[MUSIC] In the proof of this lemma, I simply repeated the argument about the Taylor expansion, which we used in the last lecture. Moreover, in this last lecture, we proved that the dimension of Jacobi cusp form of weight 8 and index 2. The smaller or equal to one. The same we prove for weight 6 and index 3, and we use only the argument about the Taylor expansion and about the number of zeros of Jacobi theta series. The Jacobi theta series. I can construct the corresponding generators. For the first space We can take the following function. We take eta (tau, zed) to the power 4. We get Jacobi form of index 2, with the character V eta to the power 12, to kill it, we're at the 12th power of the Dedekind Eta function. So, this is Jacobican form of weight 8 and index 2. The most simple form for this function. This is eta cubed theta to the power 4. Then, I can construct cusp form of weight 6 and index 3. It will be the 6th power of the product of eta and theta. So the index is 3, the weight is 6. The character is. And this form is a cusp form. So we found the corresponding generators of these spaces. So let me fix this. Let me write it once more. C times eta cube theta to the power 4 and for weight 6 and index 3. We also have only one function. Certainly, we can take the eighth power of the Jacobi theta series. We get the Jacobi form of weight 4 and index 4, with a trigot character. This is non cusp form. And this function is very, very interesting. We'll study this function later. We can find it for Y-coefficient in very, very concrete form, and we'll do it when we study the so called Jacobi Eisenstein series. But now I would like to continue the series of this example using the Jacobi theta series. Now, we constructed three cusp form. The cusp form of weight 8 and index 2, and the cusp form of weight 8, index 2, and the cusp form of weight 6 and index 3 always are even. Could we construct a Jacobi form of odd weight? Let me fix this question. Question. To construct, A Jacobi Form of odd weight. And we can do it using a Jacobi theta series. I can write you one formula immediately, after that, we can study Jacobi form of odd weight. But first of all, example. The following, Jacobi theta series has weight 11. Eta to the power 21 (tau) times theta into Zed. Jacobi cusp form of weight 11 and index 2. So you see using that Jacobi theta series, we can solve a lot of questions about Jacobi form. We can really understand how they look like, how we can construct corresponding examples. Moreover, using the fact about the divisors of Jacobi modular form, we can't really reconstruct them in terms of Jacobi theta series. But now, I would like to understand what type of transformation we will have here. Let me prove the following proposition. First, let me introduce the font notation theta. T, that is by definition, theta (tau, t, z). t here is an arbitrary natural number. So Jacobi form theta t (z) is the Jacobi form of weight 1/2 and index t squared over 2, with respect to the following multiply system. v eta cubed times vH to the power t. Second, for any Jacobi form of weight K and index m. The form phi(tau, tz) is the Jacobi form of weight K and index t to the square by m. So the multiplication of the argument that by t, change the index by t to the square. But in the case of Jacobi theta series, we also have to change the character. So I proved. I would like to prove the first property because the second is simpler. Proof. Let's analyze the functional equation of the modified Jacobi theta series. The modular equation is very simple in this case. We're at the index t is equal v eta cube of the matrix a, b, c, d, (c tau + d) to the power of one-half, the weight is one-half, e to the power pi i, but that is multiplied by t. So we have ct to the square z to the square over c tau + d. Theta in tau tz. So, you see that t to the square, we can put here in the form of the index. So the index will be t to the square over 2. If I write down the corresponding written equation, I get -1 to the power, lambda t dos plus mu t times E to the power -5 I. Then we get lambda to the square, t to the square tau, plus two lambda t t zed. And again we can put t to the square. As the index of this Jacobi form. So you see the character, [INAUDIBLE] the character vH to the power t. So we proved the first property. The second is ever simpler. But using this fact, first of all, we can construct Jacobi form of odd weight but we can also construct the a lot of new Jacobi modular form of weight 4, which will be cusp form. [SOUND]