[proofplan] We prove the theorem by using the Eichler-Shimura comparison for modular curves with algebraic local systems. First we pass from rational parabolic singular cohomology to complex cuspidal de Rham cohomology for the local system attached to $\operatorname{Sym}^{k-2}\mathbb Q^2$. Then the [Hodge decomposition](/theorems/2745) of this de Rham cohomology identifies the two Hodge summands with holomorphic weight $k$ cusp forms and their complex conjugates. Finally, functoriality of these constructions for Hecke correspondences gives compatibility with every Hecke operator away from $N$. [/proofplan] [step:Declare the comparison theorem used for the modular curve] Let $X_1(N)(\mathbb C)$ denote the compact modular curve containing $Y_1(N)(\mathbb C)$, and let $C_1(N):=X_1(N)(\mathbb C)\setminus Y_1(N)(\mathbb C)$ denote its finite cusp set. Let $\mathcal V_{k-2}$ denote the holomorphic vector bundle with flat connection on $Y_1(N)(\mathbb C)$ obtained from the algebraic representation $\operatorname{Sym}^{k-2}\mathbb C^2$ of $GL_2$ by extension of scalars from $\mathbb V_{k-2,\mathbb Q}$. We use the Eichler-Shimura comparison theorem for modular curves with algebraic local systems in the following form: for $k\ge 2$ and $N\ge 1$, the complexification of parabolic singular cohomology is naturally isomorphic to cuspidal algebraic de Rham cohomology, \begin{align*} H^1_{\mathrm{par}}(Y_1(N)(\mathbb C),\mathbb V_{k-2,\mathbb Q})\otimes_{\mathbb Q}\mathbb C &\cong H^1_{\mathrm{dR},\mathrm{cusp}}(X_1(N)(\mathbb C),\mathcal V_{k-2}), \end{align*} and this isomorphism is functorial for finite correspondences of modular curves. The hypotheses of this theorem are exactly satisfied: $k\ge 2$, $N\ge 1$, $Y_1(N)(\mathbb C)$ is the modular curve for $\Gamma_1(N)$, and $\mathbb V_{k-2,\mathbb Q}$ is the rational local system attached to $\operatorname{Sym}^{k-2}\mathbb Q^2$. [/step] [step:Apply Hodge decomposition to split cuspidal de Rham cohomology] The [Hodge decomposition](/theorems/3941) for the cuspidal de Rham cohomology of the compactified modular curve with logarithmic boundary at $C_1(N)$ gives a natural direct sum decomposition \begin{align*} H^1_{\mathrm{dR},\mathrm{cusp}}(X_1(N)(\mathbb C),\mathcal V_{k-2}) &\cong H^0(X_1(N)(\mathbb C),\omega^k(-C_1(N))) \oplus \overline{H^0(X_1(N)(\mathbb C),\omega^k(-C_1(N)))}. \end{align*} Here $\omega$ denotes the Hodge line bundle on $X_1(N)(\mathbb C)$, and $\omega^k(-C_1(N))$ denotes the line bundle whose sections are holomorphic weight $k$ modular forms vanishing at every cusp. The condition $k\ge 2$ is the range in which the coefficient system $\operatorname{Sym}^{k-2}$ is algebraic and the above Hodge filtration has the displayed two summands. [/step] [step:Identify the holomorphic Hodge summand with cusp forms] By the standard modular interpretation of the Hodge bundle, the [vector space](/page/Vector%20Space) of holomorphic sections \begin{align*} H^0(X_1(N)(\mathbb C),\omega^k(-C_1(N))) \end{align*} is canonically $S_k(\Gamma_1(N))$: a section of $\omega^k$ is a holomorphic modular form of weight $k$ and level $\Gamma_1(N)$, and the twist by $-C_1(N)$ imposes vanishing at the cusps, which is precisely the cusp condition. Therefore the Hodge decomposition becomes \begin{align*} H^1_{\mathrm{dR},\mathrm{cusp}}(X_1(N)(\mathbb C),\mathcal V_{k-2}) &\cong S_k(\Gamma_1(N))\oplus \overline{S_k(\Gamma_1(N))}. \end{align*} [/step] [step:Compose the identifications and verify Hecke equivariance] Composing the comparison isomorphism with the Hodge identification gives \begin{align*} H^1_{\mathrm{par}}(Y_1(N)(\mathbb C),\mathbb V_{k-2,\mathbb Q})\otimes_{\mathbb Q}\mathbb C &\cong S_k(\Gamma_1(N))\oplus \overline{S_k(\Gamma_1(N))}. \end{align*} For every prime $\ell\nmid N$, the Hecke operator $T_\ell$ is induced by the usual finite double-coset correspondence between modular curves of level $\Gamma_1(N)$. The comparison isomorphism is functorial for finite correspondences, and the Hodge-bundle identification of modular forms with sections of $\omega^k(-C_1(N))$ is also functorial for the same pull-push correspondence. Hence the composite isomorphism intertwines the action of $T_\ell$ on parabolic cohomology with the classical Hecke action on $S_k(\Gamma_1(N))$ and the conjugate action on $\overline{S_k(\Gamma_1(N))}$. Since the Hecke algebra away from $N$ is generated by these double-coset operators, the displayed isomorphism is compatible with Hecke operators away from $N$, as required. [/step]