Isomorphism of the $\ell$-adic Tate module of an elliptic curve with CM

Let $ E$ be an elliptic curve over $ K$ (totally real number field) with complex multiplication by the field $ L$ . Let $ \psi$ be the Grössencharacter associated to $ E$ , assume that $ \psi$ of type $ (-r,0)$ (i.e., the restriction of $ \psi$ to the archimedian part $ \mathbb{C}^\times$ of the idele group of $ K$ has the form $ z\mapsto z^{-r}$ ). Set $ $ V_{\ell}(\psi):=H^{1,\text{ét}}(E(\mathbb{C})\otimes_L\overline{\mathbb{Q}},\mathbb{Q}_\ell)^{\otimes r}\otimes_{K\otimes\mathbb{Q}_\ell}L_{\tilde{\ell}} $ $ where $ \otimes r$ is taken over $ K\otimes\mathbb{Q}_\ell$ .

My question is: How to prove that the $ \ell$ -adic Tate module $ V_\ell(E)$ is isomorphic to $ V_{\ell}(\psi)\oplus\imath V_{\ell}(\psi)$ as representations of $ G_K$ , where $ \imath\in G_K$ is the complex conjugation. (References would be appreciated).

Kernels in the category of $\ell$-adic sheaves on a noetherian scheme

In Exposé V, VI of SGA 5, Jouanolou develops the adic formalism, and in 1.1.3 of VI, he shows that the category of $ \ell$ -adic constructible sheaves on a scheme $ X$ is abelian. Let’s say $ X$ is noetherian. To construct the kernel of a map of $ \ell$ -adic sheaves $ u:\mathscr F\rightarrow\mathscr G$ , he first takes the kernel in the category $ \underline{\operatorname{Hom}}(\mathbf{N}^\circ,\mathrm{Ab}(X))$ , and calls this $ \mathscr K$ . He then remarks that $ \mathscr K$ satisfies the condition Mittag-Leffler-Artin-Rees and calls $ \mathscr K’$ its projective system of universal (i.e. stable) images. But in (3.1.3) of his Exposé V, he shows that if $ 0\rightarrow X\rightarrow Y\rightarrow Z\rightarrow 0$ is an exact sequence in a category such as $ \underline{\operatorname{Hom}}(\mathbf{N}^\circ,\mathrm{Ab}(X))$ , with $ Y$ strict and $ Z$ $ J$ -adic (here $ J=(\ell\mathbf{Z})$ ), then $ X$ is strict. So why bother forming the projective system of universal images? I must be overlooking something obvious.

Is there literature on a de Rham analogue of the Mumford-Tate group or ell-adic monodromy group?

Let $ X$ be a smooth projective variety over $ \mathbb{Q}$ . The theory of motives predicts that for each cohomology theory, there should be a distinguished Zariski closed subgroup of $ GL(H^k_{\bullet}(X))$ , the motivic Galois group. This group has conjectural descriptions for Betti and $ \ell$ -adic cohomology, and I am wondering if there is any literature on this group for de Rham cohomology.

The Betti cohomology $ H^k_B(X)$ is a rational Hodge structure, hence there is a representation $ \mathrm{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}_m\to GL(H^k_B(X)\otimes\mathbb{R})$ . The Mumford-Tate group $ MT^k(X)\leq GL(H^k_B(X))$ is defined to be the smallest Zariski-closed subgroup containing the image of the representation. The Hodge conjecture would imply that $ MT^k(X)$ has finite index in the Betti motivic Galois group.

The $ \ell$ -adic cohomology $ H^k(X;\mathbb{Q}_\ell)$ is a representation of the absolute Galois group, and the $ \ell$ -adic monodromy group $ G_\ell^k(X)\leq GL(H^k(X;\mathbb{Q}_\ell))$ is defined to be the smallest Zariski-closed subgroup containing the image of the representation. The Tate conjecture would imply that $ G_\ell^k(X)$ is the full $ \ell$ -adic motivic Galois group.

Is there any literature on an analogue of the Mumford-Tate group or the $ \ell$ -adic monodromy group inside $ GL(H^k_{dR}(X))$ ?

I believe that a conjecture of Ogus would imply that the de Rham motivic Galois group is smallest Zariski closed subgroup whose $ \mathbb{Q}_p$ points contain $ F_p$ for all sufficiently large $ p$ (where $ F_p\in GL(H^k_{dR}(X))(\mathbb{Q}_p)$ is the crystalline Frobenius). However I have some doubt about this, because André’s book on motives states the relationship between the Hodge conjecture and the Mumford-Tate group (Proposition, and the relationship between the Tate conjecture and the $ \ell$ -adic monodromy group (Proposition, but does not give an analogous statement for the Ogus conjecture.

I also believe that the period conjecture of Grothendieck would imply that the de Rham motivic Galois group is the smallest Zariski closed subgroup containing all elements $ \varphi\in GL(H^k_{dR}(X))(\mathbb{Q})$ whose image in $ GL(H^k_{B}(X))(\mathbb{C})$ under the Betti-de Rham comparison isomorphism is contained in $ GL(H^k_{B}(X))(\mathbb{Q})$ . However, I also cannot find a statement like this in André’s book.

Pullbacks on $\ell$-adic cohomology

For smooth projective varieties $ X$ over $ F := \overline{\mathbf{F}}_p$ , $ \ell$ -adic cohomology $ $ H^*(X,\mathbf{Q}_{\ell})$ $ is a Weil cohomology theory.

We denote, for a morphism of smooth projective $ F$ -varieties $ f : X\to Y$ , by $ $ f^* : H^*(Y,\mathbf{Q}_{\ell})\to H^*(X,\mathbf{Q}_{\ell})$ $ the usual pullback on $ \ell$ -adic cohomology.

Suppose now that we have another pullback $ \widetilde{f}^*$ on $ \ell$ -adic cohomology, compatible with the $ \ell$ -adic cycle map, and making it into a Weil cohomology again.

That is, $ \widetilde{f}^*$ still makes $ X\mapsto H^*(X,\mathbf{Q}_{\ell})$ into a functor to graded $ \mathbf{Q}_{\ell}$ -algebras that still satisfies the axioms of a Weil cohomology with the usual cycle map on $ \ell$ -adic cohomology.

Do we necessarily have $ \widetilde{f}^* = f^*$ ?

Due to compatibility with cycle maps, $ \widetilde{f}^* = f^*$ in degrees $ 0$ and top.

Regularity for $\ell$-adic cohomology

In algebraic topology, for a smooth manifold $ M$ and a closed submanifold $ Z\subset M$ , one sometimes defines Cech cohomology as a direct limit of Cech cohomologies of open neighborhoods of $ Z$ :

$ $ \check{H}^*(Z) = \varinjlim_{U\supset Z}\check{H}^*(U)$ $

and it usually agrees with usual Cech cohomology of $ Z$ .

I’m looking for an analog in $ \ell$ -adic cohomology.

Namely, let $ X$ be a smooth projective variety over a separably closed field $ k$ , $ Z\subset X$ a smooth closed subscheme.

Do we have:

$ $ \varinjlim_{Z\to U\to X} H^*(U_{et},\mathbf{Q}_{\ell}) = H^*(Z_{et},\mathbf{Q}_{\ell})$ $

where the direct limit ranges over all étale neighborhoods of $ Z$ in $ X$ ?

Independence of $\ell$ in $\ell$-adic cohomology

Let $ X$ be the base change of a smooth projective variety over a finite field, to a separable closure $ k$ of the ground field.

Do we expect for any endomorphism $ f$ of $ X$ to have an effect $ f^* : H^n(X,\mathbf{Q}_{\ell})\to H^n(X,\mathbf{Q}_{\ell})$ whose characteristic polynomial has coefficients in $ \mathbf{Q}$ ? or at least independent of $ \ell$ in some suitable sense?

Is there an example of an $ f$ such that its characteristic polynomial on $ \ell$ -adic cohomology does depend on $ \ell$ ?

Quaternion algebra-action on $\ell$-adic cohomology

Let $ E$ be a supersingular elliptic curve over $ \mathbf{F}_p$ , and $ H$ its endomorphism algebra $ \text{End}(E)\otimes_{\mathbf{Z}}\mathbf{Q}$ , a quaternion algebra (non split at $ p$ and $ \infty$ ).

For every prime $ \ell\neq p$ , there is a faithful $ \ell$ -adic algebra-representation:

$ $ \rho_{\ell} : H\to \text{End}_{\mathbf{Q}_{\ell}}(V_{\ell}(E))$ $

where $ V_{\ell}(E) := (\varprojlim_{n\ge 0} E_{\overline{\mathbf{F}}_p}({\overline{\mathbf{F}}_p})[\ell^n])\otimes_{\mathbf{Z}_{\ell}}\mathbf{Q}_{\ell}$ is the rational $ \ell$ -adic Tate module of $ E$ .

  • Using the natural isomorphism $ H^1(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell}) \simeq \text{Hom}_{\mathbf{Q}_{\ell}}(V_{\ell}(E),\mathbf{Q}_{\ell})$ we have an $ H\otimes_{\mathbf{Q}}\mathbf{Q}_{\ell}$ -module structure on $ H^1(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})$ , as is classically known.

  • Now we make a more naive construction. Functoriality of the étale site of $ E_{\overline{\mathbf{F}}_p}$ gives that for any endomorphism $ f : E\to E$ there is a map $ $ H^i(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})\to H^i(E_{\overline{\mathbf{F}}_p},f^{-1}\mathbf{Q}_{\ell})$ $ and since $ \mathbf{Q}_{\ell}$ is constant (here I am being imprecise about the nature of $ \mathbf{Q}_{\ell}$ , which is not a constant sheaf on the étale site, but the meaning is clear from the context) we also have an isomorphism $ H^i(E_{\overline{\mathbf{F}}_p},f^{-1}\mathbf{Q}_{\ell})\simeq H^i(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})$ , and we call $ $ f^* : H^i(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})\to H^i(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})$ $ the composition of the two. In other words, every element $ f\in\text{End}(E)$ has an effect $ f^*$ on $ H^i(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})$ . On the other hand, the effect of each element of $ \mathbf{Z}\subset\text{End}(E)$ on $ H^i(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})$ is invertible, and so the above construction defines, for every element $ f\in H$ , an effect $ f^*$ on $ \ell$ -adic cohomology.

Does the construction in the second point, for $ i=1$ , give an action of $ H$ on $ H^1(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})$ as an algebra? If so, does this action agree with the one constructed in the first point?

My expectation is that the answer to the first, and hence second, question is no.

The second construction should only define on $ H^i(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})$ the structure of representation of $ H^{\times}$ , and it should not be possible to upgrade this to an algebra action of $ H$ .

Existence of the $ \ell$ -adic Tate-module functor, and the fact that for abelian varieties $ A,B$ the map $ \text{Hom}_{\rm AV}(A,B)\otimes_{\mathbf{Z}}\mathbf{Q}_{\ell}\to \text{Hom}_{\mathbf{Q}_{\ell}}(V_{\ell}(A),V_{\ell}(B))$ is injective, should be crucial to have an algebra action of $ H$ on $ H^1(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})$ , and it feels it should not be possible to construct it just as a consequence of functoriality of the étale site, a much less deep fact.