are behind I mean the heavy weights which are behind and why is that arithmetic so obviously here we are doing arithmetic when I say arithmetic I mean arithmetic geometry but the way his conjecture has been proven by get scurry so maybe I write in blue here for l at least three and here maybe I should put dream fields here in blue again and maybe la fog in blue for l at least three and again this is a length program but that's a geometric length program so the proof relies on the geometric length program and as of today there is no other proof I mean in either case here that means the existence of eladic companions which it's a purely eladic problem which you can formulate without automatic side but yet the answer is given via the automatic side so via the length program and then some very very clever arithmetic geometry of dream fields are highly highly clever to go from dimension ones that means from function fields to higher dimensional higher dimensional varieties over finite field if they are smooth and that's the reason why we assume smooth here I can make the remarks that the delinx conjecture initially was for normal varieties but on normal varieties there is no progress so far and and then the young's conjecture here yet again it's a conjecture which is purely eladic and but the only proof we have of it as of today is the proof of a denies get scurry using the geometric length length program so once I have said it's a sort of advertisement like this so now let me formulate an easy simplified formulation of the main result so let me write in black here and let me write here simplify the theorem and then I will not make it much more I will not increase the level of difficulty of the theorem the theorem is more general than this but I will formulate the same theorem under another angle where one can at least suspect how to use the material which are already presented so what is a simplified theorem here so we do complex geometry so I apologize so where we leave the territory of arithmetic geometry and as we started the discussion before we start with x quasi-projective the quasi-projective variety which is smooth defined over c so that's the assumption and maybe there is one thing I have been said to start with when I was talking on obstruction is that there is one invariant of such a variety that's its topological fundamental group of the underlying complex manifold which consists of the complex point which is based at some complex point here so now this base point here is basically for decoration because as we know from the 19th century by the work of Riemann first and then Poincaré then as we change the base point here is then we obtain isomorphic fundamental groups they are isomorphic via an outer isomorphism so when we talk on some feature like this group is finally presented I will abbreviate by finally presented the reason is because this topological manifold here has I mean it's connected I should say everything is connected here I apologize has the homotopy type of finite CW complex and for this reason the fundamental group is finally presented so we know basically nothing on this group there are very very little knowledge we have we know it's finally presented and if X was projective in addition not only quasi-projective but really projective we would have a few obstructions for a finally presented group to be the fundamental group of complex projective variety which are coming from a billion theory and of a hot theory and even a little bit of non-Abelian hot theory a la Carlos Simpson but in general we don't know so that's the assumption and why because we don't know the in fact since basically the 19th century we were interested not in this group itself but in its complex linear representation so complex linear representations that's so I just write by one here if you like by one top two glr of c for some r and then for all r one two three in the fourth and because as we already talked on the on the base point here if you change the base point if you fix one representation is going to be changed by an automorphism by an by a nother conjugation so you look at this up I'm sorry up to a conjugation by glr of c so when you look at a representation of the fundamental group up to conjugacy in the target group then another formulation is that it is a local system that's exactly the same datum so it's a complex linear representation of the fundamental group up to conjugacy and then because this group is finally presented then there is a modular space for such room in fact finitely generated would be enough because we have finitely many generators and then you look at a parameter space of all representations so you look for each generator which you choose it's a choice you look at one matrix here in glrc and then you have relations so that defines an affine variety it's a modular space usually which is called either the character variety of pi one or the modular space the beating modular space of x here I should say x over c if you like and then in rank r and then if you fix a point here so this is a friend representation because you have you are in glr you are not in glr without a base so that means you are not on any torso under glr but you are under glr of a base and you have chosen a point here so you this is called the friend modular space and then you have an action as we discussed by a conjugation of glr and this yields a g it quotient here and the quotient here is a modular space here so a complex point here is a representation of the pi one and a complex point here is a local system that means a representation modulo conjugacy okay and we are going to be interested in irreducible l so now i'm already in the position to formulate uh the theorem so let me write so it looks like it's not a theorem but let me try to formulate it so here's a formulation so theorem so we have x and r as before and assume there is a irreducible l a frank r i forgot to say that its r is called the rank of the local system okay so it's a very weak uh it's a very weak assumption then the conclusion is going to be the following the conclusion is going to be for all prime number l there is an irreducible ql bar representation or local system so what does that mean ql bar local system i didn't introduce the terminology it means ie let me write ie there is a row which goes from the topological fundamental group so base somewhere into glr of ql bar so that means let me reduce and it is irreducible okay and but then which is defined over the l bar so here you sense that some arithmetic is coming in so you have an of course if i was just saying irreducible ql bar local system then you could immediately object but it's not a theorem because we know by logic which is not my area there is an abstract field isomorphism between c and ql bar so once you have an irreducible glrc representation and then local system then you have an irreducible glr ql bar local system because you just post compose with as the abstract field isomorphism between c and ql bar so this would be zero theorem but here the serum comes and it would be reducible because it's a field as a morphism so if the complex representation was irreducible the post compose representation would be reducible as well but here the serum says that there is a ql bar representation of the topological fundamental group and then you look at it modulo conjugacy but then modulo conjugacy is defined over the l bar that means let me write it explicitly because many of you don't do arithmetic geometry this means that up to conjugacy and then in gl bar of ql bar then this representation here goes from the topological fundamental group into glr of the l bar that means it's integral is that okay i think the formulation is clear i hope if there is an objection now question please tell me because now i'm going to give you a completely different formulation of the same thing i will give you two different formulations so elaine then that's peter hi yeah just so uh as i say the audience yeah good to see uh it's probably not that familiar with many of these things so um just to be clear yeah this is uh not true for arbitrary finite and presented say group in other words the quasi-projective feature is being used to achieve this correct yeah so uh what i want to say let me first say thank you for for the question peter so uh let me just say that uh for example let us for example let us take the simplest possible example let us take x is gm that means the complex point of x are going to be uh the units in c okay and then let us take a rank one representation as though you take a rank one representation so the topological fundamental group here we know it this is z you you fix the best point here and now you look at a rank one local system so you have a z mapping to gl1 of c let us take rank one so ccc star and then you have a free choice so that means any representation so it's rank one so it's irreducible so that means you look at your base base vector here z is spanned by a loop gamma it goes to whatever you want so for example it goes to one over l you fix an l and it goes to one over l okay so uh this one clearly that is not integral this is no way you can conjugate in c star because it's a billion now it's rank one to make it integral so the theorem is telling you that once you have an irreducible one then you have a one which is integral by l for any n but it's not the one you had at the at the origin and uh but here you can see on this example you don't do anything i mean you uh you send gamma to uh for example l it's integral but in general of course and this this is uh also uh to to answer the the remark by uh by peter then it's very specific to quasi-projective varieties if you have taken a more complicated fundamental group it will not be the case and i will come to this because this was the introduction of the lecture i was telling you that it gives an obstruction this theorem gives an obstruction so now is that okay can i go on yeah yeah thanks thank you now now let me write a second formulation which is still very mild in which glues on the formulation before so you know once you have let us take the result which i said before once you have an irreducible complex local system in a given rank then you will find for any l you will find a ql bar irreducible local system in this rank which is integral so now let us um let us remember so let us write the the result here i mean let us write this glr ql bar representation so you have one defined modulo conjugation but but it modulo conjugation it has values here so let us write it but now this group is an abstract group i'm sorry you don't see so the topological fundamental group is abstract and this group glr of the l bar it's a profanal group is that okay it inherits the topology of the l bar in fact already ql bar it was already inheriting inheriting the topology of ql bar so glr of ql bar glr of the l bar but it's a profanal group here so uh this map here which starts from an abstract group to profanal group by the very definition it factors through the profanal completion of this abstract group i hope you all agree with that so it factors here uh no sorry it factors here and here this is a topological fundamental group so the same thing here and then the profanal completion do you agree but now by the Riemann existence theorem this profanal completion of the topological fundamental group of a complex manifold this is the et al fundamental group of the underlying now scheme of a variety defined over c base at the geometric point xc that is the Riemann existence theorem which has been exploited by Grotendig to define the et al fundamental group as a unification of the theory of the galois theory which predates of Grotendig and predates Riemann Poincaré and the theory of Riemann Poincaré so that means here we obtained by the very definition a continuous representation and so that means the second formulation is if so now i write the second formulation if there exists an l over c as before which is irreducible complex local system then for any l prime number there is an irreducible and an irreducible what it's an irreducible representation of the et al fundamental group this means there is an irreducible l-addict local system okay that's a second formulation so if there is an irreducible complex local system in a given ring then for any l there is an irreducible l-addict local system then it starts to be mysterious starts to be arithmetic and arithmetic in two senses one sense is that you are looking at the l-addict local system and another sense is going to come now and which is closer maybe to the heart of peter is that you can express what what i said here on the moduli so third formulation and now it involves the petty modular space which i defined before so now the petty modular space which is also called the character variety of the fundamental group because it depends only on the topological fundamental group of the underlying complex manifold as i was explaining before then this modular space here is defined it's a fact over spec z so it's not a fine modular space and what we are interested in this year we are looking at the complex points so of course on complex points and more generally on points they find over an algebraically closed field then it is a fine it's a coarse modular space so it's fine on the field value points when the fields are algebraically closed in fact more generally when the fields are c1 and we are looking not at this whole modular space but we are looking at the open which consists of irreducible points so we write irreducible here so now we know what irreducible means when we have a representation with values in glr of a field of an algebraically closed field so that means geometrically irreducible but we don't know what it means when we have a representation here uh when we have a representation of pi one top in glr of some wing let's say ring and then the right notion in fact which had been defined by dream fed before was to say you post compose this representation by any field value point of k so it's a field of a so it's a field value point and to say that it is irreducible it means that it is irreducible after post composing with any field value point of your egg Elaine I think we have a question that chats yes under bets maybe you want to unmute and ask away directly hi and oh it was just a very kind of small small question but you resulted this uh that glr of said l bar is uh pro finite and I just like is it I just wanted to ask but why that was true because I thought oh okay okay so let me uh I was fast here and thank you for the question so I come back here I hope you can see so you have this represent so maybe I change the color here you have this representation here and because this group is in is finally presented but in in particular it's finally generated it is landing in glr of some finite extension of the egg okay let me say oh is that okay because this one is finally uh generated is enough here and once so that means now you can look at all the quotient glr of oh m r r is bad ms here where m is a maximal ideal of right yeah is that okay does it yeah okay okay so so thank thank you for the question I should have said that and it's good you tell me because uh it explains really the factorization and okay so I proceed is that okay it's great thank you yeah thank you so uh so we are here here so you you have in this modular space you have the irreducible modular space space with the properties at the points are irreducible representation in this sense which I said here and now you have inside this leaves the inside of mbxr without any reducibility property and what you can consider here is the closure here uh no not the closure um or maybe maybe I don't do that here I apologize uh you can consider what did I want to say uh so I consider the points here I mean what I was yeah in fact I gave you the definition here so uh those are for the irreducible ones and now ah I apologize so you look here at this piece of the modular here which has the properties that the characteristic zero points are irreducible and now the characteristic p points are anything and when I say p it's not good I should say l to tie up with the notation we had before so we look at this modular here and this map here uh this is uh this is a finite type over z and now I am in the position to tell you the third formulation which is of course much closer to the heart of algebraic geometers is the following is that the existence of l which is irreducible uh overseas and a Frank R it's equivalent to saying that the structure amorphism which is here is dominant because it is telling you that you have a nearly a c point here of this of this modular so it is telling you that it is dominant so that's the assumption you assume this map is dominant and now the conclusion is that for that it is subjective the structure amorphism is subjective okay so now rather than talking on the language programs here in various shapes uh how much time do I still have you said it's 50 minutes that would be that I have another 10 minutes 10 plus epsilon yeah we have a yeah we have a bit more time because of the beginning yeah because of the beginning so let me show you where the obstruction is coming from and then let me tell you in one word where what we apply where if I manage or in the discussion but let me tell you the the obstruction because of course it's a joy so uh now you have a group uh which has been uh constructed by Becker, Emmanuel Breuya, and Varju I'm going to give you a group here but no here uh which has which does not have so it's a finitely presented group and it does not have if I watch the exact property you discuss for a finitely presented group you fix are and also what I have once said here in the whole discussion I was sloppy here there is one thing I should say in all this discussion we fix we fix a torsion determinant of the local systems that means when you take the representation you look at the determinants so the modular spaces are fixing this determinant at infinity and and in addition they fix here some conjugacy classes at infinity at infinity means uh you look at the fundamental group of this quasi-projective varieties it transfinitely mini-divisors at infinity and then you look at the loops at infinity and you fix the conjugacy classes of those loops in the representation you assume they are quasi-unipotent so I'm a little bit sloppy here but this is completely irrelevant the exact formulation is anyways before and here so you mimic the story you fix an arc you have a gamma and you fix a character of the fundamental group so let's say a xy so it's a torsion character and then the theorem here if you work a lot it's going to show you that uh if this group so if gamma is pi one top of x as before then it's going to be the case that if gamma has an irreducible uh rank r representation with determinant this determinant here this xy said xy determinant then for any l it has a representation rho l from gamma to g l r of the l bar which is which is irreducible over ql bar and which has determinant this this l here determinant this character here so I should make precise what it means because here I had a torsion determinant with c value here is with ql bar value but let us forget this for a moment and they construct here because there is no discussion like this they construct a group and then they look at the character variety for sl2 so the determinant is trivial so there is no discussion here so I can be fast here and then they show that the character variety has only it's defined over q uh over q I apologize and over q bar it has only two points which are complex conjugate or if you like galois conjugate I should say galois conjugate and it's not integral in the sense I said before it's not integral by two so it's quite magnificent because suddenly you have so the conclusion is their gamma is not a by one as we said but also it says that's the serum and also the serum plus a little bit more the serum I was trying to say in very easy terms before this serum is an obstruction so sorry the xc they're giving explicit uh complex variety there no no no I say fine I say if gamma was a fundamental group of an x as before I mean the c point of an x then it would have the properties that if you have this representation here any representation then uh you we they construct a group which if it was the fundamental group here then for any representation you would find an eladic representation for any if you have an irreducible representation overseas and you would have an eladic representation and could I say it another way so this gamma they uh constructing cannot be the fundamental group cannot be given the serum we had before and as again because I was trying to be very deductical so explaining mathematics from the 19th century I didn't enter so precisely into the assumption we had we had this quasi unipotent monodromy at infinity yes yes yes right so I I apologize for being very weak on uh on demonstrating this assumption it's it's important and yes that's the reason why you have to do something because you consider all representation and you have to come back to the quasi unipotent monodromy at infinity in order to apply the serum so that means the serum is really uh this out of the serum you make a general definition for those groups a sort of weak integrality property for a finitely presented group in a given rank with a given torsion determinant and and then this is not verified by their specific group and consequently the the discussion we had in the serum and gives really an obstruction for for a finitely presented group to be a fundamental group of a quasi projective variety yeah thanks perfect perfect yeah thank you very much and in a very same way so now I have like maybe five minutes or something so let me tell you something about the proof so you have this moduli here and this is where you take the zariski closure on the finite field point so to speak I am very vague here and you look at irreducible representation in characteristic zero over z and the way it works is like this when you uh this assumption it's dominant but you do algebraic geometry so uh what is true is because this is a finite type scheme over z now it's really defined over z maybe with some holes with some gaps here so image are some gaps but you want to show it has no gaps so you go very far you take a very very large and if you take a very very large here it will be the case that this morphism when you take you look at close points in the fiber around the close points if you look at the reduced structure here then it's going to be smooth I mean in growth and digs terminology it's called generic smoothness of finite type morphism over reduced scheme so it's going to be smooth so this point be smooth here it's going to tell you that in fact I mean if we forget nilpotent structure here just we look at the underlying reduced structure it's going to tell you that the that's a formal completion of this scheme at this close point so this close point here as characteristic L is a pro smooth and now it's an easy consequence of bottleneck theories that in fact this representation whether it's smooth or not if you make p very large and you look at x mod p then this representation is going to come from x over fp bar so that means this local system which is now a niladic look because you look at L very far it's a niladic local system it's going to factor it's going to come from via a specialization morphism from a them local system on x over fp bar so that's the theory of growth and dig of specialization of the fundamental group which does that so once you are there what you want to say is that this is arithmetic that means maybe not the one you start with but when you look at this close point here which corresponds to the residual representation of your local system on x over fp bar you want to say that maybe not this local system you start with but another one with the same residual representation is going to be arithmetic that means it's going to be invariant under the action of some power of the Frobenius of fp here or fp or fq where x is defined here and that is provided so the arithmeticity the arithmetic lift is a consequence of this de Jong's conjecture which has been proven via via the geometric length length program so now you have an arithmetic ladic local system on x defined over fp bar and therefore over some fq and now you apply an id which i have been developing with michael gruschenick so sorry to disturb so to be clear your proof does use the geometric length length program is in the shape that there is a arithmetic lift of this even close point here correspond to a residual representation and p is large and l is large but you're going to tell us how you can also make l small i see yeah one second yeah one second l is very large now with a serial specialization i can go i can take p very large yeah and to to be sure that this close point here which corresponds to a glr fl bar representation but topological in fact factors through x over fp bar and now i apply de Jong of course there's some work because i have to compare this topological invariance here with measures the formation space of this representation here over x over fp bar let us do that so there's some quite some work here and then apply the young's conjecture and say but in fact maybe not this ladic local system but another one with the same residual representation over x over fp bar is arithmetic okay but now i have l large and then i have taken p large in function of this l and i am on this variety over fp bar and consequently over some fq for q finite finite power of p okay now i apply this sd with michelle which enables us to prove simpson's integrality conjecture so it was the ending it no longer works here simpson integrality's conjecture he was predicting that rigid local system over c are motivic and consequently are integral and we prove the integrality conjecture under some extracomological condition which is going to play no role here so the idea was you start with something which is now arithmetic over a finite field and now you apply the link's fantastic conjecture on the existence of companion it's going to tell you that this ladic local system which was arithmetic it admits companions for any abstract field as a morphism between ql and ql prime bar ql bar and ql prime bar for l l prime so l was a prime l prime is another prime and which is not p but now again the theory of specialization is going to tell me that this ladic local system this l prime ladic local system and in fact the theory of companions is going to tell me that it has the same monodromy at infinity up to some simplification it will be geometrically reducible it will have the same determinant etc etc this ladic l prime adic local system which is defined now over x over fp bar now i can lift it again and it's going to be term that's important it's also serum of the lean what happens at infinity it's going to lift again because it's them it's going to live to see it's going to live to characteristic zero and then to see so it live to see and that gives the answer to the problem and the existence of companions that's the arithmetic length program okay so i think i am pretty much on time with plus epsilon minutes and i thank you for your attention