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- Liberal arts and basic science courses
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Taguchi Yuichiro
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Thr5-6(H137)
- Group
- -
- Course number
- MTH.A402
- Credits
- 1
- Academic year
- 2018
- Offered quarter
- 2Q
- Syllabus updated
- 2018/3/20
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
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Course description and aims
This course follows "Advanced Topics in Algebra A" in the first quarter. One of the most remarkable results about zeta functions of Galois representations is a proof of the "modularity" (Fermat's Last Theorem and the Sato-Tate Conjecture have been proved as a result of it). In this course, after reviewing basic facts on modular forms, we prepare various tools used in the proof of the modularity of Galois representations and sketch the proof.
Student learning outcomes
To become familiar with basic notions such as Galois representations, modular forms, modularity, etc.
Building upon them, to think what is new yourself.
Keywords
Galois representation, modular form, modularity
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 accompanied by discussion sessions
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | Modular forms | Details will be provided during each class session |
| Class 2 | Modular curves | Details will be provided during each class session |
| Class 3 | Hecke algebras | Details will be provided during each class session |
| Class 4 | Deformation theory of Galois representations | Details will be provided during each class session |
| Class 5 | Serre's Modularity Conjecture and Fontaine-Mazur Conjecture | Details will be provided during each class session |
| Class 6 | R=T Theorem, Part 1 | Details will be provided during each class session |
| Class 7 | R=T Theorem, Part 2 | Details will be provided during each class session |
| Class 8 | R=T Theorem, Part 3 | Details will be provided during each class session |
Textbook(s)
None required
Reference books, course materials, etc.
Takeshi Saito, "Fermat Conjecture", Iwanami, 2009 Proceedings of "The arithmetic of l-adic Galois representations and Galois deformations" http://www.math.sci.osaka-u.ac.jp/~ochiai/ss2009proceeding/ss2009proceeding.html Proceedings of "Recent developments on Ｒ＝Ｔ" http://www.kurims.kyoto-u.ac.jp/~gokun/R=T.html
Assessment criteria and methods
By reports. Details will be announced in the course.
Related courses
- MTH.A401 ： Advanced topics in Algebra A
Prerequisites (i.e., required knowledge, skills, courses, etc.)
None required
