So I gave you the definition of modular forms of weight k, let's analyze this definition once more. So in modular form, it's holomorphic function which satisfies the modular equation M. And would behave at infinity. Certainly, this sum of two modular form is a modular form of the same weight. Therefore, the set of all modular form of a fixed weight is a space over complex numbers, a linear space. So we can denote by the notation. We denote by Mk or Mk s into z if we would like to emphasize the modular group. The space of all modular forms of fixed weight. So there's a space. Of all modular forms Of weight k. It's the linear space of a complex numbers and one of the most important property of the space then this space is finite dimension. Fact. This space is finite dimension for any k. You see then, We can analyze this space and the most important modular form, maybe the most important modular form with respect to full modular group is the following function, Constructed by Ramanujan. This is a famous delta function of Ramanujan, from which we can define as the following infinite product. This is a modular form of weight 12 with respect to the full modular group. Moreover, this is so-called cusp form, cusp form means that the constant term in a Fourier expansion, is equal to 0. So all this stuff, I mean the definition of modular form, the delta function and so on, you can find in the classical nice book of [FOREIGN], A Course, Of Arithmetic. The last chapter of this course is about modular form in one variable, and you can find all necessary information in this book. But later we discuss some theorem and some result of this book also in our course. Now, I would like to show you the formation of this delta function, or some elliptization of this delta function, which looks very nice. And which is also a motivation for construction of the theory of Jacobi form. So let's consider the following infinite products in two variables. This is q, to the power 4. r, to the power -4. The infinite product for M, 3 to 1. (1- q to the power n- 1 to the power r) to the power 8 (1- q to the power n r to the power -1) to the power 8 (1- q to the power n) to the power eight. So, you see that here is q. E to the power 2 piT. Then r, this is e to the power 2piZ. So we have the modular variable tau and the elliptic, later in the second lecture, I explain to you why this variable is elliptic. The elliptic variable z or r. So let's compare the Ramanujan product, And its elliptization when we add one more variable, r. The Ramanujan product contains 24 factor for each n. These product also has 24 terms. Through its kind of generalization of Ramanujan delta function. Let's analyze this term for n = 1. And we get 1. R to the power 8. Therefore, this function g, Vanishes for z equal to 0. Moreover the order of 0, z equal to 0 of this function is equal to 8. And you can show ,this is a simple exercise, than the 8 derivative with respect to z, of g for 4 into z. Z = 0 is up to a constant, the Ramanujan data function. This function g is a Jacobi form, Of weight, 4 and index. This is the second characterization of Jacobi modular form. So, Jacobi modular form is a function in two variables. So we have the weight, and we have the index, two characteristics. Though this is only an example, at the moment we don't see why this function is a Jacobi form, and moreover we don't have the definition of Jacobi form at the moment. But my first section mainly devotes to motivation of our subject. So now let's analyze the subject which we discussed. First of all, we construct the generating function of the arithmetic problem of calculation, Of number of representation of natural numbers by vectors, in the latest. So, it was our generating function. Theta, 12, 8. I'm sorry. Theta of the quadratic l in total. And this function is a modular form, Of some weight. Then we consider it the partition function. The partition function with m additional complex variable. So z here belongs to c to the power m. Where m is the rank of the latest a. And this function is a Jacobi modular form. In n variables, z. Moreover we consider it some pull backs of this theta function. And the pull backs, Pull backs gives us theta function, In one additional complex variable. So that is in c. And here we get Jacobi forms, From the book of Eichler, And Zagier. In the next lecture, the next lecture, we'll start by the definition of modular form of this type. [MUSIC]