2020 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Algebra A
- 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.A401
- Number of credits
- 100
- Course offered
- 2020
- Offered quarter
- 1Q
- Syllabus updated
- Jul 10, 2025
- Language
- English
Syllabus
Course overview and goals
This course covers basic topics of single-variable regular automorphic forms. Building on basic undergraduate level knowledge, basic properties of the Riemann zeta function are proven, and students are introduced to the theory of automorphic L-functions. Single-variable regular automorphic forms are then defined, and students become familiar with specific treatments of the materials through several examples. This course is followed by Advanced Topics in Algebra B.
Automorphic forms are the foundation of modern number theory, and are an important mathematical subject related to a variety of fields such as group representation theory, the geometry of numbers, and theoretical physics.
Course description and aims
The following concepts are especially important:
Riemann Zeta function (Euler product, analytic continuation, special values), elliptic automorphic form, Fourier coefficient, Eisenstein series.
Students will become familiar with these concepts, and learn the skills for calculating examples on their own.
Keywords
Modular forms, modular groups, zeta 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 | multiplivative functions | Details will be provided during each class sessions |
| Class 2 | Riemann zeta function | Details will be provided during each class sessions |
| Class 3 | analytic continuation and special values of the Riemann zeta function | Details will be provided during each class sessions |
| Class 4 | modular groups | Details will be provided during each class sessions |
| Class 5 | elliptic modular forms | Details will be provided during each class sessions |
| Class 6 | examples of modular forms (1): Eisenstein series | Details will be provided during each class sessions |
| Class 7 | examples of modular forms (2): Ramanujan's delta function | Details will be provided during each class sessions |
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.A402 ： Advanced topics in Algebra B
Prerequisites
basic undergraduate algebra and complex analysis