2020 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Algebra B
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Shin-Ichiro Mizumoto
- Class Format
- Lecture (Zoom)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-6 Thu (H137)
- Class
- -
- Course Code
- MTH.A402
- Number of credits
- 100
- Course offered
- 2020
- Offered quarter
- 2Q
- Syllabus updated
- Jul 10, 2025
- Language
- English
Syllabus
Course overview and goals
In this course the instructor explains basics topics of L-functions associated with single-variable regular automorphic forms. Knowledge of the definition and examples of single-variable regular automorphic forms is assumed, and the instructor covers the space structures formed by automorphic forms as a whole, and Hecke operators that act on them. Using Hecke operators, automorphic L-functions are then defined, and the instructor discusses Euler product representations and analytic continuation. This course follows Advanced Topics in Algebra A, which is held immediately before it.
Automorphic L-functions are a mathematical subject at the center of modern number theory research, and are even now the subject of active research.
Course description and aims
The following notions are impotant:
elliptic modular forms, graded ring of modular forms, Poincare series, Hecke operators, automorphic L-functions.
The aim of this course is help the students become acquainted with these notions through concrete examples.
Keywords
elliptic modular forms, Poincare series, Hecke operators, automorphic L-functions
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Standard lecture course
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | fundamental domains | Details will be provided during each class session |
| Class 2 | dimension of the space of modular forms | Details will be provided during each class session |
| Class 3 | structure of the graded ring of modular forms | Details will be provided during each class session |
| Class 4 | Poincare series | Details will be provided during each class session |
| Class 5 | Hecke operators | Details will be provided during each class session |
| Class 6 | automorphic L-functions (1): Euler products | Details will be provided during each class session |
| Class 7 | automorphic L-functions (2): analytic continuation | 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.
T. M. Apostol: Modular Functions and Dirichlet Series in Number Theory (Springer)
Evaluation methods and criteria
Course scores are evaluated by homework assignments. Details will be announced during the course.
Related courses
- MTH.A401 ： Advanced topics in Algebra A
Prerequisites
basic undergraduate algebra and complex analysis