= Bounded invariant norm characterization of cusp forms
{title2=$f\in S_k(\Gamma)\iff\sup_{\tau\in\mathbb H}|f(\tau)|(\operatorname{Im}\tau)^{k/2}<\infty$}
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 $Cy^{-k/2}$, 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.
Back to article page
