MATH526 MODULAR FUNCTIONS
| Course Code: | 2360526 |
| METU Credit (Theoretical-Laboratory hours/week): | 3 (3.00 - 0.00) |
| ECTS Credit: | 8.0 |
| Department: | Mathematics |
| Language of Instruction: | English |
| Level of Study: | Graduate |
| Course Coordinator: | |
| Offered Semester: | Fall Semesters. |
Course Objectives
Modular forms and functions can be defined as complex analytic functions on the upper half plane obeying a certain functional equation with respect to a certain discrete group action. From another perspective, they correspond to fairly natural forms and functions on spaces parametrizing elliptic curves, possibly with additional structure. As such, they played an important role in the development of the classical theory of abelian functions in mathematics (and physics) and for solutions of several problems in number theory. On the other hand, the subject became extremely popular when the connection of the Taniyama-Shimura-Weil conjecture to Fermat’s last theorem was discovered. Fermat’s last theorem was proven in 1994 by Wiles by resolving a special case of the conjecture. The full conjecture was proven in 2001 by Breuil, Conrad, Diamond and Taylor. The course aims to explain the significance of modular forms and the story of the conjecture, however not its proof.
Course Content
Elliptic functions, modular functions, Dedekind eta function, congruencies for the coefficients of the modular function j, Rademacher s series for the partition function, modular forms with multiplicative coefficients, Kronecker s theorem, general Dirichlet series and Bohr s equivalence theorem.
Course Learning Outcomes
By the end of this course, a student will:
- make computations regarding modular forms and functions using their complex analytic description,
- work with the description of modular forms and functions as meromorphic forms and functions on the moduli space of elliptic curves with additional structure,
- compute dimensions of spaces of modular forms using the Riemann-Roch theorem,
- deduce number theoretical results using the dimension formulae,
- understand the role of Hecke operators in the theory,
- understand the statement of the Taniyama-Shimura-Weil conjecture from different perspectives, and its relation to Fermat’s last theorem.