Difference between Frobenii on Tate modules of special and generic fibre

Question

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?

Answer 1

Yes. See Milne's text on Abelian Varieties: it shows that deg(F-n) is a polynomial in ℤ[n], and degree is preserved by reduction mod p. For elliptic curves, the characteristic polynomial has coefficients det(F)=deg(F) and Tr(F)=1+deg(F)-deg(1-F).