[SOUND]. Dear colleagues, dear participants of the course. Today, we are studying the last lecture. The last section Is about generalization Of Cohen-Kuznetsov-Zagier operator. This operator was defined by Kunetzov, by Cohen and by Zagier in different context. Let me formulate the definition of this operator in the context of Jacobi form. Let f be a modular form, of weight K, with respect to the full modular group. Then, the coin Kuznetzov-Zagier operator Transforms, maps f into the space of the Jacobi type forms. We defined the space in the last lecture. By definition this is, A function in two variables Which is equal to k- 1, factorial, where k is the weight. Then the sum of all non-negative, n 2 pi i to the power n. Over k- 1 + n. So we have to write it. (k- 1 + n) factorial, then n factorial, then the nth derivative of modular form times z to the power 2h. This is a Jacobi type form of weight k and index 1. More exactly this means that the function, the image of the Cohen-Kuznetzor-Zagir differential operator satisfies the following functional equation. This is the functional equation of a Jacobi type form, is the only modular equation, of weight K, was a Jacobi factor, to the power by IC, z to the square, over C tau + d. The function, in tau and z. We have this equation for any elements from the modular group S of 2 [INAUDIBLE]. So you see that we can construct a Jacobi tie form studying a modular form of weight k. What application do I have? This operator is very useful in different context. But now I would like to discuss only one application. More exactly, we can construct now, a special operation with modular form, the so called a Cohen Modular Bracket. Let's consider two modular form. One modular form of weight k and the second modular form of weight A. Then we can take two functions. The first coin Zagier operator related to the function F. And then we multiply this function by the coin Kuznetsov-Zagier operator of the function, G. But for that, I put the argument, E Z. Then as a functional equation for the first function, we have C Z to the square. But for the second function, -C Z to the square. Therefore, this product is Jacobi type form of weight k + l of index 0. In other words, if we write the Taylor expansion of this problem All Taylor will be modular forms, of weight k + l. See function, I'm sorry, here it's clear, then we'll have Taylor expansion only, with even power with it because our function is symmetric with respect to formation from Z to minus that. And the, then the N's Taylor expansion corresponding to the power, that was a power to 2n. Has weight k + l plus 2n. And we can write down the formula for this coefficient. This is the ranking bracket is the sum. Over E from 0 till n- 1 to the power e. Then we have two binomial coefficient. K = 1 + n e c l- 1 + n, n- i, f i times g n- i. You can check without any problem that this product is a casp form, or have no zeros for any coefficient. It's a Casp form of k + l + 2n. This operation is called a Rankin–Cohen bracket. So by two function, two modular form of k and l, we can construct the karst form of k + l + 2n using standard differential operator. Now I would like to generalize the definition of the Cohen-Kuznetzor-Zagier operator to the case of Jacobi modular forms in many variables. And first of all, I would like to give a new proof of this theorem. This theorem of Cohen-Kuznetzov-Zagier, then this series is a Jacobi type form of weight k and index 1. The next step of our lecture is to give, A new proof, a new purely algebraic proof, Of Cohen-Kuznetsov-Zagier theorem. Which works without any changes. In the case of Jacobi modular forms in many variables. Our idea is the following. First, we use the automorphic correction of Jacobi form. The automorphic correction, this is our approach to the Taylor expansions of Jacobi forms. So, if we have Jacobi form of weight k and index 1. Jacobi or Jacobi type form, this does not matter, we get the automorphic correction by multiplication by the function, e to the power -8 p to the square G2 (tau) Z to the square. And we get Jacobi type form, weight k, and index 0. Certainly, we can construct multiplying by e to the power 8 p to the square, G2 times Z to the square. This is our first idea. The second idea is to construct in some way exponential function of the differential operator, D, where D is our differential operator. 1 over 2 by I, D or D tao. This works very well in the case of the modular form. This is the case, gives the latest l is trio. But if the latest l is non-trivial, we consider exponential function of the theta variable. H is the the heat operator. In our normalization, h is equal to -1 over 8 p to the square 4 pi i d over d tao minus the square over d over dz. These two operators helps us very much. [SOUND]