[MUSIC] I would like to represent these last function E in more Linear way. So let's analyze this identity once more. This is our function which contains the iteration of the major part of the duration of the differential operator in this function as you see splits into the product of two functions. This result I can present in the following diagram. We start with the space of the modular form, forms of weight k with respect to the full modular group. If we apply the operator e, we get a power series of modular forms of weight. K plus n. K plus 2n. For the formal variable x to the formal variable. x we use the following more concrete formula. x is equal to 2 pi i z to the square or you can write it as minus 4 p to the square z to the square. Variable z and I repeat then the first power assumes transform in modular form into the space of Jacobi type form of weight k and index 0. It means the power series of module of form of different ways. But then we can use the automorphic correction. This is the inverse automorphic correction Or maybe I can call it minus automorphic correction. Automorphic correction. This, however, main idea I can repeat you, then automorphic correction. This is a transformation, which transforms this space of Jacobi type form of weight, one into the space of Jacobi type form of index, index 1 into the index 0. So minus of the inverse of automorphic correction transforms the Jacobi type form of index 0 into index 1. And this is exactly the Cohen-Kuznetsov-Zagier operator. The composition of the power series E, the analog of the exponential function of the usual differential operator. With the minus automorphic correction, gives us the Cohen-Kuznetsov-Zagier operator. And we proved that Cohen-Kuznetsov-Zagier theorem. Starting from a modular form of weight k, we get the Jacobi type form of weight K and the index, 1. Now, I can formulate some number of exercises. First of all, we can start the Cohen-Kuznetsov-Zagier operator, not from the weight K. Not from the modular form but we can start this operator from the quasi-modular form G2. For that we have to start our iteration with a modular differential operator D2 which transforms The quasi modular Eisenstein Series, G2 into the space of modular form of weight of 4. So, but we have to correct our formulas, because in this differential operator will modify the constant. This is not 2k, but simply k. Please prove that the following modification of quasi, the differential operator gives us Jacobi type form of wave 0. And index one starting from the quasimodular Eisenstein series. So in modification of this formula gives us another very nice Jacobi type form. Using the Jacobi type form, you can generalize Rankin–Cohen bracket for G2. Rankin–Cohen brackets for G2. I can ask the questions that can calculate Rankin–Cohen brackets of the square of G2. Why we use gamma function? So in many cases in our formula we use gamma function. Why? Because without any changes we can apply this formula for modular form of half integral weight. This is why I write here gamma function, not the usual factorial. So please make this modification. Try to modify all this formula for half integral k. In particular you get Rankin–Cohen brackets for half integral. And then, the same formula you can apply for non positive weight. But then, you have to be careful, because gamma function might have pull. So in the summation, we have this restriction. And the corresponding result. This is modification of Cohen-Kuznetsov-Zagier operator for non-positive weight. And then, in this summation we have this restriction. In particular, from this formula, you see that, for k smaller or equal to 0, for any metamorphic modular form of form k, we have the following identity. This is direct carry over from this identity. This is a classical Bol identity for modular form. So the modular K + 1 power of differential operator transforms modular form of weight- K into the modular form of weight K + 2. Please make this exercise. [SOUND]