For a congruence subgroup $\Gamma \subseteq \mathrm{SL}_2(\mathbb{Z})$ we have the Eichler-Shimura isomorphism $$ M_k(\Gamma) \oplus \overline{S_k(\Gamma)} \cong H^1(\Gamma,V_k) $$ with $V_k$ a certain $\Gamma$-module coming from a $(k-1)$-dimensional representation of $\mathrm{SL}_2(\mathbb{Z})$. We have $V_2 \cong \mathbb{C}$ so that for $k=2$ the isomorphism reads: $$ M_2(\Gamma) \oplus \overline{S_2(\Gamma)} \cong H^1(\Gamma,\mathbb{C}) = \operatorname{Hom}(\Gamma,\mathbb{C}) $$ My question concerns the prove of this isomorphism, in particular the special case $k=2$. As far as I know, a crucial step in the original proof is computing the dimensions of both sides and showing that they are equal. This confuses me, because I think that there should be a "geometric" proof, that moreover works in a much more general setting. I think so because group cohomology of $\Gamma$ is even based on the idea that it captures the cohomology of a certain space with fundamental group $\Gamma$. Let's focus on $k=2$. We simply need:
- the first cohomology of the open modular curve $Y_\Gamma$ is related to the left side of the equation, i.e., modular forms
- this is indeed the case, as one can see modular forms as differentials on $Y_\Gamma$, which in turn live in the first (de Rham) cohomology group of $Y_\Gamma$
- the open modular curve $Y_\Gamma$ has fundamental group $\Gamma$
- also this is true in nice situations, and generally it's quite close to true
Then by the basic principle/idea of group cohomology, this should give us a pretty straightforward proof of the isomorphism for $k=2$, no? And for $k > 2$, I've heard that there is also a correspondence between certain $\Gamma$-modules like $V_k$ and local systems on the modular curve. So in that case there should also be some general theorem relating group cohomology with sheaf cohomology with values in local systems, and then the Eichler-Shimura theorem should just be a special case of that general theorem, no?
Can anybody tell me if one can prove the isomorphism in the flavor I have sketched, and if yes, why nobody (as far as I've seen resources) does so?