2020 Advanced topics in Algebra B

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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.A402
Credits
1
Academic year
2020
Offered quarter
2Q
Syllabus updated
2020/9/18
Lecture notes updated
-
Language used
English
Access Index

Course description and aims

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.

Student learning outcomes

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.

Keywords

elliptic modular forms, Poincare series, Hecke operators, automorphic L-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 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

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.

Textbook(s)

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.A401 : Advanced topics in Algebra A

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

basic undergraduate algebra and complex analysis

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