[proofplan] We compare the Hecke correspondence defining $T_p$ with the Frobenius and Verschiebung correspondences on the special fibre of the modular curve. The key input is the moduli description of the reduction of the $p$-th Hecke correspondence: in characteristic $p$, cyclic subgroup schemes of order $p$ on an elliptic curve split into the Frobenius kernel and an étale subgroup obtained from Verschiebung. This identifies the reduced correspondence with the sum of the Frobenius and Verschiebung graphs, and hence gives the equality on the Jacobian. Finally, the relation $F_pV_p=V_pF_p=[p]$ gives the quadratic identity on every $\ell$-adic Tate module. [/proofplan]

[step:Realize $T_p$ as a correspondence on the special fibre]Let $Y_0(N)_{\mathbb{F}_p}$ denote the open modular curve over $\mathbb{F}_p$ classifying pairs $(E,C_N)$, where $E$ is an elliptic curve over an $\mathbb{F}_p$-scheme and $C_N \subset E$ is a cyclic subgroup scheme of order $N$. Since $p \nmid N$, the $\Gamma_0(N)$-level subgroup $C_N$ is finite étale over the base in characteristic $p$. Let $Y_0(N,p)_{\mathbb{F}_p}$ denote the moduli stack classifying triples $(E,C_N,D)$, where $(E,C_N)$ is as above and $D \subset E$ is a cyclic finite flat subgroup scheme of order $p$ satisfying $D \cap C_N = 0$. Because $p \nmid N$, the condition $D \cap C_N = 0$ is automatic after identifying $C_N$ as prime-to-$p$ torsion, but we keep it in the notation to record the modular interpretation. Define the two degeneracy maps \begin{align*} \pi_1: Y_0(N,p)_{\mathbb{F}_p} &\to Y_0(N)_{\mathbb{F}_p}, \\ (E,C_N,D) &\mapsto (E,C_N), \end{align*} and \begin{align*} \pi_2: Y_0(N,p)_{\mathbb{F}_p} &\to Y_0(N)_{\mathbb{F}_p}, \\ (E,C_N,D) &\mapsto (E/D,(C_N+D)/D). \end{align*} The Hecke correspondence defining $T_p$ on the special fibre is the correspondence \begin{align*} Y_0(N)_{\mathbb{F}_p} \xleftarrow{\ \pi_1\ } Y_0(N,p)_{\mathbb{F}_p} \xrightarrow{\ \pi_2\ } Y_0(N)_{\mathbb{F}_p}. \end{align*} Passing to the smooth compactification $X_0(N)_{\mathbb{F}_p}$ and then to divisors of degree zero gives the induced endomorphism $T_p$ of $J_0(N)_{\mathbb{F}_p}$.[/step]

[guided]The operator $T_p$ is not first defined by a formula on functions in this proof; it is defined geometrically by a correspondence. The point of introducing $Y_0(N,p)_{\mathbb{F}_p}$ is that its points record exactly the extra choice which appears in the Hecke operator: for a point $(E,C_N)$, one chooses a cyclic subgroup scheme $D \subset E$ of order $p$ and then replaces $E$ by the quotient $E/D$. The two maps have different roles. The map \begin{align*} \pi_1: Y_0(N,p)_{\mathbb{F}_p} &\to Y_0(N)_{\mathbb{F}_p}, \\ (E,C_N,D) &\mapsto (E,C_N) \end{align*} forgets the subgroup $D$. The map \begin{align*} \pi_2: Y_0(N,p)_{\mathbb{F}_p} &\to Y_0(N)_{\mathbb{F}_p}, \\ (E,C_N,D) &\mapsto (E/D,(C_N+D)/D) \end{align*} takes the quotient by $D$ and transports the prime-to-$p$ level structure. Since $p \nmid N$, the subgroup $C_N$ is finite étale in characteristic $p$, so quotienting by the $p$-power subgroup $D$ preserves the meaning of the $\Gamma_0(N)$-structure. Thus the correspondence \begin{align*} Y_0(N)_{\mathbb{F}_p} \xleftarrow{\ \pi_1\ } Y_0(N,p)_{\mathbb{F}_p} \xrightarrow{\ \pi_2\ } Y_0(N)_{\mathbb{F}_p} \end{align*} is precisely the reduction modulo $p$ of the usual Hecke correspondence at $p$. Extending this correspondence to the compact modular curve and applying the standard functorial action of correspondences on degree-zero divisor classes gives the endomorphism $T_p$ of $J_0(N)_{\mathbb{F}_p}$.[/guided]

[step:Decompose the characteristic $p$ Hecke correspondence into Frobenius and Verschiebung components]Let $X_0(N,p)_{\mathbb{F}_p}$ denote the compactified special fibre of the modular correspondence whose open part is $Y_0(N,p)_{\mathbb{F}_p}$, with the two proper degeneracy maps to $X_0(N)_{\mathbb{F}_p}$ extending $\pi_1$ and $\pi_2$ across the cusps. We use the Deligne-Rapoport-Katz-Mazur description of this compactified special fibre: for $p \nmid N$, the correspondence cycle $X_0(N,p)_{\mathbb{F}_p}$ is the sum, with multiplicity one, of two copies of $X_0(N)_{\mathbb{F}_p}$, and their degeneracy maps are respectively the absolute Frobenius graph and the Verschiebung graph on $X_0(N)_{\mathbb{F}_p}$. This statement applies because the level $N$ is prime to $p$, so the $\Gamma_0(N)$-structure is finite étale in characteristic $p$, the Deligne-Rapoport integral model is regular at $p$, and the compactified degeneracy maps extend over the cusps; the theorem includes the supersingular points in the cycle identity, so no pointwise étale subgroup of $E[p]$ is being chosen there. More explicitly, as correspondence cycles on $X_0(N)_{\mathbb{F}_p}$, the two components are \begin{align*} \Gamma_{F_p} &= \{(E,C_N,\ker(F_{E/\mathbb{F}_p}))\}, \\ \Gamma_{V_p} &= \text{the copy of } X_0(N)_{\mathbb{F}_p} \text{ whose degeneracy maps are } (V_p,\operatorname{id}). \end{align*} Here $F_{E/\mathbb{F}_p}: E \to E^{(p)}$ is the relative Frobenius isogeny, $\ker(F_{E/\mathbb{F}_p}) \subset E$ is its finite flat order-$p$ kernel, and $V_{E/\mathbb{F}_p}: E^{(p)} \to E$ is the Verschiebung isogeny dual to relative Frobenius. The notation for $\Gamma_{V_p}$ is a correspondence-cycle description, not a pointwise assertion that every elliptic curve in characteristic $p$ has an étale order-$p$ subgroup of $E[p]$. On $\Gamma_{F_p}$, the quotient map \begin{align*} E &\to E/\ker(F_{E/\mathbb{F}_p}) \end{align*} identifies with the relative Frobenius map \begin{align*} F_{E/\mathbb{F}_p}: E &\to E^{(p)}. \end{align*} Because the base field is $\mathbb{F}_p$, the Frobenius twist of $X_0(N)_{\mathbb{F}_p}$ is canonically identified with $X_0(N)_{\mathbb{F}_p}$, and this component induces the endomorphism $F_p$ on $J_0(N)_{\mathbb{F}_p}$. On $\Gamma_{V_p}$, the second component is defined by the degeneracy maps encoded by Verschiebung. Under the same identification of Frobenius twists over $\mathbb{F}_p$, this correspondence component induces the Verschiebung endomorphism $V_p$ on $J_0(N)_{\mathbb{F}_p}$. Therefore the reduced correspondence defining $T_p$ is the sum of the Frobenius and Verschiebung correspondences: \begin{align*} [X_0(N,p)_{\mathbb{F}_p}] = [\Gamma_{F_p}] + [\Gamma_{V_p}] \end{align*} as correspondences on $X_0(N)_{\mathbb{F}_p}$.[/step]

[guided]The essential phenomenon is special to characteristic $p$. Over a field of characteristic different from $p$, the correspondence $T_p$ records cyclic subgroup schemes of order $p$. In characteristic $p$, the compactified correspondence does not decompose by choosing an étale subgroup of $E[p]$ pointwise on every elliptic curve; that would fail at supersingular points. Instead, the correct statement is a cycle identity for the whole compactified correspondence. Let $X_0(N,p)_{\mathbb{F}_p}$ denote the compactification of $Y_0(N,p)_{\mathbb{F}_p}$, with the two degeneracy maps extending across the cusps to proper maps \begin{align*} X_0(N,p)_{\mathbb{F}_p} &\to X_0(N)_{\mathbb{F}_p}. \end{align*} We invoke the Deligne-Rapoport-Katz-Mazur description of the mod $p$ Hecke correspondence. The theorem requires the level away from $p$ to be prime to the characteristic; here this is exactly the hypothesis $p \nmid N$. Under this hypothesis, the $\Gamma_0(N)$-level subgroup is finite étale in characteristic $p$, the compactified integral model has the standard modular interpretation at the cusps, and the theorem gives the following precise conclusion: as a correspondence cycle, $X_0(N,p)_{\mathbb{F}_p}$ is the sum with multiplicity one of two copies of $X_0(N)_{\mathbb{F}_p}$, one carrying the Frobenius degeneracy map and the other carrying the Verschiebung degeneracy map. The statement includes the supersingular locus, so it avoids any invalid choice of an étale order-$p$ subgroup on a supersingular elliptic curve. The first component is \begin{align*} \Gamma_{F_p} = \{(E,C_N,\ker(F_{E/\mathbb{F}_p}))\}. \end{align*} Here \begin{align*} F_{E/\mathbb{F}_p}: E &\to E^{(p)} \end{align*} is the relative Frobenius isogeny of the elliptic curve $E$, and its kernel is a finite flat subgroup scheme of order $p$. Quotienting $E$ by this subgroup gives \begin{align*} E/\ker(F_{E/\mathbb{F}_p}) \cong E^{(p)}. \end{align*} Thus this component of the correspondence sends $(E,C_N)$ to its Frobenius twist. Since the ground field is $\mathbb{F}_p$, the Frobenius twist of the special fibre is canonically identified with the special fibre itself, so the induced endomorphism on the Jacobian is $F_p$. The second component is the correspondence-cycle component \begin{align*} \Gamma_{V_p} = \text{the copy of } X_0(N)_{\mathbb{F}_p} \text{ whose degeneracy maps are } (V_p,\operatorname{id}). \end{align*} This is the component dual to relative Frobenius in the sense of finite flat group schemes and Cartier duality. After identifying Frobenius twists over $\mathbb{F}_p$, it induces the Verschiebung endomorphism \begin{align*} V_p: J_0(N)_{\mathbb{F}_p} \to J_0(N)_{\mathbb{F}_p}. \end{align*} Hence the entire reduced Hecke correspondence is the sum of these two components: \begin{align*} [X_0(N,p)_{\mathbb{F}_p}] = [\Gamma_{F_p}] + [\Gamma_{V_p}]. \end{align*} This is the geometric content of the Eichler-Shimura relation.[/guided]

[step:Pass from the correspondence identity to the Jacobian identity]Let $X := X_0(N)_{\mathbb{F}_p}$ and let $J := J_0(N)_{\mathbb{F}_p} = \operatorname{Jac}(X)$. A finite correspondence \begin{align*} X \xleftarrow{\alpha} C \xrightarrow{\beta} X \end{align*} acts on divisor classes by the homomorphism \begin{align*} \beta_* \alpha^*: \operatorname{Pic}^0(X) &\to \operatorname{Pic}^0(X). \end{align*} This construction is additive in correspondences: if a correspondence cycle is a sum $C_1 + C_2$, then its induced endomorphism of $\operatorname{Pic}^0(X)$ is the sum of the endomorphisms induced by $C_1$ and $C_2$. Applying this additivity to \begin{align*} [X_0(N,p)_{\mathbb{F}_p}] = [\Gamma_{F_p}] + [\Gamma_{V_p}] \end{align*} gives \begin{align*} T_p = F_p + V_p \end{align*} in $\operatorname{End}(J)$.[/step]

[guided]A correspondence acts on the Jacobian through pullback followed by pushforward. More explicitly, for a finite correspondence \begin{align*} X \xleftarrow{\alpha} C \xrightarrow{\beta} X, \end{align*} the map on degree-zero line bundles is \begin{align*} \beta_* \alpha^*: \operatorname{Pic}^0(X) &\to \operatorname{Pic}^0(X). \end{align*} Since $J=\operatorname{Pic}^0(X)$, this is an endomorphism of the Jacobian. The construction is linear in the correspondence cycle. Therefore, if a correspondence decomposes as a sum of two correspondence components, then the induced endomorphism is the sum of the two induced endomorphisms. In the previous step we identified the reduced $p$-th Hecke correspondence as \begin{align*} [X_0(N,p)_{\mathbb{F}_p}] = [\Gamma_{F_p}] + [\Gamma_{V_p}]. \end{align*} The first component induces $F_p$, and the second component induces $V_p$. Hence the endomorphism induced by the full Hecke correspondence is \begin{align*} T_p = F_p + V_p \end{align*} in $\operatorname{End}(J_0(N)_{\mathbb{F}_p})$.[/guided]

[step:Derive the quadratic relation on the Tate module]Let $\ell$ be a prime with $\ell \ne p$, and define the $\ell$-adic Tate module \begin{align*} T_\ell(J) := \varprojlim_{m} J[\ell^m](\overline{\mathbb{F}}_p). \end{align*} The endomorphisms $F_p$, $V_p$, and $T_p$ act functorially on $T_\ell(J)$. For an abelian variety over $\mathbb{F}_p$, Frobenius and Verschiebung satisfy \begin{align*} F_p \circ V_p = V_p \circ F_p = [p]. \end{align*} Using $T_p = F_p + V_p$, we compute in $\operatorname{End}(T_\ell(J))$: \begin{align*} F_p^2 - T_pF_p + p &= F_p^2 - (F_p + V_p)F_p + [p] \\ &= F_p^2 - F_p^2 - V_pF_p + [p] \\ &= -[p] + [p] \\ &= 0. \end{align*} Thus \begin{align*} F_p^2 - T_pF_p + p = 0 \end{align*} on $T_\ell(J)$.[/step]

[guided]The equality $T_p=F_p+V_p$ is an equality of endomorphisms of the abelian variety $J$. Therefore it may be evaluated on any functorial realization of $J$, including the $\ell$-adic Tate module \begin{align*} T_\ell(J) := \varprojlim_m J[\ell^m](\overline{\mathbb{F}}_p). \end{align*} The restriction $\ell \ne p$ ensures that this Tate module is the usual free $\mathbb{Z}_\ell$-module attached to the prime-to-$p$ torsion of $J$. The defining relation between Frobenius and Verschiebung on an abelian variety in characteristic $p$ is \begin{align*} F_p \circ V_p = V_p \circ F_p = [p], \end{align*} where $[p]$ is multiplication by $p$. Combining this relation with $T_p=F_p+V_p$ gives a purely algebraic computation: \begin{align*} F_p^2 - T_pF_p + p &= F_p^2 - (F_p + V_p)F_p + [p] \\ &= F_p^2 - F_p^2 - V_pF_p + [p] \\ &= -[p] + [p] \\ &= 0. \end{align*} Thus $F_p$ satisfies the quadratic polynomial \begin{align*} X^2 - T_pX + p \end{align*} on the $\ell$-adic Tate module.[/guided]

[step:Specialize to a weight $2$ eigenform quotient] Let $f$ be a normalized weight $2$ Hecke eigenform of level $\Gamma_0(N)$, and let $A_f$ be the associated abelian quotient of $J_0(N)$. Assume $T_p f = a_p f$. On the $f$-isotypic quotient of $T_\ell(J_0(N)) \otimes_{\mathbb{Z}_\ell} \mathbb{Q}_\ell$, the operator $T_p$ acts as multiplication by $a_p$. Therefore the preceding relation becomes \begin{align*} F_p^2 - a_pF_p + p = 0. \end{align*} Equivalently, the characteristic polynomial of arithmetic Frobenius on the corresponding two-dimensional $\ell$-adic Galois representation is \begin{align*} X^2 - a_pX + p. \end{align*} This is the stated Eichler-Shimura relation on the Jacobian and on the associated Galois representation. [/step]