[SOUND] [MUSIC] To describe this behavior, we need the following definition. The level. Of the latest L, by definition, as this is, The minimum positive q such that, q times the joule lattice is a sub lattice of L. So, you can really analyze at this level, or in [FOREIGN], for different lattices, but using the structure of the dual lattice. The fact that this is determinant of L, you can prove that q is always a divisor of 2 determinant, L. Now, we need the Hecke congruence group, gamma 0 (q) of s into Z. A group of elements such that c is congruent to 0 modular q. And now we have the following fact. To prove it, we have to analyze very careful the modular behavior of the Jacobi theta series or general Siegel theta series, with group heuristic, so. Follows, From the modular, Properties, Of theta L, h. The fact is the following. For any h in the discriminant group, the component. f h of the vector-valued modular form satisfies the following functional equation. So maybe I need more place for this equation. Fh (a tau + b / c tau + d), that is a modular form of weight k- n0/2. With some naturally character but multiply system. L of order, 8 of the metrics. M this our a, b, c, d in gamma 0 q times the following constant, minus pi i a b h to the square. And then we have, The component with index of our vector. So we'll have the following functional equation for the component of the vector-valued modular form SL2z vector-valued modular form. But this is functional equation with respect to the Hecke congruence subgroup. If n0, the realm of a is even, then we have the following explicit formula for this function. In this case, this is not of order 8, but simply a quadratic character of the following form. The sign of d means plus minus 1 to the power n0 / 2. And then the chronic symbol, Of the discriminate of a over the absolute value of d. a, b, c, d are elements of the metrics from the Hecke congruence subgroup. So you see that in fact you finalize as a component of our vector modular form with respect to the Hecke congruence subgroup. We have this orbit a relation between the component of our lattice. So, in some simple cases, let us use then, the discriminate as a group is a sightly group of prime order. Then there was a chance, I formulate it like this, To write all components of our vector related modular form in terms of fo, and tau. So in some cases, this is really true. For example, you can can analyze the Eiglier Zigea book. And to understand, for example, the structure of the Jacobi form of index 1. It means p is equal to 2 in this case and for some other prime indices. So if this is true then we can prove sometimes that the space of Jacobi modular form, it's just isomorphic, through the space of vector-valued. I have here the symbol of vector, vector-valued modular form of weight k- n0 / 2, (tau). But in some cases, the space, if the last argument involves the component is true. And if you can write down all component in terms of the 0s component then will be isomorphic to some space. M which is subspace. In the space of modular form of weight k- n0 / 2 with respect to the congruence (gamma o (q)). This is exactly the case of Eiglier Ziga modular form. You can find this theorem in the book of factors again. Where we can prove that the space of Jacobi modular form is isomorphic to some subspace, so called Kohnen plus subspace of the modular form of half integral index. So, in this direction, you can start a realize structure of the space of Jacobi modular form for the lattice L and the main conclusion of this lecture. Then the main object, For this space is the discriminate group. Group D A defines discriminate group, and I can add. Now, I can not prove it, cannot give illustration, but in the PDF file to this lecture maybe I add some remark on this. Then you can add also the property of the finite octagonal group of this finite quadratic model. For example, the action of this group could give us a very nice decomposition of this space. With respect to some characters of this finite octagonal group. So you see then the Jacobi modular form and vector-valued modular forms are equivalent object. But sometimes Jacobi modular forms has a lot of very nice features. We can take the multiplication of Jacobi form. We can write down polynomials in term for Jacobi form. So this type of effect, Jacobi forms as Jacobi forms, but not as vector-valued modular form, I would like to study in the last part of our course. [MUSIC]