 So welcome to Paris Peking Tokyo Seminar. So it's my great pleasure to introduce the speaker today, Yixing Liu from Yale University. So he will tell us on the very some block of conjecture for ranking self ag motives. So please start. Okay, thank you very much for the introduction and the invitation. It's probably one of the largest seminar I have ever given more than 100 people. So I just realized that two years ago, Yi Chaotian also give a talk in the seminar on a similar, on the same topic, but that time was during the middle of our project. So the results at that time was not the final one. So today I have many two girls, one is to state the main results. I mean, our final results obtained from this long-term project. And then I will discuss the main strategy. And some of it, there might be a little bit overlap with each house talk, especially on the arithmetic geometry of certain shimmer variety. Then I will emphasize a new technique. I mean, in the study of Selma group using defamation theory of Galois plantation. So I will start from the main results. Okay, so this talk is based on the joint work with Yi Chaotian and Liang Xiao, who is now in picking and Wei Zhang and the Xing Wenzhu. And we started the liking 2017, I think, and finished at the end of last year. So some notation. So we will fix a CMS tension of number fields like F over F plus, we see the complex conjugation. And then we denote by gamma F, the absolute Galois group of the imaginary field F. So here's our first theorem. So our first data theorem that is more elementary to state. So let N be an integer that is at least the two. And we consider two modular epicurves over the total real field F plus A and A prime such that they are geometrically simple. So there is no CM multiplication. And some additional conditions. So we suppose that first, they are not geometrically isogenous to each other. So we consider the M minus one symmetry power of the first one and N's symmetry power of the second one. We assume they are both modular. And we also assume another technical condition is that we assume the F plus is not Q, if N is at least the three. Okay, so I will explain why we need this, I mean, little bit later, but let me finish the statement first. Then the statement is the following. So suppose the central critical value, which are writing a classical way, okay? In a classical way, N is the center of this function. So in particular, if you allow yourself to take N to be one, which I didn't, but if you allow, then this is just simply the central value of the L function of the base change of the second in the curve A prime, okay? So in general, we consider a more, I mean, more complicated L function like this. If the central value does not vanish, then the building some lower color conjecture, which is the generalization of BSD, sorry, predicts that the Selma group should vanish, okay? Then the conclusion is that the blockado Selma group, which I will recall a little bit later, of this galore plantation, okay? So this twist N is somehow echoes this N here, okay? So this is the correct twist of this galore plantation, vanishes, so this Selma group vanishes for all, but finally many rational prime L. So conjecturally, it should vanish for all L, but due to some technical complication, we only prove for all, but finally many L. By the way, this all, but finally many is effective, okay? It's not L, I mean, like some large enough, but there's a way to say how large enough. Oops, sorry. So before continuing, let me give the, recall the definition of blockado Selma group. So in this case, actually, the definition is quite easy. So for galore plantation of F, so gamma F to GLV on a finite dimensional vector space V over a finite tension of QL, I mean, here it's just QL, okay? We define this H1F, this subscript F stands for finite, okay? To be the subspace of all the first cohomology of this galore plantation consisting of those classes, whose localization belongs to the local version of this finite part for every non-archimedean place V of F. So what is this local version? So when L does not divide V, it is just that unremifed subgroup of this H1 of this local galore cohomology, which everybody knows. But when L divides V, you need a little bit, so L, I mean, actually this should be shi-addigo-Hodges theory, but since we consider L, it's the L-addigo-Hodges theory. So let me just, I mean, for those who don't know this definition, I would just give one interpretation. When V is a crystalline at this place V, then we know that if you ignore this F, then H1 parameterizes extension of this V by the trivial representation. And this H1F denotes the subspace of H1 consisting of those extensions that are still crystalline. Okay, so that's one interpretation. You can just define this in that way when V is crystalline. But in general, you need a little bit more on the L-addigo-Hodges theory. So that's the definition of block-hadocellular group. And there's a close relation of this fancier definition with the classical definition of the Selma group for an Idb curve, as this is actually the original motivation for block-hadocellular to give this more general definition. Any questions so far? Good, so. Yeah, it's fine. Yeah, so this is our first theorem which is stated in terms of symmetric power of Idb curves. So you may ask, I mean, so when we say this is a unconceivable motive, do we only obtain like things must be like symmetric power? The answer is no. We have another set of theorem which applies to more abstract automorphic representation. Now I will try to state. So since we are going to do automorphic representation, let me give the definition of those automorphic representation we are gonna consider. We call this representation relevant because they are relevant to our work. So there are three of them. So first, we require pi as the irreducible hospital automorphic representation. Second, we require pi to be conjugate self-dual. So if you compose pi with the conjugation of F, it should as equivalent to the contragradient of pi. So the third condition is pi has certain homological weight that is a minimal. So more precisely it says the following. So for every Archimedean place tau of F, this pi tau is the principal series of this N characters. So what are these N characters? These are just, this ARG means the argument to character, just the argument of a complex number which you bifomulate the z over square root of z bar. Okay, so this is the generalization of the weight two case in the classical for the classical modular form. So for those who are familiar with this Archimedean theory, this representation, so this is the admissible representation of GLNC. And this representation is exactly the standard base change of the trivial character of the definite unitary group of rank N, okay? So before we will only consider this kind of automatic representation. So note that for a coefficient field E of this kind of automatic representation and every finite place lambda of E we may attach a Galois representation for the row pi lambda. I mean, actually this is a compatible system of Galois representation from gamma F to GL and E lambda. This is achieved by a series of work I mean, for many years when Harris Taylor, Subushin and the finally, but shouldn't be Harris. And so this is some kind of common knowledge now. So you may attach Galois representation for this kind of automatic representation, okay? So now I will introduce an elementary but very important notation in this talk. So in what follows, we will consider an integer N which is at least two and we will denote by N zero and N one, the unique even and all the numbers in N and N plus one. So sometimes N zero is N plus one and sometimes N one is N plus one depending on the parity of N. So this is a, and we also denote by R alpha which is the floor of half of N alpha if you write in formulas this way, okay? So these are stupid definition but they are very important notation, okay? Because for later argument, the most important thing is not to which one is larger. It is which one is even and which one is all then. So, oops. Now we also give another technical definition. You don't need to memorize this definition, okay? So it's a technical one. We will say that a special, we will call a prime P of F plus is a special inert prime if it is of degree one over Q and inert in F and whose underlying rational prime P is and ramified in F, F is the CM field. So actually the only thing important is this word inert, okay? And this degree one is totally just technical. It's not necessary. And this underlying prime P is ramified in F which simply means P is large enough, okay? So the only thing keeping your mind is the word inert and this is already reflected in this terminology, okay? So the important thing is this P is inert in F. Now I can state the theorem in the abstract version. So we consider two relevant to the plantations I call this pi zero pi one of GLN zero AF and the GLN one AF respectively, okay? And I let E be a number field that is a coefficient field for both pi zero and the pi one. Again, there are some conditions. So first suppose that there exists a, I mean, you can ignore this very, doesn't matter. This is a special inert prime P of F plus such that the even rank one, the even rank one is Steinberg and that all the rank one at P is ramified who the Satake parameter contains one exactly one. So let me explain a little bit about this condition. So since pi one is conjugate self-dual. So it's Satake parameter is given by N1 complex numbers, invertible complex numbers that appear in pair. So it's alpha, alpha inverse, beta, beta inverse, gamma, gamma inverse. And since N1 is all the rank, so one must appear in the Satake parameter. So this condition is a genericity condition, which means that it exactly contains one once, okay? It should not contain three one or five one, just exactly once, okay? And the second condition is easier, which means that for each of them, pi zero and the pi one, it is super-cospital somewhere, okay? So for alpha is zero and one, there exists a non-archimedian place W alpha of F such that pi alpha at W alpha is a super-cospital. Condition C is same as before. We require F plus not to be Q, if N is at least the three. So the conclusion is same as before. So if the central critical value, which I write in the automorphic way, which means half is the center, is not zero, then for all but finitely many primes lambda of E, the block-cuddle-selmo group, H1F with the corrected twist vanishes, okay? So let me explain a little bit why we needed these three conditions, okay? So Y, A, B, and C. So A is needed really due to our method, okay? So the main method is so-called arithmetic level raising. So A is needed for the existence of so-called a level raising prime, which we will go with more details later. So namely, we needed those primes at which the Frobenius acts on this color representation via a special way modulo of various lambda, okay? So I'm vague now, but I will be more precise later, okay? So I mean, as already you can see, this condition must specify some particular elements in this color representation, right? And how about the B? Okay, B is easier. I mean, B is needed so that these two color representation are absolutely irreducible modulo, all but finitely many lambda, okay? So due to our technical reason, we need residual absoluteness for these two color representation. But just like, let me give a remark. So I mean, conjecturally, conjecturally B should, this sentence should automatically be satisfied even without this condition. But our current knowledge doesn't give that, okay? So we put a condition to make sure this is known under condition B. I mean, B is not really needed. It is always should be true that it is absolutely irreducible modulo, all but finitely lambda. Okay, for those experts, let me just give another important remark. You may feel that this condition might be used for the gangloss-proscillic conjecture, but actually no, okay? We have already removed this condition when we use this gangloss-proscillic conjecture, which I will say to later, okay? So this is not for GDP conjecture. This is only to ensure that the color representation is absolutely irreducible, residual absolutely irreducible for all but finitely lambda, okay? So C, so how about C? You see this condition C also in the previous theorem for NP curves. So C is needed because two ingredients we need are not available where F is Q and N is at least the three at this moment. So the first is the cohomology of the stable part of the unitary Schmure variety, which is not compact when F is Q. And somehow I cannot find the literature that is responsible for this cohomology at this moment. And the second, we need a result by Karani Schultzer on this vanishing of non-middle degree cohomology that this recent work. And some of you may know that recently Karani Schultzer extended their result to non-compact Schmure variety as well. However, they still assume F is not Q, okay? Even for their recent result. So when F is Q, there were two difficulties for all these kinds of results. First is that the involving Schmure varieties are not compact anymore. That's the first difficulty. The second is difficulties that the appearance of so-called a Cuspid subgroup. That is something complicated during this stabilization of trace formula. And it only appears in the unitary group case when F is Q, okay? So I mean, C is still some technical issue. So as long as these two things are clear, the one F is Q, then our theorems can automatically extend by removing C. In fact, when we write our paper, we just assume these two things and go through the whole process without assuming C, okay? And the questions for the main results. Okay, good. So let me give a remark, okay? Let me give a remark. We also obtain results. So you can see this is, all these results are in the rank zero case. So L value, the order of vanishing of L value is zero implies the Selma group has rank zero. So you may also ask, can we obtain something in the rank one case as Colivagan did? The answer is partially yes. We do have a partial result toward the rank one case, but I'm not gonna state because first it's technical, second, it's just a partial result, okay? So I'm not gonna state that, okay? So now I'm gonna explain some main steps. So, yeah. So first step is how to use this condition. How to use this condition. This is really some analytical condition, right? I mean, how to really use this analytic condition to obtain some arithmetic information. So this is the GGP conjecture, which is our now state first. So step one is to use Gangl's Prasad to relate this L value to certain period integral on definite Shimura set. So let's start from this condition. The central value does not vanish. So there we can obtain the following conclusion. The conclusion is quite long, okay? So, but let's do this slowly. So first we can find a unique pair of totally positive definite Hermitian space, VN and the VN plus one over F in which VN has dimension N and the VN plus one is simply VN direct sum with a unit line where E has norm one. Okay, that's the first thing. The second thing is that for alpha equals zero and one, we can find actually unique irreducible subrepentation pi alpha of this adelic unitary group contained in the, in this space of automorphic functions on UV alpha, such that, I mean, this is hospital automatically because it's definitely unitary group, okay? Such that the base change of pi alpha is the bigger pi alpha. Okay? So there's that we can find the OE value of the functions F zero in pi zero. I mean, by construction, this representation automatically realized on the space of automorphic functions. We can find the functions F zero in the first representation of F one in the pi one such that the diagonal integration is non-zero. So this is a period integral. I mean, I'm writing this in a fancy way as an integral, but in fact it's just a finite sum because as long as you fix an open complex subgroup, this double quotient is a finite set. Okay, so this is in fact a finite sum but I'm just writing it in a fancy way. Okay, that's the whole, this is the content of the Genghis-Prasad conjecture at least one direction. This is the content of the conjecture and this is known now, okay? Recently known now. So let me explain a little bit of history. So this result, which I mean this one follows from a serious work by originally the beginning by Jackie Wallace who proposed a relative trace formula approach to this conjecture. And then Zhiyue Yun established the fundamental lemma toward this trace formula. And by Wei Zhang who solved this smooth matching problem at non-archimedian places. And then by Hang Xue who solved the smooth matching for archimedian places. And I mean, there's a, and also by Vaspu Jia and the local aspect of this conjecture. And on Buzha Plessis on very important refinement and Buzha Plessis and Zhiyue and Zhiyue and Zhiyue are very important refinement on Wei Zhang's work. And up to this point, we know this conjecture but we have to assume that both Pi zero and the Pi one are super-cospital somewhere. And this is one of the most stubborn condition in all of this the trace formula stuff. Okay, it's because at some point we want to use simple trace formula on the spectral side. So we have to assume the automorphic representation is super-cospital somewhere. But until very recently, by our recent joint work with Buzha Plessis myself and Wei Zhang and Xing Wenzhu, we discovered a new technique that can remove this condition. There's a restriction that the representation is super-cospital somewhere. So we now have this full version of this theorem. So this is now unconditionally known. Okay, so that's the content of the step one, this GDP. So we have a start from the non-vanish of the L function and we obtain some period sum or period integral is non-vanishing. So the next question will naturally be how to use this condition, okay? How to use this condition, okay? So let me put a little bit more condition because apparently we don't want to write a such long expression. So I will put a S, V and R for this quotient. So this S stands for set, okay? Because this is a Schmura set, so S is for set. So we also fix the choices of F0 and F1 as above. And we also fix an open-compact subgroup of this UNVF that fixes F alpha for alpha equals zero and one and we'll carry them implicitly in the notation. So we will not write any complex subgroup in the following discussion, but please remember there are fixed the open-compact subgroups. And we also fix the finite set sigma plus of primes F plus all the side of which everything is N-RMA file, okay? So basically sigma plus will contain all the ramification that appears up to this moment. And the question so far, now we will go to step two. Okay, great. So step two, this is the point where arithmetic geometry enters. So far there's no geometry, there's only a set. So this set will be related to some quite beautiful arithmetic geometry via some better reduction of some integral model of Schumer varieties. So I guess this part, each should have explained some of them two years ago, but I guess you should probably forget when they are. So let me recollect. Oh, it's two years ago. Yeah, yeah, I understand. I mean, so, okay. So I will just hopefully you can remember, you can recall some of them. So let me fix a special inner prime P of F plus, whose underlying rational prime P is called prime to sigma plus, okay? So if you forget what the special inner mean doesn't matter, this only means basically means P is inert in F and the underlying rational prime P is sufficiently large, okay? I mean, nothing else is important. So for alpha equals zero one, we will construct a strictly semi-stable quasi-projective scheme, I would denote by the bolder face which means the integral model. The bolder face MP, P states for the sub prime P and the V and alpha states for the corresponding Hermitian space over spec ZP square. I mean, in general, it should be FP, but since we assume P has a degree one over Q, so FP is just the ZP square, okay? Over spec ZP square of relative dimension N alpha minus one. Of course, I mean, satisfy some condition. Okay, so what's the property of this integral scheme? So I will summarize them as follows. So the generic fiber of this scheme is essentially just a Schmurr variety, but not of our original Hermitian space. Our original Hermitian space is totally definite. So the corresponding Schmurr variety is just a set. It has dimension zero, okay? So this Schmurr variety is attached to a somehow slightly different Hermitian space. We call this a nearby Hermitian space. This Hermitian space is unique up to isomorphism satisfying the following condition. So the signature, instead of being totally definite, this one actually has signature N alpha minus one one at some fixed Archimedean place tau of F plus, okay? So we switch the signature at one Archimedean place. And we also require it is same as our original Hermitian space for all other places other than this Archimedean place and our chosen special inner prime P. So by Hauser principle, this will force that V prime and V are different at the P, okay? Otherwise you only change the Hauser environment by one place, it's not gonna work. So we change the Hauser environment at two places. So you get this another Hermitian space. Remember, because we have already assumed this P is co-prime to Sigma plus, which means that the original Hermitian space should be unremifed at the P, which means it should admit a self-dual lattice at the P. Now, since V prime is different from V at P, this will imply that the V prime does not admit a self-dual lattice, okay? And at the level structure at the P, we will require it is a stabilizer of an almost a self-dual lattice. I mean, if this one does not admit a self-dual lattice, then it admit an almost self-dual lattice. And we require that level at P should be the stabilizer of an almost a self-dual lattice, in which case it is a special maximal subgroup, but not hyperspecial. Oh, sorry, sorry, it is special, okay, special maximum. Okay, this is the property of its generic fiber, okay? So how about special fiber? The special fiber, which I will denote by this Roman N, I mean, this is just the base changes from Cp square to Fp square. So this is a scheme defined over Fp square. And since our original scheme is strictly semi stable, this scheme is a normal crossing divisor of the integral scheme. So how does this normal crossing cross, okay? So this is a, what's the strata of this normal crossing divisor? So I mean, actually it's quite simple. It's not very, I mean, the strata only has two layers. So it is a union of, so the top dimension strata only have two kinds of them. The first one, we call this MP-circ and the second one, we call MP-bullet, okay? So in which the MP-circ is very easy. This is a PNF minus one vibration over this Schumerer set, okay? So this Schumerer set has a canonical structure. I mean, originally it's a set. A set can always be regarded as a scheme over Fp bar. And there is a canonical Fp square structure on this set. So in fact, you can regard this set as a zero dimensional scheme over Fp square. And this strata is a PNNF minus one vibration over this zero dimensional scheme. And this scheme is smooth. And this is smooth vibration over smooth scheme. So this is smooth. I mean, of course it should be smooth. Otherwise it's not strictly semi stable. So this is a smooth strata. I mean, and there's another one. And by the way, this one is not geometrically irreducible, okay? I just use a... I mean, by the way, both of them are not geometrically irreducible. I just put them as because due to the nature of their property, okay? And the second one, this bullet one is much more mysterious, okay? We only know it's smooth. We don't know much about the structure, okay? However, inside this scheme, you can consider so-called a basic locus, okay? So this basic locus, you can have a very nice description. And this basic locus, which I didn't define, but it doesn't really matter. I mean, the point is that inside this smooth scheme, there are some very important and meaningful cycles, okay? So this basic locus is a delinuistic variety vibration of dimension R alpha. So what is R alpha? If you remember, this is the flow of half of N alpha over, also over this set, but not the exact same, but I will cheat a little bit. You can ignore this essentially, okay? Also over this set. So this means that the basic locus, I mean, strictly speaking, should be the normalization of the basic locus. So there is a... So this smooth scheme, which has dimension N alpha minus one as well, contains a bunch of a family of very special cycles of dimension flow of half N alpha, parameterized by this set. And each of them is actually a delinuistic variety, okay? And what is the intersection? The intersection should also be smooth and off dimension N alpha minus two, right? I mean, this is the property of strictly semi stable scheme. So this intersection, you can characterize either as a divisor here or as a divisor here, but since this guy is so mysterious, we don't know it's the characterization of itself. So we can characterize this intersection as a divisor in this one. I mean, this is just a projected space or its divisor should be easy. So in fact, this is a Fermat hypersurface in this vibration, okay? So I think this gives a pretty clear, clean characterization of the structure of this special fiber of this integral scheme. And then the definition of this integral scheme actually use some modular interpretation of some abelian scheme with additional structure, okay? So this basic locus in black one meet the white one. Say that again, the basic locus. Yeah, basic locus in this black one meet the circle one. You mean this one? Yeah, yeah. So this basic locus, so you have basic locus in there. This meets another component. In N zero. Oh no, I mean the dimension is half of it, right? Yeah, yeah, yeah. So for example, when, so for example, let me give an example. Suppose we consider V four, right? V four, okay? So the dimension of the, this scheme is dimension three. So dimension three, we have two parts. The first part is a P three over a set. The second part is a three-fold. It's mysterious, we don't know what that is, okay? And inside this three-fold, we have a family of divisors because it's a fiber of dimension two, okay? A family of divisors, they are certain delinquent surfaces. Yeah. And they are over this set, okay? So that's the one example, okay? Thank you. Any other questions? I mean, since, if I give a blackboard talk, I will draw some picture, but now it's a little bit hard, so, okay? So let me continue. So I mean, the notation is gonna be a little bit slightly more complicated. Just interrupt me if you want me to go back, okay? So I try to keep it as simple as possible. So now at this moment, there's no representation, right? We are purely doing geometry at this moment. So how does this representation pi alpha come in? So for simplicity, I will just pretend that this coefficient just to be q, okay, otherwise there's too many notations. So I will just pretend that all this automorphic representation are defined over q. For example, if they correspond to symmetric power of elliptic curve, then they are defined over q. So in the below, you see this t, it always mean heck algebra. That's a common notation. So we have this abstract spherical heck algebra, which I denote by t alpha of this uv alpha away from p and the sigma plus, okay? And this is representation pi alpha because it's conjugate self-dual, it will give rise to a homomorphism pi alpha, sorry, phi alpha from this heck algebra to z. I mean, c is because we assume it's q, otherwise should it be oe, okay? To z, and for, I mean, this is just given by set up, I mean, the set up a parameter, symmetric polynomial of set up a parameter. So in particular for everywhere else, we can take this homomorphism modular L. So we denote by M L for L, the kernel of the composition of this homomorphism with the quotient to map c to f L. Which is a maximal ideal of t alpha, okay? So this is, so let me explain the logic of this notation. This M is a maximal ideal, alpha goes with the parity, okay? So if M zero, this means it's responsible for the representation whose rank has parity even, has even parity. And this L, which means modular L, okay? Okay, now, so our goal is to study the local Galois homology. Let me explain the thing a little bit here. So it looks complicated. So first you have the, so let's start from the she first. So this is the nearby cycle of coefficient, constant coefficient c alpha twisted by alpha, okay? So this is the nearby cycle for the integral model and the both of these MP. So the nearby cycle is a sheaf on the special fiber, sorry, the base change of the special fiber to the FP bar, okay? And this is the first layer, then this is the middle cohomology, middle cohomology of this nearby cycle sheaf. And then since this t alpha act on this integral model, this algebra acts entirely, acts on this cohomology, okay? So this is a C alpha module, and we have a HEC algebra acting on this C alpha module. So we can localize this T alpha module at this maximal ideal. So we get a localized T alpha module, which is here, okay? But it itself receives a Galois action of QP square, because the Galois group of QP square acts on this nearby cycle cohomology. And since the action of the Galois group and the HEC algebra commutes with each other, so this entire localized HEC module will receive a Galois action of QP square. So it makes sense to take the first cohomology, okay? So I hope this explains this long expression. Okay, so I think it's clear, okay? So our goal is to study this one, okay? So study what? Study what about this one, right? I mean, what's the actual goal of this one? So, I mean, at front this point, it is really different when alpha is zero and alpha is one, okay? I mean, when alpha is zero and alpha is one, the point of studying this cohomology is completely different, okay? So let me first start from the alpha equal one case, which means this is all the dimensional, sorry, this is an even dimensional cohomology because N1 is odd. So this is an even dimensional cohomology. So when it's even dimensional, what we need to understand is to ramify the subspace of this H1, which will boils down to the competition of certain theta cycle because it's even dimensional, okay? So theta cycle on this MPV and alpha. So you may realize this is not smooth. So what does it mean by theta cycle? It actually means the theta cycle on components of this, on smooth components of this scheme and that they are interaction with each other, okay? I mean, this study rely heavily on the recent work of Shao and Zhu on this theta cycle on the special fiber of Schmer varieties, okay? So I'm not going to talk about details of this aspect anymore. I mean, rather I will talk about the even case, the even rank case. This is much more, first, it's much more difficult. Second, it's much more interesting, okay? So when alpha equals zero, what we need to understand is the singular quotient, which is H1 quotient out by its ramified subspace. This is called, usually called H1c, okay? The singular quotient, which boils down to the so-called erismatic level raising phenomenon, okay? So that's our, so in what follows, we will stop explaining further steps toward the main series, but to explain it more on this erismatic level raising phenomenon, which employs new ideas from the theory of Galois definition, okay? So any question? So, I mean, so one, so it is better. Okay, so the emphasis now is just that this local Galois convolution, okay? So it's better to keep in mind of this thing from this moment. Good, so let me continue with the erismatic level raising. So before I mentioned that there's a very special condition. I mean, there is so-called a level raising prime, which requires some special element in the modular area of its Galois plantation. So what exactly are those special things? So I will explain this at least partially toward this step, okay? So here's a definition. We say that a special inner prime P is a level raising prime with respect to L, if the underlying rational prime P is a co-priming sigma plus, which we have already assumed, okay? I just repeat. Second, I want L does not divide the P times P square minus one. This is somehow elementary. You certainly don't want to P to be one when you modular L, because I mean, it will cause some complication. Okay, the third one is a really important condition. We want to do more the L-Satake parameter of pi zero P. Recall that we have explained that since pi zero is a conjugate self-dual, it's Satake parameter comes in pairs. So it comes in pairs like L for L for inverse. So we want that the more L-Satake parameter contains the pair P P inverse exactly ones and does not contain a pair negative one negative one, okay? So here's the proposition, the main theorem of the so-called level-raising isomorphism. Okay, suppose L is effectively sufficiently large and that P is a level-raising prime with respect to L, then we have the following canonical isomorphism. So this is the following. So recall that this is the right-hand side, sorry, this is the left-hand side, is the singular part of the local Galois homology we have introduced, but now we take the quotient instead of localization, okay? So this singular part of the local Galois homology is canonical isomorphic to another space which is simply the CL-valued automorphic functions on the Schmura-Sat module, the same, this is again a hacker module, the same hacker ideal, okay? So you may see, this chooses a very different, here it's the singular Galois homology of the nearby cycle homology of some integral model of a different Hermitian space, of a different Hermitian space, but here it's the Hermitian, sorry, CL-valued automorphic form on the totally definite Hermitian space, okay? Although I use the same notation, but here it's a different Schmura-variety, okay? Of FL vector space of finite dimension, okay? Actually, there's a version modular, not just L, but modular L to a power, but for simplicity I will not state that one, okay? But I mean, actually we need to do L to some power more generally, but for the simplicity of this talk, let me just do FL version. And the proof of, I mean, if you have looked at our paper, our paper has more than 200 pages. The proof of the single isomorphism probably took more than 100 pages, okay? So this is the real technical result of the work. So now I will explain, so this is the meaning of the level raising isomorphism. And probably you can already feel the flavor of this isomorphism because remember that we stop at some non-vanishing of certain period integral, right? On this Schmura set. And how can you use that? So here is a very important link. We already linked this Schmura set to certain important arithmetic geometry of some Schmura-variety. And that period integral will also relate to certain non-vanishing, sorry, non-vanishing of certain cohomology class that lives in some kind of this local Galois cohomology, okay? So this is the most important isomorphism that you link this period integral into some arithmetic property of some cohomology class of a Schmura-variety of positive dimension, okay? So, okay, so how we prove this? So the proof of this uses two main ingredients and these two main integrants are somehow technically quite different. So the first part is we studied the geometry under the intersection theory on this special fiber. We can show that the right-hand side, which is the Schmura set side, is canonically a sub-quotient of the left-hand side with the Galois cohomology side, okay? This is through a very subtle study of some intersection theory of certain the Linn-Lustiger cycle on the special fiber of this integral model. So if you already know this guy is a sub-quotient of this guy and they are both finite sets, how can we conclude they are isomorphic? So you only need to compare their cardinality, right? I mean, they're both finite dimensional FL vector spaces and one is a sub-quotient of the other one. So to show they're same, you only need to compare cardinality, okay? So how to compare cardinality? This, we have a new idea, a new technique using Galois deformation. I don't think such kind of application has ever been used before. So this is actually a new application of Galois deformation, okay? So now I will consider more about this Galois deformation theory. So now it will be more like a number theory and less arithmetic geometry. So recall that we have associated the Galois representation associated to this pi naught and we assume it's residual absolutely irreducible, which is the case for L sufficient large by our condition in the theorem. So we denote this row bar pi naught L as the residual representation. So it's a representation with coefficient FL. So we take a special inner prime P of F that is a level raising prime with respect to L. So we need to consider a global deformation problem. We call it all mix for the polarized Galois representation. I mean, sorry. Before the details, let me give a, why this isomorphism is called the level raising, okay? So the reason is the following. So this ideal is responsible for the representation pi naught, okay? So remember this pi naught originally, it is unremifed at the P, okay? So this pi naught should not appear in this cohomology at all because this is more likely is ramified at the P, okay? So if you don't consider this, it shouldn't appear at all. I mean, this whole thing shouldn't just be zero. However, if you take this, got this, set up a parameter modular L, then we will have some extra congruence so that this pi naught will congruent to some other representation that is ramified at P. So in some sense, you raise the level of your original pi naught so that it is now ramified at P. So that's why I call it's a level raising isomorphism, okay? So this is, I mean, this is the significance of this isomorphism because our priority, this space should just be trivial if you don't have conditions like this. Because of this special kind of condition, you will have some non-trivial thing on this left hand side. So now this Galois deformation theory should really see both the original representation pi naught which is un-ramified at P and it should also be available to see the raised one, the new new automorphic representation that appear in this cohomology. And those representation will necessarily be ramified at P, okay? So we need to consider a global Galois deformation problem that encodes both the original pi zero and those that are level raised from pi zero. So that's why we call the mix because it is mixed from both ramified and un-ramified. So the deformation problem are mixed for the polarized Galois representation, classifying deformations raw with the following local restriction. So for V inside the sigma plus, for those the Galois representation could be ramified, we put a no restriction. And the second for V above L, we say that the Rho V is the fontanella fire with the correct horchate weights. Basically it says that it's still un-ramified. Okay, so I will not go to the details. Okay, the key is the following. So when V is P, we want the Rho V to be tamely ramified. So tamely ramified Galois representation were determined by two things. First is a Frobenius generator. The second is a TAM generator. So if you look at our condition here, so the TAM generator, the Frobenius generator should have two special Satake parameters, module L, which should first be P negative n zero and the second one should be P negative n zero plus two. Sorry, you see this power because it is a tated twist, okay? So this is the Frobenius and the TAM generator, you should see some little monogamy here because if this S and S prime remain to be P negative n zero and P negative n zero plus two, then it is possible that you could put a new potent X here, right? Because then you will obtain a little, you will see a small, like a Steinberger representation on this block. However, this phi and the T should subject to a relation because they subject to a relation themselves. So more precisely the relations here, so S should be a lifting of P to the negative n zero and S prime, the product should still remain this number and the restriction, the relation between phi and the T will give this relation. So X times S minus P negative n zero should be zero. So this gives you somehow a semi stable anode, right? So this is a semi stable anode in the local deformation problem. So this local deformation problem has a nodal singularity. Okay, for VLs, we require Roeby to be unremifed. Now, we, this is the mixed version because you see both the unremifed and then remifed and you can quotient out by the corresponding ideal to get the purely unremifed one and then the remifed one. You want this X to be zero, right? I mean, if X zero, of course, this is unremifed because the TX by trivial representation, trivial matrix. So which means you should the modular R are mixed by the ideal generated by X. So you recover R unremifed and then the remifed one, you should require S to, you should remain S to be its original value P to the n negative n zero. So in some sense, you need to modular by this second ideal. And the congruent ideal is the ideal that it is satisfied both condition. So you take congruent, this is so called the congruence, this congruence stands for congruence. It's the tensor product of two quotient algebra over the R mix. Okay, sorry, I have to be a little bit quick. Okay, so how do we, by the way, our original goal is to study two things, right? First thing is the certain Galois cohomology and second is certain automorphic form space. So how are they related to Galois deformation? So here's the key proposition. So there exists an R RAM module, H RAM, such that this Galois cohomology is naturally a module so there's a universal Galois representation realized on R RAM of dimension n zero and the realizing on the space, somehow the coefficient is this H R RAM. So this is a Galois representation. So this Galois representation is some multiplicity space tensor with the universal Galois representation, okay? And you can compute it's H 1 sin just simply by computing H 1 sin on this universal Galois representation. In some sense, I'm separating the Galois part and the HEC part, okay? And the computation is quite easy. So the H 1 sin, the left hand side of this isomorphism one we want to consider is actually isomorphic to this module. That's H RAM tensor, this is R RAM module but the tensor with R con and also tensor with FL, okay? Similarly, this CL module I would denote by Hn is naturally an Rn module and this quotient is actually the Hn tensor, R con, tensor FL. So you see, it's very similar. I mean, the shape is very similar. The only difference is that here it's obtained from some module over H RAM and here it's obtained from some module over Rn, okay? So you want to show that these two sides have the same cardinality. You only need to show that these two modules, H RAM over Rn and Hn over Rn, they are free modules of the same rank that you are done. And this is the case actually. So here's the proposition. So suppose this Gallo Apprentice residual episode irreducible, which we have already been supposed. The second is that this is so called a Taylor-wise condition. We want its base change to F zeta L remains absolutely irreducible. And the third one is a little bit technical, okay? So for V in sigma plus, we want every polarized local lifting of this local Gallo Apprentice is a mini mullogram effect, okay? Then both Rm and Rn are finite free modules over Rm and Rn respectively, and of the same rank, okay? Then through this proposition, we obtain the level raising isomorphism, which is the isomorphism between this one and this one. Okay, since I don't have time, I will ignore because my original plan is to explain what this mini mullogram if I mean, but I think I will not explain anymore. So because it's quite some Gallo Appreformation theory. So I think I will stop this moment. Thank you. Thank you very much. Thank you very much for the excellent lecture. So are there any questions? So let me ask one question. So you need to be able to be sufficiently large. And so it's conditioning is basically from this level raising isomorphism. Yes, yes. So the three conditions, okay, so let us go back. So the three conditions, I mean, as I explained, C is purely technical. B shouldn't be there because I mean, this is because we don't know much about the Gallo Appreformation yet. I mean, the only key condition that is special to our approach is A is this condition. So essentially, I think a B and a C should be removable in future. But A is very hard. I mean, if we continue to use this method, A cannot be removed. A can be improved, but not completely removed. Thank you. So are there more questions? So maybe you can show us the last slide, yeah. Okay, so I will be quick. So the point is, what does it mean by a lifting is minimally ramified, okay? So what is a minimally ramified polarized local lifting? So this concept was actually raised by Closier-Harris Taylor like a decade ago when they studied Gallo Appreformation for general rank. So let me give a, so I will just explain the simplified situation. Suppose that we have a Gallo Apprentice, n zero dimensional galvanization with coefficient FL. So suppose it's a terminally ramified. And so the time generator will expire a unipotent element of Jordan type m1 to ms, okay? So they are lifting to whatever a ring with residual field FL is so-called a minimally ramified. If rho v is still terminally ramified and the rho v is conjugate to a unipotent element of the same Jordan type, okay? So in fact, when you raise the, for example, if your original Jordan type is two 11111, then when you lift, it is possible that the Jordan type will be refined. It could be 1111111, okay? So the minimally ramified means that the Jordan type won't be ramified. So this Jordan type will somehow will characterize how ramified a reputation, attain a unipotent matrix could be. So this minimally ramified, which means you cannot ramify, have a further ramification when you lift, okay? So we have extended this notion to any place v plus. I mean, this is quite technical of the ramifying F. I mean, this is a technical work, but the essential idea is this one. And there's a proposition as the following. Suppose L is larger than N zero, then the local deformation problem classifying minimally ramified polarized lifting is a formal smooth over C L of pure relative dimension N zero square, okay? And for experts, you know that this is a very important local ingredient for any, like R equal T theorem, okay? So the point is that how does it, so when does, I mean, in our main theorem, we only see where L is large enough, right? We, in the theorem, we never mentioned the word minimally ramified, but here it comes an additional condition like a minimally ramified. So actually we proved a theorem which we call the automatic minimality. So if you have a compatible system of Galovar plantation coming from automotive form, so it is expected that for L large enough, or, and if it's a place V, for L large enough, or local deformation of this Galovar plantation should always be minimally ramified automatically, okay? So let me say this is a conjecture, so we call the rigidity of automotive plantation. So we fix the finite set of sigma plus of primes of A plus, therefore all but finitely many primes lambda of E, depending on sigma plus, we have, this is a relatively absolutely irreducible, these are all the conjectures, okay? The third one, for any V plus, every polarized local lifting is automatically minimally ramified, okay? So homework verified above conjecture for N equal two. This is not hard, okay? So theorem, suppose pi is super cuspid or somewhere, then the above conjecture holds, okay? So let me give a remark. So originally we also need a pi to be a twist of Steinberg and some place not above sigma plus in order to deal with the Taylor-Weier's condition. But the later, Toby G told us an argument to remove this restriction. So now the theorem is very clean. You only need to pi to be super cuspid or somewhere, okay? So this conjecture, the automatic minimality is true. For a large enough, yeah, that's it. Thank you. Thank you very much. So are there more questions? Okay, so thank you very much. Yeah, okay, great. Yeah, thank you very much for the beautiful lecture. Okay.