Let's consider the category of smooth projective varieties over a fixed characteristic $p>0$ algebraically closed field $k$. For a Weil cohomology theory with coefficient field $E$, by definition it shall satisfy finiteness property, Poincare duality, Kunneth formula, existence of cycle maps, weak and hard Lefschetz theorem.

Consider the class $\mathcal {C}$ of characterestic zero fields $E$ that support a Weil cohomology. Not all fields of zero characteristic can be a coeffiecient field, as is discussed in Serre’s example of a supersingular elliptic curve over $k$. We know

- $\mathbb R, \mathbb Q_p \notin \mathcal{C}$.
For $E_1 \hookrightarrow E_2$, $E_1 \in \mathcal {C} \Rightarrow E_2 \in \mathcal {C}$ .

$\mathbb Q_l \in \mathcal {C}$ $(l \not = p)$ .

- $W(k)[1/p] \in \mathcal {C}$.

More interestingly, we also know some pseudofinite field lie in $\mathcal C$, namely the ultraproduct of $\mathbb F_l$ $(l \not =p)$ using a non-principal ultrafilter, see "a new Weil cohomology theory" by Ivan Tomasic. And there is a dicussion of Weil II for such Weil cohomology theory, see https://webusers.imj-prg.fr/~anna.cadoret/Weil2Ultra_OberwolfachReports.pdf。

So my question is, can we describe other interesting fields in $C$ ?

notan algebraically closed field), then crystalline cohomology is a Weil cohomology with $\mathbb{Q}_p$-coefficients. There was an answer by SashaP on MO explaining why Serre's objection does not apply (and also a comment by Will Sawin, I think). $\endgroup$ – schematic_boi May 11 at 8:11notdefined over $\mathbf{F}_p$ and the ring of those that are defined over $\mathbf{F}_p$ $\endgroup$ – John P. May 11 at 15:45