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.



Reference request: The first cohomology of SL(2,Z) with coefficients in homogeneous polynomials 
The result you're looking for is contained in the following article :


Haberland, Klaus. Perioden von Modulformen einer Variabler and Gruppencohomologie I (German) [Periods of modular forms of one variable and group cohomology I], *Math. Nachr.* **112** (1983), 245-282.


Let $S\_k$ (resp. $M\_k$) be the space of holomorphic cusp forms (resp. holomorphic modular forms) for $\Gamma = SL\_2(\mathbf{Z})$. Let $\Gamma\_{\infty}$ be the stabilizer of $\infty$ in $\Gamma$. Let $V\_k$ be the space of polynomials of degree $\leq k-2$ with complex coefficients. Haberland proves an exact sequence


\begin{equation}
(\*) \qquad 0 \to S\_k \oplus \overline{S\_k} \to H^1(\Gamma,V\_k) \to H^1(\Gamma\_\infty,V\_k) \to 0.
\end{equation}
Let $T = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \in \Gamma\_{\infty}$. There is a natural map $V\_{k+1} \to H^1(\Gamma\_\infty,V\_k)$ sending a polynomial $P$ to the cocycle $c\_P$ determined by $c\_P(T) = P(X+1)-P(X)$. It is easy to check that this map induces an isomorphism $\psi : V\_{k+1}/V\_k \cong H^1(\Gamma\_\infty,V\_k)$, so that the latter space is one-dimensional.


The "Eisenstein cocycle" you're looking for is a natural map $\delta : M\_k \to H^1(\Gamma,V\_k)$ which Haberland constructs the following way (actually I learnt this construction and many other properties of $\delta$ during Zagier's 2002-2003 lectures at the Collège de France).


Let $f \in M\_k$. Let $\widetilde{f}$ be an Eichler integral of $f$, that is any holomorphic function on $\mathcal{H}$ such that


\begin{equation}
\left(\frac{1}{2\pi i} \frac{d}{dz}\right)^{k-1} \widetilde{f}(z) = f(z).
\end{equation}
Note that $\widetilde{f}$ is unique up to adding some element of $V\_k$. Since we integrate $k-1$ times, the function $\widetilde{f}$ should be thought of as a function of "weight" $k-2\cdot (k-1) = 2-k$ (of course this isn't true in the strict sense). Let us make this more precise.


For any $n \in \mathbf{Z}$, let $|\_n$ denote the weight $n$ action of $SL\_2(\mathbf{R})$ on the space of complex-valued functions on $\mathcal{H}$ (so that any $f \in M\_k$ is a fixed vector of the weight $k$ action of $\Gamma$). Note also the weight $2-k$ action gives the usual action of $\Gamma$ on $V\_k$. The crucial fact is that we have


\begin{equation}
\widetilde{f} |\_{2-k} (\gamma-1) \in V\_k \qquad (\gamma \in \Gamma).
\end{equation}
This can be proved using Bol's identity


\begin{equation}
\left(\frac{d}{dz} \right)^{k-1} (F |\_{2-k} g) = \left(\frac{d^{k-1} F}{dz^{k-1}} \right) |\_k g
\end{equation}
which holds for any holomorphic function $F$ on $\mathcal{H}$ and any $g \in SL\_2(\mathbf{R})$.


Since $\gamma \mapsto \widetilde{f} |\_{2-k} (\gamma-1)$ is obviously a coboundary in the space of functions on $\mathcal{H}$, it defines a cocycle in the space $V\_k$. Therefore we get $\delta(f) \in H^1(\Gamma,V\_k)$ and this element doesn't depend on the choice of $\widetilde{f}$. Thus we have constructed $\delta : M\_k \to H^1(\Gamma,V\_k)$.


It is not difficult to check that if $f =\sum\_{n \geq 0} a\_n e^{2i\pi nz}$ then the image of $\delta(f)$ in $H^1(\Gamma\_\infty,V\_k)$ is the image of the polynomial $\frac{a\_0 \cdot (2\pi i)^{k-1}}{(k-1)!} \cdot X^{k-1} \in V\_{k+1}$ under the isomorphism $\psi$ above. In particular $\delta$ is injective, and the exact sequence $(\*)$ gives the isomorphism you want.


Note that there is a distinguished choice of $\widetilde{f}$, namely


\begin{equation}
\widetilde{f} = \sum\_{n \geq 1} \frac{a\_n}{n^{k-1}} e^{2i\pi nz} + \frac{a\_0 \cdot (2\pi i)^{k-1}}{(k-1)!} z^{k-1}.
\end{equation}
Let $c\_f \in Z^1(\Gamma,V\_k)$ be the cocycle associated to this choice of $\widetilde{f}$. Let us compute the value of $c\_f$ on $T$ and $S= \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. First as explained above, we have


\begin{equation}
c\_f(T)=\frac{a\_0 \cdot (2\pi i)^{k-1}}{(k-1)!} ((X+1)^{k-1}-X^{k-1}).
\end{equation}
To compute $c\_f(S)$, Haberland uses the natural integral representation of $\widetilde{f}$ in terms of $f-a\_0$, and gets


\begin{equation}
c\_f(S) = \frac{(2\pi i)^{k-1}}{(k-2)!} \int\_0^{\infty} \left(f(z)-\frac{a\_0}{z^k}-a\_0 \right) (z-X)^{k-2} dz
\end{equation}
(there is a similar but more complicated formula for $c\_f(\gamma)$ for any $\gamma \in \Gamma$, see below). Then $c\_f(S)$ can be expressed in terms of the special values of $L(f,s) := \sum\_{n=1}^\infty a\_n/n^s$ at integers $s = 1,\ldots,k-1$. It is then a good exercise to compute $c\_f(S)$ when $f$ is the Eisenstein series $E\_k$, in terms of Bernoulli numbers and of $\zeta(k-1)$ (this is Satz 3 in Haberland's article, Kapitel 1).


Please tell me if something isn't clear in my explanation.


EDIT : I found the following expression for $c\_f(\gamma)$ where $\gamma \in \Gamma$. It is quite complicated (maybe it could be somewhat simplified) :


\begin{equation}
\begin{aligned}
\frac{(k-2)!}{(2\pi i)^{k-1}} c\_f(\gamma) &= \int\_{z\_0}^{\infty} (f(z)-a\_0)(z-X)^{k-2} dz + \int\_{\gamma^{-1} \infty}^{z\_0} \left(f(z) -\frac{a\_0}{(cz+d)^k} \right) (z-X)^{k-2} dz \\
& + \frac{a\_0}{k-1} \left((X-z\_0)^{k-1}-(X-\gamma z\_0)^{k-1} |\_{2-k} \gamma + X^{k-1} |\_{2-k} (\gamma-1) \right)
\end{aligned}
\end{equation}
where $z\_0 \in \mathcal{H}$ is arbitrary and $\gamma= \begin{pmatrix} a & b \\ c & d \end{pmatrix}$.


