 Thank you very much for the introduction. It's my great pleasure to speak in this seminar. I would like to thank the organizers for the invitation. Today I will talk about a group based on zeta functions arising from real and paedic groups. They share some properties with the familiar zeta functions. So here I exhibit two familiar zeta functions. The first is the Riemann zeta function, which can be expressed as a product over primes p, one over one minus p to the minus s. And the Riemann hypothesis is a big open question for the Riemann zeta function. And the second zeta function is the zeta function attached to a smoothly reducible projective variety defined over a finite field. And its zeta function counts the number of rational points over finite extensions of the base field. So when we collect these numbers together and organize in the right way, and that zeta function can be expressed also as an infinite product, the over closed points in the form of one over one minus u to the degree of the points. So in this context, the closed points plays the role of the primes. And it's well known that this infinite product converges to a rational function. So I will see that as the alternating product of polynomials with coefficients in z and the constant term one. Actually, so each polynomial P i is the reverse characteristic polynomial over the Frobenius acting on the iso-elastic cohomology of the variety. As you can see that the bottom polynomials have an index by the even integers and the top one by the odd integers. And for zeta function attached to a variety, the Riemann hypothesis is known to hold, which means that if a polynomial P i is not constant, then all the roots have the same absolute value. In this case, it's q to the minus i over two. So now we move to the zeta functions arising from groups. So the first one is the zeta function, the sub zeta function. So we know that the sl2r acts on the upper plane by fractional linear transformations. So we take a discrete torsion-free co-compact subgroup of sl2r and we look at its orbit space. So upper plane mod out by gamma. Because of the choice of the group, this orbit space is a compact Riemann surface and with its fundamental group isomorphic to the group gamma. This upper plane can be interpreted as a homogeneous space which is the pg2r mod out by its maximum compact subgroup. So on this compact Riemann surface, what we are interested in is to count the closed geodesics. But I would like to say a few words about what kind of closed geodesics we are interested in. Okay. So normally if we just pick up a closed geodesic, it has a starting point and has an orientation. Okay. But what we are interested in is that the closed geodesics such that when you change the starting point and you travel along it, it still remains a closed geodesic. And this kind of closed geodesic, we call that a geodesic cycle. So we are interested in counting the geodesic cycles. Okay. So let me explain, well, what kind of, what is a geodesic cycle? Okay. So we have a closed geodesic on the quotient. We can lift it up in the upper plane. If we get a geodesic path from a point on the upper plane to another z, to another point, gamma z for some element, gamma in the group, which is not identity. Okay. So now that's this element of gamma. Okay. Because of the choice of our group, this gamma is the hyperbolic element. As such, it has two fixed points on the reals, union infinity. Okay. So through these two fixed points, there's a unique geodesic, which is the semicircle. Okay. And the semicircle is invariant under the action of gamma. Okay. So if we pick a point z on the semicircle, then we know that a gamma z, well, lies here, and likewise a gamma square z and so on. So then this portion of the geodesic from z to gamma z, when you pass it to the quotient, we identify these two endpoints, we get a closed geodesic. But this closed geodesic has the feature that if I change the starting points, a z prime, I travel along it. Actually, I will end up with a gamma z prime, as you can see that, so it is still a closed geodesic. Okay. So then the geodesic arising from this z to gamma z for this z chosen on the semicircle gives us a desired geodesic cycle. And all the geodesic cycles arise this way. And incidentally, if I choose this z, which is outside of this part plane, and then if I do the same thing, connecting z with the gamma z, then it will, when I pass it to the quotient, it will not have this property. Okay. So all the geodesic cycles or all the geodesic things we are going to consider in the future have this feature that when I change the starting point, it remains a geodesic. Okay. So we want to count the geodesic cycles, but then because we are in this continuous situation, there are many different initial points. So we're going to ignore in starting point, we say that we can't count them up to equivalence. Okay. So well, well, the geodesic cycle is called a primitive. It is not obtained by repeating a cycle of shorter length more than once. Okay. Then the equivalence classes of primitive geodesic cycles are called prime, are called primes of this compact remand surface. So whether we are interested in counting actually either prime or prime powers. So just this two kind of geodesics. So to them, Selberg defined a zeta function, okay, which is as follows. So he did this in 1956. So you take product over all the primes, and it factors one minus e to the minus length of the geodesic, then times s. And to also include all the other prime powers, so to speak, we shift s by the positive integers k. So we have this second product. Okay. We think we're zero. Okay. So from the way I described this geodesic, we see that the geodesics that we are considering, well, corresponds to conjugacy classes of the group gamma, non-trivial conjugacy classes. Okay. And the primes will correspond to a conjugacy class, conjugacy classes generated by the so-called primitive elements. Okay. So an element is primitive if it generates its centralizer in the gamma. Okay. So there are two ways to see that this Selberg zeta function. Okay. And because our quotient x gamma is a compact remand surface, so there's a Laplacian operator acting on that, on the space of L2 functions on the remand surface. And it has discrete spectrum starting from zero, and then goes up and all the way to going to infinity. Okay. So Selberg has shown that the Selberg zeta function behaves very much like the remand zeta function. Okay. In particular, it has a simple pole at s equal to 1. Okay. And it differs from the remand zeta function in that it's zeros in the critical strip. The real part of s between zero and one are known. Okay. They are expressed in terms of the eigenvalues of the Laplacian. Okay. In other words, all the non-trivial zeros of the Selberg zeta function are the solutions to eigenvalue lambda n equal to s times 1 minus s4 and at least one. Okay. So as we see from this expression of the zeros, when this lambda n is less than one quarter, there are finitely many of them. So we obtain the real zeros of the Selberg zeta function in the critical strip. They are only finitely many of them. And the remaining ones will get a complex roots and they all have the property that they sit on the vertical line real s equal to 1 half. So the remand hypothesis holds except for finitely many real zeros. Okay. So in view of these zeros in the critical strip, so we expect that this Selberg zeta function has close relations with this product of this eigenvalue lambda m minus s times 1 minus s. Okay. Unfortunately, this infinite product does not converge. But on the other hand, because this lambda n's are the eigenvalue of delta, so this infinite product, really, I mean, if this were finite product, then this is exactly would be the determinant of the Laplacian minus s times 1 minus s. So indeed, there's a way to make this idea rigorous. So to define the determinant of the Laplacian. Using the spectral zeta function so that formally it is equal to this infinite product. Okay. Lambda times s minus times 1 minus s. Okay. So now in 1987, Sonic and independently, Volros came up with the expression which gives a relationship between the Selberg zeta function and the determinant of Laplacian. Okay. So this determinant of Laplacian is equal to Selberg zeta function times this extra factor. Okay. So as you can see that this extra factor here is e to the constant plus s times 1 minus s. And the use is the bound double gamma function squared over gamma s times 2 pi to the s. So this factor here is independent of gamma. It's raised to twice of the genus of the Riemann surface minus 2. So it's exponent here gives the topological information of this Riemann surface. Okay. So this theorem is extended to allow gamma co-finite by Efrat and Koyama in 1991. Okay. So now, so you have a look at the Selberg zeta function. And so he thinks about that it's, you see that this Riemann surface expresses a double coset of the PGO2R. Okay. So he thinks that, well, what happens if we replace R by a piatic field by Qp? So that's the completion of a Q at the acumenium, non-acumenium place. Okay. And then he studies to see what happens. So that's the what we are going to do. Okay. So we can see that replacing R by Qp, so take a PGO2Qp mod up by its maximum compact on the right. And on the left, take a discrete torsion-free co-compact subgroup. Okay. So we have this double coset. We call that x gamma. Okay. So to this double coset, you have to define a zeta function just as similar to the Selberg zeta function. It's a product of the conjugacy causes of gamma for the primitive gammas. And each factor is one over one minus u to the length of gamma. Okay. So this converges, say, for the u, absolute value is small. Okay. Okay. So now the question is that, well, so we think of this as a PGO2Qp mod its maximum compact, that's the piatic upper half plane. So what is this piatic upper half plane? Okay. So this piatic upper half plane actually can be interpreted as a P plus one regular tree. So it is a tree that each vertex has exactly P plus one neighbors. Okay. So and the vertices of this tree are the cosets of a PGO2Zp. Okay. We can interpret each coset as the equivalence class of a rank two lattice over Zp. Then two lattices adjacent, if they can be represented by lattices, L and L prime, such that the L prime is strictly between L and PL. Okay. Then PL is strictly between L prime and PL prime. So PL and L represent the same vertex. So this is an interactive graph. So in this case that this L prime has index P inside L. Okay. So when we interpret this piatic upper half plane as a P plus one regular tree, then when we mod out the gamma, so we get this X gamma, then becomes a finite P plus one regular graph. Okay. So say I'll notice this and you remark that. So this the Harris data function defined using the groups can be reinterpreted as the data function defined on the closed geodesics. Okay. For the this P plus one regular graph. Okay. It's by exactly the same same identity. And therefore it counts the closed geodesics inside this graph. Now, because we are in the graph, which is a discrete case. So when we counted the closed geodesics, the different starting point is viewed as a different closed geodesic cycles. Okay. And the prime again has the same definition as before. It's an equivalent class of primitive closed geodesic cycles. Okay. Okay. So the graph data function in this definition is similar to the data function associated to curves defined over finite field, for instance. Okay. So what's the, I mean, the property of this data function. So 1966, you have a show that the data function also converges to a rational function. Okay. So it's numerator is one minus u squared raised to the oil characteristic of the graph, which is a number of vertices minus number of edges. So that it gives the topological information about the graph. And the denominator is a determinant of identity matrix minus the adjacency matrix times u plus pu square times identity matrix. So this adjacency matrix of graph, well, it's a square matrix has rows in the columns, parameterized by the vertices. So the graph and ij entry records the number of edges from vertex i to vertex j. So as such, it is a real symmetric matrix. So all its eigenvalues are real. We can arrange it from large to small. And since the graph is P plus one regular, it's easy to show that eigenvalue sits between P plus one and the minus of P plus one. And, and it does achieve the P plus one is eigenvalue with all one function as eigenvector. Okay. So in other words, this denominator of the data function can be expressed the product of one minus the eigenvalue of the adjacency matrix times u plus pu squared. So this eigenvalues of a are called the spectrum of the graph. So denominator gives the spectrum information of the graph. So this expression is very similar to what we saw for the silver zeta function, except that this expression was proved more than 20 years earlier. Okay. So for, we know that the zeta function for nice curve satisfy the Riemann hypothesis, as I said at the beginning, then how about the zeta function for the graphs? Okay. I should also remark that this Euler characteristic for this regular graph actually is a negative integer. So this one minus u. So the numerator actually goes through denominator is a power of one minus u squared. Okay. So this zeta function actually is one over a polynomial. So the poles, this plus minus one coming from the numerator, so to speak. And the poles coming from the eigenvalues P plus one and minus P plus one, they are called the trivial poles of the zeta. And the remaining poles coming from the non-trivial eigenvalues, which are strictly between P plus one and minus P plus one, they give rise to the non-trivial poles. So we say that this silver zeta, sorry, this zeta function satisfies the Riemann hypothesis if all the poles from the non-trivial eigenvalues have the same absolute value. In this case, it's a P to the minus one half. Okay. So this happens if and only if all the non-trivial eigenvalues have absolute value bounded by a twice squared of P. And the graphs with this properties are called the Riemann gene graphs. So in other words, the zeta function satisfies the Riemann hypothesis if and only if the graph is Riemann gene graph. Okay. So in other words, for P plus one regular graphs, not everybody satisfies the Riemann hypothesis. Only special kind of graph satisfies the Riemann hypothesis. They are called the Riemann gene graphs. Okay. So now let me give you some connection between the zeta functions of graphs and the zeta functions of curves. So as we explained over here, okay. So maybe some Riemann gene, if we look at the Riemann gene graphs, they satisfy the Riemann hypothesis. Maybe they, some Riemann gene graphs could be related to the zeta function of the curves. Okay. So here's one example. Okay. So we have this tree, P plus one regular tree, we just have to choose a nice group such that after we mark out, we come up with this Riemann gene graphs and which its data is related to a curve. Okay. And so one choice of such kind of a group gamma is as follows. So we take a definite quaternion algebra over Q, which ramifies on an infinity and a prime L, which is not equal to P. And we let DL be its multiplicative group modular center. Okay. So because of our choice of quaternion algebra, so if I look at a QP points of DL, it is exactly PGL to QP. Okay. And so for the discrete co-compact subgroup, we just choose the subgroup of Q, of DL, which is integral points everywhere except that we allow P in the denominator. Okay. So that is our gamma L. Okay. And this group is torsion free if Ls come to one module of 12. Okay. So in this case, well, we get the Riemann gene graph. And also if I go back and look at its data function, we look at a denominator here. Okay. So we have the trivial eigenvalue P plus one. Okay. Let's remove the factor from the denominator. And the remaining part will be coming from the non-trivial eigenvalues. And the non-trivial eigenvalues are satisfied by the Riemann gene bound. Okay. And it turns out that using the theorem by Schumacher, if I remove the factor from the trivial eigenvalue, then the remaining part of the denominator of the zeta function of this graph is actually equal to the P1 of the zeta function of the modular curve of X naught L, Mark P. So in short, when for some choice of gamma, and this interesting part of the zeta function of the graph, it's actually is also the interesting part of the zeta function of a curve defined over a finite field. Okay. So next, well, Hashimoto expressed another way to think about the closed geodesic. Okay. So for graphs, when we describe closed geodesics, usually we think of that as a sequence of vertices. And to describe adjacency, we have the adjacency operator A. But when we compute the trace of the powers of adjacency operator, I mean A, then it also gives us a closed path, which are not geodesic. Okay. So to guarantee that we only count the geodesic cycles that we want, the Hashimoto suggests that we should express them as the sequence of the directed path. So this way, to each edge of a finite graph, we associate two opposite orientations. Okay. So then the neighbors of a directed edge from U to V are the directed edges from V to W, with a W not equal to U. Okay. So when we define the neighbors this way, and so then we can come up with the directed edge adjacency matrix. Then when we raise to the N's power, compute the trace, and that will give us as precisely the closed geodesic cycles of geodesic cycles of length N that we want. Okay. So because of this relation, we have another expression of a zeta function, which is one over the reverse characteristic polynomial of this edge adjacency matrix, T. Okay. So therefore for the graph, there are two ways to express the zeta function. One uses the vertical adjacency. The other one uses the directed edge adjacency. Okay. So this actually works for any regular graphs. So when we look at our case for the PGO2 over QP, then this A actually can be also expressed group theoretically as HEC operator, and the T is the evahoric HEC operator. So we can also consider the aughting L functions attached to graphs first and up, so one can take the finite dimensional unitary representations of the fundamental group, so which is the gamma, and then we can define the aughting L functions, and for aughting L functions also has two expressions, has the expressions in two ways, just like what we saw for the zeta function. So this was done by Harah, Hashimoto, and Stark-Tarras. Okay. And for the Riemann surface, the surfaces that are self-considered, okay. One can also do the same thing, take a finite dimensional unitary representation, and define the aughting L function. Okay. So this aughting L function in 2015, the poll showed that actually one can express this aughting L function as a determinant of 1 minus an operator, which she called a transfer operator, which depends on s and the representation. So this expression is just like a Hashimoto's expression. So what I could ask is that, well, what is the spectral expression, like the work of Sarnac-Voros, okay, using the spectrum, okay. And so what is the other side for this row, which is non-trivial? I didn't find it in the literature. Okay. So next, we go to the next stage from PGL-2 to PGL-3, okay. Again, more QP, okay. So instead of when we look at a PGL-2 part case, then we get the people's one regular tree. And for the PGL-3, then what we get is called a Bruhakti's building. So it is a contractable two-dimensional simplicial complex, okay. And the vertices are parameterized by the PGL-3 QP model by its maximum compact. And again, the two vertices adjacent, I mean, but it's defined by the same way as before. And the three mutually adjacent vertices form a chamber, although it's a triangle, okay. And the group PGL-3 QP will act on this building trans-trivial vertices and preserves the adjacency. So if we think of our trees as gluing together these infinite lines along some edges, then this two-dimensional building is obtained by gluing together Euclidean planes along some triangles, okay. So each Euclidean plane is tied by equilateral triangles, okay. So if I choose any point on this plane, okay. So it has, I mean, in this case, there are six vertices. So three vertices, well, are represented by lattice, which index P versus the original vertex. And the other three has the index P squared. So all the neighbors represented by lattices with index P, well, are represented by Hecker operator A1. And the other one is by Hecker operator A2. So the edge to the vertices, though, with index P are called a type I edges, and the other one called a type II edges, okay. So when we reverse the directions, then they will swap the types, okay. So the edges on this building are parametrized by a parahoric, cosets of parahoric subgroups, okay. So since we have this Euclidean plane, so our metric will be just a usual Euclidean metric. So when we consider the geometric, sorry, geodesic path, so I take an edge, say type I edge, I consider what are the allowable path that enables. So one is going to say to the right, so that's a type I. And it has two other type I edges, but then there's a turn. So we do not allow the turns, we have to go straightforward. So this is what happens in this plane. I mean, you can use a different plane. So as long as they put together, it gives us a straight line. So this kind of neighboring relations can be expressed by a parahoric operator, okay. We call the LE. And since type II edges are the opposite of type I, so and then the adjacency is a transpose of LE, okay. Now for the triangle, okay, so for each triangle, we associate the three directed triangles, okay. So we pick the orientation, which is such that the sides are the type I sides. So if I have these three vertices ABC, so we have ABC, then we have BCA and the CAB. So these are the three directed chambers. And the directed chambers are parameterized by the cosets of parahoric subgroups. And so if I take the directed chamber ABC, and its neighbor will be those I'm sharing the BCH edge. So the BCD with the D not equal to A, okay. So then this kind of adjacency is described by your Horry Hacker operator, LB. So all the combinatorial adjacency has a corresponding group theoretic operators to that. Okay. So we do the same thing as before, we choose the nice disputation preco-compact subgroup and mud out on the left side. We get a finite two-dimensional simplicial complex, okay. So then on this complex, we well consider the geodesics. So so it's a two-dimensional. So we have the, we also have two kinds of geodesics. So one is using the directed edges. So we have either geodesic cycles using all the type one edge or using all the type two edges. So counting those numbers is exactly the same expression. Okay. And that data function can be expressed as one over the determinant of one minus that this pair of Horry operator times U is a type one edge. If it's a type two for some reasons we are going up, then we put U squared because its neighbors well has index P squared, okay. And putting the zeta function of using the type one edges and type two edges together, we get as well, we get the edge data function for this finite quotient x gamma. So it's equal to one over that the leu and the other one, it's that one minus is transpose times U squared. Okay. So then because the two-dimensional, we also consider chamber zeta function. Okay. So we take a chamber, we just try and go and then we move it around following the allowable neighbors. Okay. So on this plane, so what it will without to give us a strip like this one to the till to the right or till to the left and the third one will be horizontal. So there are three possible strips. Okay. So and when we pass the quotient and we again count the geodesic, we call the galleries because it's two-dimensional of a given length. Okay. So as here, as we can see that when we take this strip, we have two orientations, we can go up or go down. So we take the orientation such that the boundaries are all the type two. So I mean, if we take the opposite one, it will be the same zeta. Okay. So I just want to make a remark that so if I take this strip, so I start with chamber, directed chamber x, then somewhere I have a gamma x. So in the when I pass the quotient, I will glue them together to get geodesic galleries. So I look at the boundary. Okay. It's a boundary is a type two geodesic cycle. Okay. But the way we glue it together, I mean, have two possibilities. One is that we end up with a cylinder. So it has two boundaries. And the other possibility is that we end up with a mobile strip. So in that case, we only have one boundary. So this is the interesting part. Okay. So now we have the description of the zeta functions for the edges and the zeta functions for the chambers. And put together, they also satisfy some identity similar to what we saw for the graphs. So in terms of, we can express that in terms of the operators about the edge adjacency and the chamber adjacency. Okay. Or just the other zeta. Okay. And it's equal to another the other side uses the operator of the vertices. Okay. So for the graphs, we have the denominator is a degree two polynomial in you for the PGO three is denominator is degree three polynomial in you and uses the both HEC operators. Okay. And the numerator is one minus U cube to the order characteristic of the this two dimensional simplicial complex. Okay. So the red is that we see that in terms of the zeta function. So this one over the determinant using the HEC operators. That's why they call that zeta function associated zeta zero because they use this operator on vertices. And the zeta zero actually is the length and cell function. Okay. So if I divide this data zero to the left hand side, so we do have the even index data so denominator and the top will be the zeta's index by the odd dimensions. And so one special thing I want to notice is that, well, for the chambers, one needs to change the sign. Okay. And this identity will extended to PGO M by Khan and JKU. Okay. And so there are some some subtleties, but the idea is what we see here. Can I ask a question? It's Peter. All right. When you say part two at the bottom of 20 of the page that that is a langans L function, you mean if you choose gamma, of course, most things are arithmetic once you're in a higher rank. But is that what you mean here? Or what's your definition of a langans L function in this context? So you look at the representations, which appears in L2 of the group mod of gamma on the left, and the spherical representations appear in this L function. So for each irreducible spherical representation, you have to associate the langans. So this thing is a product of all the representations, which appear there. Okay, all right. Okay. So there are three proofs to this identity. The first one is combinatorial. Okay. So to prove it, we take the logarithmic derivative of both sides and try to show that they are equal. So this one is really counting. This is how this identity was discovered because we counted the geodesic cycles and the geodesic galleries, and compare with the expression on the right-hand side. So this is hard work. Okay. So I will not say too much about it, but it's from this counting, we noticed that you see that when I define all these zeta functions, especially for the cycles, we are only defining in terms of the primes. Okay. I did not define it in terms of conjugacy classes of gamma because there's no good expression to say that which conjugacy classes of gamma, which we should pick, it certainly includes those conjugacy classes generated by primitive elements, but there are other things involved. Okay. And the second one, because of this our combinatorial objects have group backgrounds, so we use the group theory, so the use representation theory. Okay. So we regard these operators, the HEC operators and the parahoric operators and the euahoric HEC operators as the, I regard them as the operators on the L2 functions on the PGO3 QP model by gamma, which are writing variant under the maximum compact for A1A2 and the parahoric subgroup for the LE and the euahoric subgroups. Okay. So we really wanted to compute the eigenvalues to really find out the zeros of the top and the bottom and to see how they cancel and compare both sides. Okay. So when we see them, the functions is on the L2 space, so that means that we should think about the whole thing, the L2 space of the PGO3 QP model by gamma and the group PGO2 QP act on the right-hand side and decompose this space into the pieces of the unitary irreducible representations. Okay. So what of interest to us are those representations which contain the euahoric fixed vectors. Okay. So when we collect those and put it together and we can get what we want. Okay. So for such kind of representations, the work of Kasselman and Tadik have classified them into five types. Okay. So for each type, we just compute what I say, what are the eigenvalues of the euahoric HEC operators on the space of euahoric fixed vectors. Just compute them for each type. Okay. And likewise for the other operators. Okay. So this really allows us to find out the zeros of each term and then for each given representation, we can see just before our eyes how the zeros cancel with each other and to see that this identity holds. Okay. And the other proof is co-homological because this 1 minus u cube to the Euler characteristic. I mean this Euler characteristic is really very suggestive that they should have some co-homological way to prove it. Okay. And so for this, we construct a co-chain complex. Okay. And each co-chain group is the polynomial ring with coefficient c valued functions on the directed simplices like on the c0 will be on the vertices, c1 will be on the directed edges type 1 and type 2, and the c2 will be on the directed chambers. Okay. So we have and that then we come well then we have the co-boundary maps which are the some variations of the standard co-boundary maps. And the key is to construct endomorphisms on this co-chain. So we construct, find some automorphism phi 0 on the c0, the phi 1 on c1 and the phi 2 on c2. Okay. And such that that map like when I compare to another map, which is just multiplication by 1 minus u cube on each co-chain group. And these two endomorphisms of the co-chain, they are homotopic. Okay. When they are homotopic, that means that the, when I look at determinant of the first one using the phi's, phi i's on the cohomology H i is equal to the determinant of the second map 1 minus u cubed on H i. Okay. So then when I do the alternating product of say phi i on the cohomology group that's equal to the alternating product of the determinant of phi i on the cohomology group and similarly for the use 1 minus u cubed. But for 1 minus u cubed part that we can compute it, we get 1 minus u cubed times the dimension, which is the number of vertices minus twice the number of edges plus three times the number of chambers. Okay. So then we just need to compute the determinant of the phi on the left hand side. So we compute the determinant of phi on c naught is z naught to the inverse and the c1 is z1 inverse then times 1 minus u cubed to the number of edges on the c2 is z2 inverse times 1 minus u cubed of twice of numbers. So when you put it together, you'll see that I just cancel out quite right. Okay. And this cohomological method is very convenient. We can generalize it to get a similar identity for the RTL function associated to finite dimensional unitary representation of the group gamma. It's also the cohomological method that which allows the common e to prove that the z identity for the PGON case. Okay. So now how about the Riemann hypothesis? Okay. So we first we define that that this finite quotient is a rummaging complex. If all the trivial zeros coming from this determinant using the HECA operators have the same absolute value, which is a p to the minus 1. Okay. And from the representations, the already language, this means that all the non-trivial spherical representations appear in the L2 or PGO3 QP mod out by gamma while are tampered. And because we have this expressed information about the zeros, okay. So from the rummaging on the HECA operator side, we can see that that's equivalent to all the representations appear in that with the parahoric invariant containing parahoric invariant vectors that they should be tampered. And likewise, the euahoric ones should also be tampered. So they are all, I mean, one implies any other. So in this sense, we say that this finite quotient is rummaging complex if and only if it's zeta i, okay, is satisfied Riemann hypothesis for i equal to 0, 1, 2. Okay. So this Riemann hypothesis doesn't mean that they all have the same eigenvalues but I mean that the representations are tampered, but they have, they will have different bunches of eigenvalues as you can see from in each case. What are the possible eigenvalues? Okay. So next, another degree two, another dimension two building is associated to the symplatic groups. Okay. So in this case, each plane is parametrized, tiled by triangles of different shapes. But the main difference is that there are two kinds of vertices, the black ones, they are primitive vertices, they are parametrized by the pgsp4 or mod-mechanos compact, but they are the red vertices, I mean, they don't show up. Okay. So between the black vertices, there are two kinds of edges, we say that of a type spin and the standard and there are two types of the edges. So the slanted ones are the spin type and the dotted ones are the standard type. There are also two kinds of chambers and we can do the same game as before. We also came with a similar identity. Okay. So this identity is that for the one-dimensional cycles of spin type and the standard type divided by two-dimensional cycles of spin type is equal to one minus u square to the Euler characteristic, times another factor, then times the data function of vertices of spin type. Okay. So the data function of vertices of spin type is the Linen's L function of the spin type. It's a degree four in u, okay, using the two-hack operators. And we can, so this extra factor here involves the county, the number of special vertices and the non-special vertices. So the identity is more complicated. We can also swap spin and the standard to get another identity. Okay. So then the z zero of standard is a degree five in u. So that is called the Linen's L function of the standard type. Okay. And the identity is a lot more complicated, but one has that. So this identity for the simplicity group is proved using the representation theoretical method by comparing the eigenvalues of the operators. Okay. So I would like to end this talk by talking about that, the distribution of primes. So just like on the Riemann zeta function, if we know the behavior of the zeta functions, then we know that the number of primes up to x has the main term, which is a logarithmic integral of x, which comes from the pole at s equal to one. And the error term, well, it depends on the location of the non-trivial zeros. Okay. So similarly, for what we have considered here for the compare Riemann surface, let's start again, his thesis show that if we consider the number of primes of this compare Riemann surface, what length up to x, its main term comes from the pole at s equal to one of the zeta functions, which is a logarithmic integral x. And if it has a bunch of the real zeros in the critical strip between one half and one, and each of that will also contribute to a term, so logarithmic integral or x to the power is given by the location of the real zeros. And the error term is given by x to the three quarters times log x squared. Okay. And for the combinatorial zeta functions, we can see that for each dimension and each given type, and the zeta function is a product of the prime geodesics one over one minus length to the c, and that there is a suitable operator so that this zeta functions one over determinant of one minus that operator times u. Okay. So likewise, if we know the location of the poles of the zeta, in other words, we want to know the eigenvalues of this operator t. So if we can determine the largest eigenvalue in absolute value of t, that's called a lambda. And so I look at the circle absolute value u equal to lambda, and we count the number of distinct eigenvalues of t which lie on that circle, that number is called a delta. And then if we also know the multiplicity of each eigenvalue with absolute value lambda, so this multiplicity is independent of the eigenvalue with largest possible absolute value, that's called a kappa. Okay. So suppose we know these three quantities. Okay. Then this delta here has a combinatorial meaning is the greatest common device of the length of all the primes of that given dimension of that type. Okay. And then we also know that corresponding prime geodesic theorem, also for large n, we know the number of geodesics with the lengths, now will be n times this gcd delta. So it's the kappa with multiplicity times largest eigenvalue in absolute value lambda raised to the length over n. Okay. And the geodesic with lengths less than n delta is a proportion of what we saw with lengths equal to n delta. Okay. So this is known for the pg2qp, this by Hashimoto. And for pg3qp, and while we have this x representation theory actually tells you that explicitly what, well, the eigenvalues, so we know that well, we can also give a combinatorial proof and also do that for pgsp4. So this is a joint work with my ex-phd student. So in his thesis, he proved this using the group language. So then I gave a combinatorial way to see that I think it's more transparent. Okay. So likewise, for the pg2 and pg3, if the quotient is Rommnogen, then we get a good bound for the error term. Okay. So that's all I would like to say. And so apparently what's lacking in this thing is what happens if I replace periodic fields back by R for pg3 and pgon, et cetera. So thank you very much for your attention.