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I am currently trying to learn about the Eichler-Shimura isomorphism, and all the definitions seems to be somewhat cumbersome and hard to commit to memory as intuitive. On the other hand, the following map feels very intuitive/simple:

$$f(z)\to j^{k-2}(M,z)f^{(k-1)}(Mz)-f^{(k-1)}(z)$$

Here $f\in\mathcal{M}_k(\Gamma)$, $j(M,z)=cz+d$, and $f^{(k-1)}(Mz)$ denotes $k-1$ fold integration of $f$.

Since the choice of constants is arbitrary the term "$j^{k-2}(M,z)f^{(k-1)}(Mz)-f^{(k-1)}(z)$" is only defined up to a term $p^M-p$ where $p$ is a polynomial of degree $\leq k-2$ and $M$ is the action $f\to j^{k-2}(M,z)f(Mz)$. These terms are exactly the coboundaries in $H^1(\Gamma,V(k))$, so assuming that $j^{k-2}(M,z)f^{(k-1)}(Mz)-f^{(k-1)}(z)$ has the cocycle property then we have a natural map $\mathcal{M}_k(\Gamma)\to H^1(\Gamma,V(k))$.

To prove that $j^{k-2}(M,z)f^{(k-1)}(Mz)-f^{(k-1)}(z)$ has the cocyle property we simply note that the $(k-1)$-fold differentiation of $j^{k-2}(M,z)f^{(k-1)}(Mz)$ is exactly equal to $j^{-k}(M,z)f(Mz)$ (this can be seen through a simple induction) and thus the modularity property (up to polynomials) transfers.

I'm not quite sure how this extends to a map $\mathcal{M}_k(\Gamma)\oplus \overline{S_k(\Gamma)} \to H^1(\Gamma,V(k))$; one could $(f,g)$ to the sum of the natural embeddings of $f$ and the natural embedding of $g$ but this feels arbitrary. Moreover, I am not sure if I can prove Eichler-Shimura using this morphism. Is it equivalent to the already known ones? Is there some reference where people define this morphism? I find it hard to believe that I am the first one to come up with this interpretation of Eichler-Shimura.

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    $\begingroup$ You can use the same definition to associate a cocycle to $g$. But this is not the best thing to do, it's better to see $S_k \oplus \overline{S_k}$ as algebraic de Rham cohomology, in the case $k=2$ this is $H^1_{\mathrm{dR}}(X(\Gamma))$ which has dimension $2g$ where $g$ is the genus. Then the image under Eichler-Schimura of this $\mathbb{Q}$-structure will also involve quasi-periods of modular forms. You can find details here, Section 4. $\endgroup$ Jun 30, 2021 at 18:06

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