2026 (Current Year) Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Algebra F
- 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 (S3-215(S321))
- Class
- -
- Course Code
- MTH.A502
- Number of credits
- 100
- Course offered
- 2026
- Offered quarter
- 2Q
- Syllabus updated
- Mar 5, 2026
- Language
- English
Syllabus
Course overview and goals
In Classical Iwasawa theory, we study p-adic properties of ideal class groups of cyclotomic fields and zeta functions for a fixed prime number p. In this course, we explain the motivation and the fundamental part of Iwasawa theory.
Course description and aims
(1) We learn the treatment of the special values of zeta-functions
(2) We learn the p-adic analogue of zeta-functions
(3) We learn the formulation of Iwasawa Main Conjecture
Keywords
ideal class groups, Selmer groups, zeta functions, Iwasawa algebra, cyclotomic fields, p-acid L-function, Iwasawa Main conjecture
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
This is a standard lecture course.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Kubota-Leopoldt p-adic L-function and the Iwasawa construction |
Details will be provided during each class |
| Class 2 | Proof of the construction of the Kubota-Leopoldt p-adic L-function |
Details will be provided during each class |
| Class 3 | Iwasawa Main conjecture I |
Details will be provided during each class |
| Class 4 | Iwasawa Main conjecture II |
Details will be provided during each class |
| Class 5 | Galois cohomology and Selmer groups |
Details will be provided during each class |
| Class 6 | Generalization of the Iwasawa Main Conjecture I |
Details will be provided during each class |
| Class 7 | Generalization of the Iwasawa Main Conjecture II |
Details will be provided during each class |
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.
L. Washington, Introduction to Cyclotomic Fields, Springer
T. Ochiai, Iwasawa Theory and Its Perspective, Volume 1,2,3, American Mathematical Society
Evaluation methods and criteria
Course scores are evaluated by homework assignments. Details will be announced during the course.
Related courses
- MTH.A501 ： Advanced topics in Algebra E
- MTH.A301 ： Algebra I
- MTH.A302 ： Algebra II
Prerequisites
A basic knowledge of Algebraic Number theory and Class field theory is desirable.
Other
None in particular.