[proofplan] We identify weight $2$ cusp forms with holomorphic differentials on the compact modular curve by sending $f$ to $2\pi i f(z)\,dz$. The [Hodge decomposition](/theorems/2745) for a compact Riemann surface then identifies the first complex Betti cohomology with the direct sum of holomorphic and antiholomorphic differentials. Finally, the Hecke correspondences are holomorphic finite correspondences, so their pull-push action preserves type and agrees on holomorphic differentials with the classical Hecke action on $q$-expansions; complex conjugation gives the antiholomorphic eigenvalues. [/proofplan] [step:Identify weight $2$ cusp forms with holomorphic differentials on $X_0(N)$] Let $\mathbb{H} := \{z \in \mathbb{C} : \operatorname{Im}(z) > 0\}$ denote the complex upper half-plane, and let \begin{align*} \pi_N: \mathbb{H} \longrightarrow Y_0(N) := \Gamma_0(N)\backslash \mathbb{H} \end{align*} be the quotient map to the open modular curve. For a cusp form $f \in S_2(\Gamma_0(N))$, define a holomorphic $1$-form $\omega_f$ on $Y_0(N)$ by the descent condition \begin{align*} \pi_N^*\omega_f = 2\pi i\, f(z)\,dz. \end{align*} This is well-defined because, for every matrix $\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma_0(N)$, the weight $2$ transformation law gives \begin{align*} f(\gamma z)\,d(\gamma z) &= f(\gamma z)\frac{1}{(cz+d)^2}\,dz \\ &= f(z)\,dz. \end{align*} The cusp condition says exactly that this descended form extends holomorphically across the cusps of the compactification $X_0(N)$. Thus we obtain a complex-[linear map](/page/Linear%20Map) \begin{align*} \Phi: S_2(\Gamma_0(N)) &\longrightarrow H^0(X_0(N), \Omega^1_{X_0(N)}) \\ f &\longmapsto \omega_f. \end{align*} The standard identification of weight $2$ cusp forms with holomorphic differentials on the compact modular curve says that $\Phi$ is an isomorphism (citing a result not yet in the wiki: weight $2$ cusp forms are holomorphic differentials on $X_0(N)$). [guided] The reason weight $2$ appears is that $dz$ transforms with weight $-2$. If $\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma_0(N)$, then $\gamma z = \frac{az+b}{cz+d}$ and \begin{align*} d(\gamma z)=\frac{1}{(cz+d)^2}\,dz. \end{align*} A weight $2$ modular form satisfies $f(\gamma z)=(cz+d)^2 f(z)$, so the product $f(z)\,dz$ is invariant: \begin{align*} f(\gamma z)\,d(\gamma z) &=(cz+d)^2 f(z)\frac{1}{(cz+d)^2}\,dz \\ &=f(z)\,dz. \end{align*} Therefore $2\pi i f(z)\,dz$ descends from $\mathbb{H}$ to a holomorphic differential on $Y_0(N)$. The factor $2\pi i$ is the conventional normalization compatible with $q=e^{2\pi i z}$, since $dq/q=2\pi i\,dz$. The only remaining issue is what happens at the cusps. A modular form of weight $2$ gives a meromorphic differential on the compactification, and the vanishing condition in the definition of a cusp form is precisely the condition that the differential has no pole at any cusp. Hence the descended form extends to an element of $H^0(X_0(N),\Omega^1_{X_0(N)})$. Conversely, every holomorphic differential on $X_0(N)$ pulls back to a $\Gamma_0(N)$-invariant holomorphic differential on $\mathbb{H}$, 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 be cuspidal. This is the standard identification of weight $2$ cusp forms with holomorphic differentials on $X_0(N)$ (citing a result not yet in the wiki: weight $2$ cusp forms are holomorphic differentials on $X_0(N)$). [/guided] [/step] [step:Apply Hodge decomposition to first Betti cohomology] Let $X := X_0(N)$. Since $X$ is a compact Riemann surface, the [Hodge decomposition](/theorems/3941) for compact Riemann surfaces gives a complex-linear isomorphism \begin{align*} H^1_B(X,\mathbb{C}) &\cong H^{1,0}(X) \oplus H^{0,1}(X) \\ &\cong H^0(X,\Omega^1_X) \oplus \overline{H^0(X,\Omega^1_X)}. \end{align*} Here $H^{1,0}(X)$ is represented by holomorphic $1$-forms and $H^{0,1}(X)$ is represented by their complex conjugates. Composing this decomposition with the isomorphism $\Phi$ from the previous step gives a complex-linear isomorphism \begin{align*} H^1_B(X_0(N),\mathbb{C}) \cong S_2(\Gamma_0(N)) \oplus \overline{S_2(\Gamma_0(N))}. \end{align*} [guided] The Hodge decomposition is the bridge between cohomology and differential forms. For a compact Riemann surface $X$, every class in $H^1_B(X,\mathbb{C})$ has a unique harmonic representative, and every harmonic complex $1$-form splits uniquely into its type $(1,0)$ and type $(0,1)$ parts. This gives \begin{align*} H^1_B(X,\mathbb{C}) \cong H^{1,0}(X)\oplus H^{0,1}(X). \end{align*} For a Riemann surface, $H^{1,0}(X)$ is exactly the space $H^0(X,\Omega^1_X)$ of holomorphic differentials, and $H^{0,1}(X)$ is its complex conjugate: \begin{align*} H^{0,1}(X)\cong \overline{H^0(X,\Omega^1_X)}. \end{align*} Applying this with $X=X_0(N)$ and then using the isomorphism \begin{align*} \Phi: S_2(\Gamma_0(N)) \longrightarrow H^0(X_0(N),\Omega^1_{X_0(N)}) \end{align*} from the previous step gives \begin{align*} H^1_B(X_0(N),\mathbb{C}) \cong S_2(\Gamma_0(N)) \oplus \overline{S_2(\Gamma_0(N))}. \end{align*} This proves the vector-space decomposition (citing a result not yet in the wiki: Hodge decomposition for compact Riemann surfaces). [/guided] [/step] [step:Show that Hecke correspondences preserve the Hodge summands] For each $n \in \mathbb{N}$, let $C_n$ denote the Hecke correspondence on $X_0(N)$, with holomorphic finite maps \begin{align*} p_1,p_2: C_n \longrightarrow X_0(N). \end{align*} The induced Hecke operator on cohomology is the pull-push operator \begin{align*} T_n^B := (p_2)_*p_1^*: H^1_B(X_0(N),\mathbb{C}) \longrightarrow H^1_B(X_0(N),\mathbb{C}). \end{align*} Because $p_1$ and $p_2$ are holomorphic maps of compact Riemann surfaces, $p_1^*$ sends $(1,0)$-forms to $(1,0)$-forms and $(0,1)$-forms to $(0,1)$-forms, and the trace map $(p_2)_*$ has the same type-preserving property. Therefore $T_n^B$ preserves the direct sum decomposition \begin{align*} H^1_B(X_0(N),\mathbb{C}) \cong H^{1,0}(X_0(N)) \oplus H^{0,1}(X_0(N)). \end{align*} [guided] A Hecke operator on the modular curve is not a single self-map in general; it is a correspondence. For each $n \in \mathbb{N}$, write this correspondence as \begin{align*} p_1,p_2: C_n \longrightarrow X_0(N), \end{align*} where $C_n$ is the compact Riemann surface parametrizing the appropriate cyclic degree-$n$ isogeny data, and both $p_1$ and $p_2$ are finite holomorphic maps. The cohomological Hecke operator is \begin{align*} T_n^B := (p_2)_*p_1^*. \end{align*} Here $p_1^*$ pulls differential forms back along $p_1$, and $(p_2)_*$ is the trace, or pushforward, along the finite map $p_2$. We must check that this operator respects the Hodge splitting. A holomorphic map between Riemann surfaces is complex-linear on tangent spaces, so pulling back a local form of type $(1,0)$ again gives a form of type $(1,0)$, and pulling back a local form of type $(0,1)$ again gives a form of type $(0,1)$. The trace map along a finite holomorphic map is locally the sum over inverse branches; each inverse branch is holomorphic away from ramification, and the resulting trace extends across ramification. Thus the trace map also preserves type. Consequently the composition $(p_2)_*p_1^*$ preserves both $H^{1,0}(X_0(N))$ and $H^{0,1}(X_0(N))$. This proves that the Hecke action is block-diagonal with respect to the Hodge decomposition. [/guided] [/step] [step:Compare the geometric Hecke action with the classical Hecke action on cusp forms] Let $f \in S_2(\Gamma_0(N))$ and let $\omega_f := \Phi(f) \in H^0(X_0(N),\Omega^1_{X_0(N)})$. The compatibility between the geometric pull-push definition of Hecke operators on differentials and the classical double-coset definition of Hecke operators on modular forms gives \begin{align*} T_n^B(\omega_f)=\omega_{T_n f}. \end{align*} Under the identification $\Phi$, the restriction of $T_n^B$ to the holomorphic summand is therefore exactly the classical Hecke operator $T_n$ on $S_2(\Gamma_0(N))$ (citing a result not yet in the wiki: compatibility of geometric and classical Hecke operators). [guided] The holomorphic summand has already been identified with $S_2(\Gamma_0(N))$ by the map \begin{align*} \Phi: S_2(\Gamma_0(N)) &\longrightarrow H^0(X_0(N),\Omega^1_{X_0(N)}) \\ f &\longmapsto \omega_f. \end{align*} To prove Hecke equivariance on this summand, we must show that applying the Hecke correspondence to the differential $\omega_f$ gives the same result as first applying the classical Hecke operator to $f$ and then forming the associated differential. This is exactly the compatibility between the two definitions of Hecke operators. The geometric definition is the pull-push map \begin{align*} T_n^B=(p_2)_*p_1^* \end{align*} on differentials, while the classical definition is the double-coset operator on weight $2$ modular forms. In local coordinates on the upper half-plane, the inverse branches appearing in the finite correspondence are represented by the same matrices that occur in the double-coset formula. Since $\omega_f$ is represented upstairs by $2\pi i f(z)\,dz$, the pullback formula for differentials supplies exactly the weight-$2$ automorphy factor. Hence the resulting differential is represented by $2\pi i (T_n f)(z)\,dz$, which means \begin{align*} T_n^B(\omega_f)=\omega_{T_n f}. \end{align*} Thus $\Phi$ intertwines the classical Hecke action on $S_2(\Gamma_0(N))$ with the geometric Hecke action on $H^{1,0}(X_0(N))$ (citing a result not yet in the wiki: compatibility of geometric and classical Hecke operators). [/guided] [/step] [step:Use complex conjugation to obtain the antiholomorphic eigenvalues] Let \begin{align*} \overline{\omega_f} \end{align*} denote the antiholomorphic $1$-form obtained by complex conjugating the holomorphic differential $\omega_f$. Since the Hecke correspondences defining $T_n^B$ are algebraic over $\mathbb{Q}$, their pull-push action commutes with complex conjugation. Hence \begin{align*} T_n^B(\overline{\omega_f}) = \overline{T_n^B(\omega_f)} = \overline{\omega_{T_n f}}. \end{align*} If $f$ is a simultaneous Hecke eigenform with $T_n f=a_n(f)f$, then \begin{align*} T_n^B(\omega_f) &=\omega_{T_n f} =\omega_{a_n(f)f} =a_n(f)\omega_f, \end{align*} so the line $\mathbb{C}\omega_f$ in the holomorphic summand is a $T_n$-eigenline with eigenvalue $a_n(f)$. Conjugating gives \begin{align*} T_n^B(\overline{\omega_f}) &=\overline{a_n(f)\omega_f} =\overline{a_n(f)}\,\overline{\omega_f}, \end{align*} so the line $\mathbb{C}\overline{\omega_f}$ in the antiholomorphic summand is a $T_n$-eigenline with eigenvalue $\overline{a_n(f)}$. Combining the Hodge decomposition, the identification of cusp forms with holomorphic differentials, and the Hecke compatibility proves the stated Hecke-equivariant isomorphism. [/step]