Overview

Jacobi modular forms constitute a bridge between arithmetic and different mathematical subjects like algebraic and geometry, topology, mathematical physics and string theory. The theory of Jacobi forms was stated by the great book of M. Eichler and D. Zagier (1985) which was cited a thousand times. We follow another way in the theory (in particular, our main hero is the odd Jacobi theta-series and we consider Jacobi modular forms in many variables) but we try to keep the style and the spirit of this wonderful book.
The target audience of this online HSE course contains all master and post-graduate students in mathematics and physics. Basic knowledge:
1) Complex analysis (a very basic course);
2) The theory of quadratic forms (a very introductory course).
3) It is better to know the definition of modular forms with respect to the full modular group SL_2(Z). See the first part of the section 7 of J.-P. Serre «A Course in Arithmetic». In any case we discuss this definition in the course.
After this distant course students will be able to start a realisation of their individual research project or master thesis in the theory of modular forms of many variables and their applications. The results of the theory of Jacobi forms of many variables have been presented so far only in scientific articles. The course will focus primarily on various applications of Jacobi forms and will be available to students specializing in various areas of mathematics.
The course consists of 12 video lectures nearly for 90 minutes and 12 tasks with exercises and problems related to every lecture. All tasks will be evaluated.
We are planning to prepare a book based on this online course.

Syllabus

Introduction to the Course

Welcome to the course! I hope you have an opportunity to reserve some time to explore the course content, course logic and our grading policy. The course consists of 12 lectures. This course will help you to start your progress in the field of the theory of Jacobi modular forms.

Best regards, Valery Gritsenko

Jacobi modular forms: motivations

This module is devoted to motivations to study Jacobi forms. We provide some first examples including theta-functions. Also there is a peer review in the end of this module.

Jacobi modular forms: the first definition

This module is devoted to the first definition of Jacobi forms. In this module we also define Jacobi modular group. Also there is a peer review in the end of this module.

Jacobi modular group and the second definition of Jacobi forms. Special values of Jacobi modular forms

This module is devoted to the second definition of Jacobi forms. In this module we also consider special values of Jacobi forms. Also there is a peer review in the end of this module.

Zeros of Jacobi forms. The Jacobi theta-series, the Dedekind eta-function and the first examples of Jacobi modular forms

This module is devoted to zeros of Jacobi modular forms, their Taylor extensions and the first examples of Jacobi forms. Using classical Jacobi theta-series and Dedekind eta-function we construct a series of Jacobi forms. Also there is a peer review in the end of this module.

The Jacobi theta-series as Jacobi modular form. The basic Jacobi modular forms

This module is devoted to detailed study of Jacobi theta-series. We will discuss abelian, modular and some other properties of this function. Also there is a peer review in the end of this module.

Theta-blocks, theta-quarks and the first Jacobi cusp form of weight 2

This module is devoted to very important notion of theta-blocks and theta-quarks. In this module we also construct the first Jacobi form of weight 2. Also there is a peer review in the end of this module.

Jacobi forms in many variables and the Eichler-Zagier Jacobi forms

This module is devoted to Jacobi forms in many variables. In this module we also define classical Eichler-Zagier Jacobi forms in terms of Jacobi forms in many variables. Also there is a peer review in the end of this module.

Jacobi forms in many variables and the splitting principle. Theta-quarks as a pull-back. Weak Jacobi forms in many variables

In this module we continue studying Jacobi forms in many variables. Among other things we discuss splitting principle and realize theta-quarks as a pull-back. Also there is a peer review in the end of this module.

The Weil representation and vector valued modular forms. Jacobi forms of singular weight

This module is devoted to very useful notion of the Weil representation and vector-valued modular forms. In this module we also define Jacobi forms of singular weight. Also there is a peer review in the end of this module.

Quasi-modular Eisenstein series. The automorphic correction of Jacobi forms and Taylor expansions

This module is devoted to Quasi-modular Eisenstein series. In this module we also define the automorpic correction of Jacobi forms and its Taylor expansion that gives us the way to construct the series of Jacobi forms. Also there is a peer review in the end of this module.

Modular differential operators. The graded ring of the weak Jacobi modular forms

This module is devoted to Modular differential operators. In this module we also consider the Jacobi forms as the space with the structure of the bigraded ring. Also there is a peer review in the end of this module.

Jacobi type forms and the generalisation of the Cohen-Kuznetsov-Zagier operator

The last module is devoted to Jacobi type forms. In this module we also consider the generalisation of the Cohen-Kuznetsov-Zagier operator. Also there is a peer review in the end of this module.