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- Liberal arts and basic science courses
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Purkait Soma
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- Day/Period(Room No.)
- Thr5-6(M-119(H118))
- Group
- -
- Course number
- MTH.A402
- Credits
- 1
- Academic year
- 2024
- Offered quarter
- 2Q
- Syllabus updated
- 2024/3/14
- Lecture notes updated
- -
- Language used
- English
- Access Index
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Course description and aims
This course follows Advanced topics in Algebra A, 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. If time allows, we present a famous application to the congruent number problem.
Student learning outcomes
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 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 | Automorphic forms, finite dimensionality | Details will be provided during each class session |
| Class 2 | Poincaré series and Eisenstein series | Details will be provided during each class session |
| Class 3 | Modular forms for congruence subgroups, Jacobi's theta function | Details will be provided during each class session |
| Class 4 | Hecke Algebras | Details will be provided during each class session |
| Class 5 | Hecke Algebras of Modular groups, Eigenforms | Details will be provided during each class session |
| Class 6 | Automorphic L-functions, Euler product, Newforms | Details will be provided during each class session |
| Class 7 | Automorphic L-functions: Meromorphic continuation, functional equation | Details will be provided during each class session |
Out-of-Class Study Time (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
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
- ZUA.A331 ： Advanced courses in Algebra A
- ZUA.A332 ： Advanced courses in Algebra B
Prerequisites (i.e., required knowledge, skills, courses, etc.)
MTH.A401 ： Advanced topics in Algebra A
Other
None in particular.
