 First of all, thank Alina, Mike, and Philip for the invitation to speak in the number theory web. So it's a player to be here. So today, I'm going to talk about algebra cycles. But more emphasis on questions over number fields. Well, since this is a number theory talk, so that later on, I will really focus on a special class of algebra cycles, starting from diagonal cycles and few kinds of generalization of diagonal cycles. So there's really some kind of generalized diagonal cycles. So let's say we call that so algebra cycles. So we will only talk about the diagonal cycles over a variety, which is going to be assumed to be smooth-projective for the whole talk. So this will be assumed to be smooth-projective for the whole talk. So recall that this just means I'm considering linear combination, formal combination, of closed sub-variety. So z, or some irreducible ones of certain fixed co-dimension. So consider formal, finite formal combination with, well, in the beginning, we can look at z coefficient, but eventually, we'll just look at rational coefficient. So what we want to consider is, of course, the sort of invariance attached to algebra cycles, such as cycle classes in cohomology groups, or maybe the most refined relation. So we can consider the so-called a chao group. So those are defined by fixing a co-dimension. So co-dimension i, so i is a co-dimension. So those will be cycles, algebra cycles, of co-dimension i modulo, so-called a rational equivalence. So one example would be, if i is one, so here, okay, let's actually also fix a ground field, f. So if i is one, so this gives us the more well-studied, so-called divisor class group, which is actually isomorphic to picar, picar group, sorry, picar group of, oh, that's right, this is more, we have a better understanding in the co-dimension one case. In the higher co-dimension case, it's rather difficult to study. So here, so as you can see that, if now we start to impose the condition that, if f is a number field, so in that case, we see that even the divisor case is hard to study because, well, it depends what we want to know, because you see that this question of studying picar group or divisor class group already in crude, for example, if x is a curve, then we know that, well, there's a degree map taken degree to the kernel, degree zero guys, this will be sort of the modulated group, so this, or the Jacobian variety of this curve. So that's, we know that's pretty hard stuff, well, related to the Birch's one-dial conjecture. So that's in the co-dimension one case. So in higher co-dimension case, this becomes slightly, I mean, this becomes even more difficult. Nevertheless, there's a belief that in higher co-dimension case, so here we know this is related to invariance such as ill function by the Birch's one-dial conjecture. So in high-dimensional case, there's also such expectation due to Beneson or Bloch-Hockato conjecture. So there's a belief that there's a relation between, between Chao group, especially the part which has trivial co-homology class. So here this is really the co-homology cycle class in the co-homology. This is H2, this curve, let's say over C. Okay, so that's, the general picture, it seems to be hard to do anything with such generality. So now I'm going to move to the, what do we want to do in the, as we're setting the title of the talk. So I want to study various sort of diagonal cycle. So actually for a number of theoretical interest, it's, so see here, sort of a very interesting cycles. Well, for geometric reasons, geometric interest in cycles are mostly in the, in the middle, the middle co-dimension. So here, because well, at least the co-homology group, we know only the middle degree co-homology contains most information. So for arithmetic cycles, like the example of curve, we see that the co-dimension one seems to be the most interesting case. And so, so what actually wants to have a case where the total, total variety, the ambient variety has all the dimension. So this is all the dimension. So two, two and plus one where the cycle has co-dimension, or maybe two and minus one. So this is so-called arithmetic middle dimension. So any zero, any one, we get a divisor on the curve. So the next case we would like to consider, say curve on a three-fold and so on. So, so this is sort of arithmetic interesting because if you look at integral model, if you consider integral model over, so integers, then the dimension gets added by one and the integral model of the cycle has actually middle dimension on this scheme over integer range. So this has dimension, two and plus co-dimension, which is actually in the middle dimension. So that's the sort of one side remark. So we would like to really consider situation where the dimension of total, an ambient variety is actually odd where the cycle has co-dimension exactly half, slightly bigger than half. Okay, so back to the diagonal cycle. So we know the classical diagonal cycle. So now I'm going to change the notation a little bit. So now I'm going to consider X being again a smooth projective variety, but now I can consider the product. So we know the classical diagonal cycle. It's just the diagonal embedding. So the diagonal embedding of this variety to the square, to the self product. So we know this plays an important role in algebra geometry. And even in this case, we still have a lot of questions answered, such as the diagonal is supposed to decompose according to the Q-length decomposition for the cohomology. So this actually is supposed to decompose as sum of delta i where delta i acts on, well, this in a child group of the product of the appropriate degree, such that delta i acts on the cohomology being the projector to, maybe just J, being the projector to the i-th component being a projection. So this is sort of a Q-length decomposition. So this is still not known in general. So, but okay, so that's, at least in some special cases, this is known for example, if X is a curve, I think it's also known in the case of a surface, in general, it's not known. So in the case of a curve, so we have sort of good understanding about this diagonal cycle. So now, so this is the first class of diagonal cycle. Well, this is not in an arithmetic middle degree because it's actually exacting the middle, middle core dimension. So if I want to have something more arithmetic, so we can consider the so-called small diagonal. So this is a curve. This was studied by Gross and Schoen in the maybe late 80, early 90. So I can consider a curve rather than considering diagonal to the square, I can consider the embedding of, so this is the curve instead of Q, a third power. So that's a three-fold, I have a curving of three-fold. So okay, unfortunately, I'm gonna use the same notation, delta. So maybe it's not so standard notation in this case now. But nevertheless, I have this diagonal embedding. So it's interesting question to know whether this class is trivial or not in the child group. And of course, one first have to take care of the cohomology class. So the cohomology class is, so if you look at this class in the child group of X3, so I have a cycle class for the cohomology. So here let me denote, so yeah, it's a co-dimension two cycles, H4 of X3. Let's say I look at the complex fiber, the singular cohomology, baby cohomology. So it's easy to actually trivialize the cohomology. So Gross Schoen defined the modification. You can modify easily to get a cycle with trivial cohomology. So this modification, so let me recall you this modified diagonal or small diagonal of Schoen cycle. So this definition depends on a choice of a base point. I can choose a point X. Then I can define the modified for the delta E. This is gonna be the main diagonal modified by some small, so the partial diagonal. So more precisely I can write delta minus, okay, delta one, let me, let me denote, let me recall. I have those partial diagonal, namely I can look at, say delta one, this will be, so you let the first two coordinates to be the same or the third one being constant. So there's some partial diagonal on the triple product. So of course you also consider the other two similarly defined partial diagonal, but then you can also define diagonal by allowing two coordinate. So the first two coordinate being this constant point. So okay, so we should, so the correct again, so maybe correct the first step, maybe the correction is too much. So let's just add a back delta one, two, and it's switching the indices. So this turns out to be, this modified diagonal turns out to be, to have a trivial homology class. That's, it's quite easy to compute that. So one natural question will be, will actually gross on the show, in their paper, they proved that this class is trailing the child group if, so of course one wants to know if this is actually now a trivial class in the child group. So gross on the show, they proved, it is true if X is hyper, well, okay, first of all, there's also a choice of base point. So a good choice of base point, as you can imagine, the most canonical choice of the base point E will be the, well, will be the canonical class. So, so now let's see, it turns out that it's easy to see that if, if this is now, if it is zero in the child group, again, here my child group is, has a rational coefficient. If it's zero, then actually this class, so earlier I defined just using one base point, you can generate this construction to any, any divisor of degree one. So, so here I'm going to use that version. So, so I can now read and take E to be any class in the Picard of degree one. So the actually vanish of this class, but necessary condition is that this base point or base class has to be a multiple of the canonical class. So really has to be, you know, two G minus two divided by the, you see the canonical class, this, this right as, E to be the class of the canonical bundle. That's right. This says C of X. So for now I'm going to assume the base point is the base class is this multiple of canonical class. So grossly improve that. Actually this, if you do that, so delta C this modified diagonal vanishes if the curve is hyper elliptic. So in other words, if there is an evolution under the quotient of the curve by that evolution is P one. So that's when result of the proof. So when the actual question was actually can, does there exist curve which are, which have vanishing diagonal class or this modified diagonal class for non-hyper elliptical curve. So question is whether there exists non-hyper, hyper elliptic such that this class is zero. So should I also remark that this cycle class is, is more or less, more or less the same. I mean, in the sense that the simultaneous vanishing the same as the so-called Teresa cycle, which were sort of studied much earlier. So Teresa cycle is, is defined by the following. If I have a curve X, I can look at the embedding to Jacobin variety of X by choosing a base point. So now let's choose the same base class. So then you can look at the image X minus negative one pullback of X. So this, this class is called a Teresa cycle. So it's lies in a child group of one cycle of the Jacobin. So, so this was sort of a much well studied before and it was known for generic curve and this class is now zero. So over complex number. So actually for the moment, everything can be done over the complex number. So we don't have to really assume the base space is a number field. So if you think about the module space of all the curves of fixed genus G, I can assume G is at least two because it's understood what happens when G is zero and one. So if G is two or higher on the module space Teresa proved already that this cycle is actually now rationing non-trivial. If, well, if you look at the general curve. So the question is actually what about sort of the sort of so-called a special curves. So the initial question, I mean one question would be what is the most most special curve. So here what that depends on how how people approach the space of all genus curve G curve. So maybe one way for me maybe the most special ones are so-called always curve. It's like in the case of elliptical curve, we believe curves with more elliptical curve with bigger and the morphine's run namely so-called curves with complex multiplications this should be more special. So here maybe we should consider curves with the largest possible automorphine's runs as being the most special curves. I mean those are very special in the sense also they are sort of isolated. There are only finally many such curves in any given genus. So this is so-called always curve, always curve. So namely a curve of fixed genus such that the automorphine group where we know the order is bounded by 42 times the degree of colonic bundle. So it's 84 times G minus 1. So the horus curve are precisely the curve with maximum possible. I mean this equality might not be achieved for every G but anyway we know for example if G is G is 3 there's a Klein curve, genus 3 and has automorphine group of order 168 and I think in the next example is the genus 7 case. So this case is even more special because actually it's a Shimura curve. It can be realized as a Shimura curve. It's also a Shimura curve attached to certain degree 3 total real field. With a full level structure at prime 2. So in this case it's automorphine group. So I'm not going to see too much here. So here this is actually a group of order 84 times 6. It's actually PGL2F8. So it's a finite group. So Shimura noticed that this curve is actually can be written as a Shimura curve. Although Frick and McBeath this study is curve even before Shimura. So one theorem we proved. So this name is still state 1 result here. So theorem I proved with Tony and Chiu last year was actually for this curve we know the modified diagonal cycle. So if X is this harvest curve with genus 7 so called the Frick and McBeath curve then this class is 0 in the Chiu group. So this was known well if you consider not the Chiu group but the class in the intermediate Jacobin this was already proved. So the key here was actually the first cohomology so the Bayley cohomology if you try to see so with complex coefficient so this is a representation for PGL2F8 which is a finite group of a lead type. It's actually it's more or less it cannot be reducible. Composed as H1 0 using Hodgdon composition so each piece is actually irreducible representation of dimension 7 and it's actually a cuspid representation for this finite group of lead type. So you can actually exactly you can write down exactly what this representation is and you can so our theorem was actually if you check the third tensor power so I have a group action called G here so you can look at a diagonal invariance G action so if there's no invariant then our theorem shows that actually this class must be a speed failure so in other words we prove a criterion unless it's a sufficient condition for the class to be 0 using using certain sort of additional symmetry on the curve if a curve has a large autoimmune group such that this third tensor power of the representation on the the repenting of G the autoimmune group on the first cohomology if this invariant space is 0 then then you can conclude this class is actually trailing the chao guo so that's one sort of so going back here we're trying to study this diagonal cycle for curves so one example where you can you can you can pull this class of vanish so of course still we believe being vanishing is very rare so at least for a curved curve we tend to believe only finding many curved curves have this property or only finding the harvest curve so those curves they all define over a number of fields they have a very large autoimmune group but ok so we don't really have much we don't have a lot of evidence towards this conjecture but somehow the proof seems to indicate it seems to be rare to have such vanishing or for some of the most special curves on a more just curves with given genus so ok that's one thing I want to talk about so now next I want to well since we know it seems to be rare to have vanish do you know at least that if you take instead of asking about the triviality of this cycle if you ask about H1 x tensor 3 g invariance do you know that that is 0 for a for a horizontal curve we don't know we don't know ok so right we don't actually we proved something but I'm trying to recall what we exactly proved so we checked for certain replantations for certain family so yes I mean ok so we don't we haven't we haven't proved so in other words a weaker conjecture right would be this condition here how do I actually let me see ok then highlight what I want maybe this is it here yeah ok this then we haven't well of course when we conjecture this is true only for finally many words curve right so that's same to be accessible but actually we don't know how to prove it so we don't know how to prove this is true only for finally many words curve so that's maybe that can be expressed just in terms of representing you can convert it that's just but still the difficulty is this g is still the g representation is not still not so easy to to write down yeah so probably let me recall that we in fact prove this for PSL2FQ if g is a yeah so we did prove that if if a horse curve has g being PSL2 maybe FQ and you finally feel in that case we can prove it but I think not the problem is this doesn't exhaust or horrors automorphism yeah okay all right okay so that's so now I want to move to something which at least give us possibly something which you know non-trivial example right so okay so this is I'm going to specialize to the case so now so now X is a Shimura curve so in this case actually the question is related to well in this case have the third power the triple product one can actually study the quotient of the or esogenous factor of the Jacobian so I can project down if I take three esogenous factor of the Shimura curve so okay by the way when I say Shimura curve I have already implicitly fixed a division algebra over a total real field division quaternion algebra over a total real field uh and so on so I can take say for simplicity let's say let's say I have three elliptic curves which are esogenous factor of the Jacobian so that I have you know I have this map so here I'm using what in the case of if the base field is Q we know every every elliptic curve arises as a portion of Jacobian of Shimura curve so in that case you see that actually every um elliptic curve can appear in this triple product of elliptic curves um well in in general actually so um so in general if you have any abenovariety we know that every abenovariety can appear as an esogenous factor of the Jacobian of curve so we can in some sense we can so here we're just trying to generalize the idea we had before we let's say we want to project the diagonal class on the triple product of curve to certain product of abenovarieties so you see that actually in general we can always uh manage it to produce certain class on any triple product of abenovarieties so here let's just say let's test the idea in the case of elliptic curves um um for the reason that for elliptic curve we actually have sort of modularity at least conjectually over general number of it um so we can use actually Schmurocker to to realize um to realize the idea we just mentioned so so here so I can look at the projection here also look at the modified diagonal cycle so let's let's just write this as a this is abenovariety where ai's are so elliptic curves I mean for simplicity let's say I'm working with base field of being Russian number um so I can look at so then I get this class called a delta a as in it's a one cycle on this abenovariety has trivial homogenous class so by what I mentioned earlier the generalization of BSD conjecture where this class um should contribute to to the vanishing of vanishing order um of the corresponding L function so in this case so we expect so some kind of growth um so in this case we would expect this guy should have connection to to corresponding L function so this is some theorem we proved um I proved with Chen Yuan Shou Zhang so here we are not we don't have a complete result I mean we don't have a result as general as I stated here but let me say if I assume if I assume an ai the elliptic curve have all have semi-stable reduction so ai have a square free conductor square free conductor so let's assume that so then actually we prove um this class um if you compute the so-called bernison-bluho height pairing um so here I said the cohomology class is trivial so this is the cohomology class is trivial um so this actually gives you the first derivative of the Haase-V L function um out of the central point this case is two um so here when I say the Haase-V zeta function I really meant you know the part which uh comes from the primitive part uh so name each one a ai so the tensor product of the data module of the elliptic curve okay so um so that's the result um we actually could prove so this allows us at least I mean theoretically this allows us to to get a um uh a you know so conjectually this actually left hand side the height pairing is conjectually long to generate so it should actually tell us exactly when the cycle is non-zero in the Chao group well unconditionally at least we know if the height pairing is non-zero then this cycle class should be non-zero so so in that way we at least have some theoretical um way of getting non-zero cycle um diagonal cycle um using L function of this triple product of a uniform curves okay so that's one theoretical result um in this direction um so now I have spent like most of my time to study this sort of small diagonal so I want to so my abstract I wanted to give two other classes of cycles related to diagonal cycle so now let me um so let me say my next class is um well okay so let me also propose some question here maybe those questions should be natural after we saw this um triple product of a uniform curve so one would of course naturally wonder what would happen if I um look at higher power of a uniform curve or more product of more than three uniform curves right so if I take a could be more you know more than three say n where n is bigger equal to three you know say three curves so is there any way of producing uh algebra cycles in the um in um in middle middle coordination or I mean arithmetic middle coordination um so that um it's kind of strange somehow for the moment the only thing we have is actually sort of not more than three curves if you have more um seems to be hard uh seems to be um no um no ways of producing middle uh dimensional uh arithmetic middle dimensional cycle um I mean unless you have you know some degenerate cases so um alright so that's one question I will come back to this question later uh after I introduce the next two classes of example so there is some sense partially uh the some sense we give some hints how to how to attack this question at least in some special cases we have more than uh three factors of each curve so yeah so now next I want to talk about sort of another class of diagonal cycle um so it's it's sort of not a really diagonal cycle in the sense above um so this is sort of a g gangra's percent so the ggp uh one called also one might also call diagonal but you will see it's not not quite of the same um so here the general structure looks as a following if I take a say variety smooth projective variety I take a divisor see so why is the divisor so why has code dimension one um so code dimension one so divisor so why is the divisor so then I can look at what I have this map while embedding into x I can look at the graph so I can look at the graph so why it goes to uh y times x right so this is actually by design just give us a arithmetic middle dimension cycle because um right uh because y has code dimension one so so the total dimension will be let's say this has dimension n this has dimension minus one right so this will be code dimension and you know I'm being right of two minus one dimension so which is very nice so if we're trying to find some test ground for the generalization of gsc conjecture this could be perfect because I have something in the middle in the arithmetic middle dimension um so this is indeed the case for for large class of a more variety so so ggp this study sort of um so a large class of where you have this kind of code dimension one then you take the graph um so more precisely you can look at so x to be more variety um of unitary type so those are attached to some quadratic so here you have to fix some quadratic extension for unitary group so this means you fix some quadratic extension let's say f over q being imaginary quadratic then you can look at some Hermitian space with respect to this quadratic quadratic extension then you can look at unitary group so here we have to assume the signature at argument in place to be same one type so has only one negative one sign in the in the signature so it's indefinite Hermitian parent so then you can look at this more of a variety attached to the unitary group so unitary group here so the good thing the nice feature here is that you can take a Hermitian subspace of codemission one with similar signature condition so that will produce a embedding of unitary group and this embedding will give rise to embedding of more of a variety y into x which are actually codemission one so therefore give you a large supply of examples where you can apply this graph and you get a sort of middle dimensional cycle here so let's give you a large class of special cycles on product so let's give rise to y goes to x times y and one knows well in principle one knows how to compute possibly the zeta function for this product because there are Shimura varieties so people have studied in the past 30-40 years how to compute possibly zeta function therefore you can at least where those zeta functions are known to have meromorphic continuation and so on so in this case one of course would like to study the same question as in growth zagi or the theorem I pulled with Xinyu and Shoubo-Zhang where you want to relate to the height parent to special value well function so conjectually in this case so very rough description of the conjecture is that if you take any well okay so why so here I have so one should imagine decompose the motive of this product according to the heck algebra action so there's a large supply of correspondence acting on this variety so you should sort of decompose into small pieces ideally in decomposable pieces so that can be those intercomposable ones can be parameterized by sort of automorphic representations corresponding groups in definition of a Schmura variety so if you take such a representation pi so pi is the eigen space for the heck algebra so in this case you can look at the pi part this is sort of like the case earlier we are projecting down the cycle class to to some Yiddipal curves or the product of Yiddipal curve so this is supposed to relate to the derivative of L value to center up to some constant so this would be sort of the conjecture of generalization of of gross formula so okay so regarding this formula I mean that we don't have a result yet like the one for for product of Yiddipal curve rather we have something close so theorem so this is a so we have a draft I wrote with Daniel Dazzini so where we prove a p-article version so we have a p-article version so we explain what does that mean so p-article version so one p-article version pose so in the sense that on the left hand side I have to reduce I have to look at the p-article head so p-article height defined by Nikovar so there's a p-article head parent and the right hand side I have to look at the first derivative of certain p-article function p-article function attached to the automorphic world so again there's a lot of also like a square free conductor condition here also have some analog of impose the square for the conductor so it's not really as general as one would like to to have but we have this kind of result all right okay so let me then so this is one so now I start to wonder you know so this kind of construction what else can you do right so what other construction can you come up with for natural algebra cycles so so my final very quickly I'm going to mention the last class of so that last one is defined by Yifeng Liu so so he defined something which sort of sits in between those previous two so okay so in general for higher dimensional things it's a triple product it's too difficult to study if X has a hard dimension it's a dimension bigger than two this becomes much more difficult and in fact they really vanish I mean the modified diagonals like they wouldn't vanish in the Chao group unless you know very special variety X you consider so nevertheless Yifeng Liu maybe there's some kind of you can actually project down to let's say when the Jacobian maybe the Albanian variety of X if this is non-zero say it has an isotonic factor called A say A is an elliptical curve so in this case well you can you can choose a base point and compose this sort of this map so you can actually produce a cycle which is easy in the arithmetic middle dimension so in this way you see that this last class is it's also rather as delta so delta will be in this Chao group of two factor then times elliptical curve so that will be in the arithmetic middle dimension it's interesting question to study this situation when X is a Shimura variety like my case earlier so this was so in this case if X is Shimura variety for unitary group even one type as above so in this case okay one one can use well the global questions look very different from here but actually locally the computing intersecting number can be applied so when you work progress of one of my graduate students so it's who is working on analog PRD format PRD height format relating PRD height with PRDO function so I want to maybe spend maybe I'm already running over time already I have one more minute to mention what I'll go back to this question I mentioned so how to attack if you have more factors here so so what my final questions so here you can hopefully you can relate back to the more power of I mean product with many elliptical curves in the following way if suppose my just in one example if I look at the Shimura variety attached to U21 so this is the surface it has two dimension so it's co-homolog should contain the co-homolog of product of two curve elliptical curve so in this case take the conjecture of the predict you have if you take two elliptical curve well say the same elliptical curve but self product then base change to the quadrally field defining the Shimura variety so there should be some correspondence predicted by data conjecture some correspondence here so then you can hopefully transport the cycle x times x or x times y or the last two construction this are different curves you can hopefully transport any cycle well you have two factor you can put another two factor then times a so you can transport cycle actually use correspondence times correspondence times identity so there should be a way of transporting cycles from the last two construction to multiple factor product of the elliptical curves so therefore I mean the last two classes of examples potentially might give information to the questions of constructing cycles on the product of many elliptical curves although the price you have to pay is to actually to solve this to find so if you find an example of this correspondence between Shimura surface and product of two elliptical curves then you are able to produce you can apply this idea okay to wrap up so I sort of have talked about three classes of diagonal cycle the small diagonal of x into a triple product then I have this sort of ganglion process construction of divisor embed into an ambient variety then I take the graph finally have some some sort of intermediate between the triple product and the gross GDP cycle so they seem to be all interesting to me and they seem to be the only algebra cycles in the automotive world I mean related to Shimura where one can study hopefully there will be more being discovered in the future and let me stop here thank you