About this Course
4.7
11 ratings
3 reviews
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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Advanced Level

Advanced Level

Clock

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

Approx. 53 hours to complete
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English

Subtitles: English
Globe

100% online courses

Start instantly and learn at your own schedule.
Calendar

Flexible deadlines

Reset deadlines in accordance to your schedule.
Advanced Level

Advanced Level

Clock

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

Approx. 53 hours to complete
Comment Dots

English

Subtitles: 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 video
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
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

Valery Gritsenko

Laboratory of Algebraic Geometry and its Applications

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...

Frequently Asked Questions

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