[SOUND] [MUSIC] Theorem. The dimension of the space holomorphic Jacobi form of weight k and the index m is a finite dimension. Proof. In the previous lecture we construct, Special pullbacks of Jacobi modular form. More exactly, for any irrational vector. P minus Q in Q power of 2. We construct and for any Jacobi form, polymorphic Jacobi form of weight k and index m. We construct the following, function. By definition, this is e to the power 2 pi i, q to the square tau plus pq, times phi (tau) q tau plus p. This special value of the Jacobi modular form in this rational point is a modular form of weight k, with respect to a congruence of group gamma x, of the group with the character x, gamma x is subgroup, SL2. Now we can choose many rational plans. For example, we can choose 2M point X1, X2 and so on, then The map, Pi tao Z to pi xi I tao 2m Xi is not equal to Xj EQ2 then this mat is injective. Why this is true? Because we proved in the corollary 3. Then the elliptic function phi is determined uniquely up to multiplicative comfort by zero. Therefore, this function is defined uniquely by these images. Immediately, we get. Then, as a dimension. Of the space of holomorphic Jacobi form of weight k, in index m. Smaller or equal than the sum of the dimension of the corresponding spaces of modular forms. And this sum is finite. This theorem is proved. I would like to tell you that this estimation Is not good at all. Later we get better result about the dimension of the space of polymorphic Jacobi form. But, at the moment, this estimation gives us this very important result. Which we claim In our first lecture. Then for all k and m. The space of the Jacobi modular form is a fine to be matched. To get a better estimation, I would like to consider Taylor expansion to Jacobi modular form. It means that I would like to analyse Jacobi form in the neighborhood of 0 of the complex plane. Section 5.2. Taylor Expansions. Of Jacobi forms. Let i B in Jacobi form, what is corresponding Fourier expansion. In the summation, 4nm- l to the square is greater or equal to 0 q = e to the power of 2 pi i tau, r = e to the power 2 pi i z. I would like to analyze the Taylor expansion in z. Fd(tau) z to the power d, d is greater or equal to d0 is greater or equal to 0, where d0. This is order of zero, of Jacobi form. Analyzing the Taylor expansion, I would like to get a better estimation on the dimension of the space of Jacobi forms. First of all, let me analyze F0. By definition, this is a value on Jacobi form zero, and we can find it's Fourier expansion. Sum over N, great equal to zero, then it's the sum over 8 charged at 4 and NM minus L to the square is greater or equal to 0. So this sum is finite. N,l, q to the power n. This is the modular form of weight K with the respect to SL2. The function phi, the Jacobi modular form, is periodic. Is equal to five tao vector. Therefore, all Taylor coefficients are also periodic. And we can define Fourier expansion of this function. The following property is true. If d is positive then we don't have the constant in the Fourier expansion. N is strictly positive here. Why do you have this property? Let's analyze once more, the Fourier expansion, of Jacobi form. If n is equal to zero. Then the corresponding folic coefficient is zero for all A, for all non-zero A. Therefore, we can get positive power. Of D, Only from purification with M positive. It means that, any power, Z to the power D comes with some positive power of Q So this, Property is true. But in general, this is not a modular form [MUSIC]