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Let $H_k$ be the vector space of degree $k$ homogeneous polynomials in two variables.I'm looking for a reference for the fact that $H^1(SL(2,\mathbb Z);H_k)=M^0(k+2)\oplus\overline{M^0(k+2)}\oplus E_{k+2}$, where

1. $M^0(k+2)$ is the space of cuspidal modular forms of weight $k+2$.
2. $\overline{M^0(k+2)}$ is its conjugate.
3. $E_k$ is $1$-dimensional if $k\geq 4$ is even, and is zero otherwise.

I know how to get the $M^0(k+2)\oplus\overline{M^0(k+2)}$ piece. By the Eichler-Shimura isomorphism, this is the same as the cuspidal cohomology $H^1_{cusp}(SL(2,\mathbb Z);H_k)$, spanned by cocycles that vanish on the matrix $T$=[1 1 \ 0 1]. Serge Lang's book on Modular Forms has a detailed explanation of how to get this. I'd really like to know how to get the $E_{k+2}$ piece, which as far as I understand, corresponds somehow to the Eisenstein series, which is the "extra" modular form of weight $k+2$ that is not cuspidal.

It's not too hard to see that $H^1(SL(2,\mathbb Z);H_k)=H^1(PSL(2,\mathbb Z);H_k)$, with $PSL(2,\mathbb Z)$ having the nice presentation $\langle S,T\,|\, (ST)^3=S^2=I\rangle$, so one should be able to represent the $E_{k+2}$ cocycle by specifying its values on $S,T$.

So, to summarize, I'd be satisfied with any proof that $H^1(SL(2,\mathbb Z);H_k)=M^0(k+2)\oplus\overline{M^0(k+2)}\oplus E_{k+2}$, but I'd be happiest with an answer that lets me get my hands on the $E_{k+2}$ cocycle as explicitly as possible, perhaps even telling me its values on $S$ and $T$. Any comments that would help clarify this are appreciated.

Edit: I think I have a better understanding of what the "extra" Eisentstein cocycle is, based on Kevin's comments. It seems that the cuspidal cocycles vanish on $T$, whereas the Eisenstein cocycle vanishes on $S$, although I don't see how to show, for example, that there is only one dimension's worth of cocycles vanishing on $S$, up to coboundaries. (Edit: I'm not entirely sure about this.)

Edit: Shimura's Introduction to the Theory of Automorphic Forms only covers the cuspidal part of the above isomorphism.

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Let $H_k$ be the vector space of degree $k$ homogeneous polynomials in two variables.I'm looking for a reference for the fact that $H^1(SL(2,\mathbb Z);H_k)=M^0(k+2)\oplus\overline{M^0(k+2)}\oplus E_{k+2}$, where

1. $M^0(k+2)$ is the space of cuspidal modular forms of weight $k+2$.
2. $\overline{M^0(k+2)}$ is its conjugate.
3. $E_k$ is $1$-dimensional if $k\geq 4$ is even, and is zero otherwise.

I know how to get the $M^0(k+2)\oplus\overline{M^0(k+2)}$ piece. By the Eichler-Shimura isomorphism, this is the same as the cuspidal cohomology $H^1_{cusp}(SL(2,\mathbb Z);H_k)$, spanned by cocycles that vanish on the matrix $T$=[1 1 \ 0 1]. Serge Lang's book on Modular Forms has a detailed explanation of how to get this. I'd really like to know how to get the $E_{k+2}$ piece, which as far as I understand, corresponds somehow to the Eisenstein series, which is the "extra" modular form of weight $k+2$ that is not cuspidal.

It's not too hard to see that $H^1(SL(2,\mathbb Z);H_k)=H^1(PSL(2,\mathbb Z);H_k)$, with $PSL(2,\mathbb Z)$ having the nice presentation $\langle S,T\,|\, (ST)^3=S^2=I\rangle$, so one should be able to represent the $E_{k+2}$ cocycle by specifying its values on $S,T$.

So, to summarize, I'd be satisfied with any proof that $H^1(SL(2,\mathbb Z);H_k)=M^0(k+2)\oplus\overline{M^0(k+2)}\oplus E_{k+2}$, but I'd be happiest with an answer that lets me get my hands on the $E_{k+2}$ cocycle as explicitly as possible, perhaps even telling me its values on $S$ and $T$. Any comments that would help clarify this are appreciated.

Edit: I think I have a better understanding of what the "extra" Eisentstein cocycle is, based on Kevin's comments. It seems that the cuspidal cocycles vanish on $T$, whereas the Eisenstein cocycle vanishes on $S$, although I don't see how to show, for example, that there is only one dimensions dimension's worth of cocycles vanishing on $S$, up to coboundaries.