Frobenius automorphisms of cohomology of a variety

Question

Suppose $X$ is a smooth variety defined over $\mathbb{Q}$. There are (at least) two automorphisms of cohomology groups of $X$ that are called "Frobenius", and I would like to understand how they are related.

• If $p$ is a prime of good reduction for $X$, then $H^n_{dR}(X)\otimes\mathbb{Q}_p$ depends functorially on the special fiber of an integral model of $X$. The Frobenius endomorphism of the special fiber induces an automorphism $F_p\in GL\big(H^n_{dR}(X)\otimes\mathbb{Q}_p\big)$.

• Fix a prime $\ell$ and an algebraic closure $\bar{\mathbb{Q}}$ of $\mathbb{Q}$. Then $H^n_{et}(X_{\bar{\mathbb{Q}}},\mathbb{Q}_{\ell})$ comes with an action of $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ that is unramified for all but finitely many primes $p$. For unramified $p$, there is a well-defined conjugacy class $\Phi_p\subset GL\big(H^n_{et}(X_{\bar{\mathbb{Q}}},\mathbb{Q}_{\ell})\big)$ coming from the conjugacy class of the Frobenius at $p$ in $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$.

What is the relationship between $F_p$ and $\Phi_p$?

Answer 1

I think you are intending to take everything with smooth proper varieties (or else there is no good notion of a prime of good reduction).

In this case, the two Frobenius elements have the same characteristic polynomial.

In the smooth projective case, this is due to Katz-Messing. Translating this into your language requires using the isomorphism between crystalline cohomology of the special fiber and de Rham cohomology of the lift, and the isomorphism between etale cohomology of the special fiber and the generic fiber.

In the general case it is sufficient to know that both cohomology theories are pure and satisfy the Grothendieck-Lefschetz fixed point formula. Then the characteristic polynomial of Frobenius in each case will be a term of the Weil zeta function, which is independent of the cohomology, and hence will be equal in both cases. For crystalline cohomology the purity should follow from Katz-Messing by an alteration argument, as in Chiarellotto, or from Kedlaya's $p$-adic proof of the Weil conjectures.