Meromorphic modular form

A meromorphic modular form obeys the modular transformation law and is meromorphic on the upper half-plane and at the cusps.

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 $f\phi$. 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.

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.

For a nonzero meromorphic modular form $f$ of weight $k$,where the sum takes one representative from each modular-group orbit.

A nonzero weight-fourteen cusp form would have order at infinity at least one and order at $i$ at least one, since $i^{14}=-1$. The valence formula for the modular group would then give $1+1/2\le 14/12$, which is impossible.

A weight-two modular form obeys $f(i)=i^{2}f(i)=-f(i)$, so it vanishes at $i$. The valence formula for the modular group would then have left side at least $1/2$, but right side $2/12=1/6$. All orders are nonnegative by holomorphy, giving a contradiction for nonzero $f$.

For nonnegative integer $k$ and a finite-index subgroup $\Gamma$, a nonzero modular form has order at infinity, measured in a genuine periodic cusp parameter, at most $k[S{L}_2(\mathbb{Z}):\Gamma ]/12$. The product of its slash translates over the left cosets is a nonzero level-one form; the translates along the infinity-cusp orbit contribute its full local order to this product. The valence formula for the modular group gives the bound. Initial coefficients through that bound therefore give an injective linear map.

The Klein j-invariant is the weight-zero level-one modular functionIt is holomorphic on the upper half-plane and has a simple pole at infinity.

The valence formula for the modular group shows that $\Delta =(E_4^3-E_6^2)/1728$ has no zeros in the half-plane. For every $a\in \mathbb{C}$, the weight-twelve form $E_4^3-a\Delta$ has total weighted zero order one. At $a=0$ its sole zero is the order-three zero at the cubic elliptic orbit; at $a=1728$ its sole zero is the order-two zero at the quadratic elliptic orbit. Otherwise it has one simple ordinary zero orbit. Thus each value of the Klein j-invariant occurs on exactly one modular orbit and classifies one homothety of complex lattices.

The function $j(\tau )/j(2\tau )$ has weight zero and level ${\Gamma }_1(2)={\Gamma }_0(2)$. It has a simple zero at the cusp at infinity and a simple pole at the cusp zero.