# Frobenius automorphisms of cohomology of a variety

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 $\Phi_p$?

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.