[MUSIC] Let us analyze the modular and the elliptic equation for the simplest possible ellipse. [SOUND] The model equation, For the is 001 gives us the fact that our function is periodic with period one for the first variable. The elliptic equation where (lambda, mu) = (0,1) gives us the second periodicity with respect to that. Therefore we can consider the Fourier expansion of this function. Phi has a Fourier expansion, Of the following form. A (n,l), this is the Fourier coefficient of this function phi and the Fourier coefficinet has two indexes. The first index is related to the variable tau. N times tau, and the second index a is related to that. n and a are integers. But if you consider all holomorphic function with arbitrary Fourier expansion, then the space of corresponding function will be very, very large. So we would like to put some, Condition on our Fourier expansion. And now I can finish the definition of Jacobi form, The function phi is called, Holomorphic. Or cusp. Or weak. Jacobi, Form of weight, k and index, M. If the Fourier expansion of this function has the following property, zk efficient a ( n,l ) is a Fourier coefficient. a (n, phi) = 0 unless, 4nm- l to the square is greater or equal to 0. I repeat, f is called holomorphic. Jacobi form of weight k and index m if we have this property of Fourier coefficient. The function phi is called cusp. Jacobi form of weight k and in index m. If we have a stronger condition, For nm- l is strictly positive. Cusp. And our is called weak, Jacobi form of weight k and index m. If a and l is equal to zero, unless, n is not. So we have three definition. We have the definition of holomorphic Jacobi form of weight k and index m. Of cusp with Jacobi form of weight k and index m and of weak Jacobi, form of weight k and index m. So maybe I can introduce here some notations. Notations. We denote this place of holomorphic form of weight k and index m by g, k, m. Then, The space of cusp form, we denote by gkm cusp. And the space of weak Jacobi form, of weight k and the index m, we denote by gd. So this is, I repeat, the space of holomorphic. Jacobi form, the space of cusp form and the space of weak, Jacobi, Forms, Of weight k and index m. Maybe here I can add, A very clear remark. Then in the space of Jacobi cusp form is subspace of Jacobi holomorphic form. And the space of holomorphic form is subspace of Jacobi weak form of the same weight of the same index. Because the third condition is the weakest condition and the cusp condition is the strongest condition. The main fact about this space, which we prove later. [SOUND] Fact, Then all these spaces are final dimension. But now I would like to analyze this condition, the condition on the Fourier expansion, Of Jacobi form. Why do we have that? Later I'll give you many different explanation of the first condition and of the third condition. But now I would like only analyze one example. In the first lecture, we consider it, The Jacobi theta series. Theta l u into and z. u is a known zero vector of an even integral positive definite quadratic l. Then this function, By definition, this is the generating or if you like, petition function of the following form, This summation is taken of all vectors and the latest A. This is the definition of this type of function, we calculated its Fourier expansion, Where r the summation of all n and l and z where r, this coefficient is the number of vector d in l of the square, n maybe remind you. This quantity, This is a number of vector v in l such that v to the square = 2n and the scale of product of (v,u) = l. Then, you see, then this number is not 0, implies that 4 nm- l to the square is greater to pr equal to 0, why? Because the matrix, ((v,v) (v,u) (v,u) (u,u)) = ( 2n, l, l, 2n) is greater or equal to 0. Because v and u are two actors of positive definite quadratic form. [SOUND]