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Jacobi modular forms: 30 ans après

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Course Summary

Learn about modular forms and Jacobi forms in this advanced mathematics course. Dive deep into the theory and applications of these important mathematical concepts and gain a deeper understanding of topics such as automorphic forms, elliptic curves, and more.

Key Learning Points

Explore the theory and applications of modular forms and Jacobi forms
Gain a deeper understanding of automorphic forms and elliptic curves
Learn about the role of modular forms in number theory and other areas of mathematics

Learning Outcomes

Understand the theory and applications of modular forms and Jacobi forms
Deepen your knowledge of automorphic forms and elliptic curves
Gain a deeper understanding of the role of modular forms in number theory and other areas of mathematics

Prerequisites or good to have knowledge before taking this course

Familiarity with advanced mathematics concepts such as calculus, linear algebra, and abstract algebra
Experience with mathematical proofs and rigorous reasoning

Course Difficulty Level

Advanced

Course Format

Online
Self-paced
Video lectures
Quizzes and exercises

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Notable People in This Field

Andrew Wiles
John Tate
Harold Stark

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Description

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) Do you have technical problems? Write to us: coursera@hse.ru

Outline

Introduction to the Course
About the University
About the Course
About University
Rules on the academic integrity in the course
Course Overview
Grading and Logistics
Suggested Readings
About the Instructor

Jacobi modular forms: motivations
Motivations
Theta-function
Modular and abelian transformations
Pullbacks of theta-function
Modular forms

Jacobi modular forms: the first definition
Definition of Jacobi forms
Definition of Jacobi forms (part 2)
Basic properties of Jacobi forms
Jacobi modular group
Symplectic group
Jacobi modular group (part 2)

Jacobi modular group and the second definition of Jacobi forms. Special values of Jacobi modular forms
Heisenberg group
The action of Jacobi modular group
The action of Jacobi modular group (part 2)
The second definition of Jacobi forms
Special values of Jacobi forms
The first theorem

Zeros of Jacobi forms. The Jacobi theta-series, the Dedekind eta-function and the first examples of Jacobi modular forms
The zeros of elliptic functions
The zeros of Jacobi forms (part 2)
Taylor expansion of Jacobi forms
Taylor expansion of Jacobi forms (part 2)
Dimensions of some spaces of Jacobi forms
Examples of Jacobi modular forms

The Jacobi theta-series as Jacobi modular form. The basic Jacobi modular forms
Examples of Jacobi forms (part 2)
Properties of Jacobi theta-function
Properties of Jacobi theta-function (part 2)
Properties of Jacobi theta-function (part 3)
Construction of Jacobi forms using theta-series
Construction of Jacobi forms using theta-series (part 2)

Theta-blocks, theta-quarks and the first Jacobi cusp form of weight 2
Theta-blocks
Theta-blocks (part 2)
Theta-quarks
Dimension of the space of Jacobi forms of odd weight
Theta-quarks (part 2)
The proof of theorem about theta-quarks
The first Jacobi cusp form of weight 2

Jacobi forms in many variables and the Eichler-Zagier Jacobi forms
Jacobi theta-series and Jacobi forms in many variables
Jacobi theta-series and Jacobi forms in many variables (part 2)
Hyperbolic reformulation
Examples of Jacobi forms in many variables
Examples of Jacobi forms in many variables (part 2)
Examples of Jacobi forms in many variables (part 3)

Jacobi forms in many variables and the splitting principle. Theta-quarks as a pull-back. Weak Jacobi forms in many variables
Theta-quarks as pull-backs of Jacobi forms
Theta-quarks as pull-backs of Jacobi forms (part 2)
Theta-quarks as pull-backs of Jacobi forms (part 3)
The splitting principle
The splitting principle (part 2)
Weak Jacobi forms in many variables

The Weil representation and vector valued modular forms. Jacobi forms of singular weight
The Weil representation
The Weil representation (part 2)
Jacobi forms of singular weight
Jacobi forms of singular weight (part 2)
Jacobi forms of critical weight
Jacobi forms of critical weight (part 2)

Quasi-modular Eisenstein series. The automorphic correction of Jacobi forms and Taylor expansions
Automorpic correction
Differential operator D
Quasi-modular Eisenstein series
Automorphic correction and Taylor expansions
Automorphic correction of Jacobi forms in many variables
Automorphic correction of Jacobi forms in many variables (part 2)

Modular differential operators. The graded ring of the weak Jacobi modular forms
Modular differential operators for Jacobi forms
Heat operator
Heat operator (part 2)
Applications of differential operators for Jacobi forms
The graded ring of the weak Jacobi modular forms
Jacobi type forms in many variables

Jacobi type forms and the generalisation of the Cohen-Kuznetsov-Zagier operator
A new proof of Cohen-Kuznetsov-Zagier theorem
A new proof of Cohen-Kuznetsov-Zagier theorem
A new proof of Cohen-Kuznetsov-Zagier theorem (part 2)
Cohen-Kuznetsov-Zagier theorem in case of many variables

Summary of User Reviews

Discover the beauty of modular forms and Jacobi theory with this top-rated course on Coursera. Many users have praised the engaging and comprehensive content of the course, making it an ideal choice for both beginners and advanced learners.

Key Aspect Users Liked About This Course

The course provides a thorough and well-structured introduction to modular forms and Jacobi theory, making it easy to understand even for those with limited background knowledge.

Pros from User Reviews

Excellent content and presentation
Great for beginners and advanced learners alike
In-depth explanations and examples
Engaging and interactive
Well-structured and easy to follow

Cons from User Reviews

Some exercises are too difficult for beginners
More challenging than expected for some
Not enough emphasis on practical applications
Limited interaction with the instructor
No certificate offered for free users

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Jacobi modular forms: 30 ans après

Jacobi forms are holomorphic functions in two complex variables. They are modular in one variable and abelian (or double periodic) in another variable....

Save Course