[MUSIC] Now I give the definition of Jacobi modular group. The Jacobi modular group, Is the following subgroup gamma J. Of the Siegel modular group. The last line, Contains three 0 and 1. The second column contains three 0 and 1. And we have some element in all other places. Please check the gamma J is a subgroup Sp2(Z). This is so-called parabolic set group of the Siegel modular group fixing one vector. What structure has this group? First of all, we can construct the following embedding of the usual modular group, gamma J. A, b, c, d in this square bracket means the following simplistic metrics. We have a, b, c, d in these quantity, quantity of the group, a, b, c, d and 0, in all other places. This is some gamma J, because, Sl2(Z), Is Sp1(Z) which certainly is a group of Sp2(Z). If I take an arbitrary, now, an arbitrary element a, b, c, d. Take an arbitrary element in gamma J, then certainly, you see that this matrix and Sl2(Z) for the same reason. And now, if we multiply M, By a, b, c, d to the power of -1, We get the following, Matrix. In gamma J. Please check, then this element is defined by q. This follows from the definition of gamma J, gamma J is of Sp2(Z). And all matrices of this form, Are elements of the Heisenberg group, this is the seconds group of gamma J, which would I would like to define now. The Heisenberg group, H(Z) gamma J. By definition, H(Z) contains all unimportant elements, Of the Jacobi group. Pq and r, r and Z. The last calculation, Shows, That gamma J is the product of Sl2(Z) and the Heisenberg. Because any element we can represent as a product of two elements. For this, you have to find four elements in Sl2. For elements a, b, c, d, in the Sl2 part of the Jacobi group. The rest is the Heisenberg part. And now without any details, I can give you the second definition of Jacobi forms. A Jacobi form, Is a function in two variables, tau and Z. The Siegel upper half-plane H2 contains all symmetric complex matrices tau, Z, Z, omega. Such that imaginary part of Z is positive. But this is equivalent to the fact that imaginary part of tau and the imaginary part of omega are positive, so we'll have the diagonal limits are in the half plane. And the major new part of this latest expositive gives us the following quality in major part of tau times imaginary part of minus imaginary part Z to the square is strictly positive. Therefore, for any tau and Z in the direct product of H1xC, there is a [INAUDIBLE] in H1 such that, tau, Z, Z, omega is an element of the Siegel upper half-plane. And now if phi is a Jacobi form of weight k and index m, I can define the function phi tilde M at Z, which is by definition is equal phi(tau Z) times e to the power of 2 pi i m omega, where M is the index. And then you can believe me, we can see this equation in the lecture. Then the modular equation, And the elliptic equation in the definition of Jacobi form, Are equivalent. So the only modular equation, Jacobi modular equation for the Jacobi modular group gamma J. More exactly, phi(tau, Z) satisfy, The two equations, the modular equation and the elliptical equation if and only if, This modifies function phi M tilde, which is now the function in the Siegel upper half-plane satisfy only one functional equation. Phi M tilde / K operating g is equal to phi tilde M for any g in the Jacobi modular group. What's slash operator here, this is /K operating, For the Siegel modular group Sp2, there. So you see then two rather complicated modular. And the modular in the elliptic equation, I would like to emphasize that the both equation are rather complicated. We have rather non-trivial automorphic factors in there. The equivalent to the only, very clear modular equation for the subgroup of the Siegel modular group. In the next lecture, we can see that the second definition of Jacobi forms in some detail. [MUSIC]