2020　Advanced topics in Algebra A

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
Mizumoto Shin-Ichiro
Course component(s)
Lecture
Mode of instruction
ZOOM
Day/Period(Room No.)
Thr5-6(H137)
Group
-
Course number
MTH.A401
Credits
1
Academic year
2020
Offered quarter
1Q
Syllabus updated
2020/9/18
Lecture notes updated
-
Language used
English
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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 B.
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 sessions
Class 2 Riemann zeta function Details will be provided during each class sessions
Class 3 analytic continuation and special values of the Riemann zeta function Details will be provided during each class sessions
Class 4 modular groups Details will be provided during each class sessions
Class 5 elliptic modular forms Details will be provided during each class sessions
Class 6 examples of modular forms (1): Eisenstein series Details will be provided during each class sessions
Class 7 examples of modular forms (2): Ramanujan's delta function Details will be provided during each class sessions

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

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.

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.A402 ： Advanced topics in Algebra B

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

basic undergraduate algebra and complex analysis