Eichler-Shimura congruence

Question

I'm trying to understand the Eichler-Shimura congruence which relates the Hecke operator $T_p$ to Frobenius at $p$ in characteristic $p$.

Two possible ways to compute $T_p$ mod $p$ seem to be:

A) Look at the map $Div(X_0(N)) \to Div(X_0(N))$ induced by the correspondence $X_0(N) \leftarrow X_0(Np) \to X_0(N)$ where the first map forgets the subgroup at $p$ and the second mods out by it. And then just take this map and look at the induced map $Div(X_0(N)_{F_p}) \to Div(X_0(N)_{F_p})$ in characteristic $p$.

B) Look directly at the correspondence $X_0(N)_{F_p} \leftarrow X_0(Np)_{F_p} \to X_0(N)_{F_p}$ in characteristic $p$ and try to compute directly the induced map $Div(X_0(N)_{F_p}) \to Div(X_0(N)_{F_p})$.

In case (A), starting with a point in $X_0(N)$, it has $p+1$ lifts to $X_0(Np)$ and thus the induced map on divisors will result in divisors of degree $p+1$. Explicitly, I'm getting: $$ (E,C) \to (E,(C+\ker(F))/\ker(F)) + p (E,(C+\ker(V))/\ker(V)) $$ which looks at lot like the standard Eichler-Shimura relation.

But in case (B), starting with an ordinary point in $X_0(N)_{F_p}$, it has $2$ lifts to $X_0(Np)$ --- one where we pick $\ker(F)$ as our group scheme of order $p$, and another for $\ker(V)$. Thus the induced map on divisors will result in divisors of degree $2$, and it seems to yield $$ (E,C) \to (E,(C+\ker(F))/\ker(F)) + (E,(C+\ker(V))/\ker(V)). $$

What's wrong with the argument in case B?

Answer 1

There's a little more geometry here that should be accounted for in characteristic $p$. Namely, the curve $X_0(Np)_{\mathbb{F}_p}$ is reducible -- its two components are isomorphic to $X_0(N)_{\mathbb{F}_p}$ and they intersect transversally at the supersingular points (see e.g. Ribet and Stein's online notes). So the components are disjoint on the ordinary locus and account for the two subgroups you wrote down.

The left pointing map $\pi: X_0(Np)_{\mathbb{F}_p} \to X_0(N)_{\mathbb{F}_p}$ in the correspondence is an isomorphism on the component corresponding to $\ker (F)$, but it is an inseparable degree $p$ map on the second component. Roughly speaking, that this is degree $p$ and inseparable comes from the fact that there are $p$ "non-canonical subgroups" in characteristic 0 which all reduce to $\ker (V)$ in characteristic $p$.

To compute the effect of the correspondence on divisors, you need to pull back divisors along an inseparable map. So your argument in Case B is forgetting to multiply by the ramification index (which is $p$) for the unique preimage of $(E,\ker(V))$ under $\pi$ in the "$\ker (V)$ component" of $X_0(Np)_{\mathbb{F}_p}$

Edit: The OP asks how to check the inseparability rigorously and without using characteristic 0. As Eric says in the comments, this is a somewhat subtle issue which can be found in Katz-Mazur (see section 13.5), probably also in Deligne-Rapoport. The Ribet-Stein book has a more approachable discussion (read the proof of Theorem 12.6.4 carefully), but they implicitly use the fact that the Frobenius map $X_0(N) \to X_0(N)$ corresponds under the moduli interpretation to the Frobenius map $(E, C) \mapsto (E^{(p)}, C^{(p)})$, which is really the heart of the matter.

But if you're willing to ignore issues of representability of functors, then there's an elementary way to see this last fact: first note that base change reduces us to the case where $N = 1$. Then use the parameterization of $X_0(1)$ by the $j$-invariant to see that both the Frobenius $X_0(1) \to X_0(1)$ and the map $E\mapsto E^{(p)}$ can be identified with $j \mapsto j^p$.