Let $A$ be an abelian variety over a finite field $\mathbb{F}_q$ and $x_i$ the Frobenius eigenvalues on $H^1$. Does $x_i \mapsto q/x_i$ permute the $x_i$, and why? I think Poincare duality only gives us this with $q$ replaced by $g^g$, $g$ the dimension of $A$.