Let $F$ be a $p$-adic field. Let $(G\_{F}, W\_{F}, I\_{F})$ denote the (absolute Galois group, Weil group, inertia group) of $F$.
Let $X/F$ be a proper smooth variety. Let $\ell$ be a prime number $\ne p$. The $\ell$-adic cohomology $H\_{\ell}^{i} = H\_{\text{ét}}^{i}(X\_{\bar{F}}, \mathbb{Q}\_{\ell})$ is naturally endowed with a continuous Galois representation $\rho$.


The Weil–Deligne representation
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By the $\ell$-adic monodromy theorem of Grothendieck, one associates a Weil–Deligne representation with $H\_{\ell}^{i}$: It gives a nilpotent “monodromy” operator $N$. One restricts the representation $\rho$ to $W\_{F}$, and changes it to $$ \sigma (\Phi^{a}x) = \rho(\Phi^{a}x) \exp(-t(x)N), \qquad a \in \mathbb{Z}, x \in I\_{F} $$ where $\Phi \in W\_{F}$ is a Frobenius element, and $t \colon I\_{F} \to \mathbb{Z}\_{\ell}$ a projection onto the $\ell$-adic component of $I\_{F}$.


The Weil–Deligne representation associated with $H\_{\ell}^{i}$ is $(\sigma, H\_{\ell}^{i}, N)$. It is an object in $\mathrm{WDRep}\_{\mathbb{Q}\_{\ell}}(W\_{F})$.


Question
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Choose an (non-canonical, non-continuous!) embedding $i\_{\ell} \colon \mathbb{Q}\_{\ell} \to \mathbb{C}$. By extending scalars, we obtain an object $(\sigma, H\_{\ell}^{i}, N) \otimes\_{i\_{\ell}} \mathbb{C}$ in $\mathrm{WDRep}\_{\mathbb{C}}(W\_{F})$.


Let $\ell' \ne p$ be another prime. Let $i\_{\ell'} \colon \mathbb{Q}\_{\ell'} \to \mathbb{C}$ be an embedding. We can repeat the entire process to obtain an object $(\sigma, H\_{\ell'}^{i}, N) \otimes\_{i\_{\ell'}} \mathbb{C}$ in $\mathrm{WDRep}\_{\mathbb{C}}(W\_{F})$.



> 
> **Q.** Are $(\sigma, H\_{\ell}^{i}, N) \otimes\_{i\_{\ell}} \mathbb{C}$ and $(\sigma, H\_{\ell'}^{i}, N) \otimes\_{i\_{\ell'}} \mathbb{C}$ isomorphic in $\mathrm{WDRep}\_{\mathbb{C}}(W\_{F})$?
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> 
> 


From a motivic viewpoint this should certainly be true, but on the other hand maybe this is one of the strongest forms of $\ell$ independence that one can ask for.


If there is no answer in general, I would be very happy to learn about partial cases, where this is known.



Is the Weil–Deligne representations coming from $\ell$-adic cohomology independent of $\ell$?
In a [recently published paper](http://www.ams.org/mathscinet/search/publdoc.html?pg1=MR&s1=3123639), Rutger Noot has studied the case of abelian varieties. In 2011, he had conducted the [PhD Thesis of Abhijit Laskar](https://hal.archives-ouvertes.fr/tel-00644861/document) where the methods are applied in a motivic setting to obtain more general results depending on the Mumford-Tate group of the underlying motive.


