[MUSIC] Now, I would like to consider automorphic correction for holomorphic Jacobi form. Let phi be holomorphic Jacobi form of weight k and index m. First of all, if we write down the Taylor expansion of this function, or maybe I remind you first to go the full expansion of this function. Then n is on negative and 4nm- l g to the square is always non negative. So, this hyperbolic norm of the index of all known zero free coefficients are non negative. Then how we can write down, it's Taylor expansion. First of all, for the function f zero we have the following Fourier expansion, this is the sum of all non-negative n. Then this sum of all l in z, such that 4 nm- l to the square is greater or equal to 0. So for a fixed n, we have only finite sums, a (n, l) q to the pow of n. This is the function f zero plus so on. But if we try to find. Maybe if I click this here this is f zero introl, maybe here we add also a Taylor expansion of this function fn in tao, z to the power n, n greater or equal to zero. Now, if we try to find the Fourier expansion of this function, what do we get? First of all, if we analyze the Taylor expansion of any term in the Fourier expansion. Here, you have z to any posit power only if l is not zero but then n is strictly positive. I don't like to write down, it's possible to do but I don't like to write down the Fourier expansion explicitly. But we know that we have no constant term in the Fourier expansion of the Taylor coefficient fn b(n) q to the power n, n is strictly positive. So, now I would like to analyze the function of the automorphic correction. Of our holomorphic Jacobi form, so you have e to the power -8 t to the square m G tau Z to the square fi to z is equal to the sum g n and two to the power n n equal to zero. So we see that in many cases, these functions are cast form. Now, I would like to describe all these cases. For example, if our function fai is a Jacobi cast form, Then all this function will be certainly cast form because here the first function, f zero will be cast form. But now, I can formulate a better corollary. Now, the first corollary was about weak Jacobi form but the second about holomorphic Jacobi form all over two phi is a more Jacobi form of a k, in the index m. Then, All function or Taylor coefficients or automorphic corrections will be cast forms of wave k + n. If the constant term of the Taylor expansion of our function Is a cast form of weight k or if this function is identically 0. [SOUND] Certainly, if we have this property in all other Taylor coefficients of function five, we have no constant term, and then in this product, we'll have no constant term in all Taylor coefficients. It gives us the next corollary. It's very simple, but useful in many considerations. This is not explicit formula for the dimension. Explicit formula you can get using zerinminro formula, I gave you a reference to a recent white paper on the subject. But here, we have the following elementary estimation. You can find this estimation posted in the Kaiser's Idea book that is dimension. The smaller equal to the dimension of the space of all modular form of weight k. Let us assume that k is even. Plus the sum of the dimensions of the space of parabolic modular form of weight K+2i summation from 0 to m. So, this is application of automorphic correction to the classical zero Jacobi forms in one variable. So, you see that the automorphic direction is very, very useful. But the next step is to use the same idea in the case of Jacobi forms in several variables. So our next step, Is automorphic, Correction, Of Jacobi Modular forms in many variables. In this case, our method gives us very, very nice results. First of all, I would like to fix the setup for this method. Thai will be holomorphic, or weak, doesn't matter, Jacobi form of weight K and index L or the latest L. So we can consider holomorphic or weak, it's two different cases Jacobi. So then, V is a non-zero vector. In L, and L1 will be the orthogonal complement of V and L. This is the latest of corp rank one in L, now we can write arbitrary complex vector in the complexification of the latus L as a sum, all total of sum of two vectors. Z1 and usual Z times V. V where Z1 is a complex vector and is of terminal co and Z is another three complex number. Now, we are ready to write the automorphic correction of a Jacobi form in many variables. [MUSIC]