 So let me start to set up, so the set up is, oh, actually the title of the talk was about GL2 times GL3, Rock and Silver. So after I send my abstract to the conference organizers, so we start to work harder and harder, turns out during the two-month phase, we actually put the result from GL1 times GL plus one, for all of it, and several more days. So I set up, as I'm following, I have to work with the same extension, so f over zero, same extension of total real field with Galois loop, rated by C, so it's non-trivial evolution. And we'll consider, it has to be the alphamorphism of contagion, higher, it has to be the alphamorphic, like a reducible of GLN for the field f, and the same thing that we have GL plus one, and we assume this is a cohomological for the trivial coefficient, so it shows up in the cohomology, on a certain smaller variety attached to the unit, so this is also a polymerized goal, it's conjugate self-heal, part of conjugate self-heal, it's conflagrarian, it's isomorphic please, sorry, that's definitely true, so the compass can't repeat the evolution, Galois evolution, so conjugate self-healology, and we write E as a field of definition, so the head field of cohesion of opion, so actually I will bring a pair, for the moment I don't have to, but later on I will bring a pair of contagions for GLN and GL plus one respectively, so for the moment I will just work with one, so from here, by the work of many people starting from cottonweeds, Coaseo and Harris Kenner, Coaseo, Hsu, Hsu, Hsu, Hsu, Hsu, Hsu, Hsu, Hsu, Hsu, Hsu, so what kind of touch Galois representation, so for each prime, for each place E over a prime L which will be fixed throughout this book, so such a place you can touch Galois representation or raw pylons, raw lambda, so this is a representation, continuous representation of Galois on the algebraic corner of F over F, that goes to GLN, N-dimensional representation of ARD field, E lambda, so it's satisfied my local global compatibility, particularly in those good places, it's restriction to a decomposition group shouldn't be aromified, and it's the same as simplification should be given by the local, local arm and sub-hacky parameter over the recommendation pi at those places, so our, so I have to tell you, there's some certain normalization, this one, so raw and lambda, you take this confidence, if you contribute by this Galois evolution, isomorphic to its view, up to 22, so the twist, which shows is one minus N, so the weight of this representation is N minus one. In my situation, this representation shows up in the cobalmology of Shimura variety, in such a degree of cobalmology. Okay, so now I'm done, we're up here, as I said earlier, so we're all, so from pi N, so pi being a pair of pi N and pi on the past one, so the same construction works for the past one, and I get a pair of Galois conclusions, so now I have to passively take a larger peak field, so I get raw and lambda and raw and lambda. So with such data, so let me recall you the so-called, so the broadcast of Shimura group, recall the definition of Galois representation, so let me write as following, it's kind of V, so V then we'll get a tensor product, so V or E lambda is a Galois representation, continuous, Shimura group is defined as following, you write on each one F, so it can be V, this is defined as the kernel, so I restrict it, first of all, but you impose, so this is parallel to the definition of a similar group for a particular curve, for example, so this is product over all the places, and locally, so I have to consider the local Galois formula, and I have to define the local, so the so-called each one F, the local guy, so the key is I have to tell you what this guy is, this guy is defined, so in the case when the place V has characteristic, different from my coefficient L, easy to define when V is away from L, you find this has the so-called aromethyl, called aromage, then we need any of the kernel, just write it aromethyl, namely those which can be trivialized after base change to the maximum aromethyl extension, so the second line, in the case when V is above L, this is slightly more complicated to be used, so you find this as those classes which become trivialized after you extend the coefficient to decrease, so the function here is written, okay, so that's the definition of the rule of account of several rules, it's vector space will find a dimension over the coefficient of the work condition, so we apply this situation to, we apply this definition to the case, so V will be the tensor product, we apply this situation, okay, so that's the same rule, and you can, now let me just recall you that the conjecture of, a special case of a conjecture of Bennington rule of account, following for any place, lambda above L and pi V as the product as above, so we can look at the Ronkin-Several L function, so the L function pi and Ronkin-Several convolution with pi plus one, so here we're normalizing, so its center is S equal to half, look at the order of vanishing, this predicts to be equal to dimension of the rule of account of several rules, V, Ron, and Ron plus one, so it's a test by N, so I have a weight minus one, so this is a special case of Bennington conjecture, generalizing the brushes on direct conjecture to higher dimensional special case, okay, so here I should emphasize that our orientation is assumed to be called homological for the trivial thing, so it doesn't, for the moment, we haven't worked on the more higher weight situation, so I should, now maybe, well maybe I should just point out this conjecture of raw account is more general than what I stated, it's supposed to be correct with, so there are conjectures about arbitrary geometric galore orientation, or, I mean, motive, but so the point here is you can formulate this conjecture without reference to, you know, like, the actual motivic realization, so it's more flexible to just work with the galore orientation, or the compatible family of galore orientation, as long as you can attach galore orientation, and the galore function can be understood, in this case, it's a half morphic method, so maybe since we have a very broad audience here, I also just want to mention not, so a formal direction for this conjecture, if we're now sufficient, if we have sufficient knowledge about the raw account of conjecture, like I said earlier, it's about any motive, so any way it doesn't have to be to a weight of minus one, so here we're looking at a violet at the center, so the violet at off center is even easier to understand, for example, you know it's down to zero, if you are in a range of absolute convergence, so particularly if you, so let me just point out one, just one remark, which came up in the past semester stage seminar, so where I learned the following, if you assume B is a product of H1 of a certain curve, if you take any curve over a number of years, so if you look at the first call point, so you take tensor product, so N times, okay, different N, and twist by one, so the weight is going to be positive if N is bigger than two, or non-negative if N is bigger than equal to two, so by the trade-off case of raw account of conjecture, what I would expect the similar curve to be, but for this motive, it's supposed to be trade-off with raw account of conjecture, would imply this is trade-off, and actually the so-called non-billion version of Chibaldi, of Mionkin, would tell you that, if you assume this is true for, let's write it as Vn or n bigger than equal to two, if you assume this, you can actually prove, you can give a new proof of model conjecture, so the non-billion, the non-billion Chibaldi, it's going to tell you that it's going to curve some model conjecture for C. So, as I said, this is completely a follow-up to what we're doing, we're working with the central value, and this is actually for, it's a point where the L function will be, it's supposed to be converging, absolutely. So now let me state our result. So for that, I have to give a very technical definition, which is hidden here. So I have to impose, so I have to impose this, I have to choose a place L, so this is my definition. So this definition involves, so for the given time plus one, I have to assume this prime L is placed lambda, satisfied with all the conditions. Can you raise the board a little bit? Yeah, I will. It will shadow. It will shadow, so I will go over here. But for people who are familiar with the story of OER system, for the case of e-litric curve, when one applies those arguments of Koliwagen OER system, you are, in many cases, you have to impose conditions on the galore ventilation, especially on the residue galore ventilation. So here I'm imposing various conditions, but you should not worry about those conditions for the moment, because unless you know the proof, you wouldn't even care about those conditions. So it suffices to say that those conditions are, they came up in the process of a proof, and they are supposed to be harmless, if we know enough about the residue ventilation for a higher dilution of galore ventilation, which unfortunately we don't have too much information. So let me just quickly say that I want this prime to be big enough, and I want the residue to be absolutely irreducible, and I want the transfer product of the residue ventilation to contain certain non-trader scanners, and those conditions probably are very familiar to Barry. And so there is a condition, it's ideal being isolated, this is just because we're, for the moment, we haven't spent much time to improve this, this probably can be removed. So basically you want this galore, this eigenvalues, eigenvalues attached to, so the eigencharacter of the HECA algebra, for certain sphere of HECA algebra, so the eigencharacter attached to a given out of the repetition of this one to be, does not have any non-trader congruence. So there's only one lifting from the residue one to the constant zero. So, and more seriously are those local conditions, but. So the condition, every one there is for pi, for, for both, for both? For, for pi, for both, for both. So you can formulate a condition for each individual, but here it's for the pair, for both, so for pi. So you start with this pair of out of the repetition, so then you impose a condition on the attached galore. Okay. It's probably better if I give some example later where those can be, can be actually verified. So they, they don't give you an empty set of examples. So are we, are we seeing any more of those? Okay, I can say for one of the last conditions, so the, so the generic, the composable, of certain split places is to, in order to apply certain torsion freemies over carriani shots, and the cohomology, or whatever, right? So, so the service commission five is to apply certain polylogs, or a system type argument, or actually more precisely, it's the version where Borboni-Darone uses to prove certain run-zero case of Borgschisman-Dare conjecture. So we'll see those later on, those conditions. So this one, number five is something, so this condition we'll see, and we'll comment later. So for one of the, so I don't know those but the hypothesis I can state already now is, which says here we can prove is that for such pi, so this such place, well, it's customary to call such place admissible, but it's just to abuse the name, so it's admissible. So this is admissible, then, so then this conjecture holds in the case of run-zero, so it's L function. Has run-zero, so then the same number is zero. So that's the results we can put. So before giving examples, let me just give some remark related. So here, this is about the case of run-zero. So we're starting from the case the L function has order zero. So we do have a result for run-one, but it's much more, it's sort of less satisfactory, is a result, in the case of all of L function is one. Well, in a sense, you have to assume certain, so there's some algebra cycles coming from the ganglors-prasad conjecture, or arithmetic version of ganglors-prasad conjecture. So if I assume this algebra cycle doesn't vanish in the first component, so assuming, so you need to condition that for the GDP, so the ganglors-prasads, the ganglors-prasad algebra cycles, the diagonal cycles, so-called. So if this cycle has non-zero contribution to the similar group, so in that case, you can also prove, so under this stronger, well, under this condition that this cycle class doesn't vanish, then you can prove the run of the same one is one. So like, just like the case of growth-zagier-clean-wagon situation. So that's the case of run-one. I should also see that this run-one result, so there's also independent, a different prove for the run-one case by a injection and a scanner. So we have more in progress to prove the run-one case, mainly to assume that the cycle class doesn't vanish when you prove the run is actually one. There, we really construct, by constructing an anti-psychotomy system. As I will describe later, our group doesn't really construct a system in the sense of a polyblogging. Instead, we have a system in the sense of a polyblogging-down-row, so we use congruence, so we change tomorrow, right? So they only use one particular one, right? Without changes, we won't write it. Can you out there in the ranks here, at least? Do you also get this mutually? Which one do you mean? If you need one there, in the ranks here. Oh, do you get a YouTube version? For live.p or other stuff? Actually, no for the moment. So we really work with currency zero, quite a few. So, in other words, for example, we don't, actually we don't prove, at least we don't know yet, if you do a finite equation. You're supposed to get a zero similar group when the prime is dark enough. We actually don't yet. We haven't checked carefully, but it seems to be the case we don't get it very much. So that's some problem here, but we're happy enough to get this result. So now let me give examples of regarding those conditions. I have to tell you about this. You do have some long, long, dump-tier examples. So the examples that we've discussed come from in your course. So recall that that's very useful trick by taking product. So if you have one curve, you can take product. So it is very productive to take product. Like we know that the working syndrome is kind of a product where we use, you can prove it with a V-conjecture, or V to use it to prove the Riemann hypothesis for curve over a finite field. Or like in a subtle take case, you take the product over a little curve, this page. So here we're gonna take a product over a little curve, give examples, so let's take a little curve. So A and B, assume they're modular, so assume they're modular in the curve. So over F0, which is my total real field, is a total real field, so I can take a product. Well, the best thing I hope to do is to understand all the sort of the motive of this product, so A to M power, B to M power, so this seems to be very hard. So what we are really interested is a certain special case. Maybe we consider the motive of the symmetry of power, each one A, so to the N power, then the symmetry of power of each one will be M plus one. So assume A and B don't have complex multiplication, and also assume A and B are not geometric isogeny, isogen is to each other. Then we consider, so this will be, and assume this is modular, so assume it's in power. So we know this, we know some examples of modularity for symmetric power when the exponent is not too big, assume. So this is known for N, at least. Okay, so for N, it's most three, I mean they get ready this way, M minus one. And so that this is really our geo, geo plus one. Chris Barney's algorithm contains our geo plus one, so that it's known to be modular for small N, and for maybe for slightly bigger N by the work of, recent work of Clasio and the Chuck Forum. So we have such examples of modularity for symmetry of power. And those, if you take a base change, so this will give you example of pi N and pi N plus one, attached to the base change, so take a base change in those guys, base change, M minus one. And under those conditions that we don't have complex multiplication, and you're not discharging it to each other, you can, so you can choose. So when L is large enough, it's large, you can, okay, admissible in a sense, in a sense, I'll just be fine on the top. So you can, you're able to choose large prime, the large region of characteristic L, so the residue of our completion, we'll provide those, we'll satisfy those, all those conditions. So I should comment that, ideally one would like to study all possible, all possible pieces of modem appearing in this product. This amounts to study all possible symmetry of power of one of those two guys times the symmetry of power, not necessarily N plus one, but arbitrary pair. So ideally one would like to study all possible running zero instead of just GOM times GOM plus one. But for the moment, we're only able to do, we're constructing for GOM and GOM plus one. Okay, so that's the result I want to describe, then I want to describe some ingredients of the proof. It's following, so I know they all have done the vanish and on my out-of-of-repentation they all, they can all be descended, they can descend to unitary group. So I will consider a unitary group. So the condition on order of L function is zero, so that I can, first of all, I can descend the derivative into a unitary group and by the global ganglion process. Okay, I should say that here I'm also relying on, implicitly I didn't mention, I'm also relying on certain improvement version of ganglion process, which some part was still in progress of work of people in transformer, but I will omit in this talk. So this tells you that you can find a definite unitary group in a heart meeting places they are all compact, which is some form of UN unitary group M variable, M pass Y variable, such that, so the so-called ganglion process, here is integral doesn't vanish, I have a sub-group, so I have only very smaller unitary group, such that here is integral doesn't vanish. So this is the starting point. So you get at least construction. So then, you want to construct a system of called model class, which will be used to bound the size of several groups. So you want to start from here, you produce a system of called model class, see class index by, so now my notation might be a little weird, so power will be transferred by, or maybe, if it be V, so power will be transferred by integer, so which will be used to do reduction of my galore notation module of M power of L, so I will do module of M power of the galore notation. So we will take reduction module of this power, and V is in the north place, so it's like a refining version of this condition star. So from this condition, you can, of course, you can choose even many such programs, instead of just one by Chebotaric density. So those classes, they will be in H1 over F, and the rule is the test of product, module of M power. And those classes, so the localization, the local information of V, the same place of V, is controlled by, so the localization is controlled by, by the same, by this period region. So you know the local information at this place by certain so-called reciprocity law, so you know those localization in terms of the non-management of the authors. So particularly, you know those classes will get down to zero if M is big enough. So this will be the system of called most class, which we use to bound to the side of the same old group, and I should tell you where do they come from, where do those classes come from? So they come from, so here I'm starting with a definite unitary group, and one can get a nearby unitary group by changing the signature arc meaning place and changing the item of the class of the group at the place V at the same time. So given G, this is a definite unitary group, and given a unitary place V, so I can choose, I can find a nearby, nearby unitary sense, it's attached to the formation space, which has signature, group G, V. This is attached to, so this is a signature, so you change signature of the one arc meaning place into one M minus one arc meaning place of one place, and V, so very inert, and another group G, GV, G values, is unreliable. Where you choose such as the new group G, GV, and V is not isomorphic to G, oh it's not, right? So initially you can produce a similar variety attached to, well okay I'm describing really for each factor that you do the product, so you can attach a similar variety, so here there's certain technique you should buy, where you have to replace the group unitary group by the semi, the unitary group with the semi-two factors, so it's more of a variety case. So as in a GTP, a risk inversion in GTP, and here you have the diagonal cycles, so remember everything can be, should be really carried out for this subgroup H, which is definitely here you replace by, but nearby and so on, so you get an algebra cycle, this subgroup variety will give you algebra cycle, and this will provide, so from here, so I have a similar variety attached to subgroup, changing the class of V to a certain level structure, and I have to do a similar variety to G, changing at a V, so every time if you pick a place of V, you get this construction, and here, so the group at V is not harmified, it's attached to a Hermitian space, so locally at V it's attached to a Hermitian space, which does not have a self-do line, so what you do is you take a, as best as you can, so you take a line which is nearly self-do, so this guy will have battery attached to V, so we have integral model, and this integral model, which are regular and same as table reduction, table reduction, so if you think of a special case of G or two, a unit of two variable, so you would actually get the case of Dernfeld, so you would get the same as table reduction where the special fibers are two families of projective lines, but this is a high dimensional origin, and unfortunately it's not so straightforward to describe the geometry of the special fiber, unless A is two is a restation, so if A is bigger than four, bigger than two, so the only case which is easy to describe is actually the case of modular curve, this is a regular curve, for higher dimensional case it's nothing more involved, so now I'm done with those classes, so we come from this other cycle, so in the end I want to, so you have to study, so the difficulty part is to study so the localization of those converging class and to make a connection to those, to the ultimate appearance integral, on a definite unitary group, so I'm gonna describe one case, so you have to do this for each, so you have to do some work for each factor of the tuition rule of writing, so I'm taking a product of the tuition rule of writing, so I have to study certain level reason phenomenon, so you have to realize the level reason, you know, drop it to three, so you have to realize the level reason in terms of those special algebra cycles, so let me just give one case, as I said I have to do both for both factors, for both tuition rule of writing, but I can do this, I can describe this for one, for one of them, so let me recall you those tuition rule of writing, so I should say that here my remark, the first remark, in the case of rank one, so where are you, we start with a indefinite unitary group, then you switch to a nearby line which is definite, so it's a certain switch of sign or fucking equation, so I'm gonna describe a symbol case, which is actually not actually used, in this process, it's not used directly for this construction, but it's parallel, so the phenomenon is very close, so let me describe a case where, so when the tuition rule, that's a good reduction, so maybe this might sound confusing, because here I said this has a better reduction, and the same as table, and the better reduction, V, but I would describe the parallel situation, so now let me recall you that my tuition rule of writing, so now let's just consider one factor, h to unitary group of unvariable for n, and I choose V to be a inert place, so I have a level, k is a compact open, so I'm changing notation now, my g is really a gv, okay, and it's a different v, so let me consider it, so we're right h to u, u1 n minus one, and with a level, k, so we're k, so g is now the indefinite one, so I'm changing notation a little bit, find out there was some choosing a compact open, so that kv is actually hyper special, so I'm looking at a good reduction case, okay, so this guy has, well again, I'm really going to say gu, gu, and I have a PL type modular variable, which gives me a good integral model over oe, certain place above v, it's both model, so now you look at the special fiber, and you look at the super single locus, the super single locus, it's over the residue field, so this is an algebra closure, so this has a nice description in terms of the Boolean-Hartist theory of this unit-herb group, so roughly speaking, this is a distant unit of certain, certain delinistic varieties, DL variety, so the point, the key point is, those varieties, the power we tried to buy, are definitely in her group, so you shouldn't think of the example which showed up in yesterday's, in the talk of Akshay yesterday, where if you think of the Shimura kernel case, the super single points are parametrized by, but they're definitely paternal algebra, so this is a high-dimensional situation, so you have definitely a unit-herb group, so parametrized irreducible components of the super single locus, so the key point is the dimension is exactly below half, so the dimension is two to the n minus one, if you choose the floor function, well, so it's n is even, so it's really, so let me just write as, so I'm using the fact that it's even, if n is odd, if n is even, you just get n half minus one, and this total, the special fiber, the special fiber, special fiber half dimension, n minus one, so it's exactly, well, this is odd integer, so it does not have half integer, I mean it does, it's half is not integer, so this is exactly the close possibility to the half dimension, so now what do you do? So you have those odd-year cycles, so I can consider the cycle class map, so I consider all of those, I consider above-gear problem map, so attach to this place, so this is the, you look at special fiber, and you look at odd-year cycles, support it on the super-single locals, so let me just write as super-single locals, the dimension is n divided into minus one, so you take a localization, so this is a HEC module, the HEC algebra, away from this place, still acting, HEC action, so away from, for HEC algebra, away from this place, so you localize, write some maximal ideals, let's go to each one of the middle dimension of the homologation of the special fiber, okay I'm ignoring this page shift, also you localize this maximal ideal, so the conjecture, which we call hard Iharo, hard version of Iharo, is that, this is actually subjective, if this is subjective, if m is as in this definition, is admissible, so here, so my admissibility condition was for this pair, you can do the same definition by forgetting one factor, so what do you get, will be the definition of admissibility, so key fact is that, so here you only consider this condition of, where as even, so you get this condition where, the pairwise, so the Sataki parameter in this place, so the pairwise ratio will be distinct from QV, which is cardinality of this residue field, except for the one pair, if there's one pair, where this distinct pair, the ratio distinct pairs of the Sataki argument is equal to QV modulo, modulo L, such pair is unique, what's the point, it could be multiple pairs, but you want this to be unique, so then this would be subjective, so this is one, this is easy to state, but hard to prove, which we don't know how to prove, and there is a version where we use, it's hard to describe, but it's easy to prove, and we prove it, so I'm presenting a hard conjecture, but it's easier to state, so I should say that, you have to do a similar thing for two other situations, so this is in a torsion situation, because I'm looking at, really, something modulo L, and you also do, so you also have a similar version where A is R, where this guy will be exact half dimension, and the seminar, so if we change the situation to consider instead of a blue jack, you just look at the contribution of a cycle class in the space of Tate cycles, so where you have a similar conjecture, which is proved, which says that the cycle class of those irreducible components from super-singular lollies, they actually contribute to the Tate cycles in the middle dimension, generic, where generic meaning is exactly this, what I wrote here, so okay, here is for the case thing, so we have a switch and a plus one, so meaning there's only exactly one eigenvalue in the subhockey parameter, exactly one guy which give you a contribution of Tate cycles, so this was studied extensively by, by Yichao Tian and Liangxiao, and Liangxiao machine went through, so we have a full picture on those cycle class, and what I'm presenting here is a version which is some sort of torsion version, where you consider instead of cycle class one, you consider a blue jack, hope you know, and then there's a version, both version for good reduction, so there's a third question which is for the semi-stable reduction which is related to what I needed over there, so you can imagine there's a version for that, but with value that, so in some sense you can reconstruct everything if you know the principle, every case is different, you have to do different one, and what we were able to pull is the other two cases, so we call it easy yihara one, so I'm presenting a hard one for you, so maybe you can inspire people who can pull it, so those are like the ingredients in the pool, I can stop here.