Let there be an isogeny $f:A_1 \rightarrow A_2$ between two abelian varieties over a $p$-adic field $F$ and assume $f$ has degree $p^n$. By the universal property we get a moprhism $f_0: \mathcal{A}_1 \rightarrow \mathcal{A}_2 $ between Neron models over $O_F$.
When is the morphism $f_0$ etale? Is there a simple criterion in terms of the integral $p$-adic Tate modules?
Motivation: Consider an imaginary quadratic field $K$, the morphism between complex elliptic curves $f: \mathbb C/p^mO_K \rightarrow \mathbb C/(\mathbb Z +p^mO_K)$ can be defined over a large enough $p$-adic field (even over a number field) $F$ so the source and target have good reduction, then $f_0$ is etale iff $p$ splits completely over $K$. How about higher dimensional $CM$ abelian varieties?
