### 2021　Advanced courses in Algebra D

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Mathematics
Instructor(s)
Taguchi Yuichiro
Course component(s)
Lecture    (ZOOM)
Day/Period(Room No.)
Thr5-6(H137)
Group
-
Course number
ZUA.A334
Credits
1
2021
Offered quarter
4Q
Syllabus updated
2021/3/19
Lecture notes updated
-
Language used
English
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### Course description and aims

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".

### Student learning outcomes

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.

### Keywords

L-function, Galois representation, modularity, φ-module, Dwork theory

### Competencies that will be developed

 ✔ Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

### Class flow

This is a standard lecture course. There will be some assignments.

### Course schedule/Required learning

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

### Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to indulge themselves in L-functions and Galois representations.

None required.

None required.

### Assessment criteria and methods

Course scores are evaluated by homework assignments (100%). Details will be announced during the course.

### Related courses

• MTH.A407 ： Advanced topics in Algebra C1
• ZUA.A334 ： Advanced courses in Algebra C1
• ZUA.A334 ： Advanced courses in Algebra D1