This course follows "Advanced Course 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.
To become familiar with basic notions such as Galois representations, modular forms, modularity, etc.
Building upon them, to think what is new yourself.
Galois representation, modular form, modularity
✔ Specialist skills | Intercultural skills | Communication skills | ✔ Critical thinking skills | ✔ Practical and/or problem-solving skills |
Standard lecture course accompanied by discussion sessions
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 |
None required
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
By reports. Details will be announced in the course.
None required