2024 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Algebra A
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Soma Purkait
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-6 Thu
- Class
- -
- Course Code
- MTH.A401
- Number of credits
- 100
- Course offered
- 2024
- Offered quarter
- 1Q
- Syllabus updated
- Mar 14, 2025
- Language
- English
Syllabus
Course overview and goals
Modular forms are fundamental objects in mathematics, primarily a central topic in number theory, they appear in wide ranging fields like group representations, geometry, combinatorics and physics. The aim of this course together with "Advanced topics in Algebra B" is to introduce the basic notion of modular forms with a view towards both classical and modern applications.
Course description and aims
Students are expected to understand the basic notion of modular forms. Looking through concrete examples and applications, students get acquainted with the fundamental importance of modular forms in current research.
Keywords
Upper half-plane, Weierstrass ℘ function, Eisenstein series, Modular functions, Modular forms
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 | Introduction: Modular forms are ubiquitous | Details will be provided during each class sessions |
| Class 2 | Elliptic functions, Weierstrass ℘ function, Eisenstein Series | Details will be provided during each class sessions |
| Class 3 | Upper Half-plane and Fuchsian Groups | Details will be provided during each class sessions |
| Class 4 | Fundamental Domains | Details will be provided during each class sessions |
| Class 5 | Modular functions, Modular forms (Level 1), Eisenstein series | Details will be provided during each class sessions |
| Class 6 | Ramanujan's Delta function, Valence Formula and applications | Details will be provided during each class sessions |
| Class 7 | Modular j-invariant, uniformization theorem (Elliptic curves), E_2 and Delta | Details will be provided during each class sessions |
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
The 1-2-3 of Modular Forms, Universitext, Springer 2008
Evaluation methods and criteria
Course scores are evaluated by homework assignments. Details will be announced during the course.
Related courses
- MTH.A402 ： Advanced topics in Algebra B
- ZUA.A331 ： Advanced courses in Algebra A
- ZUA.A332 ： Advanced courses in Algebra B
Prerequisites
Basic undergraduate algebra and complex analysis
Other
None in particular.