In preparing a set of notes, I realize there is a simple question for which I cannot find an easy answer; likely due to simple ignorance. Any introductory text on etale cohomology notes that supersingular elliptic curves with endomorphism giving a quaternion algebra provided a representation theoretic obstruction for a Weil cohomology theory valued in $\mathbf{Q}_p$ (or $\mathbf{Z}_p$). Although such curves can be defined over $\mathbf{F}_p$, their endomorphism algebra is defined over $\mathbf{F}_{p^2}$.

Crystalline cohomology is a Weil cohomology theory for smooth projective varieties over perfect fields, which a priori includes $\mathbf{F}_p$. This seems to suggest that when the ground field is $\mathbf{F}_p$, this is Weil cohomology theory taking values in $W(\mathbf{F}_p) \cong \mathbf{Z}_p$, appearing to violate the obstruction above. However, in those examples values should probably be in $W(\mathbf{F}_{p^2})$ instead. Grothendieck's Dix Exposes article "Crystals and the de Rham Cohomology of Schemes" states that coefficient ring of a $p$-adic cohomology should be in *some* extension of $\mathbf{Q}_p$ but is unclear what extensions are forced. The Wikipedia article also suggests to work with varieties over algebraically closed fields which again resolves the issue, but in all the surveys and notes I've seen, it seems to be vague on any limitations of the choice perfect ground field to guarantee a Weil cohomology theory.

Can someone point out what if any limits should be imposed on the ground fields or why someone super cleaver can't come up with other examples, like the stated elliptic curves, which limit the ground field?

isno obstruction. The question is how to explain the appearance of one as $\mathbf{F}_p$ is not prohibited. This at least raises the question of what ground fields are allowed for crystalline cohomology to give a Weil cohomology theory, which my cursory search has shown none, but this seems to suggest that it is a $\mathbf{Z}_p$-valued Weil cohomology theory for at least some varieties. $\endgroup$ – lemiller Aug 24 '17 at 18:52limitationon the ground field. $\endgroup$ – lemiller Aug 24 '17 at 19:07