TITLE: Is there an $\mathbb{R}$-valued cohomology theory for varieties over $\mathbb{F}_p$?
QUESTION [12 upvotes]: If $E$ is a supersingular elliptic curve over $\mathbb{F}_{p^m}$ with $m\geq 2$ its endomorphism ring is a maximal order in a quaternion algebra ramified at $p$ and $\infty$ so there can't be a Weil cohomology with coefficients in $\mathbb{Q}_p$ or $\mathbb{R}$.
For varieties over $\mathbb{F}_p$ there is a $\mathbb{Q}_p$-valued cohomology theory (the crystalline cohomology).
Is there an $\mathbb{R}$-valued cohomology theory as well?

REPLY [16 votes]: Let me advertise a conjectural positive answer to a slightly different question. This actually works not only over $\mathbb F_p$ but over its algebraic closure $\overline{\mathbb F}_p$.
Consider the following $\mathbb R$-linear category, sometimes called the category of real isocrystals $\mathrm{Isoc}_{\mathbb R}$. Objects are finite-dimensional $\mathbb C$-vector spaces $V$ together with a ($\mathbb C$-linear) grading $V=\bigoplus_{i\in \mathbb Z} V_i$, and a $\mathbb C$-antilinear graded isomorphism $\alpha: V\to V$ (i.e., $\alpha(xv)=\overline{x}\alpha(v)$ for $x\in \mathbb C$ and $v\in V$, and $\alpha(V_i)=V_i$) such that $\alpha^2|_{V_i} = (-1)^i$. (Thus, $\alpha$ induces a real structure on the even part of $V$, and a quaternionic structure on the odd part of $V$.)

Conjecture (See Conjecture 9.5). There is a Weil cohomology theory for varieties over $\overline{\mathbb F}_p$ with values in $\mathrm{Isoc}_{\mathbb R}$.

The grading should correspond to the weight decomposition (always split in this case, as all motives over $\overline{\mathbb F}_p$ are pure). In particular, we see that for motives concentrated in even weights, a fibre functor with $\mathbb R$-coefficients ought to exist, even for varieties over $\overline{\mathbb F}_p$, refining the previous answer.
The conjecture is known to follow from the Tate conjecture. However, I'd believe there should be a direct way to construct it (like etale and crystalline cohomology), but I don't have any insight into how.
Why is this an analogue of isocrystals? Kottwitz has constructed for any local or global field $F$ an $F$-linear Tannakian category, that for nonarchimedean local fields reduces to isocrystals, and for $\mathbb R$ gives the above category. For function fields, it gives a category of isoshtukas; for number fields, a linear-algebraic description is unknown. Conjecturally, a Weil cohomology theory should even exist with values in Kottwitz' category for $F=\mathbb Q$. The latter Weil cohomology theory should even induce a fully faithful functor from motives over $\overline{\mathbb F}_p$ into Kottwitz' category for $F=\mathbb Q$; this is closely related to the Langlands--Rapoport conjecture.