[proofplan] The proof has two parts. First, we identify a cusp form $f$ with the differential $2\pi i f(z)\,dz$ and compute the trace of the pullback differential along the Hecke correspondence; the factor from pulling back $dz$ is exactly the automorphy factor appearing in the weight $2$ slash action, so the correspondence action is the usual Hecke operator. Second, we use the functoriality of integration under pullback and pushforward of finite correspondences to show that the dual action on periods sends the cycle functional attached to $\gamma$ to the cycle functional attached to $T_n^H\gamma$. This proves that the Abel-Jacobi quotient is equivariant and that the period lattice carries the homological Hecke action. [/proofplan]

[step:Identify cusp forms with holomorphic differentials on the modular curve] Let $\mathfrak{H}=\{z\in\mathbb{C}:\operatorname{Im}(z)>0\}$ denote the complex upper half-plane. Let \begin{align*} \Phi: S_2(\Gamma_0(N)) &\to H^0(X_0(N), \Omega^1) \\ f &\mapsto 2\pi i\, f(z)\, dz \end{align*} be the map in the statement. If $f \in S_2(\Gamma_0(N))$, then the weight $2$ transformation law gives \begin{align*} f(\delta z)\, d(\delta z) = f(z)\, dz \end{align*} for every $\delta \in \Gamma_0(N)$, because $d(\delta z) = (cz+d)^{-2}\,dz$ when $\delta=\begin{pmatrix}a&b\\ c&d\end{pmatrix}$. Hence $2\pi i f(z)\,dz$ descends from $\mathfrak{H}$ to a holomorphic differential on the open modular curve. At a cusp $s$, choose a scaling matrix $\sigma_s\in SL_2(\mathbb{R})$ sending $\infty$ to $s$, and let $w_s\in\mathbb{N}$ denote the width of $s$ for $\Gamma_0(N)$. Define the local parameter \begin{align*} q_s: \sigma_s^{-1}\mathfrak{H}/\langle z\mapsto z+w_s\rangle &\to \mathbb{C} \\ z &\mapsto e^{2\pi i z/w_s}. \end{align*} Since $dq_s=(2\pi i/w_s)q_s\,dz$ in this cusp coordinate, and since cuspidality gives an expansion \begin{align*} f(\sigma_s z)(c_sz+d_s)^{-2}=\sum_{m=1}^{\infty} a_{s,m} q_s^m \end{align*} for the corresponding scaling matrix entries $c_s,d_s$, the descended differential has the form \begin{align*} 2\pi i f(z)\,dz = w_s\sum_{m=1}^{\infty} a_{s,m} q_s^{m-1}\,dq_s \end{align*} in the parameter $q_s$. This is holomorphic at $s$ because the Fourier expansion has no constant term. Thus $\Phi(f)$ extends to $X_0(N)$. Conversely, every holomorphic differential on $X_0(N)$ restricts to a $\Gamma_0(N)$-invariant differential on the upper half-plane, hence has the form $2\pi i f(z)\,dz$ for a weight $2$ modular form $f$, and holomorphy at the cusps forces $f$ to vanish at every cusp. Therefore $\Phi$ is the canonical identification. [/step]

[step:Compute the trace of the pulled back differential along the Hecke correspondence]Let $n \in \mathbb{N}$. Choose double-coset representatives $\alpha_1,\dots,\alpha_r \in M_2(\mathbb{Z})$ for the Hecke correspondence defining $T_n$ on $\Gamma_0(N)\backslash \mathfrak{H}$, where each $\alpha_j$ has positive determinant and acts on $z \in \mathfrak{H}$ by fractional linear transformation. On the analytic uniformization, the correspondence is represented locally by the branches \begin{align*} z \mapsto \alpha_j z, \qquad 1 \le j \le r. \end{align*} Let $f \in S_2(\Gamma_0(N))$ and write $\omega_f := \Phi(f)=2\pi i f(z)\,dz$. The pullback along the $j$-th branch is \begin{align*} (\alpha_j)^*\omega_f &= 2\pi i\, f(\alpha_j z)\, d(\alpha_j z). \end{align*} If \begin{align*} \alpha_j=\begin{pmatrix}a_j&b_j\\ c_j&d_j\end{pmatrix}, \end{align*} then \begin{align*} d(\alpha_j z)=\frac{\det(\alpha_j)}{(c_jz+d_j)^2}\,dz. \end{align*} Therefore \begin{align*} (\alpha_j)^*\omega_f =2\pi i\,\det(\alpha_j)(c_jz+d_j)^{-2} f(\alpha_j z)\,dz. \end{align*} By definition of the trace map on differentials for the finite map $\pi_1$, the differential $(\pi_1)_*(\pi_2)^*\omega_f$ is obtained locally by summing these pulled-back branch differentials. Hence \begin{align*} T_n^\Omega(\omega_f) &=(\pi_1)_*(\pi_2)^*\omega_f \\ &=2\pi i\left(\sum_{j=1}^r \det(\alpha_j)(c_jz+d_j)^{-2} f(\alpha_j z)\right)dz. \end{align*} With the convention that the usual weight $2$ Hecke operator attached to the same double-coset representatives is \begin{align*} (T_n f)(z)=\sum_{j=1}^r \det(\alpha_j)(c_jz+d_j)^{-2} f(\alpha_j z), \end{align*} the expression in parentheses is exactly $T_n f$. Thus \begin{align*} T_n^\Omega(\Phi(f))=\Phi(T_n f). \end{align*}[/step]

[guided]The point of the convention \begin{align*} T_n^\Omega=(\pi_1)_*(\pi_2)^* \end{align*} is that differentials are first pulled back along the target branch of the correspondence and then traced back to the source. We compute this on the upper half-plane, where the correspondence is represented by finitely many fractional linear maps \begin{align*} z \mapsto \alpha_j z. \end{align*} For a fixed branch, write \begin{align*} \alpha_j=\begin{pmatrix}a_j&b_j\\ c_j&d_j\end{pmatrix}. \end{align*} The associated fractional [linear map](/page/Linear%20Map) is \begin{align*} z \mapsto \frac{a_jz+b_j}{c_jz+d_j}. \end{align*} Differentiating this map gives \begin{align*} d(\alpha_j z)=\frac{\det(\alpha_j)}{(c_jz+d_j)^2}\,dz. \end{align*} Therefore the pullback of the differential $\omega_f=2\pi i f(z)\,dz$ along this branch is \begin{align*} (\alpha_j)^*\omega_f &=2\pi i\, f(\alpha_j z)\,d(\alpha_j z) \\ &=2\pi i\,\det(\alpha_j)(c_jz+d_j)^{-2}f(\alpha_j z)\,dz. \end{align*} The trace map for $\pi_1$ sums the contributions of all local branches lying above the same point. Hence \begin{align*} (\pi_1)_*(\pi_2)^*\omega_f =2\pi i\left(\sum_{j=1}^r \det(\alpha_j)(c_jz+d_j)^{-2}f(\alpha_j z)\right)dz. \end{align*} The weight $2$ slash factor for this correspondence normalization is exactly the factor \begin{align*} \det(\alpha_j)(c_jz+d_j)^{-2}. \end{align*} Thus, under the convention \begin{align*} (T_n f)(z)=\sum_{j=1}^r \det(\alpha_j)(c_jz+d_j)^{-2} f(\alpha_j z), \end{align*} the function multiplying $2\pi i\,dz$ is the usual Hecke transform $T_n f$. Consequently \begin{align*} T_n^\Omega(\Phi(f))=\Phi(T_n f). \end{align*}[/guided]

[step:Relate the differential action to the homology action by integration] Let $\gamma \in H_1(X_0(N),\mathbb{Z})$ and let $\omega \in H^0(X_0(N),\Omega^1)$. By functoriality of integration under pullback, \begin{align*} \int_\gamma (\pi_1)_*(\pi_2)^*\omega = \int_{(\pi_1)^*\gamma}(\pi_2)^*\omega. \end{align*} Applying functoriality again, now for pushforward of chains under $\pi_2$, gives \begin{align*} \int_{(\pi_1)^*\gamma}(\pi_2)^*\omega = \int_{(\pi_2)_*(\pi_1)^*\gamma}\omega. \end{align*} By the definition \begin{align*} T_n^H=(\pi_2)_*(\pi_1)^*, \end{align*} we obtain \begin{align*} \int_\gamma T_n^\Omega\omega = \int_{T_n^H\gamma}\omega. \end{align*} Thus the action on homology is adjoint to the action on holomorphic differentials under the period pairing. [/step]

[step:Show that the Abel-Jacobi period lattice is preserved] Define the period embedding \begin{align*} \Lambda: H_1(X_0(N),\mathbb{Z}) &\to H^0(X_0(N),\Omega^1)^\vee \\ \gamma &\mapsto \left(\omega \mapsto \int_\gamma \omega\right). \end{align*} Let $(T_n^\Omega)^\vee$ denote the dual endomorphism of $H^0(X_0(N),\Omega^1)^\vee$, defined by \begin{align*} ((T_n^\Omega)^\vee \ell)(\omega)=\ell(T_n^\Omega\omega) \end{align*} for $\ell \in H^0(X_0(N),\Omega^1)^\vee$ and $\omega \in H^0(X_0(N),\Omega^1)$. For $\gamma \in H_1(X_0(N),\mathbb{Z})$, the previous step gives \begin{align*} ((T_n^\Omega)^\vee\Lambda(\gamma))(\omega) &=\Lambda(\gamma)(T_n^\Omega\omega) \\ &=\int_\gamma T_n^\Omega\omega \\ &=\int_{T_n^H\gamma}\omega \\ &=\Lambda(T_n^H\gamma)(\omega). \end{align*} Since this equality holds for every $\omega \in H^0(X_0(N),\Omega^1)$, we have \begin{align*} (T_n^\Omega)^\vee\Lambda(\gamma)=\Lambda(T_n^H\gamma). \end{align*} Therefore $(T_n^\Omega)^\vee$ preserves the lattice $\Lambda(H_1(X_0(N),\mathbb{Z}))$ and its induced action on that lattice is exactly $T_n^H$. Passing to the quotient \begin{align*} J_0(N)(\mathbb{C}) \cong H^0(X_0(N),\Omega^1)^\vee / \Lambda(H_1(X_0(N),\mathbb{Z})), \end{align*} the Abel-Jacobi description is equivariant for the Hecke correspondence. This proves both asserted compatibilities. [/step]