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?



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


