[MUSIC] Now I can rewrite this formula in terms of the vector valued theta series, theta L (tau + 1, z) = matrix U (T) theta L, h (tau z), where U(T) =diag (unitary matrix). T here, certainly, this is a translation in the modular group. This formula is simple, but now I would like to show you without proof, the next formula. We can calculate, The vector valued theta series, the vector valued Jacobi theta series after the modular transformation. We get tau to the power n0 over 2 (- i) to the power n0 over 2 (det L) to the power -one-half (matrix e to the power 2pi i (gh)) gh in L stop over L. This is the matrix of dimension determinant of n times, now maybe I have no place, so later we'll add it. Then we have the standard Jacobi vector e to the power pi i z to the square over tau, theta L (tau z). So you see that our vector valued Jacobi theta series transforms like modular form like Jacobi form of weight n0 over 2 and there's this Jacobi factor. But we get a vector valued Jacobi form. This matrix we denote by U(S). S is usual, it's a modular [INAUDIBLE] One can check and this is a simple exercise, that U(S), Is, A unitary matrix. This is a unit matrix of rank determined of L. So, we'll have two function equation, the first one is very simple. The second I give you without proof, but it's verifiable if you explicit reference to the proof of this fact. But you can do it using the the parcel summation formula. But in the both function equation, we have, 2, Unitary matrix. We know then the translation and the evolution generate the full modular group SL2(Z). Therefore, we have the following result for the vector valued Jacobi theta series. For any M=a,b,c,d SL2 (z) theta L (M tau z over c tau + d) = (c tau + d) to the power n0 over 2 U(M) theta L (tau z), where, U(M) is a unitary. Matrix. So, for, Even n0 for all lattices of even rank. We get a representation U, Of the modular group, in the unitary group, Of order, Determinant. This is true if the rank of the lattice n is even. If, The rank of L = n0 is ought. The result is true, but U will be a representation of the double cover of this group. In both cases, this representation is called, The Weil, Representation Of the lattice, L, but for ought n0, we have to consider the double cover of SL2 because certainly, This square root depends on the choice of the square root. The choice of this square root, Will determine and project, the representation of the group, SL2. And this is the second factor in the splitting formula of an arbitrary Jacobi form. Let's analyze the first formula of our lecture today. Jacobi theta series. Jacobi function in many variables is a product of two factors, a vector valued function in tau and a vector valued Jacobi theta series. And the last result we can formulate as follows, so, This functional equation implies that theta L (tau, z) is a vector valued, Jacobi modular form, Of weight n0 over 2 for the lattice L. But what can we tell about the first vector? In fact, we prove the following theorem. Let phi be holomorphic Jacobi form of weight K. Then phi (tau z) = vector valued function times the vector valued Jacobi form, and, Phi L (1) is a vector-valued modular form with respect, To the the full modular group S L(z), more exactly this function satisfies the following functional equation. Phi L (a tau + b over c tau + d) = (C tau + d) to the power of k- n0 over 2 x U(M). The conjugation, it's a complex conjugation of the value representation of the matrix M x iL(tau). For any, M, And SL2(Z). This is direct corollary from the functional equation of the Jacobi vector-valued theta series. Please write down the functional equation for the Jacobi form and then you get this functional equation for the vector valued modular form. The only fact which we use here, this is a fact that our matrix U is a unitary matrix. Moreover, the corresponding vector-valued modular form is holomorphic at infinity. L tau is holomorphic, At infinity because for any component I have seen this formula today. We have this, Type of Fourier expansion where n is strictly is non-negative. hL* over L. From this formula falls then our vector valued modular form is parabolic, It means that here the condition will be a little bit stronger. We get strictly positive n in Fourier expansion if Jacobi form phi is Jacobi cusp form. This is the result of the splitting principle considered in the last lecture, together with the fact than the vector valued theta series of all Jacobi theta characteristic is a vector valued modular form. [MUSIC]