I'm studying Drinfeld modules to study elliptic curves over function fields.
(We assume the characteristic $p$ is large enough if we need.)
Simply, we consider rank $2$ Drinfeld $A$-module $\rho$ over $K$ where $A = \mathbb{F}_{p} \left[T\right]$ and $K = \mathbb{F}_{p}\left(T\right)$. Let $E/K$ be the elliptic curve corresponding $\rho$. I'm interesting on the $N$-torsion part $E\left[N\right]$ for $N$ which is not divided by $p$.
I read "for $a \in A$, $x\mapsto \rho_{a}\left(x\right)$ is analog of $P\mapsto\left[n\right] P$".
My question is that what is $a_{N} \in A$ corresponding to the endomorphism $\left[N\right]: E \to E$. I think, intuitively, $a_{N} = N$ in $\mathbb{F}_{p}$. However, $\left(p+1\right)$-torsion structure and $\left(2p+1\right)$-torsion structure of $E$ is distinct but $a_{p+1} = a_{2p+1}$.
I think this question is related to what is the (natural?) faithful $\mathbb{Z}$-module sructure of $A$.