2025/2026
Модулярные формы Якоби: 30 лет спустя
Лучший по критерию «Новизна полученных знаний»
Статус:
Дисциплина общефакультетского пула
Кто читает:
Факультет математики
Где читается:
Факультет математики
Когда читается:
3, 4 модуль
Онлайн-часы:
56
Охват аудитории:
для своего кампуса
Преподаватели:
Гриценко Валерий Алексеевич
Язык:
английский
Кредиты:
3
Контактные часы:
6
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
- Активная работа на сессиях семинара
- ИНдивидуальная работа над темами НИСа
- Доклад на семинаре или реферат
Interim Assessment
- 2025/2026 4th module0.2 * ИНдивидуальная работа над темами НИСа + 0.2 * Активная работа на сессиях семинара + 0.6 * Доклад на семинаре или реферат
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