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 E, 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
|Intercultural skills||Communication skills||Specialist 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|
|Class 3||structure of the graded ring of modular forms|
|Class 4||Poincare series|
|Class 5||Hecke operators|
|Class 6||automorphic L-functions (1): Euler products|
|Class 7||automorphic L-functions (2): analytic continuation|
|Class 8||supplements and prospects|
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