### 2016　Advanced topics in Algebra E

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Instructor(s)
Mizumoto Shin-Ichiro
Class Format
Lecture
Media-enhanced courses
Day/Period(Room No.)
Mon5-6(H116)
Group
-
Course number
MTH.A501
Credits
1
2016
Offered quarter
1Q
Syllabus updated
2016/12/14
Lecture notes updated
-
Language used
Japanese
Access Index

### Course description and aims

This course covers basic topics of single-variable regular automorphic forms. Building on basic undergraduate level knowledge, basic properties of the Riemann zeta function are proven, and students are introduced to the theory of automorphic L-functions. Single-variable regular automorphic forms are then defined, and students become familiar with specific treatments of the materials through several examples. This course is followed by Advanced Topics in Algebra F.
Automorphic forms are the foundation of modern number theory, and are an important mathematical subject related to a variety of fields such as group representation theory, the geometry of numbers, and theoretical physics.

### Student learning outcomes

The following concepts are especially important:
Riemann Zeta function (Euler product, analytic continuation, special values), elliptic automorphic form, Fourier coefficient, Eisenstein series.
Students will become familiar with these concepts, and learn the skills for calculating examples on their own.

### Keywords

Modular forms, modular groups, zeta functions

### 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

### Course schedule/Required learning

Course schedule Required learning
Class 1 multiplivative functions Details will be provided during each class session
Class 2 Riemann zeta function
Class 3 analytic continuation and special values of the Riemann zeta function
Class 4 modular groups
Class 5 elliptic modular forms
Class 6 examples of modular forms (1): Eisenstein series
Class 7 examples of modular forms (2): Ramanujan's delta function
Class 8 Fourier expansion of Eisenstein series

None required

### Reference books, course materials, etc.

T. M. Apostol: Modular Functions and Dirichlet Series in Number Theory (Springer)

### Assessment criteria and methods

Course scores are evaluated by homework assignments. Details will be announced during the course.

### Related courses

• MTH.A502 ： Advanced topics in Algebra F
• MTH.C301 ： Complex Analysis I
• MTH.C302 ： Complex Analysis II
• MTH.A201 ： Introduction to Algebra I
• MTH.A202 ： Introduction to Algebra II

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

basic undergraduate algebra and complex analysis