# Eichler-Shimura isomorphism and mixed Hodge theory

Let $Y(N),N>2$ 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 structure of weight $k$. So the cohomology $H^1(Y(N),V_k)$ is equipped with a mixed Hodge structure (the structure will be mixed despite the fact that $V_k$ is pure because $Y(N)$ is not complete). The complexification $H^1(Y(N),V_k\otimes\mathbf{C})$ splits

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

There is a natural way to get cohomology classes $\in H^1(Y(N),V_k\otimes\mathbf{C})$ 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 $H^1(Y(N),V_k\otimes\mathbf{C})$ 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.

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Notation: it is much more common to use X(N) for the compactified modular curve, and Y(N) for the non-complete version you refer to here. – JSE Dec 23 '09 at 16:02
Yes, indeed, thanks, JSE! Will correct this. – algori Dec 23 '09 at 17:42

The argument given in Elkik, Le théorème de Manin-Drinfelʹd, seems to generalize also to nontrivial coefficient systems. The "usual" Manin-Drinfel'd theorem claims that the difference of two cusps has finite order in the Jacobian of the modular curve. Elkik reformulates this as asserting that the mixed Hodge structure on $H^1$ of the open modular curve is the direct sum of pure Hodge structures. It is this fact that she then proves by realizing the weight spaces as eigenspaces for the action of the Hecke operators.