The fact which we analyzed and proved is very important. And you can find the application of this fact, for example in the theory of automorphic Borcherds product. And now I would like to analyze it once more. So let me compare holomorphic Jacobi form and Jacobi weak form. We know that's the space of Jacobi. Form of weight K and index M I can see that I clear Jacobi form, is the sub space of the space of Jacobi weak form. And now we saw the difference if it was the first space. We have non zero 48 K efficiency. >> Only for indices with the positive, non-negative hyperbolic norm. Then for the second Space. We see that is a no, Is greater or equal to minus m to the square. So we have finite number of possibilities. Now I would like to analyze this effect for The latest is, more exactly is, I would like to consider, uni modular lattice. Let's analyze the following example. We can see that in theorem unimodular lattice, L2 is equal to L. In this case, we mention this, we have only one class and the splitting principle Gives us the following. Factorization, Jacobi form, for example, let us consider weak Jacobi form, [INAUDIBLE] K, [INAUDIBLE] Then we have the simplest possible splitting. because we have no characteristic at all. The Fourier expansion was a function f0 is the following. [SOUND] And we see that n here is non-negative. Because Fourier coefficients of weak Jacobi form has this property. Moreover, we see that in this case, we have no difference between Weak Jacobi form, and holomorphic Jacobi form. Because invariant mu, of the latest l, is equal to 0. It's very, strange from the point of view of the seer of Jacobi form and one over [INAUDIBLE] effect. Then the space of weak Jacobi form coincide with the space of holomorphic Jacobi form. If the lattice L is unimodular. Moreover, we can define this space. The value of Jacobi theta series at 0 is equal to f0 tao theta l tao 0 and theta l tao 0. Is the usual theta series of the lattice, L, so modular form of weight N0 over 2 with respect to s into that. If the rank of A is equal to N 0, so you see that this is a modular form. Of weight. K with respect of SL2 Zed as any Value of zero of Jacobi modular form, holomorphic or with Jacobi form, so it proves the following fact. Proposition. That L is uni modular. Then, the space of Jacoby weak form, coincide with the space of Jacobi automorphic form is isomorphic. As linear space to the space of modular form of weight k minus n 0 over 2 with respect to s i 2 z. This is because Jacobi, the special value of Jacobi. That this series is of modular form and we have the [INAUDIBLE] for the [INAUDIBLE] of the space of the cast form very similar right? If we take the space of then the space is holomorphic to the space of parabolic modular form with respect to the group sh to z. This because our theta series, the first of the 0s for every coefficient of theta series is equal to 1, so our theta series starts from 1. This is a result for the unimodular lattices and this fact explains why the problem To define the structure of the graded ring of the weak Jacobi form for unimodular lattice is more complicated than non unimodular one. For example, if we take the problem to define The structure, the algebraic structure of the following degraded ring. J star E8 star, weak, it is structure, k, E8, sorry. K m k mm z. Let's put this [INAUDIBLE]. Certainly, I would like to consider only symmetric function. Without any explanation I add here [INAUDIBLE] the group generated by reflection in this lattice. Then this problem is very complicated and this is due to the fact that we have no weak Jacobi form. It's better to say, no special weak Jacobi form, it better to say then we have this very simple description of the space of weak Jacobi form for the latest N. This is weak Jacobi form of index one but if you put the same question for example for the latest d8, this problem will be much easier and in principle using the method which we describe in the next lectures you can solve this problem without great difficulties. But for the next lecture I would like to analyze, The splitting principle from the following point of view. So, this is our main result, the main result of the second part of our lecture. And in the next lecture, I would like to analyze this splitting, so if we represent Jacobi modular form as the product of vector value, modular form. Of this vector in the one variable times the vector, of [INAUDIBLE] series, then the [INAUDIBLE] of [INAUDIBLE] series. Has very well defined behavior, with respect to the action of modular transformation. This is so called the Weil representation and it mean that this function in one grade will be really a vector valued modular form, defined by the Weil representation. So, this transformation property we consider in the next lecture. [SOUND] [MUSIC]