A comparison theorem between crystalline cohomology and étale cohomology Suppose $X/\mathbb F_q$ is a smooth projective variety. Katz-Messing (eudml) shows that the characteristic polynomial of the Frobenius on $H^i_{et}(\overline{X},\mathbb Q_\ell)$ and $H^i_{crys}(X)$ are equal. Suppose now that $G \subset Aut(X)$ is a finite group. Can we identify the two cohomologies as $G$-representations (over an algebraically closed field $K$ containing both $\mathbb Q_\ell$ and $W(\mathbb F_q)$)?
 A: This is OK when  $X$ is projective and smooth, at least when the Kunneth
components of the diagonal are algebraic. Let $k$ be the algebraic closure of
a finite field. Define $M_{1}$ and $M_{2}$ to be the categories of motives
over $k$ for homological equivalence (w.r.t. some Weil cohomology $H$) and
numerical equivalence respectively. They are rigid tensor categories, and we
have a tensor functor $M_{1}\rightarrow M_{2}$ (which preserves traces). An
automorphism $a$ of $X$ defines automorphisms $h_{1}^{r}X$ and $h_{2}^{r}X$,
and $Tr(a|H^{r}(X))=Tr(a|h_{1}^{r}X)=Tr(a|h_{2}^{r}X)$, which is a rational
number (not depending on $H$). When applied to the powers of $a$, this shows
that the characteristic polynomial of $a$ on $H^{r}(X)$ has rational
coefficients independent of $H$ (by the Newton identities).
The following argument may prove the general case (or may not). Let $M(k)$ be a (good) triangulated category of motives with $\mathbb{Q}{}$-coefficients, and let $f\colon M(k)\rightarrow P$ be the universal determinant functor of $M(k)$ (Breuning 2011, 3.1). The determinant of an automorphism $a$ of an object $X$ of $M(k)$ is an element of $\pi _{1}(P)$, which contains $\mathbb{Q}^{\times}$. The result of Bondarko/Olsson should show that, for all integers $n$, $\det(1-na)\in \mathbb{Q}^{\times}\subset\pi_{1}(P)$. If so, then it follows that there exists a polynomial in $\mathbb{Q}[T]$ that is equal to characteristic polynomial of $a$ on the $l$-adic realization of $X$ for all $l$ (including $l=p$) (see Proposition 3.1 of arXiv:1311.3166).
