Let $E$ be the elliptic curve over $\mathbf{Q}\_3$ with Weierstrass equation $y^2 = x^3-x$.
It has complex multiplication by $\mathbf{Z}[\sqrt{-1}]$, with $\sqrt{-1}$ acting as $(x,y)\mapsto(-x,iy)$.


Call $R:=\mathbf{Z}\_3[\sqrt{-1}] = W(\mathbf{F}\_9)$.


Let $\mathcal{E}$ be the Néron model over $\mathbf{Z}\_3$, an abelian scheme (e.g. by Néron-Ogg-Shafarevich). Then the endomorphism $[-3] : \mathcal{E}\otimes\_{\mathbf{Z}\_3}R\to\mathcal{E}\otimes\_{\mathbf{Z}\_3}R$ is a lift of the $9$-power map on $\mathcal{E}\_{\mathbf{F}\_9}\to \mathcal{E}\_{\mathbf{F}\_9}$.


In the example by Serre discussed in the [answer](https://mathoverflow.net/q/106246) to this [question](https://mathoverflow.net/q/106238), a conjectural $\mathbf{Q}$-linear cohomology theory is discussed for varieties over $p$-adic fields, and it is argued that such theory cannot exist in light of the fact that a Frobenius action $F$ on $H^1(E\_{\overline{\mathbf{Q}}\_3})$ would satisfy $F^2 = -3=[-3]^\*$, $F[\sqrt{-1}]^\*=-[\sqrt{-1}]^\*F$, and of course $([\sqrt{-1}]^\*)^2=-1$, and these equations cannot be solved in $2\times 2$ matrices over $\mathbf{R}$ (at least, this is how I understood the question and answer).


My question only asks for a clarification on the linked question and answer.



> 
> **question**  
> 
> It seems to me the linked question asks for a cohomology theory for varieties over $p$-adic fields. How does $F$ on $H^1$ arise, then?
> 
> 
> 


I would expect such theory to be functorial *only with respect to morphisms of varieties over $p$-adic fields* (as it seems to be required in the linked question), and so $H^1(E\_{\overline{\mathbf{Q}}\_3})$ would not necessarily carry the effect of endomorphisms of $\mathcal{E}\_{\overline{\mathbf{F}}\_3}$, except those that are liftable.


The $3$-rd power map on $\mathcal{E}\_{\mathbf{F}\_3}$ is not liftable, I believe. Otherwise I'd expect the example to apply to the rational Betti cohomology of $E(\mathbf{C})$ for any $\sigma :\overline{\mathbf{Q}}\_3\simeq\mathbf{C}$.


**expectation** I expect the linked question asks for a theory that is *assumed* to carry some such $F$ (probably in the last sentence of the linked question - "taking value in Weil-Deligne representations over $\mathbf{Q}$"), and Serre's example shows $F$ cannot exist on any such $\mathbf{Q}$-linear theory. I'd just like to double-check this understanding is correct, to be sure. I'll leave the question here for other's benefit.



An example of Serre on the cohomology of some CM elliptic curves
Yes. There exists a "nice" cohomology theory for $p$-adic varieties, taking values in vector spaces over $\mathbb Q$, defined by fixing an embedding $\mathbb Q\_p \to \mathbb C$, base-changing along this embedding, and taking singular cohomology. Every morphism of varieties over $\mathbb Q\_p$ induces a map on cohomology groups in this theory (by the usual property of singular cohomology). So no argument of this type can rule out such a theory.


It is a cohomology theory valued in $\mathbb Q$-rational Weil-Deligne representations, which by assumption admit an action of Frobenius, that this argument rules out.


