Modular forms are characterized by certain transformation laws.
In general, these properties are not preserved by (holomorphic)
differentiation. I will discuss several ways to overcome this trouble:
Analytically one can use non-holomorphic differential operators
(‘‘Maaß-Shimura operators’’) or use bilinear holomorphic differential
operators (‘‘Rankin brackets’’). More arithmetically, one can show that
derivatives of modular forms are still congruent to modular forms
(‘‘Ramanujan’s $\theta$-operators’’). I will describe the connection between the analytic
and the arithmetic aspects with focus on some recent results on $\theta$-operators.