[MUSIC] Now we'll consider some applications of this modular differential operator. Or let me call them examples. First of all, I would like to consider the case of Jacobi form in one variable. So, first of all, we consider Eichler-Zagier's Jacobi forms. The Jacobi form of Eichler–Zagier of weight K and index m are in our notation, Jacobi form of weight k. For one dimensional lattice 2m. The lattice generated by a vector was square 2m. If I write down the Laplacian of any positive definite. Lattice, this is the square of the covector e i star d dz i from 1 to n0 its scalar square. But e i star, a basis of the dual lattice. So we have here S to the power minus 1 d dz where S, this is the Gram matrix of the lattice L, and d over dz, we write as a column vector. In our particular case, in the case of Eichler–Zagier, the lattice is 2m and we get here 1 over 2m d squared over dz to the square. This is Laplacian part of our operator. So the operator Hk, in our case, let's analyze its definition once more in the general case for the operator. H we use the following normalization. 1 over 2 pi i and then 1 over 8p to the square. So now we have 1 over 2 pi i delta over delta tau plus 1 over 8p to the square n 2m gives us 1 over 16 p to the square m. This is the operator Hk which transform Jacobi form of weight k and index m into the space of Jacobi form of weight K+2 and index m. Now, I would like to make some concrete calculation. Let's consider the space of Jacobi form, weak form of weight -2 and index 1. The space is generated by Jacobi form of weight -2 and index 1. And for this function, we found the explicit formula in terms of the Jacobi theta series. This is the square, the Jacobi theta series over six powers of the Dedekind eta function. And this function starts by the following Fourier expansion, r minus 2 + r to the power -1 + q multiplied by something, when r, this is e to the power 2 pi i z. Let us calculate H -2 at phi -2, 1. And then, this is equal to 1 over 2 pi i d d tau plus 1 over 16 p to the square d square dz to the square, phi -2,1 plus the weight is 2k-n0 G2 times F -2,1. But k is equal to -2 and n0 equal to 1. So here we have to put minus 5 in our case. So n0 is equal to 1. Now, I would like to calculate the first, it's better to say the zeroth term, the zeroth Fourier coefficient, in terms of the variable q of this function. Because according to our theorem, this is a modular form of weight zero, a Jacobi modular form of weight zero. The first term will be related to this operator. So we get. 2 pi i to the square over 16 p to the square, e to the power 2 pi i z, and the similar factor for -z. Then. The next terms will come from this multiplication, from the operator of multiplication by G2, by G2z. Quasimodular Eisenstein series starts with constant term minus 1 over 24. So that means in here, I have plus 5 over 24 times e to the power 2 pi i z. Then plus minus 2 times 5 over 24. Plus the similar symmetric terms for powers of z. So I simply add plus for -z. We get. So here we can simplify pi, we can simplify 2 by 4, and we get here minus. So 1 minus 1 over 4, it gives us 6 over 24. So we have minus 1 over 24 r minus 10 over 24, and by symmetry, 1 minus 1 over 24 r to the power minus 1 + q something. What do we get? We get then -24 times H -2 phi -2, 1 is equal to phi 0, 1 tau, z. This is a function, starts with the following Fourier expansion r + 10 + r to the power -1 + q something. And this is a weak Jacobi form of weight 0 and index 1. This is the function which we constructed using the modular differential operator H -2. [SOUND] [MUSIC]