TITLE: Minimal vs characteristic polynomial of geometric Frobenius
QUESTION [7 upvotes]: 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.

REPLY [6 votes]: It's conjectured -- see e.g. this question -- that the Frobenius is always semisimple, so its minimal polynomial is the radical of its characteristic polynomial (the product of its distinct linear factors, each with multiplicity 1). So $m_{F_i}$ should be independent of $\ell$.
This also shows that $m_{F_i}$ is different from $P_i$ iff $P_i$ has a root of multiplicity $> 1$. This can certainly occur, e.g. consider a supersingular elliptic curve over $\mathbf{F}_{p^2}$.