[proofplan] We regard a Hecke operator as a finite algebraic correspondence on the modular curve $X_0(N)$. The two projection maps of the correspondence give pullback and pushforward operations on divisors, and the compatibility of pullback and norm with principal divisors lets the operation descend to divisor classes of degree zero, hence to the Jacobian. Since the correspondence and its projections are defined over $\mathbb{Q}$, the resulting endomorphism is defined over $\mathbb{Q}$. Finally, the standard double-coset multiplication law for Hecke correspondences gives commutativity. [/proofplan]

[step:Realize the Hecke operator as a finite correspondence over $\mathbb{Q}$] Fix an integer $n \geq 1$. By the moduli-theoretic construction of the [Hecke correspondence](/page/Hecke%20Correspondence) on $X_0(N)$, there is a finite $\mathbb{Q}$-scheme $C_n^{\mathrm{raw}}$ equipped with two finite projection morphisms to $X_0(N)$ whose geometric points parametrize the usual cyclic degree-$n$ isogeny data compatible with the $\Gamma_0(N)$-level structure. Let $A_n$ be the finite set of irreducible components of the [normalization](/page/Normalization) of $C_n^{\mathrm{raw}}$, and write \begin{align*} C_n = \bigsqcup_{a \in A_n} C_{n,a} \end{align*} for the disjoint union of the corresponding smooth projective curves over $\mathbb{Q}$. The two projections extend uniquely across normalization and give finite morphisms over $\mathbb{Q}$ \begin{align*} \pi_1,\pi_2 : C_n \to X_0(N). \end{align*} Thus every later use of divisors, rational functions, Picard varieties, and norm maps is made componentwise on the smooth projective components $C_{n,a}$. When $n=p$ with $p \mid N$, this same normalized finite correspondence is the degeneracy correspondence at level $N$; the induced operator is the usual [Atkin-Lehner $U_p$ operator](/page/Atkin-Lehner%20Operator). Thus $T_n$ is the finite correspondence \begin{align*} X_0(N) \xleftarrow{\ \pi_1\ } C_n \xrightarrow{\ \pi_2\ } X_0(N) \end{align*} defined over $\mathbb{Q}$. [/step]

[step:Define the action on divisors by pullback followed by pushforward]Let $\operatorname{Div}(X_0(N))$ denote the free abelian group on the closed points of $X_0(N)$. Since $\pi_1$ is finite, it induces a pullback homomorphism \begin{align*} \pi_1^* : \operatorname{Div}(X_0(N)) \to \operatorname{Div}(C_n). \end{align*} Since $\pi_2$ is finite, it induces a pushforward homomorphism \begin{align*} (\pi_2)_* : \operatorname{Div}(C_n) \to \operatorname{Div}(X_0(N)). \end{align*} Define the divisor-level Hecke operator \begin{align*} T_n^{\operatorname{div}} : \operatorname{Div}(X_0(N)) &\to \operatorname{Div}(X_0(N)) \\ D &\mapsto (\pi_2)_*(\pi_1^*D). \end{align*} Both pullback by a finite morphism and pushforward by a finite morphism preserve degree up to multiplication by the degree of the corresponding finite map. Therefore, if $D \in \operatorname{Div}^0(X_0(N))$, then $\pi_1^*D$ has degree $0$, and consequently $(\pi_2)_*(\pi_1^*D)$ has degree $0$. Hence $T_n^{\operatorname{div}}$ restricts to a group homomorphism \begin{align*} T_n^{\operatorname{div}} : \operatorname{Div}^0(X_0(N)) \to \operatorname{Div}^0(X_0(N)). \end{align*}[/step]

[guided]The correspondence gives two maps, and the operator on divisors is obtained by moving a divisor through the correspondence. First we pull the divisor back along the first projection, and then we push it forward along the second projection. Let $\operatorname{Div}(X_0(N))$ be the free abelian group generated by the closed points of $X_0(N)$. Because $\pi_1 : C_n \to X_0(N)$ is finite, pullback of Weil divisors is defined and gives \begin{align*} \pi_1^* : \operatorname{Div}(X_0(N)) \to \operatorname{Div}(C_n). \end{align*} Because $\pi_2 : C_n \to X_0(N)$ is finite, pushforward of Weil divisors is defined and gives \begin{align*} (\pi_2)_* : \operatorname{Div}(C_n) \to \operatorname{Div}(X_0(N)). \end{align*} We therefore define \begin{align*} T_n^{\operatorname{div}} : \operatorname{Div}(X_0(N)) &\to \operatorname{Div}(X_0(N)) \\ D &\mapsto (\pi_2)_*(\pi_1^*D). \end{align*} The Jacobian uses divisor classes of degree zero, so we must check that this operation preserves degree zero. Pullback by a finite morphism multiplies degrees by the degree of the morphism, and pushforward by a finite morphism preserves total degree after accounting for residue field degrees. In particular, a divisor of degree $0$ pulls back to a divisor of degree $0$, and the pushforward of a degree-zero divisor again has degree $0$. Hence $T_n^{\operatorname{div}}$ restricts to \begin{align*} T_n^{\operatorname{div}} : \operatorname{Div}^0(X_0(N)) \to \operatorname{Div}^0(X_0(N)). \end{align*}[/guided]

[step:Show that principal divisors are sent to principal divisors]For every smooth projective curve $Y$ over $\mathbb{Q}$, let \begin{align*} \operatorname{div}_Y : \mathbb{Q}(Y)^\times \to \operatorname{Div}(Y) \end{align*} denote the principal-divisor homomorphism from nonzero rational functions on $Y$ to Weil divisors on $Y$. For the disjoint union $C_n=\bigsqcup_{a \in A_n} C_{n,a}$, a rational function is a tuple $g=(g_a)_{a \in A_n}$ with $g_a \in \mathbb{Q}(C_{n,a})^\times$, and \begin{align*} \operatorname{div}_{C_n}(g)=\sum_{a \in A_n}\operatorname{div}_{C_{n,a}}(g_a). \end{align*} Let $f \in \mathbb{Q}(X_0(N))^\times$ be a nonzero rational function on $X_0(N)$. Pullback of rational functions along $\pi_1$ gives the tuple \begin{align*} f \circ \pi_1 = (f \circ \pi_{1,a})_{a \in A_n} \in \prod_{a \in A_n}\mathbb{Q}(C_{n,a})^\times, \end{align*} where $\pi_{1,a}:C_{n,a}\to X_0(N)$ is the restriction of $\pi_1$. Compatibility of Weil divisor pullback with rational functions gives \begin{align*} \pi_1^*(\operatorname{div}_{X_0(N)}(f)) = \operatorname{div}_{C_n}(f \circ \pi_1). \end{align*} For a tuple $g=(g_a)_{a \in A_n}$, define the norm along $\pi_2$ by the product of the componentwise field norms \begin{align*} \operatorname{Nm}_{\pi_2}(g)=\prod_{a \in A_n}\operatorname{Nm}_{\pi_{2,a}}(g_a) \in \mathbb{Q}(X_0(N))^\times, \end{align*} where $\pi_{2,a}:C_{n,a}\to X_0(N)$ is the restriction of $\pi_2$. The standard [divisor-norm compatibility](/page/Norm%20Map) for finite morphisms of smooth projective curves gives \begin{align*} (\pi_2)_*(\operatorname{div}_{C_n}(g)) = \operatorname{div}_{X_0(N)}(\operatorname{Nm}_{\pi_2}(g)). \end{align*} Applying this with $g=f\circ \pi_1$, we obtain \begin{align*} T_n^{\operatorname{div}}(\operatorname{div}_{X_0(N)}(f)) &= (\pi_2)_*(\pi_1^*(\operatorname{div}_{X_0(N)}(f))) \\ &= (\pi_2)_*(\operatorname{div}_{C_n}(f\circ \pi_1)) \\ &= \operatorname{div}_{X_0(N)}(\operatorname{Nm}_{\pi_2}(f\circ \pi_1)). \end{align*} Thus $T_n^{\operatorname{div}}$ sends principal divisors to principal divisors.[/step]

[guided]To descend from divisors to divisor classes, the operation must respect linear equivalence, which is generated by principal divisors. The only extra point here is that the Hecke correspondence may have several components, so all rational-function and norm notation must be read componentwise. For every smooth projective curve $Y$ over $\mathbb{Q}$, let \begin{align*} \operatorname{div}_Y : \mathbb{Q}(Y)^\times \to \operatorname{Div}(Y) \end{align*} be the homomorphism sending a nonzero rational function to its principal Weil divisor. Since $C_n=\bigsqcup_{a \in A_n} C_{n,a}$ is a disjoint union of smooth projective curves, a nonzero rational function on $C_n$ is a tuple $g=(g_a)_{a \in A_n}$ with $g_a \in \mathbb{Q}(C_{n,a})^\times$, and its divisor is \begin{align*} \operatorname{div}_{C_n}(g)=\sum_{a \in A_n}\operatorname{div}_{C_{n,a}}(g_a). \end{align*} Let $f \in \mathbb{Q}(X_0(N))^\times$. Pulling $f$ back along the first projection means pulling it back on each component: \begin{align*} f \circ \pi_1 = (f \circ \pi_{1,a})_{a \in A_n} \in \prod_{a \in A_n}\mathbb{Q}(C_{n,a})^\times. \end{align*} The compatibility of pullback of Weil divisors with pullback of rational functions, applied on every component $C_{n,a}$, gives \begin{align*} \pi_1^*(\operatorname{div}_{X_0(N)}(f)) = \operatorname{div}_{C_n}(f \circ \pi_1). \end{align*} Now we use finite pushforward. For $g=(g_a)_{a \in A_n}$, define \begin{align*} \operatorname{Nm}_{\pi_2}(g)=\prod_{a \in A_n}\operatorname{Nm}_{\pi_{2,a}}(g_a) \in \mathbb{Q}(X_0(N))^\times. \end{align*} Each $\pi_{2,a}$ is finite, so the usual field norm is defined for the finite extension of function fields induced by $\pi_{2,a}$. The [divisor-norm compatibility](/page/Norm%20Map), applied component by component and then summed, gives \begin{align*} (\pi_2)_*(\operatorname{div}_{C_n}(g)) = \operatorname{div}_{X_0(N)}(\operatorname{Nm}_{\pi_2}(g)). \end{align*} Taking $g=f\circ \pi_1$, we compute \begin{align*} T_n^{\operatorname{div}}(\operatorname{div}_{X_0(N)}(f)) &= (\pi_2)_*(\pi_1^*(\operatorname{div}_{X_0(N)}(f))) \\ &= (\pi_2)_*(\operatorname{div}_{C_n}(f\circ \pi_1)) \\ &= \operatorname{div}_{X_0(N)}(\operatorname{Nm}_{\pi_2}(f\circ \pi_1)). \end{align*} The final divisor is principal on $X_0(N)$, so $T_n^{\operatorname{div}}$ respects linear equivalence.[/guided]

[step:Descend the correspondence to a morphism of the Jacobian]Since $T_n^{\operatorname{div}}$ preserves degree-zero divisors and sends principal divisors to principal divisors, it induces a group homomorphism on divisor classes in the [degree-zero Picard variety](/page/Picard%20Variety). The hypotheses for Picard functoriality are satisfied: $X_0(N)$ is a smooth projective modular curve over $\mathbb{Q}$, and $C_n=\bigsqcup_{a \in A_n}C_{n,a}$ is a finite disjoint union of smooth projective curves over $\mathbb{Q}$ by construction. For a finite morphism of smooth projective curves $\varphi : Y \to X$ over $\mathbb{Q}$, [functoriality of the Picard variety](/page/Picard%20Variety) gives a pullback morphism \begin{align*} \varphi^* : \operatorname{Pic}^0_{X/\mathbb{Q}} \to \operatorname{Pic}^0_{Y/\mathbb{Q}} \end{align*} and a [norm morphism](/page/Norm%20Map) \begin{align*} \operatorname{Nm}_{\varphi} : \operatorname{Pic}^0_{Y/\mathbb{Q}} \to \operatorname{Pic}^0_{X/\mathbb{Q}}. \end{align*} For the disjoint union $C_n$, this means \begin{align*} \operatorname{Pic}^0_{C_n/\mathbb{Q}}=\prod_{a \in A_n}\operatorname{Pic}^0_{C_{n,a}/\mathbb{Q}}, \end{align*} and the norm along $\pi_2$ is the product of the componentwise norms followed by addition in $\operatorname{Pic}^0_{X_0(N)/\mathbb{Q}}$. On divisor classes, these morphisms are represented respectively by pullback of divisors and pushforward of divisors. Applying this to $\pi_1$ and $\pi_2$, the composition \begin{align*} \operatorname{Nm}_{\pi_2} \circ \pi_1^* : \operatorname{Pic}^0_{X_0(N)/\mathbb{Q}} \to \operatorname{Pic}^0_{X_0(N)/\mathbb{Q}} \end{align*} is a morphism of abelian varieties whose action on divisor classes is exactly $D \mapsto (\pi_2)_*(\pi_1^*D)$. Using the identification of the [Jacobian](/page/Jacobian) with the degree-zero Picard variety, \begin{align*} J_0(N)=\operatorname{Pic}^0_{X_0(N)/\mathbb{Q}}, \end{align*} this morphism is an endomorphism \begin{align*} T_n:J_0(N) \to J_0(N). \end{align*} Because $C_n$, $\pi_1$, and $\pi_2$ are all defined over $\mathbb{Q}$, the Picard pullback morphism $\pi_1^*$, the norm morphism $\operatorname{Nm}_{\pi_2}$, and their composition are defined over $\mathbb{Q}$. Hence the induced endomorphism of $J_0(N)$ is defined over $\mathbb{Q}$.[/step]

[guided]The divisor calculation already gives a well-defined map on degree-zero divisor classes. We now verify that this map is algebraic, meaning that it is induced by a morphism of abelian varieties rather than only by an abstract group homomorphism. The relevant functoriality theorem for the [Picard variety](/page/Picard%20Variety) requires finite morphisms between smooth projective curves over the base field. These hypotheses are met here. The modular curve $X_0(N)$ is smooth and projective over $\mathbb{Q}$, and in the first step we replaced the raw Hecke correspondence by the disjoint union \begin{align*} C_n=\bigsqcup_{a \in A_n}C_{n,a} \end{align*} of the smooth projective normalized components. The restrictions $\pi_{1,a},\pi_{2,a}:C_{n,a}\to X_0(N)$ remain finite over $\mathbb{Q}$ because finite morphisms are stable under normalization of the source in this curve setting. For a finite morphism $\varphi:Y\to X$ of smooth projective curves over $\mathbb{Q}$, Picard functoriality gives \begin{align*} \varphi^* : \operatorname{Pic}^0_{X/\mathbb{Q}} \to \operatorname{Pic}^0_{Y/\mathbb{Q}} \end{align*} and the norm construction gives \begin{align*} \operatorname{Nm}_{\varphi} : \operatorname{Pic}^0_{Y/\mathbb{Q}} \to \operatorname{Pic}^0_{X/\mathbb{Q}}. \end{align*} On divisor classes, $\varphi^*$ is represented by pulling back divisors, and $\operatorname{Nm}_{\varphi}$ is represented by pushing forward divisors. For the disconnected curve $C_n$, the identity \begin{align*} \operatorname{Pic}^0_{C_n/\mathbb{Q}}=\prod_{a \in A_n}\operatorname{Pic}^0_{C_{n,a}/\mathbb{Q}} \end{align*} reduces the construction to the componentwise finite morphisms $\pi_{1,a}$ and $\pi_{2,a}$. Therefore the composition \begin{align*} \operatorname{Nm}_{\pi_2} \circ \pi_1^* : \operatorname{Pic}^0_{X_0(N)/\mathbb{Q}} \to \operatorname{Pic}^0_{X_0(N)/\mathbb{Q}} \end{align*} is a morphism of abelian varieties. Its action on divisor classes is exactly the divisor operation already constructed: \begin{align*} D \mapsto (\pi_2)_*(\pi_1^*D). \end{align*} Finally, the [Jacobian](/page/Jacobian) of $X_0(N)$ is its degree-zero Picard variety, \begin{align*} J_0(N)=\operatorname{Pic}^0_{X_0(N)/\mathbb{Q}}. \end{align*} Thus the correspondence induces an endomorphism \begin{align*} T_n:J_0(N) \to J_0(N). \end{align*} All schemes and morphisms used in the construction are defined over $\mathbb{Q}$, so the resulting Picard pullback, norm morphism, and composite endomorphism are defined over $\mathbb{Q}$.[/guided]

[step:Apply the double-coset multiplication law to prove commutativity]For integers $m,n \geq 1$, the compositions $T_m \circ T_n$ and $T_n \circ T_m$ are represented by the fiber-product compositions of the normalized finite correspondences on $X_0(N)$. Normalizing the fiber products does not change the induced correspondence on divisors or on the Jacobian, because pullback and pushforward of divisors are computed componentwise on the normalizations and agree with the usual correspondence multiplicities. The [Hecke double-coset multiplication formula](/page/Hecke%20Algebra) for $\Gamma_0(N)$ states that these standard normalized correspondences satisfy \begin{align*} T_m \circ T_n = \sum_{d \mid \gcd(m,n),\, \gcd(d,N)=1} d\,T_{mn/d^2}. \end{align*} This is the formula for the same geometric correspondences used above: the correspondence attached to each double coset is represented by its normalized modular-curve correspondence, and the coefficient $d$ records the multiplicity with which the component occurs in the fiber-product composition. When $p \mid N$, the operator denoted $U_p$ is the normalized degeneracy correspondence $T_p$, and the condition $\gcd(d,N)=1$ excludes factors divisible by $p$ from the summation. Thus the formula includes the $U_p$ operators and their composites at level $N$. The displayed expression is symmetric in $m$ and $n$, because $\gcd(m,n)=\gcd(n,m)$ and $mn/d^2=nm/d^2$ for each admissible divisor $d$. Hence the two correspondence compositions satisfy \begin{align*} T_m \circ T_n = \sum_{d \mid \gcd(m,n),\, \gcd(d,N)=1} d\,T_{mn/d^2} = T_n \circ T_m. \end{align*} Passing from correspondences to their induced morphisms of $J_0(N)$ preserves composition, since each correspondence acts by the functorial pullback-norm construction established above. Therefore the induced endomorphisms of $J_0(N)$ satisfy \begin{align*} T_m \circ T_n = T_n \circ T_m. \end{align*} Since $m$ and $n$ were arbitrary, the Hecke endomorphisms of $J_0(N)$ commute. This completes the proof.[/step]

[guided]The final input is the multiplication law in the [Hecke algebra](/page/Hecke%20Algebra). We must use it for the same correspondences that acted on the Jacobian above. Those correspondences were represented by normalized modular curves, and this is compatible with the double-coset law: composing two correspondences means forming the fiber product over $X_0(N)$, then taking the corresponding cycle with its natural multiplicities. Passing to the normalization only replaces that cycle by smooth projective components on which pullback, pushforward, and norm maps are defined; it does not change the resulting divisor correspondence or the induced morphism of the Jacobian. For $m,n \geq 1$, the double-coset multiplication formula for $\Gamma_0(N)$ gives \begin{align*} T_m \circ T_n = \sum_{d \mid \gcd(m,n),\, \gcd(d,N)=1} d\,T_{mn/d^2}. \end{align*} Here each $T_r$ denotes the normalized finite correspondence already used in the proof, and the coefficient $d$ is the correspondence multiplicity coming from the double-coset decomposition. If $p \mid N$, then $T_p$ is the degeneracy correspondence usually denoted $U_p$. The restriction $\gcd(d,N)=1$ is exactly what distinguishes the level-$\Gamma_0(N)$ multiplication rule from the prime-to-level formula and ensures that the displayed identity includes the $U_p$ operators and their composites. The right-hand side is unchanged when $m$ and $n$ are interchanged. Indeed, \begin{align*} \gcd(m,n)=\gcd(n,m), \qquad mn/d^2=nm/d^2 \end{align*} for every divisor $d$ appearing in the sum. Therefore the normalized correspondence compositions satisfy \begin{align*} T_m \circ T_n = \sum_{d \mid \gcd(m,n),\, \gcd(d,N)=1} d\,T_{mn/d^2}=T_n \circ T_m. \end{align*} The construction of the induced map on $J_0(N)$ is functorial in correspondences: composition of correspondences becomes composition of the associated pullback-norm morphisms. Hence the induced endomorphisms satisfy \begin{align*} T_m \circ T_n = T_n \circ T_m. \end{align*} Since $m$ and $n$ were arbitrary, all Hecke endomorphisms of $J_0(N)$ commute.[/guided]