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.
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.
Modular forms, modular groups, zeta functions
|✔ Specialist skills||Intercultural skills||Communication skills||✔ Critical thinking skills||✔ Practical and/or problem-solving skills|
Standard lecture course
|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|
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.
T. M. Apostol: Modular Functions and Dirichlet Series in Number Theory (Springer)
Course scores are evaluated by homework assignments. Details will be announced during the course.
basic undergraduate algebra and complex analysis