Regular-cusp valence formula on a torsion-free modular curve (${\sum }_P{\mathrm{ord}}_{P}f=kd/12$)

For a torsion-free effective modular group with genuine, untwisted modular cusp periods, a meromorphic weight-$k$ form has integer local orders. The tensor differential $f^{12}(d\tau {)}^{6k}$ has interior order $12\mathrm{ord}f$ and modular cusp order $12\mathrm{ord}f-6k$. Its canonical degree and the genus formula for a modular curve yield the displayed valence formula, with $d$ the projective index. Regular modular cusp periods are important in odd weights.