= Modular forms as meromorphic differentials on X(1)
{title2=$f(\tau)(d\tau)^k\in H^0(X(1),K^{\otimes k}(D))$}
A weight-$2k$ <modular form> descends as an invariant tensor differential. At an elliptic point of order $e$, a local invariant coordinate is $w=u^e$, so its descended coefficient has a pole of order at most $\lfloor k(e-1)/e\rfloor$. At infinity, $d\tau=dq/(2\pi iq)$ gives a pole of order at most $k$. Thus $D=k[\infty]+\lfloor k/2\rfloor[i]+\lfloor2k/3\rfloor[\rho]$ bounds the poles. Conversely these bounds give holomorphic pullbacks. The <Riemann-Roch theorem> on the compactified modular curve proves finite-dimensionality; since $X(1)$ is the <Riemann sphere>, the dimension is $\max(0,1-k+\lfloor k/2\rfloor+\lfloor2k/3\rfloor)$.
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