# Is there an $\mathbb{R}$-valued cohomology theory for varieties over $\mathbb{F}_p$?

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?

The answer is no. If $$A$$ is a simple abelian variety over $$\mathbb{F}_p$$, then $$End(A)\otimes\mathbb{R}$$ cannot act on a real vector space of dimension $$2dim(A)$$ if the center of the endomorphism algebra of $$A$$ has a real embedding. Let $$E=End(A)\otimes\mathbb{Q}$$. A theorem of Tate shows that $$2dim(A)=[E:F]^{1/2}[F:\mathbb{Q}]$$ where $$E$$ is a division algebra with center $$F$$. For a field $$L$$, $$E\otimes L$$ will act on an $$L$$ vector space of dimension 2dim(A) if and only if $$L$$ splits $$E$$, but $$\mathbb{R}$$ doesn't split $$E$$ (part of Tate's theorem). See Thm 1 of Tate's 1968 Bourbaki talk.

In fact, the only such abelian variety (up to isogeny) $$A$$ over $$\mathbb{F}_p$$ corresponds to the Weil number $$p^{1/2}$$. It has dimension 2 and becomes a product of two elliptic curves over a quadratic extension of $$\mathbb{F}_p$$. Moreover $$E$$ is a division algebra of degree 4 over $$F=\mathbb{Q}(p^{1/2})$$ ramified only at the two infinite places.

In a sense this is the only obstruction: if you assume the Tate conjecture, then it is known that the category of motives of weight zero over $$\mathbb{F}_p$$ does have an $$\mathbb{R}$$-valued fiber functor.

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.