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 D" is to introduce the basic notion of modular forms with a view towards both classical and modern applications.
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.
Upper half-plane, Eisenstein series, Modular functions, Modular forms.
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Standard lecture course
Course schedule | Required learning | |
---|---|---|
Class 1 | Introduction: Modular forms are ubiquitous | Details will be provided during each class session |
Class 2 | Doubly Periodic functions, Eisenstein Series | Details will be provided during each class session |
Class 3 | Upper Half-plane and Fuchsian Groups | Details will be provided during each class session |
Class 4 | Fundamental Domains | Details will be provided during each class session |
Class 5 | Modular functions and Modular forms | Details will be provided during each class session |
Class 6 | Ramanujan's Delta function and j-invariant | Details will be provided during each class session |
Class 7 | Modular forms for congruence subgroups | Details will be provided during each class session |
To enhance effective learning, students are encouraged to explore references provided in lectures and other materials.
None required.
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
Course scores are evaluated by homework assignments. Details will be announced during the course.
Basic undergraduate algebra and complex analysis
None in particular.