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.
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
|Intercultural skills||Communication skills||Specialist 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 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|
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