Bounded invariant norm characterization of cusp forms ($f\in S_k(\Gamma )⟺{\sup }_{\tau \in \mathbb{H}}|f(\tau )|(\mathrm{Im}\tau {)}^{k/2}<\infty$)

For $k>0$, a meromorphic function satisfying the modular transformation law is a cusp form exactly when its invariant norm is bounded. The bound first excludes interior poles. At each cusp a genuine periodic slash translate is bounded by $C{y}^{-k/2}$, which tends to zero. Its punctured-disk expression therefore has a removable singularity with zero value. Conversely cusp Fourier expansions decay exponentially, and finitely many compact core regions control the rest. Positive weight is essential: a nonzero constant in weight zero satisfies the bound but is not a cusp form.