Offered By

National Research University Higher School of Economics

About this Course

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This is a master course given in Moscow at the Laboratory of Algebraic Geometry of the National Research University Higher School of Economics by Valery Gritsenko, a professor of University Lille 1, France.
Jacobi forms are holomorphic functions in two complex variables. They are modular in one variable and abelian (or double periodic) in another variable. The theory of Jacobi modular forms became an independent research subject after the famous book of Martin Eichler and Don Zagier “Jacobi modular forms” (Progress in Mathematics, vol. 55, 1985) which was cited more than a thousand times in research papers. This is due to many applications of Jacobi forms in arithmetic, topology, algebraic and differential geometry, mathematical and theoretical physics, in the theory of Lie algebras, etc. The list of mentioned subjects shows that my course might be useful for master and Ph.D. 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).
The main hero of the course is the Jacobi theta-series. Using it we will construct a lot of concrete examples of Jacobi forms in one or many abelian variables, in particular, Jacobi forms for root systems.
For some of you, who will be successful with the theoretical exercises of the course, I am ready to formulate research problems for Master or Ph.D. thesis. (Ph.D. support might be available at CEMPI in Lille or at the Faculty of Mathematics of National Research University Higher School of Economics in Moscow)

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Suggested: 12 weeks of study, 3-5 hours per week...

Subtitles: English

Start instantly and learn at your own schedule.

Reset deadlines in accordance to your schedule.

Suggested: 12 weeks of study, 3-5 hours per week...

Subtitles: English

Week

1Welcome 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...

1 video (Total 14 min), 4 readings

Course Overview10m

Grading and Logistics10m

Suggested Readings10m

About the Instructor10m

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....

5 videos (Total 62 min), 1 quiz

Motivations12m

Theta-function12m

Modular and abelian transformations12m

Pullbacks of theta-function12m

Modular forms12m

Week

2This 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....

6 videos (Total 75 min), 1 quiz

Definition of Jacobi forms (part 2)13m

Basic properties of Jacobi forms12m

Jacobi modular group11m

Symplectic group11m

Jacobi modular group (part 2)13m

Week

3This 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....

6 videos (Total 86 min), 1 quiz

The action of Jacobi modular group14m

The action of Jacobi modular group (part 2)10m

The second definition of Jacobi forms15m

Special values of Jacobi forms11m

The first theorem20m

Week

4This 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....

6 videos (Total 76 min), 1 quiz

The zeros of Jacobi forms (part 2)11m

Taylor expansion of Jacobi forms12m

Taylor expansion of Jacobi forms (part 2)13m

Dimensions of some spaces of Jacobi forms10m

Examples of Jacobi modular forms14m

Week

5This 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....

6 videos (Total 84 min), 1 quiz

Properties of Jacobi theta-function9m

Properties of Jacobi theta-function (part 2)13m

Properties of Jacobi theta-function (part 3)15m

Construction of Jacobi forms using theta-series15m

Construction of Jacobi forms using theta-series (part 2)13m

Week

6This 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....

7 videos (Total 96 min), 1 quiz

Theta-blocks14m

Theta-blocks (part 2)13m

Theta-quarks14m

Dimension of the space of Jacobi forms of odd weight14m

Theta-quarks (part 2)11m

The proof of theorem about theta-quarks16m

The first Jacobi cusp form of weight 210m

Week

7This 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....

6 videos (Total 93 min), 1 quiz

Jacobi theta-series and Jacobi forms in many variables (part 2)15m

Hyperbolic reformulation14m

Examples of Jacobi forms in many variables14m

Examples of Jacobi forms in many variables (part 2)16m

Examples of Jacobi forms in many variables (part 3)16m

Week

8In 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....

6 videos (Total 90 min), 1 quiz

Theta-quarks as pull-backs of Jacobi forms (part 2)16m

Theta-quarks as pull-backs of Jacobi forms (part 3)13m

The splitting principle15m

The splitting principle (part 2)16m

Weak Jacobi forms in many variables13m

Week

9This 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....

6 videos (Total 85 min), 1 quiz

The Weil representation (part 2)15m

Jacobi forms of singular weight14m

Jacobi forms of singular weight (part 2)15m

Jacobi forms of critical weight15m

Jacobi forms of critical weight (part 2)12m

Week

10This 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....

6 videos (Total 84 min), 1 quiz

Differential operator D15m

Quasi-modular Eisenstein series15m

Automorphic correction and Taylor expansions13m

Automorphic correction of Jacobi forms in many variables12m

Automorphic correction of Jacobi forms in many variables (part 2)12m

Week

11This 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....

6 videos (Total 90 min), 1 quiz

Heat operator14m

Heat operator (part 2)15m

Applications of differential operators for Jacobi forms11m

The graded ring of the weak Jacobi modular forms17m

Jacobi type forms in many variables14m

Week

12The 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....

4 videos (Total 52 min), 1 quiz

A new proof of Cohen-Kuznetsov-Zagier theorem13m

A new proof of Cohen-Kuznetsov-Zagier theorem (part 2)9m

Cohen-Kuznetsov-Zagier theorem in case of many variables14m

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