[MUSIC] Hi there, participants of my course. Today, we are studying the lecture number seven. The title of this lecture, Jacobi theta series. And Jacobi modular form in several variables. Jacobi forms in many variables. This is very important for our cause. Today we extend our subject to the case of many billion variables. Jacobi theta series. Was the hero of the last two lectures. This is a very important function for our study. I remind you the Fourier expansion of the definition of this function. The connector character, summation of all integers. Q to the power of 8R to the power N over two. Or for the current lecture I will use more explicit notations. Let me change this term by e to the power by i n square over 4 tau plus n zeta where tau is in the usual upper plane and zed is an arbitrary complex number. Using this function we construct very many interesting Jacobi forms I recall what function would be constructed. First of all, the which Jacobi form of weight minus two and index one is defined as a potion of the square of Jacobi Theta series over the power six of the Dedekind function. This function generates The space of the weak of Jacobi form of weight minus 2 and index 1. Then, we defined the first Jacobi cast form. This function generates the space of Jacobi cast form, Of weight 10 and index 1. This is the first Jacobi cast form. Moreover, we constructed the first Jacobi 4 of odd weight. [SOUND] This is theta to the power 21, times Jacobi 30 series into that. The eighth power of the Jacobi theta series Is a Jacobi modular form, non-cusp form of weight 4 and index 4. Now I would like to put the following question. What is about the following product? Let me introduce the following notation to make our formula shorter. I would like to show you only the second argument of the Jacobi series. Then I would like to consider the following function. Theta in (z1) times theta (z2) so on times theta in (z8). So I would like to analyze the function in eight Abelian were able. This is caligraphoc is that, this is the vector of frank eight. So what function do we have? This is the first example of Jacobi modular forms in many variables. Now I would like to understand the property of this action. [SOUND] First of all, any Jacobi form is a holomorphic function satisfying two functional equations, the modular equation and the Abelian equation, or quasiperiodic condition. Now [COUGH] let us analyze the modular behavior of this new function. We have a new Theta in eight Abelian variables. Now I would like to take the modular transformation from the special linear group sl2z. So here we use another variable the vector z in eight coordinate. What do we get? The original Jacobi theta series is a modular form Jacobi modular 4 of weight one half and index one half with respect to the multiplied system of the Theta function to the power 3 times the character of the heights in the group. In the new function, we have eight factors. It means that the product will have the weight, 4. 8 x 1/2 is 4. So the first factor, (c tau + d) to the power 4. Then we have no character or multiplier factor because we have 8 factors. And for each factor their multiplier system is V eta to the power cube, then Z 8 power of this multiplies system, is true. So have no character, then we have E, Ti because theta has index one half and C then the sum of the squares of our variable Over C plus D. And then Theta function, In eight variables. So now let's analyze this formula. The sum of the square, which we have here, we can write as the scalar square of the, Vector z. So this is the modular behavior of our function. Now I would like to analyze Its periodic property. So, we change z by lambda tau + mu where lambda and mu are vectors. There's a eight vectors with eight integral cordon. So using the function equation or the usual that the series, I guess the full increase. Minus one to the power is the sum of Lambda e plus mu e [SOUND] e, from 1 till 8. This is the character of the Heisenberg group. Times e to the power minus pi. And now I can use the scalar product. It will be Lambda. Scalar product, it means lambda 1 to the square plus lambda 2 to the square and so on, tau + 2 (the scalar product of lambda and z) means lambda 1 z1 plus lambda 2 z2 and so on. Times Theta 2 z. This is a functional equation of our function. Now I would like to have a function with triple character. In what case, we have no plus minus one in this functional equation. So, I would like to have one here. To have one, we have to take Lambda and mu in the lattice D8. By the definition, the lattice D8, this is the lattice of all integral vectors. With even sum of the coordinate. Sublatens of index two is an incladen latest of frank 8. So, in this case if lambda and mu have these properties. It means then, the sum of the coordinate lambda e and, the sum of the coordinate of mu j is equal to 0, modular 2. Under this condition, we have always the factor one in this equation. We get the first example of the so-called Jacobi Form for the latest D8 or a Jacobi form was 8 billion variable that that is a vector. It was 8 complex coordinate. And this is I repeat this is a Jacobi form with respect to the latest D8. Now I would like to give you the general definition of Jacobi modular form in many variables. [MUSIC]
