Let $E$ be elliptic curve over $\mathbb Q$ and $p$ a prime of good reduction for $E$. Fix $\ell \neq p$.
If $E_p$ is ordinary then we have Frobenius $F_p$ on $E_p$. Assume $F_p$ lifts to endomorphism $F$ of $E$ (over field extension, so that $E$ has CM). Does characteristic polynomial of $F_p$ acting on Tate module $T_\ell E_p$ equal characteristic polynomial of $F$ acting on Tate module of $E$?
If yes, is this true in higher dimensions?