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- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Mizumoto Shin-Ichiro
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Mon5-6(H116)
- Group
- -
- Course number
- MTH.A502
- Credits
- 1
- Academic year
- 2016
- Offered quarter
- 2Q
- Syllabus updated
- 2016/12/14
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
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Course description and aims
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 E, 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.
Student learning outcomes
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 that will be developed
| ✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Class flow
Standard lecture course
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | fundamental domains | Details will be provided during each class session |
| Class 2 | dimension of the space of modular forms | |
| Class 3 | structure of the graded ring of modular forms | |
| Class 4 | Poincare series | |
| Class 5 | Hecke operators | |
| Class 6 | automorphic L-functions (1): Euler products | |
| Class 7 | automorphic L-functions (2): analytic continuation | |
| Class 8 | supplements and prospects |
Textbook(s)
None required
Reference books, course materials, etc.
T. M. Apostol: Modular Functions and Dirichlet Series in Number Theory (Springer)
Assessment criteria and methods
Course scores are evaluated by homework assignments. Details will be announced during the course.
Related courses
- MTH.A501 ： Advanced topics in Algebra E
- MTH.C301 ： Complex Analysis I
- MTH.C302 ： Complex Analysis II
- MTH.A201 ： Introduction to Algebra I
- MTH.A202 ： Introduction to Algebra II
Prerequisites (i.e., required knowledge, skills, courses, etc.)
basic undergraduate algebra and complex analysis
