Modular forms

Modular forms are complex functions that have significant importance in number theory, algebra, and various areas of mathematics. More specifically, they are a type of analytic function that are defined on the upper half of the complex plane and exhibit certain transformation properties under the action of the modular group. ### Definitions and Properties 1. **Holomorphic Functions**: Modular forms are typically required to be holomorphic (complex differentiable) on the upper half-plane, which consists of all complex numbers with positive imaginary parts.