课程信息
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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100% 在线课程

立即开始,按照自己的计划学习。
Advanced Level

高级

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完成时间大约为54 小时

建议:12 weeks of study, 3-5 hours per week
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English

字幕:English
Globe

100% 在线课程

立即开始,按照自己的计划学习。
Advanced Level

高级

Clock

完成时间大约为54 小时

建议:12 weeks of study, 3-5 hours per week
Comment Dots

English

字幕:English

Syllabus - What you will learn from this course

1

Section
Clock
1 hour to complete

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...
Reading
1 video (Total 14 min), 5 readings
Video1 videos
Reading5 readings
Pre-Course Survey10m
Course Overview10m
Grading and Logistics10m
Suggested Readings10m
About the Instructor10m
Clock
3 hours to complete

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....
Reading
5 videos (Total 62 min), 1 quiz
Video5 videos
Theta-function12m
Modular and abelian transformations12m
Pullbacks of theta-function12m
Modular forms12m

2

Section
Clock
3 hours to complete

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....
Reading
6 videos (Total 75 min), 1 quiz
Video6 videos
Definition of Jacobi forms (part 2)13m
Basic properties of Jacobi forms12m
Jacobi modular group11m
Symplectic group11m
Jacobi modular group (part 2)13m

3

Section
Clock
3 hours to complete

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....
Reading
6 videos (Total 86 min), 1 quiz
Video6 videos
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

4

Section
Clock
3 hours to complete

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....
Reading
6 videos (Total 76 min), 1 quiz
Video6 videos
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

5

Section
Clock
3 hours to complete

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....
Reading
6 videos (Total 84 min), 1 quiz
Video6 videos
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

6

Section
Clock
4 hours to complete

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....
Reading
7 videos (Total 96 min), 1 quiz
Video7 videos
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

7

Section
Clock
4 hours to complete

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....
Reading
6 videos (Total 93 min), 1 quiz
Video6 videos
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

8

Section
Clock
3 hours to complete

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....
Reading
6 videos (Total 90 min), 1 quiz
Video6 videos
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

9

Section
Clock
3 hours to complete

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....
Reading
6 videos (Total 85 min), 1 quiz
Video6 videos
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

10

Section
Clock
3 hours to complete

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....
Reading
6 videos (Total 84 min), 1 quiz
Video6 videos
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

11

Section
Clock
3 hours to complete

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....
Reading
6 videos (Total 90 min), 1 quiz
Video6 videos
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

12

Section
Clock
3 hours to complete

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....
Reading
4 videos (Total 52 min), 1 quiz
Video4 videos
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
4.7

Top Reviews

By DZDec 29th 2016

Very good course. I have really learned a lot from this course. The excercises are carefully selected.

Instructor

About National Research University Higher School of Economics

National Research University - Higher School of Economics (HSE) is one of the top research universities in Russia. Established in 1992 to promote new research and teaching in economics and related disciplines, it now offers programs at all levels of university education across an extraordinary range of fields of study including business, sociology, cultural studies, philosophy, political science, international relations, law, Asian studies, media and communications, IT, mathematics, engineering, and more. Learn more on www.hse.ru...

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