2022 Students Enrolled in or before 2015 School of Science Mathematics
Advanced courses in Algebra D
- Academic unit or major
- Mathematics
- Instructor(s)
- Soma Purkait
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-6 Thu (H114)
- Class
- -
- Course Code
- ZUA.A334
- Number of credits
- 100
- Course offered
- 2022
- Offered quarter
- 4Q
- Syllabus updated
- Jul 10, 2025
- Language
- English
Syllabus
Course overview and goals
This course follows Advanced courses in Algebra C, building on the topics covered there, we define automorphic forms on Fuchsian groups and study algebraic structures formed by them. We introduce Hecke operators and the theory of Automorphic L-functions - analytic continuation, functional equation and Euler product, and present a famous application to the congruent number problem.
Course description and aims
Students are expected to understand basic notions of automorphic forms, Hecke operators and automorphic L-functions. Looking through concrete examples and applications, students get acquainted with the fundamental importance of modular forms in current research.
Keywords
Automorphic forms, Hecke operators, automorphic L-functions, Newforms, Theta-functions.
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Standard lecture course
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Theta functions | Details will be provided during each class session |
| Class 2 | Automorphic forms, finite dimensionality | Details will be provided during each class session |
| Class 3 | Hecke operators | Details will be provided during each class session |
| Class 4 | Automorphic L-functions: analytic continuation, functional equation | Details will be provided during each class session |
| Class 5 | Eigenforms and Euler product of associated L-function | Details will be provided during each class session |
| Class 6 | Newforms and old forms, multiplicity-one | Details will be provided during each class session |
| Class 7 | Congruent number problem | Details will be provided during each class session |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to explore references provided in lectures and other materials.
Textbook(s)
None required.
Reference books, course materials, etc.
Neal Koblitz, Introduction to Elliptic Curves and Modular forms, GTM 97, Springer-Verlag, New York, 1993
Toshitsune Miyake, Modular Forms, english ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin 2006
Evaluation methods and criteria
Course scores are evaluated by homework assignments. Details will be announced during the course.
Related courses
- ZUA.A333 ： Advanced courses in Algebra C
Prerequisites
Basic undergraduate algebra and complex analysis
Other
None in particular.