Mini-course: The legacy of Ramanujan.
Lecturers: Bill Duke and Paul Jenkins. TA: Yingkun Li.
Description
Singular moduli are the values of the modular j-function at the points in the upper half plane that satisfy a quadratic equation. These values have been studied by number theorists since the 19th century. They are algebraic and generate class fields of imaginary quadratic fields. Both their norms and traces are integers. The work of Gross and Zagier in the 1980s gave explicit factorizations of the norms and led to the celebrated Gross-Zagier formula. The traces, which were studied by Zagier in 2003, turn out to be related to Fourier coefficients of modular forms.
In this course, we will explore the connections between singular moduli and modular forms, Borcherds product and harmonic Maass forms.
Syllabus
- Definition and Properties of Singular Moduli.
- Half-integral weight modular forms and their Fourier coefficients.
- Borcherds Products.
- Harmonic Maass forms and their Fourier coefficients.
Prerequisites
Basic knowledge of modular forms.
Notes and problems
Lecture notes and exercises 1 and 2.