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This question is based on Milne "canonical models of Shimura varieties and automorphic vector bundles"

Let $(G,X)$ be a Shimura datum, $(V,\xi)$ be a rational representation of $G$ (I guess it means a representation over $\mathbb{Q}$), satisfied some good conditions. Chapter II Prop. 3.3 in Milne's paper says there are

(1) $\mathbb{Q}_\ell$-coefficient local system $V_\ell$ on $Sh(G,X)_{et}$, for each prime $\ell$.

(2) a vector bundle $\mathcal{V}(\xi)$ on $Sh(G,X)$, with a flat connection $\nabla$.

Is there any comparison theorem relating $H^n_{et}(Sh(G,X),V_\ell)$ with $H^n_{dR}(Sh(G,X),\mathcal{V}(\xi),\nabla)$?

More precisely, these objects have canonical models over some suitable number field, say $L$. I want to study the $Gal(\bar{L}_v/L_v)$ representation $H^n_{et}(Sh(G,X)\times_L\bar{L}_v,V_\ell)$, for a place $v$ of $L$ above $\ell$. Moreover, the variation of Hodge structures should be considered, which will induce useful filtrations. I want to compare it with the algebraic de Rham cohomology $H^n_{dR}(Sh(G,X)/L,\mathcal{V}(\xi),\nabla)$

For example, if $(V,\xi)$ is the trivial representation, and $\ell$ is "good", then we can apply Falting's comparison theorem.

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    $\begingroup$ I am not sure this requires $p$-adic Hodge theory. $\Ker(\nabla)$ is $\mathbf C$-local system of which the De Rham complex of $\mathscr V(\xi)$ is a resolution. Probably, there exists a $\Q$-structure $V$ on $\Ker(\nabla)$ such that $V\otimes\mathbf Q_\ell=V_\ell$; in any case, via an embedding of $\mathbf Q_\ell$ into $\mathbf C$, $V_\ell\otimes\C$ is isomorphic to $\Ker(\nabla)$. So comparison follows from the classical comparison theorems between singular and étale topologies. $\endgroup$
    – ACL
    Commented Oct 12, 2016 at 19:22
  • $\begingroup$ I edit the question a little bit, to make it more meaningful. $\endgroup$
    – user99616
    Commented Oct 13, 2016 at 2:53

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