# An example of Serre on the cohomology of some CM elliptic curves

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).

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.

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.