Let $X$ be a quasi-projective variety over a number field $E$. Let $v$ be a place in $E$ and let $E_v$ denote the $v$-adic completion. Let $p$ be the prime number determined by $v$, and let $\Lambda = \mathbb Z_{\ell}/\ell^k\mathbb Z_{\ell}$, where $\ell$ is a prime number different from $p$ and $k\geq 1$.
Under what conditions do we have an isomorphism between the etale cohomology of $X/E$ and the etale cohomology of $X_v/E_v$, both with coefficients in $\Lambda$?
In particular, I am interested in the case where $v$ is non-archimedean.