2025 (Current Year) Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Algebra F1
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Tadashi Ochiai
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-6 Mon
- Class
- -
- Course Code
- MTH.A506
- Number of credits
- 100
- Course offered
- 2025
- Offered quarter
- 2Q
- Syllabus updated
- Mar 19, 2025
- Language
- English
Syllabus
Course overview and goals
This course follows Advanced topics in Algebra E1.
Modular forms, their L-functions and Galois representations, modular curves appear in many areas of number theory, and are studied very actively. This course hopes to provide solid background for students intending to learn advanced topics on these objects. Based on Advanced topics in Algebra E1, we study more general L-functions.
Course description and aims
Students are expected to:
-- understand fundamental notions and methods of modular forms and arithmetic geometry.
-- be familiar with modern tools and concepts in the theory of modular forms, their L-functions and Galois representations.
Keywords
L-functions, Galois representation, modular curves, p-adic 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 | Modular forms | Details will be provided during each class session. |
| Class 2 | Modular forms (continuation) | Details will be provided during each class session. |
| Class 3 | L-functions associated to modular forms | Details will be provided during each class session. |
| Class 4 | Modular curves | Details will be provided during each class session. |
| Class 5 | Eichler-Shimura isomorphism and the Galois representations associated modular forms | Details will be provided during each class session. |
| Class 6 | p-adic L-functions associated to modular forms | Details will be provided during each class session. |
| Class 7 | p-adic L-functions associated to modular forms (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)
Unspecified.
Reference books, course materials, etc.
Modular forms, Springer, T.Miyake
Elementary Theory of L-functions and Eisenstein Series, Cambridge University Press, H. Hida.
Evaluation methods and criteria
Learning achievement is evaluated by reports (100%).
Related courses
- MTH.A301 ： Algebra I
- MTH.A302 ： Algebra II
- MTH.A331 ： Algebra III
- MTH.A505 ： Advanced topics in Algebra E1
Prerequisites
Basic knowledge of undergraduate algebra, geometry and complex analysis, the subjects of Advanced topics in Algebra E1