Let $ f \in S_k(\Gamma)$ be a weight k modular cusp form of level $\Gamma$, with modular curve $Y_{\Gamma}$. Let $V^{k-2}$ be the homogenous polynomials in X and Y of degree k-2 with complex coefficients. Let $\Omega^1$ be the sheaf of differential 1 forms. We define the map:

$\phi : S_k(\Gamma) \to H^0(Y_{\Gamma}, \Omega^1 \otimes V^{k-2})$

$\phi(f) : = f(z)(X-zY)^{k-2}dz$

**Question: How do I show that this defines a global section of $Y_{\Gamma}$?** There must be some argumentation about the coordinate charts of the modular curve.

Note: This helps give an answer to my previous question (Eichler-Shimura Isomorphism) Proving c(f) is not a boundary) via the following observations:

By Dolbeaut's theorem, $H^0(Y_{\Gamma}, \Omega^1 \otimes V^{k-2} )\cong H^1(Y_\Gamma, V^{k-2})$. Then, since $Y_{\Gamma}$ is a $K(\Gamma, 1)$ surface, we know $H^1(Y_{\Gamma}, V^{k-2}) \cong H^1(\Gamma, V^{k-2})$, which is group cohomology, and the subject of my previous post. By showing that this $f$ defines a global section, it is trivial to show it is not a boundary in cohomology, via the isomorphisms.