Let $k$ be an algebraically closed field. Let $V$ be a smooth projective variety over $k$. For a map $\phi:V\to V$, do the coefficients of the characteristic polynomial of $\phi^\*:H^ i\_{dR}(V/k)\to H^ i\_{dR}(V/k)$ lie in the prime subfield of $k$?



Coefficients of the characteristic polynomial of the map on algebraic de Rham cohomology
Suppose that $\mathrm{char}\, k=p>0$. It is easy to give an example of a stack $V$ with an endomorphism that violates this property. Take $V=B\alpha\_p$, the scaling action of $\mathbb{G}\_m$ on $\alpha\_p\subset \mathbb{G}\_a$ induces an action on $B\alpha\_p$ so, in particular, the group $k^{\times}$ acts on $V$. By Proposition 4.12 of <https://arxiv.org/pdf/1909.11437.pdf> an element $\lambda\in k^{\times}$ acts on the $1$-dimensional space $H^1\_{dR}(V/k)$ by $\lambda^p$ so any $\lambda\in k^{\times}\setminus\mathbb{F}\_p$ gives an example of an endomorphism with characteristic polynomial with coefficients outside of $\mathbb{F}\_p$.


Let's now approximate this example by an actual smooth projective variety. Choose a finite subgroup $\Gamma\subset k^{\times}$ not contained in $\mathbb{F}\_p^{\times}$ and consider the group scheme $G=\Gamma\ltimes \alpha\_p$ where the semi-direct product is formed using the scaling action of $\Gamma\subset \mathbb{G}\_m(k)$ used above. There exists a smooth complete intersection $X$ of dimension $3$ with a free action of $G$. Put $V=X/\alpha\_p$ which is equipped with the remaining action. The $\Gamma$-equivariant morphism $V\to B\alpha\_p$ induces a $\Gamma$-equivariant isomorphism $H^1\_{dR}(B\alpha\_p)\simeq H^1\_{dR}(V)$ by weak Lefschetz so any $\varphi\in\Gamma\setminus \mathbb{F}\_p^{\times}$ gives a counterexample. For a reference see e.g. proof of Theorem 1.2 in <https://arxiv.org/pdf/1909.11437.pdf> , the original reference for the construction of $X$ is Proposition 4.2.3 in <http://www.numdam.org/article/AST_1979__64__87_0.pdf>


These cohomology classes come from torsion in crystalline cohomology and this is in fact necessary to get a counterexample: the characteristic polynomial of $\phi$ on the subquotient $$(H^i\_{cris}(V/W(k))/\text{torsion})/p$$ of the space $H^i\_{dR}(V/k)$ coincides with that on $H^i\_{cris}(V/W(k))/\text{torsion}$ which has integer coefficients by Theorem 2(ii) of Katz-Messing's "Consequences of the Riemann Hypothesis".


