Multiplication by the modular discriminant
= Multiplication by the modular discriminant
{title2=$M_k\xrightarrow{\;f\mapsto\Delta f\;}S_{k+12}$}
The <modular discriminant> has a simple zero at infinity and no zero in the <upper half-plane>, by the <valence formula for the modular group>. Multiplication by it identifies weight-$k$ <modular forms> with weight-$(k+12)$ <cusp forms>. Conversely dividing a cusp form by it preserves interior holomorphy and removes one order of vanishing at infinity.