The $p$-adic étale cohomology of algebraic varieties over $p$-adic fields is a fundamental subject in the study of $p$-adic representations. Moreover, thanks to the comparison theorems in $p$-adic Hodge theory, the resulting $p$-adic representations that emerge in this way can be controlled as linear objects, such as weak admissible filtered φ-modules. It is now well-known that the corresponding filtered φ-module can be realized as cohomology, such as the crystalline cohomology of the algebraic variety X.
On the other hand, it is also well-known that one can construct étale (\phi, \Gamma)-modules from $p$-adic representations, which is also helpful in understanding $p$-adic representations. Therefore, my question is, when a $p$-adic representation comes from an algebraic variety X as described above, is it possible to "directly" construct the corresponding ($\phi$, $\Gamma$)-module from X?