2025 (Current Year) Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Special lectures on advanced topics in Mathematics G
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Yoshinori Mizuno / Masatoshi Suzuki
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Class
- -
- Course Code
- MTH.E531
- Number of credits
- 200
- Course offered
- 2025
- Offered quarter
- 4Q
- Syllabus updated
- Mar 19, 2025
- Language
- Japanese
Syllabus
Course overview and goals
I will lecture on the fundamentals of Jacobi forms. A Jacobi form is a two-variable holomorphic function that combines elliptic functions and modular forms. It is an interesting subject in itself. On the other hand, its aspect as a tool is important. In particular, it has deep relationships with modular forms of degree 2 and of half-integer weight, and is used to study them. The proof of the Saito-Kurokawa conjecture is particularly impressive. I will overview the fundamentals based on the textbook by Eichler and Zagier. I hope that when similar objects arise in each person's research, they will be able to make use of this knowledge.
The textbook by Eichler and Zagier is a basic literature on Jacobi forms. Advanced concepts are not used, and the theory is constructed through grounded calculations. There are many technical points that one might not come up with on their own, but by following the calculations, we want to re-experience the fun of calculations and provide a starting point for enhancing each person's calculation skills.
Course description and aims
・Acquire the basic concepts of Jacobi forms.
・Understand the usefulness of Jacobi forms.
・Be able to perform specific calculations.
Keywords
Jacobi forms, Jacobi-Eisenstein series, Hecke operators, relations with modular forms
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
This is a standard lecture course. There will be some assignments.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | ・Jacobi forms and Jacobi groups ・Eisenstein series and cusp forms ・Taylor expansion ・Hecke operators ・Relationship with half-integer weight modular forms ・Relationship with Siegel modular forms ・Jacobi theta series and Waldspurger's theorem | Details will be provided during each class session |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
None required
Reference books, course materials, etc.
(1) M. Eichler-D. Zagier, The theory of Jacobi forms, Birkhauser (1985)
(2) 伊吹山知義, 保型形式特論, 共立出版 (2018)
Evaluation methods and criteria
Assignments (100%).
Related courses
- MTH.A301 ： Algebra I
- MTH.A302 ： Algebra II
- MTH.A331 ： Algebra III
Prerequisites
None required