[MUSIC] Dear students, dear colleagues, I invite you to follow my master course on Jacobi modular forms. 30 years ago, Martin Eichler and Don Zagier published their book, The Theory of Jacobi Forms. The book was published in the Progress in Math in 1985. This is a research book, and one of the most successful book in this Birkhäuser series. You can find more than 1,000 reference from research papers to this book. The reason for this and the theory of Jacobi forms is very successful not only in number theory but also in algebraic geometry, in particular in the theory of modular spaces surfaces. In the theory of Lie algebras, the denominator functions, the denominator functions function of affine Lie algebras are written in terms of Jacobi forms. Moreover, the data for product is a very strong modern construction in the theory of automorphic form is also based on Jacobi forms in some sense. Then you can find application of Jacobi modular forms in topology, elliptic genus and generalized elliptic genus of Calabi-Yau varieties are weak Jacobi forms of weight 0. Then you can find application of Jacobi forms in physics, in string theory, and more particular in quantum gravitation. So in integrable models. In combinatorics, this is very, very successful subject. Moreover, from my point of view, the book of Eichler Zagier may be the best second course in modular forms. So you need to know only a little bit about modular forms. The definition and two examples or three examples like the Eisenstein series of weight 4, 6, and Ramanujan theta function. If you know them, you are ready for my course. But I prepare this course in 2015, three years after Eichler and Zagier book. What are the difference between their book and my course? First of all, my course is the introduction, or it's better to say the invitation in Eichler Zagier theory. But in 2015, the protagonist of my course is the Jacobi theta series, theta tau z. So this is really our main hero. Unfortunately, you cannot find this Jacobi theta series, due to these theories, certainly Eichler and Zagier put this name, Jacobi forms, but you cannot find Jacobi theta series in the book of Eichler Zagier. The reason for this is the following. Jacobi theta series is Jacobi form of weight one-half. Our approach to this function is as follows. So now I would like to tell you, what is my course about. And now my remarks and this very, very short introduction before the course. And mainly for the specialist or for the students who know a little bit, something about Jacobi forms. So my approach to Jacobi theta series is the following. We can analyze Jacobi theta series as a function on the Siegel upper-half plane. So it means we add the set variable omega in the definition of Jacobi form. And we get a function on the Siegel upper half-plane, which is modular form with respect to a parabolic subgroup, gamma infinity is parabolic subgroup of the Siegel modular group. And this is our model for Jacobi modular group. So Jacobi modular group for us is a parabolic subgroup of Siegel modular group. Then, in this term, Jacobi theta series in three variables is a modular form of weight one-half and with respect to the multiplier system v eta cubed, certainly this is multiplier system of the Dedekind eta function times the binary quadratic character of the so-called Heisenberg group. Using Jacobi theta series, we can construct a lot of very interesting and sometimes nontrivial examples of Jacobi form. First of all, we can construct the main generator, this is the function phi minus 2 1 from the Eichler Zagier book, this is the main generator of the of the weak Jacobi form. The formula is very simple. We take the square of Jacobi theta series over the six powers of the Dedekind theta function. But we can construct less trivial example. Our next example is so-called theta-quarks. This is a product of three theta series over one eta. We can prove that this is a Jacobi form of weight one of the index a squared plus ab plus b to the square with respect to a character of order three. If we take the product of three theta-quarks, we get a Jacobi form of weight three. In this way, we get the first Jacobi form of weight three and index 30. In a similar way, but a more complicated procedure, we can construct the first Jacobi form of weight two. This function has index 37. In the Eichler Zagier book, you can find a table for some Fourier coefficient of this modular form, and we can get the explicit formula for this function. This idea of theta series brings us to the idea of automorphic correction. The automorphic correction, this is of the Taylor expansion of Jacobi form. So from this point of view, our approach is very different from the approach in the Eichler Zagier book. In the automorphic correction, this is multiplication by the exponential function with the quasimodular Eisenstein series of genus two. On this page, I give you two different explanations of the rather strange automorphic factor of Jacobi modular form. The first explanation is related with the fact that we can treat the Jacobi modular form as modular form with respect to a parabolic subgroup of Siegel modular group. This fact explains and gives us explicit Jacobi factor. The second explanation related to this automorphic correction. It means the Jacobi automorphic factor is related to the quasimodular Eisenstein series. The first part of our course is about Jacobi form in one variable. But the second part about the Jacobi form in many variables. And again, our main hero, the Jacobi theta-series, gives us the main idea and some main examples. For example, the product of eight theta series is the first Jacobi form, that means the Jacobi form of the minimal possible weight for the lattice D8. In the same way, we can construct the Jacobi form of so-called critical weight for the lattice A7. Jacobi form in many variables are Jacobi form with respect to some positive definite lattice of rank n0. It means that the second variable, the Abelian variable in the Jacobi form is a vector in the of the positive definite lattice, so a vector of dimension n0. And now, using Jacobi form in many variables, we explain this standard series, the standard Eichler Zagier series Jacobi form in one variable. For example, we can represent any Jacobi form for any lattice as the product of vector valued modular form in one variable and the vector of theta series. This subject related to the so-called representation. Then the subject to automorphic correction is very much related to the theory of modular differential operator. And we construct modular differential operator for usual modular form and for Jacobi forms in many variables. In this way, we give a new proof of Zagier theorem about existence of some differential operator which transform the space of modular form of weight k Into the space of Jacobi form of weight k and index 1. This is a very useful operator. Using this operator, we can construct Rankin–Cohen brackets and prove many nice results. We give a new proof of this theorem, and our proof, our method works very well in the case of Jacobi forms in many variables. It's better to say we have no changes in the proof for usual modular form and for Jacobi forms, because our proof is very algebraic. So instead of usual differential operator, we use the corresponding operator, and we construct a differential operator which transform a Jacobi modular form into Jacobi form. Using the construction, we can construct without any problem Rankin-Cohen brackets, and so on. So we can really formulate a lot of simple and non-simple research question. And again, for automorphic corrections and for differential operator, the theta function plays the most important role, because our theta function is this Jacobi theta series with simple zero for z equal to 0. Its derivative is cube of the Dedekind theta function. The logarithmic derivative of Dedekind theta function is the quasimodular Eisenstein series. And using this Eisenstein series, we can construct the differential operator for modular forms or for Jacobi forms in many variables. So you see that the Jacobi theta series is really the main tool of our course, or as I told you, it is the protagonist of our course. So I hope with the Jacobi theta series, which is really the kernel function of the theory of Jacobi modular form, we can construct this theory is very natural and very simple way. So you are very welcome into my course. [SOUND] [MUSIC]
