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.
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.
Automorphic forms, Hecke operators, automorphic L-functions, Newforms, Theta-functions.
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Standard lecture course
Course schedule | Required learning | |
---|---|---|
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 |
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.