Jacobi modular forms: Introduction to the Course

JACOBI MODULAR FORMS: 30 ans après.

Valery Gritsenko

Professor of the University Lille 1 in France and Chief Researcher at the international Laboratory of Algebraic Geometry of High School of Economics (HSE) in Moscow

Our course is recommended for all master students working in different directions. Motivated undergraduate students can also study this subject. To follow the course one has to know only elementary basic facts from the theory of modular forms (for example, the paragraphs 1-4 of the chapter VII of Serre’s “A Course in Arithmetic” are enough). One of the main heroes of the course is the Jacobi theta-series. Using the theta-series we construct a lot of concrete examples of Jacobi forms from the book “Jacobi modular forms” of M. Eichler and D. Zagier. In the second part of the lectures we study Jacobi forms in many abelian variables. Basic examples of such Jacobi forms related to root systems will be defined with the help of the Jacobi theta-series.

The course contains 12 lectures. In the first six lectures we consider the theory of Jacobi forms in one variable of Eichler-Zagier type. We repeat that the main hero of this part is the classical Jacobi theta-series modular with the full special linear group SL_2(Z). In the second part of the course we consider Jacobi modular forms in many abelian variables. In particular we define modular differential operators and Jacobi type forms in this context. We give applications of them in Eichler-Zagier’s case. Some constructions of the lectures are not presented in the textbooks. The last lecture is an introduction in research.

The plan of the course is explained below.

I. Jacobi modular forms: motivations. Exercises: Even integral lattices.

II. Jacobi modular forms: the first definition. Exercises: Definition of Jacobi forms.

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

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

V. The Jacobi theta-series as Jacobi modular form. The basic Jacobi modular forms. Exercises: Jacobi theta-series.

VI. Theta-blocks, theta-quarks and the first Jacobi cusp form of weight 2. Exercises: Jacobi theta-blocks.

VII. Jacobi forms in many variables and the Eichler-Zagier Jacobi forms. Exercises: Examples of Jacobi forms in many variables.

VIII. Jacobi forms in many variables and the splitting principle. Theta-quarks as a pull-back. Weak Jacobi forms in many variables. Exercises: Pullback of Jacobi forms.

IX. The Weil representation and vector valued modular forms. Jacobi forms of singular weight. Exercises: Jacobi form of singular and critical weights.

X. Quasi-modular Eisenstein series. The automorphic correction of Jacobi forms and Taylor expansions. Exercises: Differential operators and automorphic correction.

XI. Modular differential operators. The graded ring of the weak Jacobi modular forms. Exercises: Differential operators for Jacobi forms.

XII. Jacobi type forms and the generalisation of the Cohen-Kuznetsov-Zagier operator. Exercises: Jacobi type forms.

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