[SOUND] [MUSIC] So I would like to give you the formal definition Of Jacobi modular form in many variables. First of all, for this definition, we have to fix a lattice a. In this case, in the case of the first example, our lattice was D8. Now we fix An even integral, positive definite, Lattice L of rank n zero. Is that right? Certainly is positive. What does it mean? A lattice means a free zed model. As a model, l is holomorphic to z to the power n0. Moreover, we have a symmetric bilinear form on l, a quadratic form, with integral value. This explains the term, integral. Moreover, this bilinear form, symmetric bilinear form, is even. Even means that for arbitrary l, in the lattice l, it's scalar square which we denote also by l to the square, belong to the even numbers, it's even. Positive definite, it's clear then for any number 0 element of l, it's scalar. Square is strictly positive. So instead of the lattice, D8 now would take another three even integral positive definite lattice of rank n0. Tau is in H1, and Zed, Zed [INAUDIBLE] variable. Now, this is a vector in the complexification of the lattice L. The dimension, the complex dimension, of this linear space is equal to n0. A Jacobi form Of weight, k, k is usually integral in our consideration. But as in the previous part of our course, sometimes the weight will be half integral, but I don't like to fix the formal definition in this case. So a Jacobi form of weight k and index. M, m is non-negative integral. With respect for the lattice. L is a holomorphic. Function Phi on the product of the usual upper half-plane. And the complex space of dimension n 0, which satisfies two functional equation. The modular equation For any S L 2 Zed, element a, b, c, d. We get, ct plus d to the power k. K is a weight, E to the power BIC, the a scalar square of Z, over C tau, plus D. I would like to emphasize that on the exponent we'll have PII, not two PII. Then phi tau zed. And this is true for other 3 a, b, c, d, in S L 2 zed. This is the first modular equation. Then we have the second equation. Function with respect to the translation lambda tau plus mu. Where lambda and mu are two vectors in the lattice L. This quasi periodic function with a factor of quasi periodicity minus pi. Lambda to the square, tau + 2 lambda Zed. This is the first part of our definition, so a Jacobi modular form is first of all holomorphic function satisfying two functional equations. In the second part of the definition, I have to put some restriction on the Fourier expansion of this function. But to define Fourier expansion, we need to consider the dual lattice. L stop. By definition, the dual lattice. This lattice of all vectors v in the rational quadratic space L times of Q, such that for any vector a in the lattice a; v, l is integral. Our lattice is integral. So L is a sub lattice of the dual lattice. Now I would like to analyze the second equation. Our function is periodic with respect to Zed. Moreover, our function is periodic with respect to tau. It follows from the first equation. So now, [SOUND] according to the first two equation Our function is periodic with respect to tau, is period one. Certainly, this property follows from the first modular equation. If for the lattice a, b, c, d, you take the metrics 1, 1, 0, 1. It's usual periodicity of modular form. Moreover, our function is periodic with respect to zed, where mu is an arbitrary vector in the lattice L. The lattice of period is L. Due to this, we can write down Fourier expansion. Our holomorphic function. It has the following form. Of the sum, any Fourier coefficient has two indices, n and l. N is in z, but l is in the dual lattice. E to the power two by I N tau plus the scalar product l and zed. You see that our modular form is periodic, with respect to the lattice of period L. So if you change that by Zed plus mu, we have the variable vector in this free expansion. So this short remark for those participants who are not very sure about the property of Fourier expansion or function in many variables. But you can check it in any textbook. So this is the fall of the Fourier expansion of this periodic function. And now, to finish the definition of the Jacobi modular form, I can put three additional restrictions on the free expansion of Jacobi form. The function which satisfies, polymorphic function which satisfies two functional equations, a modular equation and a medial equation, is called weak Jacobi form. If we have non zero Fourier coefficients, only for non-negative n. This is the first condition. The same function is called holomorphic. Jacobi form. Not holomorphic function, but holomorphic Jacobi form. If its Fourier expansion satisfies the following condition. The Fourier coefficient is not equal to 0, then 2 n m minus the normal l is not negative. The same function is called parabolic or. Parabolic Jacobi form. If we have a more stronger condition, 2nm, minus ll, is strictly positive. So have three spaces and we use the full equation for these spaces. For the first space, weak Jacobi form of weight k, for the lattice l and index m, this is a larger space than for the space of holomorphic Jacobi form. We use the letter J and the space of cusp form. It's possible to prove that all these spaces are finite dimensions. [SOUND]