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.



Minimal vs characteristic polynomial of geometric Frobenius
It's conjectured -- see e.g. [this question](https://mathoverflow.net/questions/104102/semisimplicity-of-frobenius-operation-on-etale-cohomology?noredirect=1&lq=1) -- 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 $\mathbf{F}\_{p^2}$.


