In this course the instructor explains basics topics of L-functions associated with single-variable regular automorphic forms. Knowledge of the definition and examples of single-variable regular automorphic forms is assumed, and the instructor covers the space structures formed by automorphic forms as a whole, and Hecke operators that act on them. Using Hecke operators, automorphic L-functions are then defined, and the instructor discusses Euler product representations and analytic continuation. This course follows Advanced Topics in Algebra A, which is held immediately before it.
Automorphic L-functions are a mathematical subject at the center of modern number theory research, and are even now the subject of active research.
The following notions are impotant:
elliptic modular forms, graded ring of modular forms, Poincare series, Hecke operators, automorphic L-functions.
The aim of this course is help the students become acquainted with these notions through concrete examples.
elliptic modular forms, Poincare series, Hecke operators, automorphic L-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||fundamental domains||Details will be provided during each class session|
|Class 2||dimension of the space of modular forms||Details will be provided during each class session|
|Class 3||structure of the graded ring of modular forms||Details will be provided during each class session|
|Class 4||Poincare series||Details will be provided during each class session|
|Class 5||Hecke operators||Details will be provided during each class session|
|Class 6||automorphic L-functions (1): Euler products||Details will be provided during each class session|
|Class 7||automorphic L-functions (2): analytic continuation||Details will be provided during each class session|
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