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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 to this question, 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.

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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.

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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.

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