Master 2021/2022

Jacobi modular forms: 30 ans apres

Type: Elective course (Mathematics)

Area of studies: Mathematics

Delivered by: Faculty of Mathematics

Where: Faculty of Mathematics

When: 2 year, 4 module

Mode of studies: distance learning

Online hours: 56

Open to: students of one campus

Instructors: Valery Gritsenko

Master’s programme: Mathematics

Language: English

ECTS credits: 3

Contact hours: 2

Course Syllabus

Abstract

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 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

Learning Objectives

• Acquaintance with the basic notions, methods and problems of Jacobi modular forms. Acquiring the ability for independent study of topical mathematical literature

Expected Learning Outcomes

• Knowledge of the basic notions, methods and problems Jacobi modular forms.

Course Contents

• Modular and abelian transformations
• Pullbacks of theta-function
• Modular forms
• Heisenberg group
• The action of Jacobi modular group
• The second definition of Jacobi forms
• Special values of Jacobi forms
• The zeros of elliptic functions
• The zeros of Jacobi forms
• Taylor expansion of Jacobi forms
• Dimensions of some spaces of Jacobi forms
• Examples of Jacobi modular forms

Assessment Elements

• cumulative
• final exam
• cumulative
• final exam

Interim Assessment

• 2021/2022 4th module
  The formula for marking is 0.3 cumulative + 0.7 (offline) final exam, where cumulative is proportional to the number of tasks solved. The final mark is rounded to the nearest integer (half-integers are rounded upwards).

Bibliography

Recommended Core Bibliography

• Zagier, D., & Skoruppa, N.-P. (1988). Jacobi forms and a certain space of modular forms. Zagier, Don; Skoruppa, Nils-Peter: Inventiones Mathematicae. 94 1988. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsdzs&AN=edsdzs.GDZPPN002105705

Recommended Additional Bibliography

• Gritsenko, V., & Wang, H. (2018). Graded rings of integral Jacobi forms. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.1810.09392