[MUSIC] We'll continue our study of the Heisenberg rule, H vec. We can calculate the center of the group H. The center, it is the group of all element of this type. And now, we can understand the structure of [INAUDIBLE] group. First of all, We have the map of the adjective group that in the group H [INAUDIBLE] defined as follows. The image of r is equal to the center element 0, 0, r. Then we have the following map for any p q, r with a node by p q, its image in Z 2. This map is certainly surjective and the first map is injective, so we see that the Heisenberg group H [INAUDIBLE] is the central extension Of the abelian group Z times. I would like to add one exercise. It's rather interesting and important property of the Heisenberg rule. We can define the following binary character other highs in their group. By definition, VH (PQ) R=-1 to the power P+Q +p*q+r. These prove that we really have a binary character then These find as a Kernel of this [INAUDIBLE] Moreover, you can analyze this sub group and try to study other subgroups of H debt. I can formulate all exercise in the appendix to this lecture, but now, I would like to underline [SOUND] the action of the modular group on the Heisenberg group. S to z act by conjugation on the hazard group more exactly for any M in S h2 (Z). We have that [M] this is the image of M in the Jacobi group, times P, q, r times m minus one is equal to m times p, q, r. Please check this. In particular, we see that H is a normal subgroup of the Jacobi modular group. And the Jacobi modular group is a semi direct product of the modular group s into z and we will identify s into z with image In the Jacobi group. Phi, H, Zeta, this is, the structure, of the Jacobi modular group. You see that the Jacobi modular group contains put a simple S into that part. And another factor which is unit group, so this modular or group as a group S but with an additional factor. Now I would like to study, the action, section 3.4. The action of gamma g on the [INAUDIBLE] upper half plane, H2. First of all, we take, we can analyze first of all, all elements in M in S. So and we try to understand this action Then, we'll do the same for h and h(z). And we'll see, that this action is related to the modular equation in the definition of Jacobi modular form. And the second action is related to the elliptic modular equation in the definition. Jacobi forms but we can align this action not only for interval Jacobi group but in principle we can consider the real. Jacobi group is a group of real points because it's group which is a semi derived product of. And H over R and to analyze the action of this group we have to analyze the action of it H sub. [SOUND] Now I would like to calculate the action of the to z sub ru of the on the plane. So Z, this is and then of the and M, Symmetric and Sl2 R We have to calculate the action of the following symplectic matrix on tau, z, z this is equal, a Tao plus b, a Zed, Zed Omega times c tau plus d, c Zed, 0,1, to the power -1. In the notation of the selected group. Here we have the block AZ plus B and here CZ plus D to the power minus 1. This is no problem with calculation of the inverse matrix. -1, -6Z over C tau + d 0, 1 and now, we can calculate the result. [SOUND] a tau + b over c 2 + d, then z over c2 + d. Here, by symmetry, we have the same element. And the last element is equal to omega- cz^2 over c2+d. Length is the action of the sl2 element on the upper half plane. You see? That this term is very close to the exponential term in the definition of j core b modular four, so how we can formulate this. Let me analyze the action of this matrix on the following functions. This is a slash operator 4 ASP 2. Then, we get C top plus d to the power minus k. This is factor. Then e to the power minus 2 Pi r M because we'll have M here CZ squared. Times function A was B over C + d Z over [INAUDIBLE] + d e to the power 2pim omega. So you see that omega here and omega here are really the same, but we have this additional term which we have seen in the definition of the [MUSIC]