= Regular-cusp valence formula on a torsion-free modular curve
{title2=$\sum_P\operatorname{ord}_P f=kd/12$}
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 $12\operatorname{ord}f$ and <modular cusp> order $12\operatorname{ord}f-6k$. 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.
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