[SOUND] [MUSIC] Hi there colleagues and dear participants of my course. Today, we start the ninth section. It's titled the Weil, Representation. This is closely related to the splitting principle we analyzed in the last lecture. Let phi be in Jacobi form of weight k with a lattice L, a holomorphic Jacobi form. According to the splitting principle, we can represent our Jacobi form as a product of a function, Depending only on tau. The summation of all elements in the dual group, in the dual lattice, modulo Z lattice times, The Jacobi theta series with characteristic h. So we can write it as, The vector-valued function, phi(tau) times the vector-valued theta series L. So phi L(tau), this is the vector (fh(tau)), the vector of lengths determinant of L. The same for the vector of the theta series. This is the splitting principle we analyzed, In the last lecture. Moreover, we got an explicit formula for the Fourier expansion of the function f(tau). This is the sum of the Fourier coefficients of the Jacobi form phi, the following form (N + h to the square / 2, h) e to the power 2 pi i capital N tau, where capital N is a non-negative rational number such that N is congruent to -h to the square / 2 modulo Z. And here, the capital N is non-negative because, 2N = the hyperbolic norm, Of the index of the corresponding, Fourier coefficient. And this, the hyperbolic norm of its index is non-negative. This is because phi is holomorphic. In this definition, we see that we compare N with h squared / 2 modulo Z. And this number, the square of h, is well defined modulo 2Z, modulos are even numbers. So the next small section is the discriminant, Group, Of the lattice L. My definition, as a group, as a discriminant group, D(L), this is the quotient of the dual lattice modulo lattice. The order of the group is the determinant of L. But the square of any vector h in L star is well defined, Modulo 2Z because if we compare this product, we get h to the square + 2 (h,v) + v to the square. This number is even integral. And v to the square is even integral because the lattice L is even. So we always work with even lattice. Moreover, the usual scalar product for any h and g in L star ( h + v, g + u) = (h,g) + (v,g) + (h,u) + (v,u) is congruent, Of (h,g) modulo Z. Therefore, on the discriminant group, we have a symmetric by linear pairing, With the values in the field of rational numbers modulo the integral lattice. And this pairing is simply induced by our quadratic formula. Moreover, For any h bar in D(L) in class, its square with respect to this pairing is well defined modulo 2Z. So if we analyze the formula from the splitting principle, now you see that this number is well defined, Modulo Z. And our formula in this splitting principle is really determined only on the discriminant group. So we can see them only determinant of L, classes in this discriminant group. [SOUND] Now I would like to analyze [SOUND] the modular behavior of vector-valued theta series. Modular properties, Of our vector-valued theta series, with determinant L components. Let me start with one component. I would like to calculate it for tau + 1. By definition, we'll have the theta series with characteristic e to the power pi i, then the scalar square of ((v + h, v + h) tau + 2(v + h, z)), the summation of all v in L. And we analyze its behavior under this transformation. So we add here + 1. What do we get? The square, Is well defined modulo 2Z. So we get our function up to the vector e to the power pi i h to the square. [SOUND] [MUSIC]