[MUSIC] Dear students, welcome to the second lecture of my course. The title of this second section is the First Definition of Jacobi Form. Certainly, in the title, I would like to emphasize the word adjust. Because there in this lecture, you will see also the second definition of Jacobi form. Let me explain to you the plain of this legend. First of all, we consider the concept of Jacobi forms. Then I defined the so called Jacobi module group. We'll have two points of view on this group. First of all this group is an extension of the usual modular group s to z, which will be a group of this group. But from another side, itself will be parabolic sub group of the so called Zed module group. Or the integral group of genus two. This group is very interesting for many points of view, particularly this group, is the semi direct product of the group s into z, and the unit per 10 group, h which is the Heisenburg group. Of the latest that it's better to say of the direct product of zed by. So the concept of Jacobi modular group is very useful, because at the end of this lecture, we come to the second definition the second, more factorial definition of Jacobi form But let me start by the first definition. [SOUND] In this definition. Tau is an element of the upper half plane. It means the imaginary power of tau is positive. Then Zed is a complex number. K is an integer. And m is non-negative integer. We need this data for the definition of Jacobi form. Consider a holomorphic function, p into two variables tau and z. These holomorphic on the direct product of the upper half plane and the field of complex standards. This function satisfies two functional equation. The first equation, the so called modular equation, phii, a, tau plus b over c theta plus d, zeta over c-theta plus d is equal to c-theta plus d to the power k. K, this is an integer. Times e to the power 2 fi im c times z to the square or 1 c tau plus d times v tau zed. M is the second integer, which we have fixed for the definition, and we have this functional equation the modular equation. For any metrics, A, B, C, D in the modular group added to zed. The second functional equation is the so-called elliptic equation. Now, We take an arbitrary, Abelian translation of the variable zed. Where lambda and nu two integers. Then I get e to the power -2 t i m lambda to the square tau plus 2 lambda zed phi tau zed, and we have this elliptic equation for arbitrary lambda and mu. This is two functional equations. The modular equation, which we have for arbitrary element also modular group. And the second elliptic equation, which were have of all translations of zed by this element. So from this point of view the first module equation described our function phi as a modular form with respect to the variable, This factor is the standard at a morphic factor, which we have seen in the definition of fusion modular form, was a modular group, but in this definition, we have the second rather complicated factor related to the number M. In the definition of modular form, of Jacobi modular form. K is the weight of this Jacobi modular form, and m is the index. So you see index m in this rather complicated exponential factor in the module equation, and we see the index in the second equation as well. In the second equation, we describe our modular form as a double periodic Function in zed, it's not really double periodic, but this is double periodic up to this Data factor. But nevertheless, we can consider the variable z as an element. The complex field modular [INAUDIBLE] z, [INAUDIBLE] plus z. Which we just as well know, I hope you know that this is so called elliptic curve. Determined by the period tau. This is not the full definition of Jacobian modular form. In the moment, we consider only holomorphic function on the product of the upper half plane ends the complex number. A direct product of [INAUDIBLE] plane C, and this function such as [INAUDIBLE] two functional equation. We, to finish this definition we'll have to put a special condition on the free expansion of this function [MUSIC]