I'm reading the paper *"Higher Hida and Coleman theories on the modular curve"* by G.Boxer and V.Pilloni. But I'm confused with the different views towards Hecke operators.

$N$ is an integer and $p$ is a prime such that $(p,N)=1$. Let $X\rightarrow \mathrm{Spec} \mathbb{Z}_p$ be the compactified modular of level $\Gamma_1(N)$ defined by P.Deligne and M.Rapoport. And let $X_0(p)$ be the modular curve of level $\Gamma_1(N)\cap \Gamma_0(p)$. We have two projections:

$$ p_1\colon X_0(p)\rightarrow X,~(E,H)\mapsto E $$

and

$$ p_2\colon X_0(p)\rightarrow X,~(E,H)\mapsto E/H, $$ where $H\subset E[p]$ is a subgroup of order $p$.

I have two problems:

- Why are those two maps finite flat? It's quite easy to check for $p_1$. But I have no ideas with $p_2$.
- Let $\omega$ be the sheaf of invariant diffrential. What is the map $p_2^{\ast} \omega^k\rightarrow p_1^{\ast} \omega^k$ and how it relates to Hecke operators?