[proofplan]
We prove the relation by comparing correspondences on the good special fibre at $p$. The hypothesis $p \nmid N$ gives a smooth integral model of $X_0(N)$ at $p$, so the Hecke correspondence $T_p$ has a well-defined reduction modulo $p$. On the special fibre, we invoke the Eichler-Shimura congruence relation for the Deligne-Rapoport/Katz-Mazur integral model: the reduced $p$-isogeny correspondence acts on $\ell$-adic cohomology as geometric Frobenius plus $p$ times inverse geometric Frobenius. The compatibility of specialization, trace maps, and finite correspondences then identifies this special-fibre action with the original Hecke operator.
[/proofplan]
[step:Reduce the Hecke operator to its special fibre correspondence]
Let $k := \overline{\mathbb F}_p$, and let
\begin{align*}
C := X_0(N)_k
\end{align*}
denote the smooth proper modular curve over $k$ obtained as the geometric special fibre. Let
\begin{align*}
H := H^1_{\mathrm{\acute{e}t}}(C,\mathbb Q_\ell)
\end{align*}
denote its first $\ell$-adic étale cohomology group. Since $p \nmid N$, the modular curve $X_0(N)$ has good reduction at $p$, so $C$ is smooth and proper over $k$, and the usual $p$-th Hecke correspondence extends to the integral model over $\mathbb Z_p$.
Let $Y_0(N,p)_k$ be the special fibre over $k$ of the modular curve classifying triples $(E,G,H_p)$, where $E$ is an elliptic curve over a $k$-scheme, $G \subset E$ is a cyclic subgroup of order $N$, and $H_p \subset E$ is a finite flat subgroup of order $p$. Define the two degeneracy maps
\begin{align*}
\alpha,\beta: Y_0(N,p)_k &\to C
\end{align*}
by
\begin{align*}
\alpha(E,G,H_p) &= (E,G), \\
\beta(E,G,H_p) &= (E/H_p,(G+H_p)/H_p).
\end{align*}
The action of the reduced Hecke correspondence on $H$ is the pull-push operator
\begin{align*}
T_p := \beta_*\alpha^*: H \to H,
\end{align*}
where $\alpha^*$ is pullback in étale cohomology and $\beta_*$ is the trace pushforward for the finite morphism $\beta$.
The equality of the generic Hecke action with this special fibre action follows from the [proper smooth base change theorem](/page/Proper%20Smooth%20Base%20Change): the model is proper and smooth at $p$, and $\ell \ne p$, so $H^1_{\mathrm{\acute{e}t}}$ is identified after specialization in a way compatible with finite correspondences.
[guided]
We first put every object on the same geometric fibre. Let $k := \overline{\mathbb F}_p$, and set
\begin{align*}
C := X_0(N)_k.
\end{align*}
Thus $C$ is the geometric special fibre appearing in the theorem. We also name the cohomology group on which all operators act:
\begin{align*}
H := H^1_{\mathrm{\acute{e}t}}(C,\mathbb Q_\ell).
\end{align*}
The condition $p \nmid N$ is used here: it is the good-reduction condition for $X_0(N)$ at $p$, so the special fibre $C$ is a smooth proper curve over $k$ and the $p$-th Hecke correspondence extends over $\mathbb Z_p$.
The correspondence defining $T_p$ is represented on the special fibre by the modular curve $Y_0(N,p)_k$. For a $k$-scheme, its points classify triples $(E,G,H_p)$, where $E$ is an elliptic curve, $G \subset E$ is a cyclic subgroup of order $N$, and $H_p \subset E$ is a finite flat subgroup of order $p$. The two maps in the correspondence are
\begin{align*}
\alpha,\beta: Y_0(N,p)_k &\to C,
\end{align*}
with
\begin{align*}
\alpha(E,G,H_p) &= (E,G), \\
\beta(E,G,H_p) &= (E/H_p,(G+H_p)/H_p).
\end{align*}
The associated operator on cohomology is the pull-push operator
\begin{align*}
T_p := \beta_*\alpha^*: H \to H.
\end{align*}
Here $\alpha^*$ is the usual pullback on étale cohomology, while $\beta_*$ is the trace pushforward for the finite morphism $\beta$.
Why is it legitimate to compute on the special fibre? The [proper smooth base change theorem](/page/Proper%20Smooth%20Base%20Change) applies because the model is proper and smooth at $p$, and the coefficient prime satisfies $\ell \ne p$. It identifies the relevant $\ell$-adic cohomology groups before and after specialization and is functorial for finite correspondences. Therefore proving the correspondence identity on $C$ proves the stated operator identity on $H$.
[/guided]
[/step]
[step:Invoke the Eichler-Shimura congruence relation for the reduced $p$-isogeny correspondence]
Let $C_0$ denote the smooth proper model of $X_0(N)$ over $\mathbb F_p$, so that $C = C_0 \times_{\mathbb F_p} k$. Let
\begin{align*}
\varphi_p: C &\to C
\end{align*}
denote the automorphism of the étale topos induced by geometric Frobenius on the $\mathbb F_p$-model $C_0$ after base change to $k$. We write
\begin{align*}
\operatorname{Frob}_p := \varphi_p^*: H &\to H
\end{align*}
for its action on $H$.
We use the Eichler-Shimura congruence relation for the Deligne-Rapoport/Katz-Mazur regular model of $X_0(N)$ at a prime $p \nmid N$. It states that the special fibre of the finite flat $p$-isogeny correspondence, with the degeneracy maps oriented as $(\alpha,\beta)$ and with the trace-pullback convention $\beta_*\alpha^*$, induces on
\begin{align*}
H^1_{\mathrm{\acute{e}t}}(C,\mathbb Q_\ell)
\end{align*}
the operator
\begin{align*}
\operatorname{Frob}_p + p\operatorname{Frob}_p^{-1}.
\end{align*}
The hypotheses required by this congruence are exactly those in force here: $p \nmid N$ gives good reduction and a finite flat extension of the $p$-isogeny correspondence to the integral model, and $\ell \ne p$ allows the correspondence to act on $\mathbb Q_\ell$-cohomology by trace maps. Thus
\begin{align*}
\beta_*\alpha^* = \operatorname{Frob}_p + p\operatorname{Frob}_p^{-1}
\end{align*}
as endomorphisms of $H$.
[guided]
The point requiring a precise theorem is the geometry of the special fibre of the $p$-isogeny correspondence. It is not enough to say that the correspondence is the sum of two ordinary graphs: at supersingular points the special fibre has nontrivial component and multiplicity behaviour, and the orientation of the two degeneracy maps determines whether the resulting cohomological operator is a pullback or a trace pushforward.
We therefore state the input in the form needed for this proof. Let $C_0$ be the smooth proper model of $X_0(N)$ over $\mathbb F_p$, and write $C = C_0 \times_{\mathbb F_p} k$. Define
\begin{align*}
\varphi_p: C &\to C
\end{align*}
to be the Frobenius automorphism of the étale topos obtained from the geometric Frobenius element of $\operatorname{Gal}(k/\mathbb F_p)$ acting on the base of $C_0$. This avoids treating geometric Frobenius as a naive $k$-morphism. Its induced action on cohomology is
\begin{align*}
\operatorname{Frob}_p := \varphi_p^*: H \to H.
\end{align*}
Because $H$ is a finite-dimensional $\mathbb Q_\ell$-[vector space](/page/Vector%20Space) and $\varphi_p$ is an automorphism of the étale topos, $\operatorname{Frob}_p$ is invertible.
The Eichler-Shimura congruence relation for the Deligne-Rapoport/Katz-Mazur integral model says that, for $p \nmid N$, the reduced $p$-isogeny correspondence acts on $H^1_{\mathrm{\acute{e}t}}(C,\mathbb Q_\ell)$ by
\begin{align*}
\operatorname{Frob}_p + p\operatorname{Frob}_p^{-1}
\end{align*}
when the correspondence is oriented by the degeneracy maps
\begin{align*}
\alpha,\beta: Y_0(N,p)_k &\to C
\end{align*}
and acts through the trace-pullback operator $\beta_*\alpha^*$. The theorem includes the component multiplicities on the special fibre and the Verschiebung contribution; in cohomological form that contribution is already the term $p\operatorname{Frob}_p^{-1}$. Thus it gives directly
\begin{align*}
\beta_*\alpha^* = \operatorname{Frob}_p + p\operatorname{Frob}_p^{-1}.
\end{align*}
This is the exact operator identity needed, with no separate graph-action convention or invalid multiplication-by-$p$ map on the modular curve.
[/guided]
[/step]
[step:Conclude from specialization compatibility and the congruence relation]
By definition of the reduced Hecke correspondence in the first step,
\begin{align*}
T_p = \beta_*\alpha^*: H &\to H.
\end{align*}
The congruence relation from the previous step gives
\begin{align*}
\beta_*\alpha^* = \operatorname{Frob}_p + p\operatorname{Frob}_p^{-1}.
\end{align*}
Combining the two equalities yields
\begin{align*}
T_p = \operatorname{Frob}_p + p\operatorname{Frob}_p^{-1}
\end{align*}
on $H = H^1_{\mathrm{\acute{e}t}}(C,\mathbb Q_\ell)$. Since $C = X_0(N)_{\overline{\mathbb F}_p}$ by construction, this is the asserted Eichler-Shimura relation.
[guided]
The first step identified the Hecke operator on the special fibre with the trace-pullback operator
\begin{align*}
T_p = \beta_*\alpha^*: H \to H.
\end{align*}
This identification uses proper smooth base change and compatibility of finite correspondences with specialization: $p \nmid N$ gives the proper smooth model at $p$, and $\ell \ne p$ is the coefficient condition under which étale cohomology specializes correctly.
The previous step then supplied the precise special-fibre computation:
\begin{align*}
\beta_*\alpha^* = \operatorname{Frob}_p + p\operatorname{Frob}_p^{-1}.
\end{align*}
Substituting this computation into the definition of $T_p$ gives
\begin{align*}
T_p = \operatorname{Frob}_p + p\operatorname{Frob}_p^{-1}.
\end{align*}
The space $H$ was defined as
\begin{align*}
H = H^1_{\mathrm{\acute{e}t}}(C,\mathbb Q_\ell),
\end{align*}
and $C$ is exactly the geometric special fibre $X_0(N)_{\overline{\mathbb F}_p}$. Therefore the equality is the claimed identity on
\begin{align*}
H^1_{\mathrm{\acute{e}t}}(X_0(N)_{\overline{\mathbb F}_p},\mathbb Q_\ell).
\end{align*}
[/guided]
[/step]
