2 fixed the notation

Let $X(N),N>2$ Y(N),N>2$be the quotient of the upper half-plane by$\Gamma(N)$(which is formed by the elements of$SL(2,\mathbf{Z})$congruent to$I$mod$N$). Let$V_k$be the$k$-th symmetric power of the Hodge local system on$X(N)$tensored by$\mathbf{Q}$(the Hodge local system corresponds to the standard action of$\Gamma(N)$on$\mathbf{Z}^2$).$V_k$is a part of a variation of polarized Hodge structures structure of weight$k$. So the cohomology$H^1(X(N),V_k)$H^1(Y(N),V_k)$ is equipped with a mixed Hodge structure (the structure will be mixed despite the fact that $V_k$ is pure because $X(N)$ Y(N)$is not complete). The complexification$H^1(X(N),V_k\otimes\mathbf{C})$H^1(Y(N),V_k\otimes\mathbf{C})$ splits

$$H^1(X(N),V_k\otimes\mathbf{C})=H^{k+1,0}\oplus H^1(Y(N),V_k\otimes\mathbf{C})=H^{k+1,0}\oplus H^{0,k+1}\oplus H^{k+1,k+1}.$$

There is a natural way to get cohomology classes $\in H^1(X(N),V_k\otimes\mathbf{C})$ H^1(Y(N),V_k\otimes\mathbf{C})$from modular forms . for$\Gamma(N)$. Namely, to a modular form$f$of weight$k+2$one associates the secion $$z\mapsto f(z)(ze_1+e_2)^k dz$$ of $$Sym^k(\mathbf{C}^2)\otimes \Omega^1_{\mathbf{H}}.$$ Here$\mathbf{H}$is the upper half plane and$(e_1,e_2)$is a basis of$\mathbf{C}^2$coming from a basis of$\mathbf{Z}^2$. This pushes down to a holomorphic section of$V_k\otimes \mathbf{C}$. Deligne had conjectured (Formes modulaires et repr\'esentations l-adiques, Bourbaki talk, 1968/69) that the above correspondence gives a bijection between the cusp forms of weight$k+2$and$H^{k+1,0}$. (This was before he had even constructed the Hodge theory, so strictly speaking this can't be called a conjecture, but anyway.) Subsequently this was proved by Zucker (Hodge theory with degenerating coefficients, Anns of Maths 109, no 3, 1979). See also Bayer, Neukirch, On automorphic forms and Hodge theory, (Math Ann, 257, no 2, 1981). The above results concern cusp forms and it is natural to ask what all modular forms correspond to in terms of Hodge theory. It turns out that all weight$k+2$modular forms give precisely the$k+1$-st term of the Hodge filtration on$H^1(X(N),V_k\otimes\mathbf{C})$H^1(Y(N),V_k\otimes\mathbf{C})$ i.e. $H^{k+1,0}\oplus H^{k+1,k+1}$.

The proof of this is not too difficult but a bit tedious. So I would like to ask: is there a reference for this?

upd: The original posting contained non-standard notation; this has been fixed.

1

# Eichler-Shimura isomorphism and mixed Hodge theory

Let $X(N),N>2$ be the quotient of the upper half-plane by $\Gamma(N)$ (which is formed by the elements of $SL(2,\mathbf{Z})$ congruent to $I$ mod $N$). Let $V_k$ be the $k$-th symmetric power of the Hodge local system on $X(N)$ tensored by $\mathbf{Q}$ (the Hodge local system corresponds to the standard action of $\Gamma(N)$ on $\mathbf{Z}^2$).

$V_k$ is a part of a variation of polarized Hodge structures of weight $k$. So the cohomology $H^1(X(N),V_k)$ is equipped with a mixed Hodge structure (the structure will be mixed despite the fact that $V_k$ is pure because $X(N)$ is not complete). The complexification $H^1(X(N),V_k\otimes\mathbf{C})$ splits

$$H^1(X(N),V_k\otimes\mathbf{C})=H^{k+1,0}\oplus H^{0,k+1}\oplus H^{k+1,k+1}.$$

There is a natural way to get cohomology classes $\in H^1(X(N),V_k\otimes\mathbf{C})$ from modular forms. Namely, to a modular form $f$ of weight $k+2$ one associates the secion

$$z\mapsto f(z)(ze_1+e_2)^k dz$$

of $$Sym^k(\mathbf{C}^2)\otimes \Omega^1_{\mathbf{H}}.$$

Here $\mathbf{H}$ is the upper half plane and $(e_1,e_2)$ is a basis of $\mathbf{C}^2$ coming from a basis of $\mathbf{Z}^2$. This pushes down to a holomorphic section of $V_k\otimes \mathbf{C}$.

Deligne had conjectured (Formes modulaires et repr\'esentations l-adiques, Bourbaki talk, 1968/69) that the above correspondence gives a bijection between the cusp forms of weight $k+2$ and $H^{k+1,0}$. (This was before he had even constructed the Hodge theory, so strictly speaking this can't be called a conjecture, but anyway.) Subsequently this was proved by Zucker (Hodge theory with degenerating coefficients, Anns of Maths 109, no 3, 1979). See also Bayer, Neukirch, On automorphic forms and Hodge theory, (Math Ann, 257, no 2, 1981).

The above results concern cusp forms and it is natural to ask what all modular forms correspond to in terms of Hodge theory. It turns out that all weight $k+2$ modular forms give precisely the $k+1$-st term of the Hodge filtration on $H^1(X(N),V_k\otimes\mathbf{C})$ i.e. $H^{k+1,0}\oplus H^{k+1,k+1}$.

The proof of this is not too difficult but a bit tedious. So I would like to ask: is there a reference for this?