The current (25/03/2023 12:30 UTC) version of the Wikipedia article on modular forms has a section "As sections of a line bundle" where it claims the following:
I think there are two mistakes in this section:
- It uses the canonical bundle $\omega$. But this realizes only modular forms of even weight, and you have to take the $\frac{k}{2}$th power of it. That is, a weight $k$ modular form with $k$ even can be seen as a $\frac{k}{2}$-fold differential form, i.e., a section of $\omega^{\otimes (k/2)}$.
(One might suspect that Wikipedia just uses the definition that weight $k$ modular forms in their sense are weight $2k$ modular forms in my sense, but looking just a few lines above you can see that weight $k$ means it satisfies the transformation rule $f(\gamma \,z) = (cz + d)^k f(z)$, i.e., its my definition of weight $k$.) - It writes an equality $M_k(\Gamma) = H^0(X_\Gamma, \omega^{\otimes k})$. But the "holomorphic at every point" condition of a modular form does not transform to "holomorphic at every point" for the associated differential. That is because the chart at cusps and elliptic points is not the identity, which adds additional poles for the differential. In particular, the differential associated to a modular form is generally not holomorphic, but can have certain orders of poles at the cusps and elliptic points.
Could somebody confirm these mistakes? Or am I wrong?