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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.

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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 $\mathbf{F}_{p^2}$.

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  • $\begingroup$ Thank you! In particular, the "semisimplicity conjecture" depends on $\ell$. Am I understanding this correctly? If one proves Frobenius is semisimple on $H^i(X,\overline{\mathbf{Q}}_{\ell})$, then $m_{F_i} = \text{rad}(P_i)$, the radical, so I guess for every $\ell\neq \ell',p$, this $m_{F_i}$ will divide the minimal polynomial of Frobenius on $H^i(X,\overline{\mathbf{Q}}_{\ell'})$, but nothing more can be said. Or is it somehow known by some nontrivial argument that the semisimplicity conjecture is "independent of $\ell$"? $\endgroup$
    – user178246
    Commented Jul 10, 2021 at 11:46
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    $\begingroup$ When I wrote "should be", I meant that if semisimplicity holds for $\ell$-adic cohomology for all $\ell$ (as is conjectured), then it implies $m_{F_i}$ is independent of $\ell$. But I don't think this is known unconditionally; as far as I'm aware, semisimplicity for one $\ell$ doesn't imply it for other $\ell$. $\endgroup$
    – David Loeffler
    Commented Jul 10, 2021 at 12:38
  • $\begingroup$ Thanks. This is exactly what I was looking for $\endgroup$
    – user178246
    Commented Jul 10, 2021 at 12:39
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    $\begingroup$ @user178246 In the case of abelian varieties, this is a theorem of Tate, so, for example, the supersingular curve case is unconditional. $\endgroup$
    – Will Sawin
    Commented Jul 10, 2021 at 12:46

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