The theory of modular forms plays an essential role in various aspects of the number theory. Based on the basic facts on modular curves and modular forms explained in "Advanced Course in Algebra D1" in 3Q", we deal with the basics of the applications to the L-function and the Galois representation associated with a modular form.
Students are expected to understand applications of the theory of modular forms and their relation to some other areas.
modular form, modular curve, L-function, Galois representation, automorphic representation
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Standard lecture course
Course schedule | Required learning | |
---|---|---|
Class 1 | Galois representation associated to a cuspform: part 1 | Details will be provided during each class session |
Class 2 | Galois representation associated to a cuspform: part 2 | Details will be provided during each class session |
Class 3 | Modular forms and automorphic representations: part 1 | Details will be provided during each class session |
Class 4 | Modular forms and automorphic representations: part 2 | Details will be provided during each class session |
Class 5 | Automorphic representations and the Langlands correspondence | Details will be provided during each class session |
Class 6 | Advanced topics: part 1 | Details will be provided during each class session |
Class 7 | Advanced topics: part 2 | Details will be provided during each class session |
To enhance effective learning, students are encouraged to explore references provided in lectures and other materials.
None required.
Neal Koblitz, Introduction to Elliptic Curves and Modular forms, GTM 97, Springer-Verlag, New York, 1993
Toshitsune Miyake, Modular Forms, english ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin 2006
Course scores are evaluated by homework assignments. Details will be announced during the course.
Some knowledge of Algebraic Geometry might be assumed, but try to talk so that the participants can follow the contents without this knowledge
None in particular.