[MUSIC] Now I would like to generalize all this subject. The [INAUDIBLE] calls itself [INAUDIBLE] differential operators. All these exercises which will formulate about modular forms and differential operators, about rank and coin brackets I would like to generalize all of this stuff. To the case of modular forms, Jacobi modular forms in many variables. So, let me show how we can change the notion without any changes, in the proof. We start in the case of modular form with. So when we can see there's a modular form, our lattice is 3. This is 0. Now, we consider any positive definite lattice of rank n0 and the variable, Zed belong to the complexification of our lattice. So, this is a complex vector of rank n0. In the case of the three lattice 0, the modular differential operator has this form. This is d plus 2k [INAUDIBLE] the operator of multiplication by the quasi modular [INAUDIBLE] series [INAUDIBLE]. In the case of Jacobi modular form, modular forms and many variables we have to change the differential operator d by the operator h with the correspondent normalization. And we got the modular differential operator can, in the case of Jacobi form in many variables. So what changes? We have to make in all formulas, first of all we have to change the differential operator d. By the heat operator h and we have to change the weight k by the weight k minus n0 over 2. n0 over 2, this is the singular weight. So, we can define the following differential operator eknh. This is original operator with corresponding changes. So, we change k and d by h. But in the proof, [SOUND] in the proof of the corresponding theorem, we have to use only this congruence, which is still true for the heat operator, because the main [SOUND] modular part of the heat operator is the same. This is 1 over 2 pi i delta over delta tau. So, this operator, e k n h, is a modular differential operator. This operator, x, on the space of Jacobi modular form of weight, k. And then certainly, this operator change the weight from k to k + 2n. Like in the previous case, we can construct the power series of this operator. Certainly we have to cancel the numerator. Then after that we have the convolution of the two power series. The exponential function of the quasi-modular series, and this variant of the operator. We have to put here h instead of d and we have to change, also the weight. Please be careful since the weight of modular form is greater or equal to n0 over 2, this is certainly true for all modular form. We have proved this result when it was started as a representation. Now, like in the previous case, we have the following diagram, which explains the nature of our operator. First of all, we construct the Jacobi form of weight k and index 0. This is a power series of usual Jacobi forms. We construct the Jacobi form using the modular differential operator, then we take the time-morphic correction, which works very well in the case of many variables and the result, is the modular operator which we need. So, we proved the following theorem. [SOUND] Let phi be a modular form, holomorphic modular form of weight k for the latest l of rank and 0 over 2, h will be the heat operator. Then the following function depending on tau, that the calibrate of zed belongs to the complexification of the latest l and small Zed is the usual complex number. Then these function is a Jacobi type form of weight k for the lattice l and index 1. So this result which we have and the proof is identical to our proof of the [INAUDIBLE] theorem. Maybe I would like to remind you then this fact that we have a Jacobi type form means, then the correspondent function satisfies the following function equation. So we have the usual functional equation of Jacobi form, then Jacobi type part, and the result has weight k and we have two Jacobi factors. The first, this is the factor of Jacobi form for the latest l Pi i c Zed to the square c tau plus d times the factor of Jacobi type, 2 pi i c Zed to the square c tau plus d, in our function. And we have this functional equation for any a, b, c, d in SL2(Z). In particular, Some remarks. If L is triple lattice, then this theorem, is the last theorem, is equivalent to the of the result of [INAUDIBLE]. Then I can formulate, [SOUND] the same series of exercises. In the case of Jacobi modular forms, So please define a generalization of coin, bracket for two Jacobi forms for different lattices. I don't see any problem with this. So, We have two Jacobi forms for two different lattices. You can construct the corresponding functions. And, like in the case of trivial lattices, you have to analyze, The product. What are they? To have a Jacobi type form, Jacobi type form of weight I hope, K plus l for the lattice. The direct sum L and M and index 0. So, taking the Taylor expansion in Zed, you can get the generalization of rank and coin brackets. In this case, then you can analyze the bracket with G2. Let me write it like this. Phi and G2. Then you can analyze it in the case of half integral weight. In particular you can consider the Jacobi Taylor series is constructed by Jacobi functions. Jacobi theta. Tau Zed. You can construct a lot of nice example and a lot, you can prove a lot of nice theorems. Then I can put a question about k which are smaller than no over 2. Why this case is important? Because in this case, we could have a pole in gamma function. So this question is, [SOUND] very similar to the question four I put for modular forms. [SOUND] So, you see, we can formulate a lot of interesting research questions. Some of them are simple, some of them are more complicated, so, I would like to finish my course, end this lecture with [SOUND] the following proposition: if this subject is interesting for you, please Ask for Master and phD, subjects. I'm sure that we can found 10, 12, 15 different subjects for different students. I hope for the possible contacts with you. [SOUND] [SOUND]