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Let $T = [1,1 ; \begin{pmatrix} 1 & 1 \\ 0 , & 1 ] \end{pmatrix} \in \Gamma_{\infty}$. 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)$, so that the latter space is one-dimensional. \bigl(\frac{1}{2\pi left(\frac{1}{2\pi i} \frac{d}{dz}\bigr)^{k-1} frac{d}{dz}\right)^{k-1} \widetilde{f}(z) = f(z).(\frac{d}{dz})^{k-1} \left(\frac{d}{dz} \right)^{k-1} (F |_{2-k} g) = (\frac{d^{k-1} F}{dz^{k-1}}) \left(\frac{d^{k-1} F}{dz^{k-1}} \right) |_k gLet$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 $S=[0,-1;1,0]$. S= \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. First as explained above, we have

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) (f(z)-\frac{a_0}{z^k}-a_0)(z-X)^{k-2} z-X)^{k-2} dz\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} (f(z) \left(f(z) -\frac{a_0}{(cz+d)^k}) \frac{a_0}{(cz+d)^k} \right) (z-X)^{k-2} dz \end{equation}& + \frac{a_0}{k-1} ((X-z_0)^{k-1}-(X-\gamma \left((X-z_0)^{k-1}-(X-\gamma z_0)^{k-1} |{2-k} _{2-k} \gamma + X^{k-1} |{2-k} _{2-k} (\gamma-1))where $z_0 \in \mathcal{H}$ is arbitrary and $\gamma= [a,b;c,d]$.\begin{pmatrix} a & b \\ c & d \end{pmatrix}$. 5 added 36 characters in body The result you're looking for is contained in the following article : Haberland, Klaus. Perioden von Modulformen einer Variabler and Gruppencohomologie I (German) [Periods of modular forms of one variable and group cohomology I], Math. Nachr. 112 (1983), 245-282. Let$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 $$(*) \qquad 0 \to S_k \oplus \overline{S_k} \to H^1(\Gamma,V_k) \to H^1(\Gamma_\infty,V_k) \to 0.$$ Let$T = [1,1 ; 0, 1] \in \Gamma_{\infty}$. 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 $$\bigl(\frac{1}{2\pi i} \frac{d}{dz}\bigr)^{k-1} \widetilde{f}(z) = f(z).$$ 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 $$\widetilde{f} |_{2-k} (\gamma-1) \in V_k \qquad (\gamma \in \Gamma).$$ This can be proved using Bol's identity $$(\frac{d}{dz})^{k-1} (F |_{2-k} g) = (\frac{d^{k-1} F}{dz^{k-1}}) |_k g$$ 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 $$\widetilde{f} = \sum_{n \geq 1} \frac{a_n}{n^{k-1}} e^{2i\pi nz} + \frac{a_0 \cdot (2\pi i)^{k-1}}{(k-1)!} z^{k-1}.$$ 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$S=[0,-1;1,0]$. First as explained above, we have $$c_f(T)=\frac{a_0 \cdot (2\pi i)^{k-1}}{(k-1)!} ((X+1)^{k-1}-X^{k-1}).$$ To compute$c_f(S)$, Haberland uses the natural integral representation of$\widetilde{f}$in terms of$f-a_0$, and gets $$c_f(S) = \frac{(2\pi i)^{k-1}}{(k-2)!} \int_0^{\infty} (f(z)-\frac{a_0}{z^k}-a_0)(z-X)^{k-2} dz$$ (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) : \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} (f(z) -\frac{a_0}{(cz+d)^k}) (z-X)^{k-2} dz \end{equation} \begin{equation} + \frac{a_0}{k-1} ((X-z_0)^{k-1}-(X-\gamma z_0)^{k-1}|z_0)^{k-1} |{2-k} \gamma + X^{k-1} |{2-k} (\gamma-1)) where$z_0 \in \mathcal{H}$is arbitrary and$\gamma= [a,b;c,d]$. 4 added 567 characters in body; deleted 14 characters in body 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$S$. S=[0,-1;1,0]$. First as explained above, we have

(there should be is a similar but more complicated formula for $c_f(\gamma)$ for any $\gamma \in \Gamma$). 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 \in {1,\ldots,k-1}$. = 1,\ldots,k-1$. 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).

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}\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} (f(z) -\frac{a_0}{(cz+d)^k}) (z-X)^{k-2} dz + \frac{a_0}{k-1} ((X-z_0)^{k-1}-(X-\gamma z_0)^{k-1}|{2-k} \gamma + X^{k-1} |{2-k} (\gamma-1))where $z_0 \in \mathcal{H}$ is arbitrary and $\gamma= [a,b;c,d]$.

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