In number theory, L-functions are associated with various arithmetic objects. In this course, we explain the interpretation of L-functions based on Galois representations and discuss important results/conjectures on them. This course together with "Advanced Course in Algebra C1" given in the 3Q forms one set of contents. Advanced topics (Dwork theory, modularity) are dealt with in the latter half "D1".
Students are expected to:
- understand the proof of analyticity of the L-functions of φ-modules by Dwork theory.
- understand the notion of modularity of an L-function, and the proof of the modularity in the 1-dimensional case.
L-function, Galois representation, modularity, φ-module, Dwork theory
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
This is a standard lecture course. There will be some assignments.
Course schedule | Required learning | |
---|---|---|
Class 1 | L-functions of φ-sheaves (Part 1) | Details will be provided during each class session |
Class 2 | L-functions of φ-sheaves (Part 2) | Details will be provided during each class session |
Class 3 | L-functions of φ-sheaves (Part 3) | Details will be provided during each class session |
Class 4 | Algebraic Hecke characters | Details will be provided during each class session |
Class 5 | L-functions of CM Abelian varieties (Part 1) | Details will be provided during each class session |
Class 6 | L-functions of CM Abelian varieties (Part 2) | Details will be provided during each class session |
Class 7 | L-functions of CM Abelian varieties (Part 3) | Details will be provided during each class session |
To enhance effective learning, students are encouraged to indulge themselves in L-functions and Galois representations.
None required.
None required.
Course scores are evaluated by homework assignments (100%). Details will be announced during the course.
Basic knowledge in undergraduate algebra