Say $A_0$ is an ordinary abelian variety over ${\mathbf{F}}_q$. Call $\mathcal{A}$ the canonical lift of $A_0$ over $R := W({\mathbf{F}}_q)$. It carries a lift of the $q$-th power map on $A_0$. We call $\phi : \mathcal{A}\to\mathcal{A}$ this lift. It exists by functoriality of the canonical lift.

Call $K = \text{Frac}(R)$ and choose a field embedding $K\subset\mathbf{C}$.

Call $A$ the complex torus $(\mathcal{A}\times_K\mathbf{C})(\mathbf{C})$ and $F$ the endomorphism of $A$ induced by the Frobenius lift on $\mathcal{A}$, i.e. $$F = (\phi \times_{\text{Spec}(R)}\text{id}_{\mathbf{\text{Spec}(C)}})^{\rm an}:A\to A.$$

$F$ acts on the Betti cohomology $H^*(A,\mathbf{C})$. By the Weil conjectures (or by an argument of Serre, in this case), its eigenvalues are of the form $q^{*/2}\zeta$ for algebraic numbers $\zeta$ of complex absolute value $1$.

Suppose $v \in H^{2m}(A,\mathbf{C})$ is an eigenvector of $F$ whose eigenvalue is of the form $q^m\zeta$ for $\zeta$ a root of unity. Is $v$ a class of type $(m,m)$?