= Cusp-form divisor presentation
{title2=$S_k(\Gamma)=f\mathcal L(\operatorname{div}_{\mathrm{mod}}f-C)$}
For a torsion-free modular curve with regular <modular cusps>, let $f$ be a nonzero meromorphic weight-$k$ form and $C$ the reduced sum of <modular cusp> points. The local-order divisor of $f$, minus $C$, imposes exactly holomorphy in the interior and vanishing at every <modular cusp> on the product $f\varphi$. Dividing any <cusp form> by $f$ gives the reverse identification with a <Riemann-Roch space>. The <regular-cusp valence formula on a torsion-free modular curve> and <Riemann-Roch theorem> compute the dimension when the resulting divisor has degree greater than the canonical degree.
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