 Well, thank you for the introduction. Thank you very much to the organizers for giving me the chance to speak here. And thanks to the audience for coming to my talk. So today I want to talk about this construction of Galois representations. So everything I'll talk about today is joint work with Sakwu Shin. So there's also a preprint on the archive. So if you're interested, you can read more details there. So the F. So I'll start with the setup. Then state my theorem or our theorem. And I'll give some ideas that go into the proof. So F here is a totally real number field. We take the integer n, at least three. Basically, because for n is equal to two, better theorems are known already. And I take gamma equal to the Galois group of F. I take L, some prime number. And I will fix throughout the talk some isomorphism of C with QL bar. And this group GSP. So this is the group such that for all rings, all commutative rings R, we have GSP to n of R is equal to the set of all matrices of size 2n by 2n, such that G transpose, such that it fixes the standard alternating form up to a scalar. So this an is the matrix of size n with zeros everywhere, except on the anti-diagonal. Oh, yes. Thank you. So it's this one. OK, so that's the general symplectic group. So you see here this x. So we're preserving the form up to a scalar. And that's somehow the point of my talk. So if you ask that it preserves the pairing on the nose, you get the symplectic group. And in this case, everything that I'm about to say is already known. So the question is really about generalizing the results for the symplectic group to the group GSP. OK, so we want to construct Galois representations. And they take value in the, so in general, they should take values in the dual group. So in the case of GSP to n, the dual group is the so-called general spin group. So this group is a double cover. So if I didn't have the G, we have the spin group, which is a double cover of SO, like this. But if you add the G, you get G0. And here G spin 2n plus 1. So the point of this dual group is that if you have, so say pi is some automorphic representation of GSP to n, then you can decompose it into a restricted tensor product of local representations. So here I have GSP to n of fv. And these things, they will be unremovified for almost all places, for almost all v. And in those cases, the representation is classified by some semi-simple conjugacy class in this dual group. So that's how this dual group arises first in the picture. So given an automorphic representation, you get some kind of infinite sequence of conjugacy classes in this dual group. OK, so this is what my theorem is about. So we take pi, hospital, automorphic representation of GSP to n af, satisfying some two technical conditions. So the first one is called the homological. So I wanted the representation at infinity, twisted by some character, is homological, which means it's the algebraic homology doesn't vanish. And the Steinberg assumption is that there exists a finite f place vst such that pi is isomorphic at this place to the Steinberg representation up to twist. So for some smooth character chi. So these are the conditions. And then the statement is that there exists some Galois representation, which depends on this isomorphism. But I'll suppress that in a notation that comes from the absolute Galois group and takes values in G spin 2n plus 1 ql bar such that for all v away from some explicit set of places, we have that rho pi of the Frobenius semi-simplified is conjugate to this matrix phi v. And then I want it in ql bar, so I take the iot of it. And this s, these are all finite f places above l, the discriminant of f, and all primes p such that pi p is unremifed. So this is an unremifed representation of Gsp 2n. OK, so this s, it's slightly larger than just the set of unremifed places, but it is completely explicit. It was a set of places where pi ramifies, but it's completely explicit. OK, so we prove more properties of this representation, but in this talk, I want to focus just on the Frobenius conjugacy classes. So one of the things we prove, for instance, if pi l is unremifed, then this rho pi satisfies the expected state property that it's crystalline. So that's one of the properties we also prove. But I will just focus on this. So to prove this result, we proceed in three steps. So this step one is to prove the theorem under some extra condition, some auxiliary condition, which is this condition called spin regularity. So this condition asks, on top of the conditions that we already have, that there exists some infinite f place, v0, such that if I look at the infinitesimal character of pi at this place, so that's some real algebraic morphism from c star into the spin group. So I can compose this with the spin representation. And I want that this composition defines a regular co-character. So that's this condition spin regularity. So this is quite a strong condition. So it's a very few, quite a few representation satisfy it. But the idea is that in later steps, I'm also going to talk about this. We can remove this condition again using an eigen variety argument. OK, but let's still focus on step one. OK, maybe I can put this on top. So in step one, the idea is to try to combine two big results. So the first big result we try to use is Arthur's description of the discrete spectrum of the symplectic group now. So this theorem will certainly apply to the things we get from our kind of representations. And the second result is to use the proof of Kizin and Paziu, proof of the Rapaport, the Langlands Rapaport conjecture for similar varieties of a billion type, plus upcoming work of Kizin, Shin, and Zhu to deduce the Kotwitz point counting formula from this conjecture. And it's stabilization, point counting formula, the stabilization. So it's only this point counting formula that I need. So there is a small caveat that actually the full statement to conjecture is not known. But there is enough known to be able to deduce this point counting formula. So that's good enough for what I want to do. OK, so now the idea is to try to use these two sources of information. So we have some, so take some pi as in the theorem, but I add additionally this condition. So it's also spin regular. So the first thing you can do is just extract from pi an automorphic representation of the symplectic group. So this symplectic group, it's some subgroup. And I just take the restriction like this. And OK, then I get some representation. It will be some huge direct sum of irreducible representations to symplectic group. But I just choose one. I don't really care which. And I call this thing pi flat. OK? And pi flat is a representation to which I can apply Arthur. So using these results of Arthur, what you can do is you can actually transfer pi flat to an automorphic representation on GL2N plus 1. And on this group, you can use the known constructions of Galois representations which are already known. So what you get is basically from Arthur for the transfer and then Shin and other people for the Galois representations. What you get is a representation of the absolute Galois group that takes values in SO2N plus 1. I want to make more space. Sorry. So I'm going to S SO2N plus 1 UL bar. OK, so that's what you get from this. So this group SO2N plus 1, it is covered by this spin group. And like I stated in the theorem, the expected representation should take values in this group. You see the kernel here is just GM. So there's no problem to lift the representation. So what you could do is first just take an arbitrary lift, just something. But then, of course, you run into problems because you do not know that this arbitrary lift satisfies this condition on the Frobenius conjugacy classes. So you can try to get some more information using the central character of pi. But this will tell you only things. So if you take more than one lift, several lifts, they will all differ by characters with values in QL bar star. But using the central character of pi, you only get information on the square of those characters. So it seems like just by doing this and then trying to lift, you do not have enough information to construct the right representation. But you can lift. It is possible to lift. It's just you don't know how to assure this condition. OK, so that's what you get from RTIR. So now, because we have this Langlis-Rappelport conjecture for Schmura varieties of a obedient type, I can take some appropriate inner form of gsp2n over f and look at the Galois representation in the cohomology of this Schmura variety. So let me just write it h star Schmura. And then you want to take some isotypical component. And this will give you some Galois representation. In, well, first, it will be true to the n dimensional. But then you may also get some automorphic multiplicity, for which you do not know if it's one. So from RTIR, you get something here. From them, you get something there. And now you want to combine the information. So the first thing you can do is you can look at I just embed gl2 to the n by blocks in here. And the first thing you want to show is that this representation here is actually the eighth power of some representation. So that's actually not obvious in general. But there is a lemma in the book of Harrison Taylor, which says that if you can show that the Hodge state numbers appear with this multiplicity, and the traces at Frobenius elements also appear with the right multiplicity, then actually it factors. So here already you can get the factorization thanks to this spin regularity assumption. So then you have some representation in here. And well, the idea is to try to compare it with that one. So here, of course, we have spin. So the idea is to go to pgl2 to the nql bar. And then here we have some induced representation. Let's just call it spin bar. So I have a square like this. And now I have two ways to get into pgl2 to the n via Arthur and via the Shimura variety. And what you will know at this point, well, certainly you control the representation of Arthur at the unremifed places. You know much more, but certainly at the unremifed places it's OK. So what you can show quite easily is that these two things are conjugate on the Frobenius elements. So spin bar of rho pi flat at the Frobenius semi-simplified is conjugate to let's just call it rho pi Shimura for the induced representation here. Bar, because I want to go to pglm. So we know that this one is conjugate to that one as projective matrices. It's not true, but let's just suppose that I could draw from this the implication that now the representations are conjugate. Suppose I could deduce that spin bar rho pi flat is conjugate to rho pi Shimura bar. Well, then I claim then it's done. Because if the two things are conjugate here, so conjugate really in pgl 2 to the n ql bar, well, then I just conjugate to one to the other so that they are now equal. And then I just say this diagram is Cartesian. So now I have two things taking, right? So I just have a Cartesian square, so I get the induced map. And for this induced map, you know that at the spin, the Frobenius elements are of the right form. And then there is a theorem of Steinberg that actually the Frobenius elements are conjugate to the correct conjugacy class. So the issue is really at this question mark. So, yeah, so let's think about that. So in GLN, it's of course a well-known result that if you have two n-dimensional Galois representations that are conjugate on the Frobenius elements, then they are conjugate. Well, at least if they are semi-simple. But for other groups, this is just completely false, OK? So for PGL-M, there are known counter examples. So in general, you cannot make this implication, OK? But the point is that the spin regularity saves us. So there's the following, let's say lemma, just some lemma with group theory, so which says that if I have two projective Galois representations where R1 from V is conjugate to R2 from V semi-simple for all V away from some finite set of places and R1 strongly irreducible. So this is kind of the extra thing you have to assume. Then R1 and R2 are conjugate. So what does strongly irreducible mean? So it means that it's a representation that's irreducible. But it also stays irreducible when restricted to every open subgroup of the absolute Galois group. So this kind of kills the finite image examples. So in the lemma, then gives you that the two things are conjugate as projective representations. So going back to this board, apparently it's enough to show that one of these two things is strongly irreducible. And so we can actually show that spin bar. So actually, in fact, under spin regularity, we can show that the image of the representation of R2 in SO2 and plus 1 is this subgroup of SO2 and plus 1 is Zariski-Denz. No one to explain all the details, but the rough idea is that we have this Steinberg assumption. So there is one finite place where the automorphic representation is the Steinberg representation. And then you can show that this restricted guy is also the Steinberg, and then you use local global compatibility to see that the Galois representation has in its image regular unipotent element. And subgroups of SO2N, so just algebraic subgroup of SO2N containing such elements, they are classified. And there are only very few examples. And I just look at the examples. And I see that none of them can contain. So if the representation would factor over one of these examples, you would just see that the parameter cannot be spin regular, OK? So that's how you see that this is Zariski-Denz image. Well, and then, of course, spin bar is irreducible representation as algebraic group. So then when I compose, I get something strongly irreducible here, and then the argument works. OK, is that clear? Kind of. OK, so that's step one. So at this point, we know this theorem above. Under this extra assumption, and in step two, we want to remove the condition. So step two. So here the idea is to use the eigen variety to remove the spin regularity again. But then, actually, it turns out that this argument only works for us in certain cases. So we are basically replacing this technical condition with some other technical condition. But that other technical condition can be removed using patching and base change. So I don't want to talk about this third step, because here I really just apply known results. So yeah, we do very little here. So I just want to focus on step two. And I'll just do it for an example to kind of give the flavor how the argument works. OK, so eigen variety, variety. So I'll just take f over q real quadratic. Of course, that's not enough for the general case. But I'll also assume that this time replace does not divide l. This is also not the condition in our theorem. And when it does divide, you get some technical problems. And I'll also assume that pi l has invariant vectors for the eva-hori subgroup. So in general, you can try to put yourself in this situation by using the space change. The argument explained before used that Vsp that had the vector, to show that the image in S of the first one contains a regular element. Well, if it does divide, you have to use local global compatibility at l. So you have to show that then still you can make some attach, some wider linear representation to this pi flat at those places. And then see that the unipotent matrix that you get satisfies the conditions that you need. So the eigen variety, let me first give a vague idea what we try to do. So now we start with pi, as in the theorem, but without this condition. So then the rough idea is you have this eigen variety, which is some l-addict space. And this pi, it defines a point on it, z pi. And besides this one particular point, you have many classical points. And they should somehow be dense. And then among those classical points, I have some subset of points for which I know the theorem, right? So these are the points that satisfy spin regularity and Steinberg. So the game is that you want to show that you can approximate this point on the eigen variety with points for which you know the theorem. So let me make this a little bit more precise. So first, what properties do I need of the eigen variety? So by Leuphler, Sien-Nivier, and Buzzard, and other people, we know that there exists some triple, e psi c, where e over ql is a reduced, rigid analytical space over ql. And psi is a map from some Hecker algebra to the rigid sections of e. So this Hecker algebra is just a Hecker algebra of functions that take values in some sufficiently large finite extension of ql. And I put, oh, I forgot to define the group. So I take gc inner form. So I work with eigen varieties for some compact inner form of g. So I take the inner form of gsp, which is compact mode center at infinity. So because it's real quadratic, there are two places at infinity. And I want it to be quasi-split at all finite places. So there is just gsp itself. So then this Hecker algebra will be the Hecker algebra of functions away from my bad set at infinity. And if I'm not at these places, the group will be isomorphic to the symplectic group. So I have some integral models, and I just use the integral model that I get from gsp. And then I take just a hyper-special group. So this is the Hecker algebra. And c is some subset of e, which is Zariski dense, and accumulation. An accumulation and Zariski dense subset. Okay, and they don't have space here, but satisfy some condition. So such that, so this triple is such that. So if I have my eigen variety, I can take points in there. I can make a map to the ring morphisms from this unremovified Hecker algebra with values in ql bar. Because, okay, so if I have my point, what I can do is when given an element in the algebra, I can put it in psi. Right, and then I get some rigid analytic function, and I look what it is at x. So this gives me a map from the eigen variety to this home. So in the eigen variety, I have my c, and they are supposed to be the classical points. Right, so they shoot the map to all ring morphisms f from the Hecker algebra into ql bar. Such that there exists tau, some automorphic representation of this compact form. Okay, plus a condition on the central character, which I prefer not to talk about, such that f is basically the trace character of tau. So such that for all h in the Hecker algebra, I have that f of h. So that's an L-addict number is equal to the trace of h against tau, which is a complex number. So I take iota of that to get ring morphism to ql bar. So this is some subset, and what we ask is that this c is projecting precisely on this subset of classical things. And because the classical things there are zariski dense and accumulates, now this c really knows a lot about e, right? What do you mean by that? Is it different from a dense seeking? Well, you have two topologies on e, right? So you have the zariski topology, and you have the rigid topology. So accumulate means that if I take some point here, then there is a basis of affinoid opens, such that for each of these affinoid opens, when I intersect it with c, I get something that is still zariski dense. But this is really some technical thing. Okay, so this is the eigen variety, or these are some properties of the eigen variety. So of course more things are known. For instance, what description of the fiber of this map, but I don't really need that. Okay, so we have this space. So that's basically this box. So now I want to use the version of the theorem, which is already known, right? So I want to use these points. But now, of course, you are worried maybe they are not dense, right? So the classical points, they are dense, but maybe these points for which I know the theorem, maybe they are not dense. So we have to be worried about that. Okay, but before saying that, so I have my f pi, right? So I have my automorphic representation pi. Okay, I'm sweeping under the rock that I can transfer it to this compact inner form, but this is possible. So this defines some function in here. And I choose my point above it. So this is z pi, so that's the point that I isolated. Okay, so now we want to deal with the issue of non-density. So the idea is to alternate the eigen variety to some space that covers it, in which you do know that the theorem is true for the classical points. So the following proposition, which says that there exists some map f from some space to e, such that following things hold. So f is a finite composition of open immersions and finite morphisms. Okay, so the map is not too crazy. The image of this f still contains the point that we are interested in. And for all classical points lying in the image, we know that the condition Steinberg holds for tau c, by which I just mean the automorphic representation corresponding to this tau corresponding to this c. And last, there is some family of representations, rstd, that take value in gl or rig. Okay, so I also wanted to give some properties of this x. So x is affinoid reduced and irreducible. Okay, and then basically this x is a better thing for us to work with instead of the variety e. So I want to give a quick sketch of the proof. So sketch. So the construction of Galois representations for this symplectic group is known in much more cases, right? There you do not notice Steinberg assumption or these regularity assumptions. So actually we can show that for all c in c, there exists some beta c, which takes values in gl 2m plus 1ql bar, such that beta c, such that the trace of the Frobenius against beta c is what it should be. So that's the trace of this conjugacy class. So the standard representation of this conjugacy class and an iota of this for all v away from my set of bad places. So actually this uses a result of Tybee plus Shin and so on for the construction of the Galois representation. This is because now I am on this inner form of the symplectic group. So I actually would need Arthur's theorem for this inner form, which is not written down yet. But Tybee, he did it for some certain inner forms of which the one I took is an example. So actually by Tybee we can still make the Galois representations in this case. So now we can interpolate. So that's some result of Chez Neuvier. There exists a pseudo character to the rigid analytic sections on E, such that for all classical points and for all v not in s, I have that t standard evaluated at the Frobenius is the same, evaluated at the point c, is the same as the trace of this beta c at the Frobenius. So now we have this family. And then another result, Chez Neuvier, is that once you have such a family, then you can look at the locus in E where the local representations are absolutely irreducible. And he shows that this is open, some lemma in this paper. So this is an open sub of E. And by Taylor-Yoshida, we know that at our point, I have this Steinberg assumption, I know that the standard representation will be, this beta c will be irreducible. So this will tell me that this point that I took lies inside this irreducibility locus. Okay, so in the end I wanted f to be some finite composition of open immersions and finite morphisms. So up to changing notation, I can now just assume actually that E irreducibility is just equal to E. I just make E smaller. And I also can take an affinoid and irreducible, just take some irreducible component that passes through my point. Okay, so now if this smaller thing, and then what you do is you look at this algebra, A O-Rig of this smaller thing, modulo the kernel of this standard representation, and then you have this theorem of Ruckier. Precisely because I'm over the irreducibility locus which says that A is an azomaya algebra. Okay, so I have an azomaya algebra. So I can split it. So choose, take some f, let's say from some space y to E, some etal map, such that algebra is split. Right, and then, so then I get on this y really a family of representations. Right, because the family is just, I just have the standard embedding of gamma s into this algebra, which then is isomorphic to the matrix algebra over y. So that's how I get this family. And then? Choosing if it's possible to make what it's azomaya. Yeah, yeah, so otherwise this is some crazy algebra. That's what you will say, we'll find out in a second. Okay, so we have this thing, so then you have this theorem of Bella Is, which is basically the Grotendieg monodromy theorem in families. So here I'm using this assumption that this VST does not divide L, otherwise you have to work with results of Berger and Comes, but I don't want to talk about that. So we now just have this Grotendieg monodromy theorem, which says that there is really some family of nilpotent operators on this space y, such that for all c and y, so for all classical points, this nc is, yeah, the usual is the monodromy of beta c at the place VST. Okay, so it's saying that these operators somehow are nicely behaved when they vary in a nice family over this variety y. So I can, so now note that, so observe that. Okay, so sorry, I take x just the space where this n is the maximal operator, so where n to the 2n is not zero in y. So this is some open, right, this is an open condition. And moreover, I will know that, so I know still that I haven't lost my point, that if I look at this map here, still hits my point z pi. Okay, and that's just because I know what the unipotent operator of z pi is at VST, so I just see that it is in x. And then I have satisfied all the conditions. Okay, so now I should wrap up. Okay, so now we have this space x, so we can define c spin reg in c. So these are just all the classical points that satisfy this extra condition on the weights, that they are spin regular, and it's not difficult to see that this is still a Zariski dance and accumulation subset. So now I can do this interpolation again. So again by this result of Chez Neuvier, there exists some pseudo-character t-spin from gamma s to o-rig of x now, such that for all c in c spin regular, which is a Zariski dance and accumulates for all c in x with f of c in c spin regular, we have that for all v not in s t-spin of the Frobenius at c is equal to the trace of the spin of rho c. So here I use the theorem, which I know. So then I have this family of representations, and I can specialize it at a point. So specialization of t-spin at my point now gives the spin of rho pi. So this constructs the two to the n dimensional representation. So this is only part of the story because the next thing you have to do is actually show that this factors over the spin group. And then so to do that we again use arguments in families to kind of show that in families this thing is contained in the spin group. But I don't have time to explain it, so I'll stop here. I have a lot of questions about an announcement. This is good news, since you have quoted at least two of the chapters of the book project, one of these synergies, article in the Sorenson's article on touching. The news is that it's actually almost finished, both of you who were born when this project began. Book one, maybe some people don't even know that it actually was published some years ago, but without a lot of publicity. Book two is now waiting for the revised version of its last chapter, and it's going to be published by Cambridge University Press. Probably it will be available some point next year. What's the title of it? Oh, whatever. Whatever the title is. Something in it. So something like stabilization of the trace formula for more varieties and a lot more things. We'll have a title. Is that proposition on the right, is that very hard to prove? Well, it would have been very hard for me if I didn't have the papers of Genevieve and Bel-Aish, because for me it was basically just kind of collecting ideas of them. Well, I'm thinking of the locally equivalent implying globally equivalent. Is that the proposition you've got there? Ah, the previous one. No, that's not so difficult. Does it work for other groups, other than PGLN? I've been looked at in that case. Yeah, I've tried, but I've failed, but I don't want to claim that it's hard. So maybe you just have to try a bit harder. So can you prove anything without this time there? You expect this one to be true without that? Yeah, so there are many problems, so we have been trying and each time we run into problems. Also, one of the annoying things is, so we have GSP to end over this totally real field, right? Actually, the Shimura variety we use is not for this group. We have to pass through some inner form. You really want that this inner form is compact at all but one of the infinite places, right? Because if you do more, then you do not get the spin representation, but some tensor product. So this means that you want all infinite places but one to be compact, which actually means that if the dimension of F is even over Q, you have to introduce some finite place where the group is also non-split. And this basically means that you cannot see all the automorphic representations appear. So right now for us this wasn't the problem because we had the Steinberg place anyway. So then we can put the non-trivial thing precisely at the Steinberg component, but if you try to remove it, then you also get problems there. I'm sorry, it wasn't that proposition, it was the lemma that you proved about two... Your remarks were applying to the lemma. Yeah, yeah, not a rigid thing, just the elementary, yeah. Could you replace Steinberg by square to global? It has to transfer to GLM to something that's square to global. Yes, so you want at least something that comes from the non-trivial inner form? Yeah, but yeah, then I have to think very carefully. Maybe you can do something, but... Can you remove that parity condition by using an endoscopic lift to some larger group? Yeah, so with Sargu, we have tried many of these punctualities, but so far we didn't find anything that helped. And also in this Eigen variety argument, I had that F over Q. So in the paper we always have that this F over Q is of even dimension, and this is for two reasons. So one reason is this inner form, so hopefully that can be removed, but the other reason is also when VST divides L, to kind of deal with the technical problems that arise then, we need that this dimension of F over Q is even. So it seems like for the Eigen variety argument you want F to be even dimensional, but then for the first part about the Shimura variety you want F to be odd dimensional. So they are kind of contradicting each other. And do you have an idea what to do to make the ropy unique? Because the ropy is not unique as directed at here. Yeah, so this is not unique. So you can put some very weak... So for instance it would be unique if you put spin regular, but you can also put much weaker regularity assumption. But in general, without this assumption? So without this Steinberg assumption. So if you do not assume Steinberg, then I don't know. But if you have this regular unipotent element in the image, then it's enough to have some very weak regularity condition.