I have seen a couple of questions related to the Eichler-Shimura Isomorphism, but almost all of them have to do with hodge theory (things I am unfamiliar with) and seem, to me, different/unrelated.

Let $S_k(\Gamma)$ denote the space of modular cusp forms of level $\Gamma \subset SL_2(Z)$ and let $V_{k-2} \subset \mathbf{C}[X,Y]$ be the homogenous polynomials of degree $k-2$ with coefficients complex numbers. $SL_2(Z)$ acts on $V_{K-2}$ as follows. If $\gamma \in SL_2(Z)$ has first row $(a_1 b_1)$ and second row $(a_2 b_2)$, then $\gamma( P(X,Y)) := P(a_1X + b_1Y, a_2X + b_2Y)$

Define a map from the cusp forms to the first group cohomology of $\Gamma$ with coefficients in $V_{k-2}$

$\phi: S_k(\Gamma) \to H^1(\Gamma,V_{k-2})$

$\phi(f)[g] := c_f(g) = \int_0^{g(0)}f(z)(X-zY)^{k-2}dz$

I have already verified that $c_f$ is a cocycle, hence embedded in $Z^1(\Gamma, V_{k-2})$ but what remains is my question:

How do I prove that if $f\neq0$ then $c_f$ is not a boundary?


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