Assume $X$ is a smooth projective variety over $\overline{\mathbf{F}}_p$ and fix a prime $\ell\neq p$.

Let $F_i$ be the geometric Frobenius on $\ell$-adic cohomology

$$H^i_{\rm ét}(X,\overline{\mathbf{Q}}_{\ell})$$ for fixed $i\ge 0$.

- What relation is expected to hold between the minimal polynomial $m_{F_i}(T)$ of $F_i$ and the characteristic polynomial $P_i(T)$ of $F_i$? (apart from $m_{F_i}\mid P_i$)
- Are $m_{F_i}(T)$ and $P_i(T)$ conjectured to agree? If so, is this a known result?
- Does $m_{F_i}(T)$ depend on the Weil cohomology theory chosen to compute $P_i(T)$?

We know from Deligne's work on the Weil conjectures, that we have $P_i(T)\in\mathbf{Z}[T]$, $P_i(T)$ does not depend on the chosen Weil cohomology, and its roots are of the form $q^{-i/2}\rho$ for $\rho$ an algebraic number whose complex absolute value, for any complex embedding $\overline{\mathbf{Q}}\subset\mathbf{C}$, is one.

I'm mostly interested to understand to what extent $m_{F_i}(T)$ is, or expected to be, intrinsic.