2025 (Current Year) Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Algebra E1
- 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.A505
- Number of credits
- 100
- Course offered
- 2025
- Offered quarter
- 1Q
- Syllabus updated
- Mar 19, 2025
- Language
- English
Syllabus
Course overview and goals
In this course, we will explain the fundamental of the algebraic number theory. Especially, we explain the ideal class groups, the group of units, the zeta function of algebraic number fields. We also explain the fundamental of the elliptic curves.
Especially, we explain the Mordell-Weil group, the Tate-Shafarevich group and the Hasse-Weil L-function of an elliptic curve.
This course is followed by Advanced topics in Algebra F1.
Zeta- and L-functions 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 zeta- and L-functions. We begin with the classical Riemann zeta function.
Course description and aims
Students are expected to:
-- understand fundamental notions and methods of the algebraic number theory and the elliptic curves.
-- be familiar with modern tools and concepts in the zeta-functions and the Galois representations.
Keywords
zeta function, Galois representation, Selmer group
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 | Riemann zeta function | Details will be provided during each class session. |
| Class 2 | Analytic continuation and functional equation | Details will be provided during each class session. |
| Class 3 | Special values | Details will be provided during each class session. |
| Class 4 | Partial summation formula | Details will be provided during each class session. |
| Class 5 | Prime Number Theorem | Details will be provided during each class session. |
| Class 6 | Zero-free region | Details will be provided during each class session. |
| Class 7 | Proof of the Prime Number Theorem | 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.
「Daisuteki Seisuron」Morikita Syuppan, Ishida Makoto (no english translation)
「Algebraic Number Theory」Springer, J. Neukirch
「The Arithmetic of Elliptic Curves」Springer, J.H.Silverman
Ohter course materials are provided during class.
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.A506 ： Advanced topics in Algebra F1
Prerequisites
Basic knowledge of undergraduate algebra and complex analysis