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 $$f^* : H^i(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})\to H^i(E_{\overline{\mathbf{F}}_p},f^{-1}\mathbf{Q}_{\ell})\to H^i(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})$$ 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 $f\in \text{End}(E)\otimes_{\mathbf{Z}}\mathbf{Q} = H$, 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 question is **no**, and hence so is the answer to the second.

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

(2) I expect the problem with the second construction is that, for two elements $f,g\in H$, we may have $(f+g)^*\neq f^*+g^*$, where the first $+$ is in $H$ and the second is in $H^i$.

(3) 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. Am I right?