[MUSIC] Dear students, I am very glad that you decided to start my course, so very welcome to my first lecture. I am planning, first of all, to give you some motivations for the theory of Jacobi form. So the title of the first section, Motivations. The plan of the first section is the following. First of all, I would like to analyze the arithmetic generating function. Generating, Functions was arithmetic problem of the representation of numbers by what's rated for. Then we consider partition function of the same problem. After that, I would like to give you such definition of the classical modular form in one variable. And after that, we construct an elliptization of the famous Ramanujan delta function. So elliptization, Of Ramanujan delta function delta and tau. More exactly, I construct here an elliptization of some infinite product. Let me start by the first definition. This is the definition of the latest because I would like to construct the generating function, Of a quadratic lattice. So let, L be an integral, Even positive definite, Quadratic. Lattice. What is it? First of all, Lattice, this means that L is a free Z module of rank n. So n is isomorphic as integral model to Z to the power n. Then, Integral quadratic. It means then there is this symmetric bilinear pairing, defined under direct product of L with integrally. In another [INAUDIBLE] we have here an integral quadratic form. Even, what is it? It means that the no of any vector in L is even, for any v in L. And the last property, which is more or less evident, positive definite, Means that for any nonzero for v in L, its scalar square is strictly positive. This is a object, which we would like to work with. So let me repeat a, b, and even integral positive definite quadratic lattice of rank n. Here certainly, where u that n is strictly positive. So now I would like to define generating function for this lattice. Or in another word, the data function of this lattice. Let's consider the following arithmetic question. I denote by rL(2n) the number of vectors in L of the fixed scalar square 2m. I use here 2m because the latest is even. Moreover, the lastest is positive definite. Therefore, this number is finite. Now we can construct the generating function for this quantity, or we can define the theta function of the latest a. This is a falling formal sum, rL(2n) q to the power n. And we have, The summation, over all non-negative, integral n. Certainly the first term, of this generated function, is 1. Because the latest is positive definite. This is a formal function, but now, I would like to construct a holomorphic function related to this generating function. We put q equal to the e to the power 2 by i tau, where tau is a point in the so-called upper half-plane. This is a set of complex numbers with positive definite imaginary part. And we have the following expression for the same generating function. E to the power pi v to the square tau, the summation is taken over all the v in L. Another notation for the same function, theta a tau. This function is holomorphic, On the upper half-plane. Moreover, How to, Understand this transformation from tau to q. So tau is an element from the upper half-plane. And, q, Is an element of the unit disc, Because the absolute value of q, Is equal to e to the power- pi y, circle this, then what? What do I have here at 0? This is the image of the infinite point e, infinity of, The upper half-plane. So we can trade the same function, the same generated function as a function on the unit open disc. Or is a function in the variable tau on the upper half-plane. [MUSIC]