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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy